1.2 Material properties of importance in heat transfer
Before understanding heat transfer laws, we have to understand various properties of the material. This section is devoted to a brief discussion of some of the important properties of the material.
Before understanding heat transfer laws, we have to understand various properties of the material. This section is devoted to a brief discussion of some of the important properties of the material.
1.2.1 Thermal conductivity
As discussed earlier, the heat conduction is the transmission of energy by molecular action. Thermal conductivity is the property of a particular substance and shows the ease by which the process takes place. Higher the thermal conductivity more easily will be the heat conduction through the substance. It can be realized that the thermal conductivity of a substance would be dependent on the chemical composition, phase (gas, liquid, or solid), crystalline structure (if solid), temperature, pressure, and its homogeneity.
The thermal conductivity of various substances is shown in table1.1 and table 1.2.
As discussed earlier, the heat conduction is the transmission of energy by molecular action. Thermal conductivity is the property of a particular substance and shows the ease by which the process takes place. Higher the thermal conductivity more easily will be the heat conduction through the substance. It can be realized that the thermal conductivity of a substance would be dependent on the chemical composition, phase (gas, liquid, or solid), crystalline structure (if solid), temperature, pressure, and its homogeneity.
The thermal conductivity of various substances is shown in table1.1 and table 1.2.
Table1.1: Thermal conductivities of various substances at 0oC
Table1.2: Thermal conductivity of mercury at three different phases
The general results of the careful analysis of the table1.1 and 1.2 are as follows,
 Thermal conductivity depends on the chemical composition of the substance.
 Thermal conductivity of the liquids is more than the gasses and the metals have the highest.
 Thermal conductivity of the gases and liquids increases with the increase in temperature.
 Thermal conductivity of the metal decreases with the increase in temperature.
 Thermal conductivity is affected by the phase change.
These differences can be explained partially by the fact that while in gaseous state, the molecules of a substance are spaced relatively far away and their motion is random. This means that energy transfer by molecular impact is much slower than in the case of a liquid, in which the motion is still random but in liquids the molecules are more closely packed. The same is true concerning the difference between the thermal conductivity of the liquid and solid phases. However, other factors are also important when the solid state is formed.
Solid having a crystalline structure has high thermal conductivity than a substance in an amorphous solid state. Metal, crystalline in structure, have greater thermal conductivity than nonmetal (refer table1.1). The irregular arrangement of the molecules in amorphous solids inhibits the effectiveness of the transfer of the energy by molecular impact. Therefore, the thermal conductivity of the nonmetals is of the order of liquids. Moreover, in solids, there is an additional transfer of heat energy resulting from vibratory motion of the crystal lattice as a whole, in the direction of decreasing temperature.
Many factors are known to influence the thermal conductivity of metals, such as chemical composition, atomic structure, phase changes, grain size, temperature, and pressure. Out of the above factors, the temperature, pressure, and chemical composition are the most important. However, if we are interested in a particular material then only the temperature effects has to be accounted for.
As per the previous discussion and the table it is now clear that the thermal conductivity of the metal is directly proportional to the absolute temperature and mean free path of the molecules. The mean free path decreases with the increase in temperature so that the thermal conductivity decreases with the temperature. It should be noted that it is true for the pure metal, and the presence of impurity in the metal may reverse the trend. It is usually possible to represent the thermal conductivity of a metal by a linear relation k = k_{o}(1 + bT), where k_{o} is the thermal conductivity of the metal at 0^{o}C, T is the absolute temperature, and b is a constant.
In general the thermal conductivity of the liquids is insensitive to the pressure if the pressure is not very close to the critical temperature. Therefore, in liquids (as in solids) the temperature effects on the thermal conductivity are generally considered. Liquids, in general, exhibit a decreasing thermal conductivity with temperature. However, water is a notable exception. Water has the highest thermal conductivity among the nonmetallic liquids, with a maximum value occurring at 450^{o}C.
The thermal conductivity of a gas is relatively independent of pressure if the pressure is near 1 atm. Vapours near the saturation point show strong pressure dependence. Steam and air are of great engineering importance. Steam shows irregular behaving rather showing a rather strong pressure dependence for the thermal conductivity as well as temperature dependence.
The above discussions concerning thermal conductivity were restricted to materials composed of homogeneous or pure substances. Many of the engineering materials encountered in practice are not of this nature like building material, and insulating material. Some material may exhibit nonisotropic conductivities. The nonisotropic material shows different conductivity in different direction in the material. This directional preference is primarily the result of the fibrous nature of the material like wood, asbestos etc.
1.2.2 Specific heat capacity
Now we know that the thermal conductivity facilitates the heat to propagate through the material due to the temperature gradient. Similarly, specific heat capacity or specific heat is the capacity of heat stored by a material due to variation in temperature. Thus the specific heat capacity (unit: kJ/kg·^{o}C) is defined as the amount of thermal energy required to raise the temperature of a unit amount of material by 1^{o}C. Since heat is path dependent, so is specific heat. In general, the heat transfer processes used in the chemical process plant are at constant pressure; hence the specific heat capacity (c_{o}) is generally used.
Now we know that the thermal conductivity facilitates the heat to propagate through the material due to the temperature gradient. Similarly, specific heat capacity or specific heat is the capacity of heat stored by a material due to variation in temperature. Thus the specific heat capacity (unit: kJ/kg·^{o}C) is defined as the amount of thermal energy required to raise the temperature of a unit amount of material by 1^{o}C. Since heat is path dependent, so is specific heat. In general, the heat transfer processes used in the chemical process plant are at constant pressure; hence the specific heat capacity (c_{o}) is generally used.
Q.1. What is the basic difference among conduction, convection, and radiation?
Q.2. Define thermal conductivity.
Q.3. What is the order of thermal conductivity of gas, liquid, and metal in general?
Q.4. What should be the approach to select a good thermal insulator?
Q.5. Discuss the effect of temperature on thermal conductivity.
Q.6. What is the difference between thermal conductivity and specific heat capacity?
Q.2. Define thermal conductivity.
Q.3. What is the order of thermal conductivity of gas, liquid, and metal in general?
Q.4. What should be the approach to select a good thermal insulator?
Q.5. Discuss the effect of temperature on thermal conductivity.
Q.6. What is the difference between thermal conductivity and specific heat capacity?
Conduction: One Dimensional
The fundamentals of heat conduction were established over one and a half century and its contribution goes to a French mathematician and physicist, Jean Baptiste Joseph Fourier. You may be aware that any flow whether it is electricity flow, fluid flow, or heat flow needs a driving force. The flow is proportional to the driving force and for various kinds of flows the driving force is shown in the table 2.1.
Table 2.1. Various flows and their driving forces
Thus the heat flow per unit area per unit time (heat flux, ) can be represented by the following relation,
where, proportionality constant k is the thermal conductivity of the material, T is the temperature and x is the distance in the direction of heat flow. This is known as Fourier’s law of conduction.
The term steadystate conduction is defined as the condition which prevails in a heat conducting body when temperatures at fixed points do not change with time. The term onedimensional is applied to a heat conduction problem when only one coordinate is required to describe the distribution of temperature within the body. Such a situation hardly exists in real engineering problems. However, by considering onedimensional assumption the real problem is solved fairly upto the accuracy of practical engineering interest.
2.1 Steadystate conduction through constant area
A simple case of steadystate, onedimensional heat conduction can be considered through a flat wall as shown in the fig.2.1.
A simple case of steadystate, onedimensional heat conduction can be considered through a flat wall as shown in the fig.2.1.
Fig.2.1: Steadystate conduction through a slab (constant area)
The flat wall of thickness d_{x} is separated by two regions, the one region is at high temperature (T_{1 }) and the other one is at temperature T_{2 }. The wall is very large in comparison of the thickness so that the heat losses from the edges are negligible. Consider there is no generation or accumulation of the heat in the wall and the external surfaces of the wall are at isothermal temperatures T_{1 }and T_{2} _{}. The area of the surface through which the heat transfer takes place is A. Then the eq.2.2 can be written as,
The negative sign shows that the heat flux is from the higher temperature surface to the lower temperature surface and is the rate of heat transfer through the wall.
Now if we consider a plane wall made up of three different layers of materials having different thermal conductivities and thicknesses of the layers, the analysis of the conduction can be done as follows.
Consider the area (A) of the heat conduction (fig.2.2) is constant and at steady state the rate of heat transfer from layer1 will be equal to the rate of heat transfer from layer2. Similarly, the rate of heat transfer through layer2 will be equal to the rate of heat transfer through layer3. If we know the surface temperatures of the wall are maintained at T_{1 }and T_{2} as shown in the fig.2.2, the temperature of the interface of layer1 and layer 2 is assumed to be at T' and the interface of layer2 and layer3 as T".
Fig.2.2: Heat conduction through three different layers
The rate of heat transfer through layer1 to layer2 will be,
and,
The rate of heat transfer through layer 2 to layer 3 will be,
The rate of heat transfer through layer 2 to layer 3 will be,
and,
The rate of heat transfer through layer 3 to the other side of the wall,
The rate of heat transfer through layer 3 to the other side of the wall,
On adding the above three equations,
Where, R represents the thermal resistance of the layers. The above relation can be written analogous to the electrical circuit as,
Fig 2.3: Equivalent electrical circuit of the fig.2.2
The wall is composed of 3different layers in series and thus the total thermal resistance was represented by R (= R_{1 }+ R_{2 }+ R_{3 }). The discussed concept can be understood by the illustrations shown below.
The unit of the various parameters used above is summarized as follows,
Illustration 2.1
The two sides of a wall (2 mm thick, with a crosssectional area of 0.2 m2) are maintained at 30^{o}C and 90^{o}C. The thermal conductivity of the wall material is 1.28 W/(m·^{o}C). Find out the rate of heat transfer through the wall?
Solution 2.1
Assumptions
1. Steadystate onedimensional conduction
2. Thermal conductivity is constant for the temperature range of interest
3. The heat loss through the edge side surface is insignificant
4. The layers are in perfect thermal contact
Given,
The two sides of a wall (2 mm thick, with a crosssectional area of 0.2 m2) are maintained at 30^{o}C and 90^{o}C. The thermal conductivity of the wall material is 1.28 W/(m·^{o}C). Find out the rate of heat transfer through the wall?
Solution 2.1
Assumptions
1. Steadystate onedimensional conduction
2. Thermal conductivity is constant for the temperature range of interest
3. The heat loss through the edge side surface is insignificant
4. The layers are in perfect thermal contact
Given,
Fig. 2.4: Illustration 2.1
Illustration 2.2
Solution 2.2
Assumptions:
1. Steadystate onedimensional conduction.
2. Thermal conductivity is constant for the temperature range of interest.
3. The heat loss through the edge side surface is insignificant.
4. The layers are in perfect thermal contact.
Assumptions:
1. Steadystate onedimensional conduction.
2. Thermal conductivity is constant for the temperature range of interest.
3. The heat loss through the edge side surface is insignificant.
4. The layers are in perfect thermal contact.
On putting all the known values,
Fig. 2.5: Illustration 2.2
Thus,
The previous discussion showed the resistances of different layers. Now to understand the concept of equivalent resistance, we will consider the geometry of a composite as shown in fig.2.6a.
The wall is composed of seven different layers indicated by 1 to 7. The interface temperatures of the composite are T_{1} to T_{5} as shown in the fig.2.6a. The equivalent electrical circuit of the above composite is shown in the fig 2.6b below,
Fig.2.6. (a) Composite wall, and (b) equivalent electrical circuit
The equivalent resistance of the wall will be,
where,
Therefore, at steady state the rate of heat transfer through the composite can be represented by,
where, R is the equivalent resistance.
2.2 Thermal contact resistance
In the previous discussion, it was assumed that the different layers of the composite have perfect contact between any two layers. Therefore, the temperatures of the layers were taken same at the plane of contact. However, in reality it rarely happens, and the contacting surfaces are not in perfect contact or touch as shown in the fig.2.8(a). It is because as we know that due to the roughness of the surface, the solid surfaces are not perfectly smooth. Thus when the solid surfaces are contacted the discrete points of the surfaces are in contact and the voids are generally filled with the air. Therefore, the heat transfer across the composite is due to the parallel effect of conduction at solid contact points and by convection or probably by radiation (for high temperature) through the entrapped air. Thus an apparent temperature drop may be assumed to occur between the two solid surfaces as shown in the fig.2.8b. If T_{I} and T_{II} are the theoretical temperature of the plane interface, then the thermal contact resistance may be defined as,
In the previous discussion, it was assumed that the different layers of the composite have perfect contact between any two layers. Therefore, the temperatures of the layers were taken same at the plane of contact. However, in reality it rarely happens, and the contacting surfaces are not in perfect contact or touch as shown in the fig.2.8(a). It is because as we know that due to the roughness of the surface, the solid surfaces are not perfectly smooth. Thus when the solid surfaces are contacted the discrete points of the surfaces are in contact and the voids are generally filled with the air. Therefore, the heat transfer across the composite is due to the parallel effect of conduction at solid contact points and by convection or probably by radiation (for high temperature) through the entrapped air. Thus an apparent temperature drop may be assumed to occur between the two solid surfaces as shown in the fig.2.8b. If T_{I} and T_{II} are the theoretical temperature of the plane interface, then the thermal contact resistance may be defined as,
where R_{c}represents the thermal contact resistance.
The utility of the thermal contact resistance (R_{c} ) is dependent upon the availability of the reliable data. The value of R_{c} depends upon the solids involved, the roughness factor, contact pressure, material occupying the void spaces, and temperature. The surface roughness of a properly smooth metallic surface is in the order of micrometer. The values of R_{c} generally obtained by the experiments. However, there are certain theories which predict the effect of the various parameters on the R_{c}.
It can be seen in the fig.2.8, that the two main contributors to the heat transfer are (i) the conduction through entrapped gases in the void spaces and, (ii) the solidsolid conduction at the contact points. It may be noted that due to main contribution to the resistance will be through first factor because of low thermal conductivity of the gas.
Fig.2.8 (a) Contacting surfaces of two solids are not in perfect contact, (b) temperature drop due to imperfect contact
If we denote the void area in the joint by A_{v} and contact area at the joint by A_{c}, then we may write heat flow across the joint as,
where, thickness of the void space and thermal conductivity of the fluid (or gas) is represented by l_{g}and k_{f}, respectively. It was assumed that l_{g}/2 is the thickness of solidI and solidII for evenly rough surfaces.
2.3 Steadystate heat conduction through a variable area
It was observed in the previous discussion that for the given plane wall the area for heat transfer was constant along the heat flow direction. The plane solid wall was one of the geometries but if we take some other geometry (tapered plane, cylindrical body, spherical body etc.) in which the area changes in the direction of heat flow. Now we will consider geometrical configuration which will be mathematically simple and also of great engineering importance like hollow cylinder and hollow sphere. In these cases the heat transfer area varies in the radial direction of heat conduction. We will take up both the cases one by one in the following sections.
It was observed in the previous discussion that for the given plane wall the area for heat transfer was constant along the heat flow direction. The plane solid wall was one of the geometries but if we take some other geometry (tapered plane, cylindrical body, spherical body etc.) in which the area changes in the direction of heat flow. Now we will consider geometrical configuration which will be mathematically simple and also of great engineering importance like hollow cylinder and hollow sphere. In these cases the heat transfer area varies in the radial direction of heat conduction. We will take up both the cases one by one in the following sections.
2.3.1 Cylinder
Consider a hollow cylinder as shown in the fig.2.9a. The inner and outer radius is represented by r_{i}and r_{o }, whereas T_{i} and T_{o} (T_{i} > T_{o }) represent the uniform temperature of the inner and outer wall, respectively.
Consider a hollow cylinder as shown in the fig.2.9a. The inner and outer radius is represented by r_{i}and r_{o }, whereas T_{i} and T_{o} (T_{i} > T_{o }) represent the uniform temperature of the inner and outer wall, respectively.
Fig. 2.9. (a) Hollow cylinder, (b) equivalent electrical circuit
Consider a very thin hollow cylinder of thicknessd_{r} in the main geometry (fig.2.9a) at a radial distance r. If d_{r} is small enough with respect to r, then the area of the inner and outer surface of the thin cylinder may be considered to be of same area. In other words, for very small d_{r} with respect to r, the lines of heat flow may be considered parallel through the differential element in radial outward direction.
We may ignore the heat flow through the ends if the cylinder is sufficiently large. We may thus eliminate any dependence of the temperature on the axial coordinate and for one dimensional steady state heat conduction, the rate of heat transfer for the thin cylinder,
Where dT is the temperature difference between the inner and outer surface of the thin cylinder considered above and k is the thermal conductivity of the cylinder.
On rearranging,
On rearranging,
To get the heat flow through the thick wall cylinder, the above equation can be integrated between the limits,
On solving,
Where , and the careful analysis of the above equation shows that the expression is same as for heat flow through the plane wall of thickness (r_{o}–r_{i} ) except the expression for the area. The A_{LM} is known as log mean area of the cylinder, whose length is L and radius is r_{LM} (= ). The fig.2.9b shows the equivalent electrical circuit of the fig.2.9b.
Now we have learnt that how to represent the analogous electrical circuit for the cylindrical case. It will provide the building block for the composite cylinders similar to the plane composite we have learnt earlier. The following fig.2.10a shows a composite cylinder with 4layers of solid material of different inner and outer diameter as well as thermal conductivity. The equivalent electrical circuit is shown below in fig.2.10b.
(a)
(b)
Fig.2.10.(a) Four layer composite hollow cylinder, (a) equivalent electrical circuit
The total heat transfer at steadystate will be,
where R_{1} , R_{2 }, R_{3} , and R_{4} are represented in the fig.2.10b.
2.3.2 Sphere
The rate of heat transfer through a hollow sphere can be determined in a similar manner as for cylinder. The students are advised to derive the following expression shown below.
The rate of heat transfer through a hollow sphere can be determined in a similar manner as for cylinder. The students are advised to derive the following expression shown below.
The final expression for the rate of heat flow is,
2.4 Heat conduction in bodies with heat sources
The cases considered so far have been those in which the heat conducting solid is free of internal heat generation. However, the situations where the internal heat is generated are very common cases in chemical industries for example, the exothermic reaction in the solid pallet of a catalyst.
The cases considered so far have been those in which the heat conducting solid is free of internal heat generation. However, the situations where the internal heat is generated are very common cases in chemical industries for example, the exothermic reaction in the solid pallet of a catalyst.
We have learnt that how the Fourier equation is used for the steadystate heat conduction through the composites of three different geometries that were not having any heat source in it. However, the heat generation term would come into the picture for these geometries. It would not be always easier to remember and develop heat conduction relations for different standard and nonstandard geometries. Therefore, at this point we should learn how to develop a general relation for the heat conduction that should be applicable to the entire situation such as steadystate, unsteady state, heat source, different geometry, heat conduction in different direction, etc. Again here we will consider that the solid is isotropic in nature, which means the thermal conductivity of the material is same in all the direction of heat flow.
To get such a general equation the differential form of the heat conduction equation is most important. For simplicity, we would consider an infinitesimal volume element in a Cartesian coordinate system. The dimensions of the infinitesimal volume element are d_{x }, d_{y }, and d_{z }in the respective direction as shown in the fig.2.11.
Fig.2.11. Volume element for deriving general equation of heat conduction in cartesian coordinate
The fig.2.11 shows that the heat is entering into the volume element from three different faces of the volume element and leaving from the opposite face of the control element. The heat source within the volume element generates the volumetric energy at the rate of
According to Fourier’s law of heat conduction, the heat flowing into the volume element from the left (in the xdirection) can be written as,
According to Fourier’s law of heat conduction, the heat flowing into the volume element from the left (in the xdirection) can be written as,
The heat flow out from the right surface (in the xdirection) of the volume element can be obtained by Taylor series expansion of the above equation. As the volume element is of infinitesimal volume, we may retain only first two element of the Taylor series expansion with a reasonable approximation (truncating the higher order terms). Thus,
The left side of the above equation represent the net heat flow in the xdirection. If we put the value of in the right side of the above equation,
In a similar way we can get the net heat flow in the y and zdirections,
As we know some heat is entering, some heat is leaving and some heat in generating in the volume element and as we have not considered any steady state assumption till now, thus because of all these phenomena some of the heat will be absorbed by the element. Thus the rate of change of heat energy within the volume element can be written as,
where, c_{p} is the specific heat capacity at constant pressure (J/(kg·K)), ρ is the density (kg/m^{3}) of the material, and t is the time (s).
We know all the energy term related to the above problem, and with the help of energy conservation,
On putting all the values in the above equation,
or,
As we have considered that the thermal conductivity of the solid is isotropic in nature, the above relation reduces to,
or,
where is the thermal diffusivity of the material and its unit m^{2}/s signifies the rate at which heat diffuses in to the medium during change in temperature with time. Thus, the higher value of the thermal diffusivity gives the idea of how fast the heat is conducting into the medium, whereas the low value of the thermal diffusivity shown that the heat is mostly absorbed by the material and comparatively less amount is transferred for the conduction. The called the Laplacian operator, and in Cartesian coordinate it is defined as
Equation 2.19 is known as general heat conduction relation. When there is no heat generation term the eq.2.19 will become,
and the equation is known as Fourier Field Equation.
General heat conduction relation in cylindrical coordinate system (fig. 2.12) is derived (briefly) below.
Fig.2.12. Cylindrical coordinate system (a) and an element of the cylinder
The energy conservation for the system is written as,
Ӏ + ӀӀ = ӀӀӀ + ӀV (2.21)
where,
I : Rate of heat energy conducted in
II : Rate of heat energy generated within the volume element
III : Rate of heat energy conducted out
IV : Rate of energy accumulated (ӀV)
and the above terms are defines as,
where,
I : Rate of heat energy conducted in
II : Rate of heat energy generated within the volume element
III : Rate of heat energy conducted out
IV : Rate of energy accumulated (ӀV)
and the above terms are defines as,
Thus,
On putting the values in equation 2.21,
Thus the Laplacian operator is,
Fig.2.13. Spherical coordinate system (a) and an element of the sphere
In a similar way the general expression for the conduction heat transfer in spherical body with heat source can also be found out as per the previous discussion. The Laplacian operator for the spherical coordinate system (fig.2.13) is given below and the students are encouraged to derive the expression themselves.
Convective Heat transfer: One dimension
The rate of heat transfer in a solid body or medium can be calculated by Fourier’s law. Moreover, the Fourier law is applicable to the stagnant fluid also. However, there are hardly a few physical situations in which the heat transfer in the fluid occurs and the fluid remains stagnant. The heat transfer in a fluid causes convection (transport of fluid elements) and thus the heat transfer in a fluid mainly occurs by convection.
3.1 Principle of heat flow in fluids and concept of heat transfer coefficient
It is learnt by daytoday experience that a hot plate of metal will cool faster when it is placed in front of a fan than exposed to air, which is stagnant. In the process, the heat is convected away, and we call the process convective heat transfer. The term convective refers to transport of heat (or mass) in a fluid medium due to the motion of the fluid. Convective heat transfer, thus, associated with the motion of the fluid. The term convection provides an intuitive concept of the heat transfer process. However, this intuitive concept must be elaborated to enable one to arrive at anything like an adequate analytical treatment of the problem.
It is learnt by daytoday experience that a hot plate of metal will cool faster when it is placed in front of a fan than exposed to air, which is stagnant. In the process, the heat is convected away, and we call the process convective heat transfer. The term convective refers to transport of heat (or mass) in a fluid medium due to the motion of the fluid. Convective heat transfer, thus, associated with the motion of the fluid. The term convection provides an intuitive concept of the heat transfer process. However, this intuitive concept must be elaborated to enable one to arrive at anything like an adequate analytical treatment of the problem.
It is well known that the velocity at which the air blows over the hot plate influences the heat transfer rate. A lot of questions come into the way to understand the process thoroughly. Like, does the air velocity influence the cooling in a linear way, i.e., if the velocity is doubled, will the heat transfer rate double. We should also suspect that the heattransfer rate might be different if we cool the plate with some other fluid (say water) instead of air, but again how much difference would there be? These questiones may be answered with the help of some basic analysis in the later part of this module.
The physical mechanism of convective heat transfer for the problem is shown in fig.3.1.
Fig. 3.1: Convective heat transfer from a heated wall to a fluid
Consider a heated wall shown in fig.3.1. The temperature of the wall and bulk fluid is denoted by respectively. The velocity of the fluid layer at the wall will be zero. Thus the heat will be transferred through the stagnant film of the fluid by conduction only. Thus we can compute the heat transfer using Fourier’s law if the thermal conductivity of the fluid and the fluid temperature gradient at the wall is known. Why, then, if the heat flows by conduction in this layer, do we speak of convective heat transfer and need to consider the velocity of the fluid? The answer is that the temperature gradient is dependent on the rate at which the fluid carries the heat away; a high velocity produces a large temperature gradient, and so on. However, it must be remembered that the physical mechanism of heat transfer at the wall is a conduction process.
nal It is apparent from the above discussion that the prediction of the rates at which heat is convected away from the solid surface by an ambient fluid involves thorough understanding of the principles of heat conduction, fluid dynamics, and boundary layer theory. All the complexities involved in such an analytical approach may be lumped together in terms of a single parameter by introduction of Newton’s law of cooling,
where, h is known as the heat transfer coefficient or film coefficient. It is a complex function of the fluid composition and properties, the geometry of the solid surface, and the hydrodynamics of the fluid motion.
If k is the thermal conductivity of the fluid, the rate of heat transfer can be written directly by following the Fourier’s law. Therefore, we have,
where, is the temperature gradient in the thin film where the temperature gradient is linear.
On comparing eq.3.1 and 3.2, we have,
It is clear from the above expression that the heat transfer coefficient can be calculated if k and δare known. Though the k values are easily available but the δ is not easy to determine. Therefore, the above equation looks simple but not really easy for the calculation of real problems due to nonlinearity of k and difficulty in determining δ. The heat transfer coefficient is important to visualize the convection heat transfer phenomenon as discussed before. In fact, δ is the thickness of a heat transfer resistance as that really exists in the fluid under the given hydrodynamic conditions. Thus, we have to assume a film of δ thickness on the surface and the heat transfer coefficient is determined by the properties of the fluid film such as density, viscosity, specific heat, thermal conductivity etc. The effects of all these parameters are lumped or clubbed together to define the film thickness. Henceforth, the heat transfer coefficient (h) can be found out with a large number of correlations developed over the time by the researchers. These correlations will be discussed in due course of time as we will proceed through the modules. Table 3.1 shows the typical values of the convective heat transfer coefficient under different situations.
Table3.1: Typical values of h under different situations
3.2 Individual and overall heat transfer coefficient
If two fluids are separated by a thermally conductive wall, the heat transfer from one fluid to another fluid is of great importance in chemical engineering process plant. For such a case the rate of heat transfer is done by considering an overall heat transfer coefficient. However, the overall heat transfer coefficient depends upon so many variables that it is necessary to divide it into individual heat transfer coefficients. The reason for this becomes apparent if the above situation can be elaborated as discussed in the following subsections.
If two fluids are separated by a thermally conductive wall, the heat transfer from one fluid to another fluid is of great importance in chemical engineering process plant. For such a case the rate of heat transfer is done by considering an overall heat transfer coefficient. However, the overall heat transfer coefficient depends upon so many variables that it is necessary to divide it into individual heat transfer coefficients. The reason for this becomes apparent if the above situation can be elaborated as discussed in the following subsections.
3.2.1 Heat transfer between fluids separated by a flat solid wall
As shown in fig.3.2, a hot fluid is separated by solid wall from a cold fluid. The thickness of the solid wall is l, the temperature of the bulk of the fluids on hot and cold sides are T_{h} and T_{c}, respectively. The average temperature of the bulk fluid is T_{1} and T_{4}, for hot and cold fluid, respectively. The thicknesses of the fictitious thin films on the hot and cold sides of the flat solid are shown by δ_{1} and δ_{2}. It may be assumed that the Reynolds numbers of both the fluids are sufficiently large to ensure turbulent flow and the surfaces of the solid wall are clean.
As shown in fig.3.2, a hot fluid is separated by solid wall from a cold fluid. The thickness of the solid wall is l, the temperature of the bulk of the fluids on hot and cold sides are T_{h} and T_{c}, respectively. The average temperature of the bulk fluid is T_{1} and T_{4}, for hot and cold fluid, respectively. The thicknesses of the fictitious thin films on the hot and cold sides of the flat solid are shown by δ_{1} and δ_{2}. It may be assumed that the Reynolds numbers of both the fluids are sufficiently large to ensure turbulent flow and the surfaces of the solid wall are clean.
Fig.3.2. Real temperature profile
It can be seen that the temperature gradient is large near the wall (through the viscous sublayer), small in the turbulent core, and changes rapidly in the buffer zone (area near the interface of sublayer and bulk fluid). The reason was discussed earlier that the heat must flow through the viscous sublayer by conduction, thus a steep temperature gradient exists because of the low temperature gradient of most of the fluids.
The average temperatures of the warm bulk fluid and cold bulk fluids are slightly less than the maximum temperature T_{h} (bulk temperature of hot fluid) and slightly more than the minimum temperature T_{c} (bulk temperature of cold fluid), respectively. The average temperatures are shown by T_{1} and T_{4}, for the hot and cold fluid streams, respectively.
Figure 3.3 shows the simplified diagram of the above case, where T2 and T3 are the temperatures of the fluid wall interface.
Figure 3.3 shows the simplified diagram of the above case, where T2 and T3 are the temperatures of the fluid wall interface.
Fig.3.3. Simplified temperature profile for fig.3.2
If the thermal conductivity of the wall is k, and the area of the heat transfer is A, the electrical analogy of the fig.3.3 can be represented by fig.3.4, where h_{1} and h_{2} are the individual heat transfer coefficient of the hot and cold side of the fluid.
Fig.3.4. Equivalent electrical circuit for fig. 3.3
Considering that the heat transfer is taking place at the steadystate through a constant area and the heat loss from other faces are negligible, then the rate of heat transfer on two sides of the wall will be represented by eq. 3.43.6.
Rate of heat transfer from the hot fluid to the wall,
Rate of heat transfer through the wall,
Rate of heat transfer from the wall to cold fluid,
At steady state, the rate of heat transfers are same and can be represented by . Therefore,
On adding equations (3.7 to 3.9)
where,
Thus,
The quantity is called the overall heat transfer coefficient (can be calculated if the are known). Thus from the system described is established that the overall heat transfer coefficient is the function of individual heat transfer coefficient of the fluids on the two sides of the wall, as well as the thermal conductivity of the flat wall. The overall heat transfer coefficient can be used to introduce the controlling term concept. The controlling resistance is a term which possesses much larger thermal resistance compared to the sum of the other resistances. At this point it may be noted that in general the resistance offered by the solid wall is much lower. Similarly, if a liquid and a gas are separated by a solid wall the resistance offered by the gas film may generally be high.
3.2.2 Heat transfer between fluids separated by a cylindrical wall
In the above section we have seen that how the rate of heat transfer is calculated when the two fluids are separated by a flat wall. Another commonly encountered shape in the chemical engineering plant is the heat transfer between fluids separated by a cylindrical wall. Therefore, we will see them to understand the overall heat transfer coefficient in such a system. Consider a double pipe heat exchanger which consists of two concentric pipes arrange as per the fig. 3.4.
In the above section we have seen that how the rate of heat transfer is calculated when the two fluids are separated by a flat wall. Another commonly encountered shape in the chemical engineering plant is the heat transfer between fluids separated by a cylindrical wall. Therefore, we will see them to understand the overall heat transfer coefficient in such a system. Consider a double pipe heat exchanger which consists of two concentric pipes arrange as per the fig. 3.4.
Fig. 3.5: Schematic of a cocurrent double pipe heat exchanger
The purpose of a heat exchanger is to increase the temperature of a cold fluid and decrease that of the hot fluid which is in thermal contact, in order to achieve heat transfer.
The fig. 3.5 shows that the hot fluid passes through the inner tube and the cold fluid passes through the outer tube of the double pipe heat exchanger. The inner and the outer radii of the inner pipe are , respectively, whereas the inner radius of the outer tube is . The heat transfer coefficient of the fluid in the inner pipe is and the heat transfer coefficient of the fluid over the inner pipe is are the inner and outer wall temperatures of the inner pipe. The bulk fluid temperatures of the hot and cold fluids are , respectively, at steady state condition and assumed to be fairly constant over the length of the pipe (say L). The construction in fig. 3.6 provides a better understanding.
Fig. 3.6: Crosssection of the double pipe heat exchanger shown in fig. 3.5
The rate of heat transfer from the hot fluid to the inner surface which is at temperature
(3.12)

The rate of heat transfer through the pipe wall is,
(3.13)

(Refer to the section, heat conduction through varying area.)
The rate of heat transfer from the outer surface of the inner pipe to the cold fluid is,
The rate of heat transfers will be same, thus
Thus on rearranging above equations,
where,
If we compare the overall heat transfer coefficient shown above with the overall heat transfer coefficient discussed in eq.3.11 (for flat plate). It can be seen that due to the different inside and outside radii of the pipe, the overall heat transfer coefficient will be different. Therefore, the overall heat transfer coefficient can be defined either by U_{i} (overall heat transfer coefficient based on inside surface area) or U_{o} (overall heat transfer coefficient based on outside surface area). But it should be noted that the rate of heat transfer and the driving force remain the same. Therefore, we have
(3.19)

where,
or,
Similarly,
In terms of thermal resistance, we can use eq. 3.19
3.3 Enhanced heat transfer: concept of fins
In the previous discussion, we have seen that the heat transfer from one fluid to another fluid needs a solid boundary. The rate of heat transfer depends on many factors including the individual heat transfer coefficients of the fluids. The higher the heat transfer coefficients the higher will be the rate of heat transfer. There are many situations where the fluid does not have a high heat transfer coefficient. For example, the heat lost by conduction through a furnace wall must be dissipated to the surrounding by convection through air. The air (or the gas phase in general) has very low heat transfer coefficient, since the thermal conductivities of gases are very low, as compared to the liquid phase. Thus if we make heat transfer device for gas and a liquid (of course separated by a heat conducting wall), the gas side film will offer most of the thermal resistance as compared to the liquid side film. Therefore, to make the heat transfer most effective we need to expose higher area of the conductive wall to the gas side. This can be done by making or attaching fins to the wall of the surface. A fin (in general) is a rectangular metal strip or annular rings to the surface of heat transfer. Thus, a fin is a surface that extends from an object to increase the rate of heat transfer to or from the environment by increasing convections. Fins are sometimes known as extended surface. Figure 3.7 shows photographs of an electric motor with the fins on the motor body and a computer processor with the fins to dissipate the generated heat into the environment. Figure 3.8 shows the different types of finned surfaces.
In the previous discussion, we have seen that the heat transfer from one fluid to another fluid needs a solid boundary. The rate of heat transfer depends on many factors including the individual heat transfer coefficients of the fluids. The higher the heat transfer coefficients the higher will be the rate of heat transfer. There are many situations where the fluid does not have a high heat transfer coefficient. For example, the heat lost by conduction through a furnace wall must be dissipated to the surrounding by convection through air. The air (or the gas phase in general) has very low heat transfer coefficient, since the thermal conductivities of gases are very low, as compared to the liquid phase. Thus if we make heat transfer device for gas and a liquid (of course separated by a heat conducting wall), the gas side film will offer most of the thermal resistance as compared to the liquid side film. Therefore, to make the heat transfer most effective we need to expose higher area of the conductive wall to the gas side. This can be done by making or attaching fins to the wall of the surface. A fin (in general) is a rectangular metal strip or annular rings to the surface of heat transfer. Thus, a fin is a surface that extends from an object to increase the rate of heat transfer to or from the environment by increasing convections. Fins are sometimes known as extended surface. Figure 3.7 shows photographs of an electric motor with the fins on the motor body and a computer processor with the fins to dissipate the generated heat into the environment. Figure 3.8 shows the different types of finned surfaces.
Fig. 3.7. Cooling fins of (a) electric motor, (b) computer processor
Figure 3.9 shows a simple straight rectangular fin on plane wall. The fin is protruded a distance lfrom the wall. The temperature of the plane wall (in fact the base of the fin) is T_{w} and that of the ambient is T_{∞}. The distances of the fin are: length l; thickness t; and the breadth b. The heat is conducted through the body by conduction and dissipates to the surrounding by convection. The heat dissipation to the surrounding occurs from both top, bottom, and side surfaces of the fin. Here, it is assumed that the thickness of the fin is small and thus the temperature does not vary in the Ydirection. However, the fin temperature varies in the Xdirection only.
Fig. 3.9. 1D heat conduction and convection through a rectangular fin
Consider a thin element of thickness d_{x} of the fin at a distance x from the fin base. The energy balance on the fin element at steady state is discussed below.
where, P is the perimeter [2(b+t)] of the element, T is the local temperature of the fin, h is the film heat transfer coefficient, and bt is the fin area (A) perpendicular to the direction of heat transfer.
Thus, at steady state,
Rate of heat input – Rate of heat output – Rate of heat loss = 0
Rate of heat input – Rate of heat output – Rate of heat loss = 0
However, the other boundary conditions depend on the physical situation of the problem. A few of the typical cases are,
Case I: The fin is very long and thus the temperature at the end of the fin is same as that of the ambient fluid.
Case II: The fin is of finite length and looses heat from its end by convection.
Case I: The fin is very long and thus the temperature at the end of the fin is same as that of the ambient fluid.
Case II: The fin is of finite length and looses heat from its end by convection.
Case III: The end of the fin is insulated so that at
3.3.1 Analytical solution of the above cases
Case I:
The boundary conditions will be
Case I:
The boundary conditions will be
Using boundary conditions, the solution of the equation 3.23 becomes,
All of the heat lost by the fin must have conducted from the base at x=0. Thus, we can compute the heat loss by the fin using the equation for temperature distribution,
Similarly, for Case – II, the boundary conditions are:
The second boundary condition is a convective boundary condition which implies that the rate at which heat is conducted from inside the solid to the boundary is equal to the rate at which it is transported to the ambient fluid by convection.
The temperature profile is,
or we can write,
Therefore, the boundary conditions led to the following solution to the eq.3.23.
Thus, the heat loss by the fin, using the equation for temperature distribution can be easily found out by the following equation,
In a similar fashion we can solve the case – III also.
The boundary conditions are,
The boundary conditions are,
Thus, on solving eq.3.23,
Thus the heat loss by the fins, using the equation for temperature distribution,
It is to be noted that the general expression for the temperature gradient (eq.3.23) was developed by assuming the temperature gradient in the xdirection. It is really applicable with very less error, if the fin is sufficiently thin. However, for the practical fins the error introduced by this assumption is less than 1% only. Moreover, the practical fin calculation accuracy is limited by the uncertainties in the value of h. It is because the h value of the surrounding fluid is hardly uniform over the entire surface of the fin.
3.3.2 Fin efficiency
It was seen that the temperature of the fin decreases with distance x from the base of the body. Therefore, the driving force (temperature difference) also decreases with the length and hence the heat flux from the fin also decreases. It may also be visualized that if the thermal conductivity of the fin material is extremely high. Its thermal resistance will be negligibly small and the temperature will remain almost constant (T_{w}) throughout fin. In this condition the maximum heat transfer can be achieved and ofcourse it is an ideal condition. It is therefore, interesting and useful to calculate the efficiency of the fins.
It was seen that the temperature of the fin decreases with distance x from the base of the body. Therefore, the driving force (temperature difference) also decreases with the length and hence the heat flux from the fin also decreases. It may also be visualized that if the thermal conductivity of the fin material is extremely high. Its thermal resistance will be negligibly small and the temperature will remain almost constant (T_{w}) throughout fin. In this condition the maximum heat transfer can be achieved and ofcourse it is an ideal condition. It is therefore, interesting and useful to calculate the efficiency of the fins.
The fin efficiency may be define as,
Thus depending upon the condition, the actual heat transfer can be calculated as shown previously. As an example, for case – III (end of the fin is insulated), the rate of heat transfer was
The maximum heat would be transferred from the fin in an ideal condition in which the entire fin area was at T_{w}. In this ideal condition the heat transferred to the surrounding will be,
Therefore, under such conditions, the efficiency of the fin will be;
If the fin is quite deep as compared to the thickness, the term 2b will be very large as compared to 2t, and
The equation shows that the efficiency (from eq.3.30) of a fin which is insulated at the end can be easily calculated, which is the caseIII discussed earlier. The efficiency for the other cases may also be evaluated in a similar fashion.
The above derivation is approximately same as of practical purposes, where the amount of heat loss from the exposed end is negligible.
It can be noted that the fin efficiency is maximum for the zero length of the fin (l = 0) or if there is no fin. Therefore, we should not expect to be able to maximize fin efficiency with respect to the fin length. However, the efficiency maximization should be done with respect to the quantity of the fin material keeping economic consideration in mind.
Sometimes the performance of the fin is compared on the basis of the rate of heat transfer with the fin and without the fin as shown,
3.4 Thermal insulation
We have seen how heat transfer is important in various situations. Previous discussion indicates that we are all the time interested in the flow of the heat from one point to another point. However, there are many systems; in fact it is a part of the system, in which we are interested to minimize the losses through heat transfer. For example, in a furnace we want to have high heat transfer inside the furnace; however we do not want any heat loss through the furnace wall. Thus to prevent the heat transfer from the furnace to the atmosphere a bad heat conductor or a very good heat insulator is required. In case of furnace the wall is prepared by multiple layers of refractory materials to minimize the heat losses. Therefore, wall insulation is required in various process equipment, reactors, pipelines etc. to minimize the heat loss from the system to the environment or heat gain from the environment to the system (like cryogenic systems). However, there are situations in which we want to maximize the losses for example, insulation to electric wires.
We have seen how heat transfer is important in various situations. Previous discussion indicates that we are all the time interested in the flow of the heat from one point to another point. However, there are many systems; in fact it is a part of the system, in which we are interested to minimize the losses through heat transfer. For example, in a furnace we want to have high heat transfer inside the furnace; however we do not want any heat loss through the furnace wall. Thus to prevent the heat transfer from the furnace to the atmosphere a bad heat conductor or a very good heat insulator is required. In case of furnace the wall is prepared by multiple layers of refractory materials to minimize the heat losses. Therefore, wall insulation is required in various process equipment, reactors, pipelines etc. to minimize the heat loss from the system to the environment or heat gain from the environment to the system (like cryogenic systems). However, there are situations in which we want to maximize the losses for example, insulation to electric wires.
The petroleum conservation research association (PCRA) provides a good database on the properties and applications of industrial thermal insulations (http://www.pcra.org/English/education/literature.htm). The table 3.2 shows some common insulations used in chemical process industries for various process equipment and pipelines.
Table3.2: Thermal properties of a few of the insulations being used in the chemical process industries
Material

Temperature
(°C) 
Approximate thermal
conductivity(W/(m°C)) 
Density
(kg/m^{3}) 
Asbestos  200 to 0  0.074  469 
Glass wool  −7 to 38 38 to 93  0.031 0.041  64 64 
Fibre insulating board  21  0.049  237 
Hard rubber  0  0.151  2000 
Polyurethane foam  −170 to 110  0.018  32 
An interesting application of the heat loss from a surface of some practical significance is found in the case of insulation of cylindrical surfaces like small pipes or electrical wires. In many a cases we desire to examine the variation in heat loss from the pipe with the change in insulation thickness, assuming that the length of the pipe is fixed. As insulation is added to the pipe, the outer exposed surface temperature will decrease, but at the same time the surface area available to the convective heat dissipation will increase. Therefore, it would be interesting to study these opposing effects.
Fig. 3.10: Heat dissipation from an insulated pipe
Let us consider a thick insulation layer which is installed around a cylindrical pipe as shown in fig. 3.10 (equivalent electrical circuit is shownin figure 3.11). Let the pipe radius be R and the insulation radius is r. This (rR) will represent the thickness of the insulation. If the fluid carried by the pipe is at a temperature T and the ambient temperature is T_{a}. The insulation of the pipe will alter pipe surface temperature T in the radial direction. That is the temperature of the inner surface of the pipe and the outer surface (below insulation) of the pipe will be different. However, if the thermal resistance offered by the pipe is negligible, it can be considered that the temperature (T) is same across the pipe wall thickness and it is a common insulation case (please refer previous discussion). It can also be assumed that the heat transfer coefficient inside the pipe is very high as compared to the heat transfer coefficient at the outside of the insulated pipe. Therefore, only two major resistances in series will be available (insulation layer and gas film of the ambient).
Fig.3.11: Resistance offered by the insulation and ambient gas film
Therefore,
where, k is the thermal conductivity of the material.
On differentiating above equation with respect to r will show that the heat dissipation reaches a maximum,
On differentiating above equation with respect to r will show that the heat dissipation reaches a maximum,
So it is maxima, where the insulation radius is equal to
where, r_{c} denotes the critical radius of the insulation. The heat dissipation is maximum at r_{c} which is the result of the previously mentioned opposing effects.
Fig. 3.12: The critical insulation thickness of the pipe insulator
Therefore, the heat dissipation from a pipe increases by the addition of the insulation. However, above r_{c} the heat dissipation reduces. The same is shown in fig. 3.12.
The careful analysis of the r_{c} reveals that it is a fixed quantity determined by the thermal properties of the insulator. If R <r_{c }, then the initial addition of insulation will increase the heat loss until r =r_{c}and after which it begins to decrease. The same heat dissipation which was at bare pipe radius is again attained at r*. The critical insulation thickness may not always exist for an insulated pipe, if the values of k and h are such that the ratio k/h turns out to be less than R.
It is clear from the above discussion that the insulation above rc reduces the heat dissipation from the cylindrical surface. However, if we keep on increasing the insulation the cost of insulation also increases. Thus again there are two opposing factors that must be considered to obtain the optimum thickness. It should be calculated that what is the payback period, that is in how many years the cost of insulation is recovered by the cost of energy saving.
Fig. 3.13: Optimum insulation thickness
The optimum insulation thickness (fig. 3.13) can be determined at which the sum of the insulation cost and the cost of the heat loss is minimum.
Forced Convective Heat Transfer
4.1 Principle of convection
Till now we have understood and solved the problem where heat transfer coefficient was known. Now, we would emphasize on finding the convective heat transfer coefficient h. Finding out the heat transfer coefficient is not so easy in a given situation as it is a very complex term and depends in many physical properties of the fluid like, thermal conductivity, viscosity, density etc. Moreover, the flow field has a great impact on the convective heat transfer. The convective heat transfer requires a good knowledge of fluid dynamics, and we expect that the reader has adequate knowledge of that.
Till now we have understood and solved the problem where heat transfer coefficient was known. Now, we would emphasize on finding the convective heat transfer coefficient h. Finding out the heat transfer coefficient is not so easy in a given situation as it is a very complex term and depends in many physical properties of the fluid like, thermal conductivity, viscosity, density etc. Moreover, the flow field has a great impact on the convective heat transfer. The convective heat transfer requires a good knowledge of fluid dynamics, and we expect that the reader has adequate knowledge of that.
We have seen that it is easy to form a differential heat balance equations. However, for practical cases, it becomes tedious or impossible to solve the differential equations. Therefore, for practical situations, the heat transfer coefficient is calculated using the empirical correlations, which are developed over the years. Though these correlations have a good amount of error in most of the situations but these are indispensible for the estimation of heat transfer coefficients.
The convective heat transfer may be
 forced convection: in which the motion in the fluid medium is generated by the application of an external force, e.g. by a pump, blower, agitator etc.
 natural convection: in which the motion in the fluid is generated due to a result of density difference caused by the temperature difference.
Moreover, in many applications, heat is transferred between fluid streams without any phase change in the fluid. However, in some situations, the heat transfer is accompanied by phase change.
In this chapter we will consider the forced convection. Before we analyse the forced convection we need to know its physical mechanism.
In this chapter we will consider the forced convection. Before we analyse the forced convection we need to know its physical mechanism.
4.2 Forced convection mechanism: flow over a flat horizontal plate
Consider the flow of a fluid over an immersed flat horizontal plate of very large width, as shown in fig. 4.1. The fig. shows that the flow is fully developed with no entrance losses. The bulk flow or free stream velocity is represented by v. Velocity of the fluid at the surface of the flat plate is zero at noslip condition. The velocity of the fluid in the ydirection from the plate increases from zero to free stream velocity. The region above the plate surface within which this change of velocity from zero to the free stream value occurs is called the boundary layer. The thickness of this region is called the boundary layer thickness and is generally denoted by δ. It can be seen in the fig.4.1 that the boundary layer thickness increased with the distance x from the body edge. The boundary layer thickness, δ, usually is defined as the distance from the surface to the point where the velocity is within 1% of the free shear velocity. It should be noted that the yaxis of the fig. 4.1 is exaggerated greatly to have a clear picture.
Consider the flow of a fluid over an immersed flat horizontal plate of very large width, as shown in fig. 4.1. The fig. shows that the flow is fully developed with no entrance losses. The bulk flow or free stream velocity is represented by v. Velocity of the fluid at the surface of the flat plate is zero at noslip condition. The velocity of the fluid in the ydirection from the plate increases from zero to free stream velocity. The region above the plate surface within which this change of velocity from zero to the free stream value occurs is called the boundary layer. The thickness of this region is called the boundary layer thickness and is generally denoted by δ. It can be seen in the fig.4.1 that the boundary layer thickness increased with the distance x from the body edge. The boundary layer thickness, δ, usually is defined as the distance from the surface to the point where the velocity is within 1% of the free shear velocity. It should be noted that the yaxis of the fig. 4.1 is exaggerated greatly to have a clear picture.
Fig.4.1. Boundary layer flow past a flat plate
The velocity profile merges smoothly and asymptotically in the free shear and the boundary layer thickness is difficult to measure. However, the theoretical laminar boundary layer thickness can be calculated using the eq.4.1.
where, Re_{x} is the local Reynold number at a distance x. The derivative of δ can be found in any stated boundary.
The turbulent boundary layer thickness can be calculated using eq.4.2.
The turbulent boundary layer thickness can be calculated using eq.4.2.
Figure 4.2 shows the velocity boundary layer for the turbulent and laminar zone.
Fig.4.2: Boundary layer flow past a flat surface (a) turbulent, and (b) laminar
In continuation to the above discussion and understanding, if the solid surface is maintained at a temperature, T_{s }, which is different from the fluid temperature, T_{f }, measured at a point far away from the surface, a variation of the temperature of the fluid is observed, which is similar to the velocity variation described. That is, the fluid temperature varies from T_{s }at the wall toT_{f} far away from the wall, with most of the variation occurring close to the surface.
IfT_{s} >T_{f} , the fluid temperature approaches asymptotically and the temperature profile at a distance x is shown in fig.4.3. However, a thermal boundary may be defined (similar to velocity boundary) as the distance from the surface to the point where the temperature is within 1% of the free stream fluid temperature (T_{f }). Outside the thermal boundary layer the fluid is assumed to be a heat sink at a uniform temperature ofT_{f }. The thermal boundary layer is generally not coincident with the velocity boundary layer, although it is certainly dependant on it. That is, the velocity, boundary layer thickness, the variation of velocity, whether the flow is laminar or turbulent etc are all the factors which determine the temperature variation in the thermal boundary layer. The thermal boundary layer and velocity boundary layer are related by the Prandtl number, is called the momentum diffusivity and is called the thermal diffusivity; is less than unity, the momentum boundary layer (or velocity boundary layer) remains within the thermal boundary layer. If P_{r} >1, the boundary layers will be reversed as shown in the fig.4.4. The thermal boundary layer and velocity boundary layer coincides at P_{r} =1.
Fig.4.4: The relation of two boundary layers at different Pr numbers
The above boundary layer theory will be helpful to understand the heat transfer in the process. Through the boundary layers heat transfer is covered in a separate chapter, but the detailed derivation and development of all the relationships having engineering importance for the prediction of forced convection heat transfer coefficient is beyond the scope of the course. The reader may consult any standard fluid mechanics and heat transfer books for detailed knowledge.
The purpose of this chapter is to present a collection of the most useful of the existing relations for the most frequently encountered cases of forced convection. Some of these relations will be having theoretical bases, and some will be empirical dimensionless correlations of experimental data. In some situations, more than one relation will be given.
The discussion on heat transfer correlations consists of many dimensionless groups. Therefore, before we discuss the importance of heat transfer coefficients, it is important to understand the physical significance of these dimensionless groups, which are frequently used in forced convection heat transfer. The table 4.1 shows some of the dimensionless numbers used in the forced convection heat transfer.
Table4.1: Some important dimensionless numbers used in forced heat transfer convection
4.3 Flow through a pipe or tube
4.3.1 Turbulent flow
A classical expression for calculating heat transfer in fully developed turbulent flow in smooth tubes/pipes of diameter (d) and length (L) is given by Dittus and Boelter
A classical expression for calculating heat transfer in fully developed turbulent flow in smooth tubes/pipes of diameter (d) and length (L) is given by Dittus and Boelter
(4.3)

where,
n = 0.4, for heating of the fluid
n = 0.3, for cooling of the fluid
The properties in this equation are evaluated at the average fluid bulk temperature. Therefore, the temperature difference between bulk fluid and the wall should not be significantly high.
Application of eq. 4.3 lies in the following limits
Gnielinski suggested that better results for turbulent flow in smooth pipe may be obtained from the following relations
When the temperature difference between bulk fluid and wall is very high, the viscosity of the fluid and thus the fluid properties changes substantially. Therefore, the viscosity correction must be accounted using Sieder – Tate equation given below
However, the fluid properties have to be evaluated at the mean bulk temperature of the fluid except μ_{w} which should be evaluated at the wall temperature.
The earlier relations were applicable for fully developed flow when entrance length was negligible. Nusselt recommended the following relation for the entrance region when the flow is not fully developed.
The earlier relations were applicable for fully developed flow when entrance length was negligible. Nusselt recommended the following relation for the entrance region when the flow is not fully developed.
(4.7)

where, L is the tube length and d is the tube diameter.
The fluid properties in eq. 4.7 should be evaluated at mean bulk temperature of the fluid.
Applicability conditions, .
As different temperature terms will appear in the course therefore to understand these terms see the following details.
As different temperature terms will appear in the course therefore to understand these terms see the following details.
Bulk temperature/mixing cup temperature: Average temperature in a crosssection.
Average bulk temperature: Arithmetic average temperature of inlet and outlet bulk temperatures.
Wall temperature: Temperature of the wall.
Film temperature: Arithmetic average temperature of the wall and free stream temperature.
Free stream temperature: Temperature free from the effect of wall.
Log mean temperature difference: It will be discussed in due course of time
4.3.5 Flow across cylinders and spheres
4.3.5.1 Flow across a cylinder
The heat transfer coefficient can be found out by the correlations given by many researchers
The heat transfer coefficient can be found out by the correlations given by many researchers
Applicability of eq. 4.19: 10^{2} < Re < 10^{7}, and Re Pr >0.2 .
However, the following equation (eq. 4.20) is more accurate for the condition where 20,000 < Re < 4,00,000 and Re Pr > 0.2.
4.3.5.2. Flow across a sphere
The above correlation is applicable to both gases and liquids.
4.3.5.3 Flow over a bank of tubes
Flow over bank of tubes is one of the very important phenomena in chemical process industries. Heat exchanger, air conditioning for cooling and heating etc. involve a bank or bundle of tube over which a fluid flows. The two most common geometric arrangements of a tube bank are shown in fig. 4.5.
Flow over bank of tubes is one of the very important phenomena in chemical process industries. Heat exchanger, air conditioning for cooling and heating etc. involve a bank or bundle of tube over which a fluid flows. The two most common geometric arrangements of a tube bank are shown in fig. 4.5.
Fig.4.5: Tube banks: (a) aligned; (b) staggered
In any of the arrangements, D is the diameter of tube, S_{L} is the longitudinal spacing, and S_{T} is the transverse tube spacing.
The flow over a tube is quite different than the flow over bank of tubes. In case of bank of tube, the flow is influenced by the effects such as the “shading” of one tube by another etc. Moreover, the heat transfer for any particular tube thus not only determined by the incident fluid conditions, v_{∞} and T_{∞}, but also by D, S_{L }, and S_{T} and the tube positions in the bank. It is now clear that the heat transfer coefficient for the first row of tubes is much like that for a single cylinder in cross flow. However, the heat transfer coefficient for the tubes in the inner rows is generally larger because of the wake generation by the previous tubes.
For the heat transfer correlations, in tube banks, the Reynolds number is defined by
where v_{m} is the maximum fluid velocity occurring at the minimum vacant area of the tube bank.
For the aligned tube arrangement,
For the aligned tube arrangement,
In case of bank of tubes, generally we are interested for a single tube but interested to know the average heat transfer coefficient for the entire bank of tubes.
Zukauskas has summarized his extensive for the heat transfer coefficients for fluid past a bank of tubes,
Zukauskas has summarized his extensive for the heat transfer coefficients for fluid past a bank of tubes,
(4.27)

The applicability of eq. 4.27: , and number of tubes are atleast 20.
The constants C and m of corelation 5.26 can be found out from any standard book on heat transfer. It may be noted that the above relation is for the inner rows of bank, or for banks of many rows.
The constants C and m of corelation 5.26 can be found out from any standard book on heat transfer. It may be noted that the above relation is for the inner rows of bank, or for banks of many rows.
4.4 Momentum and heat transfer analogies
Consider a fluid flows in a circular pipe in a laminar low (fig.6.6). The wall of the pipe is maintained at T_{w} temperature, which is higher than the flowing fluid temperature. The fluid being in relatively lower temperature than the wall temperature will get heated as it flows through the pipe. Moreover, the radial transport of the momentum in the pipe occurs as per the Newton’s law of viscosity. For a circular pipe momentum transport and heat transport may be written in a similar way as shown in the eq. 4.28,
Momentum flux = momentum diffusivity × gradient of concentration of momentum
4.28(a)

It may be noted that the fluid velocity is a function of radius of the pipe.
Heat flux= thermal diffusivity × gradient of concentration of heat energy
Now, the question comes, why are we discussion about the similarities? The answer is straight forward that it is comparatively easy to experimentally/theoretically evaluate the momentum transport under various conditions. However, the heat transport is not so easy to find out. Therefore, we will learn different analogies to find the heat transport relations.
Equation 4.28 is for the laminar flow but if the flow is turbulent, eddies are generated. Eddy is a lump/chunk of fluid elements that move together. Thus it may be assumed that the eddies are the molecules of the fluid and are responsible for the transport of momentum and heat energy in the turbulent flow. Therefore, in turbulent situation the momentum and heat transport is not only by the molecular diffusion but also by the eddy diffusivities.
Thus, turbulent transport of momentum and turbulent transport of heat may be represented by eq. 4.29a and 4.29b, respectively.
The terms represent the eddy diffusivities for momentum and heat, respectively.
At the wall of the pipe, the momentum equation (eq. 4.29a) becomes,
At the wall of the pipe, the momentum equation (eq. 4.29a) becomes,
Where f is the fanning friction factor (ratio of shear force to inertial force) and is the average fluid velocity.
Equation eq.4.30 can be rearranged as,
The eq.4.32 is the dimensionless velocity gradient at the wall using momentum transport. We may get the similar relation using heat transport as shown below.
Wall heat flux can be written as,
Wall heat flux can be written as,
Where T_{av} is the wall temperature and the T_{av} is the average temperature of the fluid. Thus, the dimensionless temperature gradient at the wall using heat transfer will be,
Where the heat transfer coefficient is represented by h and dimensionless temperature is represented by .
Based on the above discussion many researchers have given their analogies. These analogies are represented in the subsequent section.
4.4.1 Reynolds analogy
Reynolds has taken the following assumptions to find the analogy between heat and momentum transport.
Reynolds has taken the following assumptions to find the analogy between heat and momentum transport.
1. Gradients of the dimensionless parameters at the wall are equal.
2. The diffusivity terms are equal. That is
2. The diffusivity terms are equal. That is
Thus if we use the above assumptions along with the eq.4.32 and 4.33,
Thus if we use the above assumptions along with the eq.4.32 and 4.33,
Equation 4.34 is known as Reynolds’s analogy.
The above relation may also be written in terms of the Darcy’s friction factor (fD) instead of fanning friction factor (f_{D} = 4f)
Where Stanton number (St) is defined as,
The above relation may also be written in terms of the Darcy’s friction factor (fD) instead of fanning friction factor (f_{D} = 4f)
Where Stanton number (St) is defined as,
The advantage of the analogy lies in that the h may not be available for certain geometries/situations however, for which f value may be available as it is easier to perform momentum transport experiments and then to calculate the f. Thus by using the eq.4.34 the h may be found out without involving into the exhaustive and difficult heat transfer experiments.
4.4.2 The ChiltonColburn analogy
The Reynolds analogy does not always give satisfactory results. Thus, Chilton and Colburn experimentally modified the Reynolds’ analogy. The empirically modified Reynolds’ analogy is known as ChiltonColburn analogy and is given by eq.4.35,
The Reynolds analogy does not always give satisfactory results. Thus, Chilton and Colburn experimentally modified the Reynolds’ analogy. The empirically modified Reynolds’ analogy is known as ChiltonColburn analogy and is given by eq.4.35,
It can be noted that for unit Prandtl number the ChiltonColburn analogy becomes Reynolds analogy.
4.4.3 The Pradntl analogy
In the turbulent core the transport is mainly by eddies and near the wall, that is laminar sublayer, the transport is by molecular diffusion. Therefore, Prandtl modified the above two analogies using universal velocity profile while driving the analogy (eq. 4.36).
In the turbulent core the transport is mainly by eddies and near the wall, that is laminar sublayer, the transport is by molecular diffusion. Therefore, Prandtl modified the above two analogies using universal velocity profile while driving the analogy (eq. 4.36).
4.4.4 The Van Karman analogy
Though Prandtl considered the laminar and turbulent laminar sublayers but did not consider the buffer zone. Thus, Van Karman included the buffer zone into the Prandtl analogy to further improve the analogy.
Heat Transfer by Natural Convection
5.1 Introduction
In the previous chapter, we have discussed about the forced convective heat transfer when the fluid motion relative to the solid surface was caused by an external input of work by means of pump, fan, blower, stirrer, etc. However, in this chapter we will discuss about the natural or free convection. In natural convection, the fluid velocity far from the solid body will be zero. However, near the solid body there will be some fluid motion if the body is at a temperature different from that of the free fluid. In this situation there will be a density difference between the fluid near the solid surface and that far away from the system. There will be a positive or negative buoyancy force due to this density difference. Hot surface will create positive buoyancy force whereas the cold surface will create the negative buoyancy force. Therefore, buoyancy force will be the driving force which produce and maintain the free convective process. Figure 5.1 shows the natural convective process for a hot and cold vertical surface.
Fig.5.1: Free convection boundary layer for vertical (a) hot surface and (b) cold surface
Consider a vertical flat plate with contact of a fluid (say liquid) on one side of the plate. Now assume that we raise the temperature of the plate to Ts, a natural convective boundary layer forms as shown in fig. 5.2. The velocity profile in this boundary layer is slightly different as compared to forced convection boundary layer. At the wall the velocity is zero because of no slip condition. The velocity increases to maximum and then reduces to zero at the end of the boundary layer because the fluid is at rest in the bulk. Initially the laminar flow is achieved in the boundary layer, but at some distance from the leading edge, depending on the fluid properties and the temperature difference between plate and bulk fluid, turbulent eddies are found thus laminar to transition region comes. On further away from the leading edge the boundary layer may become turbulent and the boundary layer instability comes in to picture. Instability of the boundary layer is quite complex and does not fall into the scope of this study material.
Fig. 5.2: Boundary layer on a hot vertical flat plate (T_{s}: surface temperature; T_{b}: bulk fluid temperature)
It has been found over the years that the average Nusselt number (or the average heat transfer coefficient) for convective heat transfer can be represented by the following functional dependence (say viscous flow past a hard body).
Nu = f(Re,Gr,Ec,Pr)

(5.1)

The Reynolds number (Re) is the ratio of inertia forces in the fluid to the viscous forces. The Grashof number (Gr) is the ratio of buoyant forces to the viscous forces. The Eckert number (Ec) is a measure of the thermal equivalent of kinetic energy of the flow to the imposed temperature differences. The Eckert number arises due to the inclusion of viscous dissipation. Thus Ec is absent where dissipation is neglected. The Prandtl number, Pr, is the ratio of the momentum diffusivity (kinematic viscosity) to the thermal diffusivity. In other words, Prandtl number is a measure of the relative magnitude of the diffusion of momentum, through viscosity, and the diffusion of heat through conduction, in the fluid.
In case of perfect naturalconvection and in absence of heat dissipation, the eq. 5.1 reduces to,
Nu = f(Gr,Pr)

(5.2)

It is to be noted that in case of perfect natural convection, the main fluid stream is absent, thus Reynolds number is no longer significant.
The dimensionless numbers involved in eq. 5.2 evaluated at the average film temperature, It can be easily found that in case of the forced convection and in absence of heat dissipation the function for average heat transfer will be,
Nu = f(Re,Pr)

(5.3)

On comparing eq. 5.2 and 5.3, one can see that the Grashof number will perform for free convection in a same way as the Reynolds number for forced convection.
Another parameter, the Rayleigh number is also used for perfect naturalconvection is defined as,
Ra = Gr . Pr

(5.4)

Thus the functional relation is eq. 5.2 can be written as,
Nu = f(Ra,Pr)

(5.5)

As discussed earlier that all free convection flows are not limited to laminar flow. If instability occurs, the problem becomes complex. A general rule one may expect that transition will occur for critical Rayleigh number of
(5.6)

The Grashof number is defined as
where,
g =  acceleration due to gravity  
β =  coefficient of volume expansion =  
T_{s} =  surface temperature  
T_{b} =  bulk fluid temperature  
L =  characteristic lenght  
v =  Momentum diffusivity (kinematic viscosity)
5.2 Empirical relations for naturalconvective heat transfer
5.2.1 Natural convection around a flat vertical surface
Churchill and Chu provided the correlation for average heat transfer coefficient for natural convection for different ranges of Rayleigh number.
It should be noted that the eq. 5.7 and 5.8 are also applicable for an inclined surface upto less than inclination from the vertical plane.
The above relations can be used for the vertical cylinder if the boundary layer thickness is quite small as compared to the diameter of the cylinder. The criteria to use the above relation for vertical cylinder is,
where, is the diameter and is the height of the cylinder.
5.2.2 Natural convection around a horizontal cylinder
Churchill and Chu has provided the following expression for naturalconvective heat transfer.
Condition of applicability of the eq.5.10:
5.2.3 Natural convection around a horizontal flat surface
In the previous case of vertical flat surface, the principal body dimension was inline with the gravity (i.e. vertical). Therefore, the flow produced by the free convection was parallel to the surface regardless of whether the surface was hotter or cooler compared to the bulk fluid around. However, in case of horizontal flat plate the flow pattern will be different and shown in fig. 5.3.
Fig. 5.3: A representative flow pattern (natural convection) for (a) hot surface down, (b) hot surface up, (c) cold surface down, and (d) cold surface up
Thus from fig. 5.3 it is understood that there are in fact two cases (i) when the heated plate facing up or cooled plate facing down, and (ii) heated plate facing down or cooled plate facing up.
where, L_{c} is characteristic length defined as below.
5.2.4 Natural convection around sphere
Churchill proposed,
Condition for applicability: Pr ≥ 0.7; Ra ≤ 10^{11}
^{}
5.2.5 Natural convection in enclosure
It is another class of problems for which there are many cases and their corresponding correlations are also available in the literature. Here two cases will be discussed, (i) in which a fluid is contained between two vertical plates separated by a distance d, (ii) the other where the fluid is in an annulus formed by two concentric horizontal cylinders.
In the case first, the plates are at different temperature, T_{1} and T_{2}. Heat transfer will be from higher temperature (T_{1}) to lower temperature (T_{2}) through the fluid.
The corresponding Grashof number will be
McGregor and Emery proposed the following correction for free convection heat transfer in a vertical rectangular enclosure, where the vertical walls are heated or cooled and the horizontal surfaces many be assumed adiabatic,
Applicability conditions for the above equation are,
or,
Applicability conditions are,
Here L/d is known as the aspect ratio.
At steady state condition, the heat flux (q_{x}) is equal thus,
where, k_{c}(= Nu_{x}k) is known as the apparent thermal conductivity.
In the second case the heat transfer is involved in the enclosure formed by two concentric cylinders in horizontal position, the correlation given by Raithby and Holland,
is the modified Rayleigh number given by,
where, d_{i} and d_{0} are the outer and inner diameter of the inner and outer cylinders, respectively. The enclosure characteristic length l is defined as (d_{0}  d_{i}).
The applicability of the eq. 5.17 is 10^{2} < > 10^{7}.
It should be noted that the rate of heat flow by natural convection per unit length is same as that through the annular cylindrical region having effective thermal conductivity k_{e} for the case,
where, T_{1} and T_{2} are the temperatures of the inner and outer cylindrical walls, respectively.
Heat Transfer in Boiling and Condensation
Heat transfer in boiling and condensation
In the previous chapter we have discussed about the convective heat transfer in which the homogeneous single phase system was considered. The heat transfer processes associated with the change of fluid phase have great importance in chemical process industries. In this chapter, we will focus our attention towards the phase change from liquid to vapour and viceversa.
6.1 Heat transfer during boiling
The conversion of a liquid into a vapour is one of the important and obvious phenomena. It has been found that if water (say) is totally distilled and degassed so that it does not have any impurity or dissolved gases, it will undergo liquid to vapour phase change without the appearance of bubbles, when it is heated in a clean and smooth container. However, in normal situation, as can be understood, the presence of impurities, dissolved gases, and surface irregularities causes the appearance of vapour bubble on the heating surface, when the rate of heat input is significantly high.
The boiling may be in general of two types. The one in which the heating surface is submerged in a quiescent part of liquid, and the heat transfer occur by free convection and bubble agitation. The process is known as pool boiling. The pool boiling may further be divided into subcooled or local boiling and saturated or bulk boiling. If the temperature of the liquid is below the saturation temperature, the process is known as subcooled, or local, boiling. If the liquid is maintained at saturation temperature, the process is known as saturated or bulk boiling.
The other form of the boiling is known as forced convective boiling in which the boiling occurs simultaneously with fluid motion induced by externally imposed pressure difference. In this chapter, we will mostly consider the pool boiling.
As generally the bubbles are formed during boiling, we will first refresh the following basic information. Consider a spherical bubble of radius in a liquid as shown in fig. 6.1
Fig. 6.1: Force balance on a submerge spherical bubble in a liquid
The pressure of vaporisation inside the bubble, P_{vap}, must exceed that in the surrounding liquid, P_{liq}, because of the surface tension (σ) acting on the liquidvapour interface.
The force balance on the equatorial plane
The eq. 6.1 shows that to create a bubble of small radius, it would be necessary to develop very large pressure in the vapour. In other word, a high degree super heat is necessary for the generation of a tiny bubble (or nucleus) in the bulk liquid. This is the reason, the bubble are usually formed at bits existing in the surface irregularities, where a bubble of finite initial radius may form, or where gasses dissolved in the system of the liquid come out of the solution.
6.2 Boiling of saturated liquid
In this section, we will discuss about the boiling curve which is a loglog plot between heat flux (q/A) or heat transfer coefficient (h) and excess temperature (ΔT). Excess temperature (ΔT = T_{w}  T_{sat}) is the temperature difference between heating surface (T_{w}) and saturated temperature of the liquid (T_{sat}).
Figure 6.2 shows a typical representative pool boiling curves for water contained in a container where the water is heated by an immersed horizontal wire. Consider we are measuring the heat flux (thus, h) and the temperature difference (ΔT) between the boiling water (T_{sat}) and the wall temperature of the heater wire (T_{w}). The temperature of the heater wire is gradually raised while measuring the heat flux between heated surface and boiling water until a large value of ΔT reaches. The corresponding plot is prepared at the loglog scale. The plot shows six different sections in the boiling curve shown in the fig.6.2.
Fig. 6.2: Saturated water boiling curve
The different regimes of the boiling plot (fig. 6.2) have different mechanism. We will see those mechanisms inbrief in the following section.
Section PQ: In section PQ, initially when the wire temperature is slightly above the saturation temperature of the liquid, the liquid in contact with the heating surface get slightly superheated. The free convection of this heated fluid element is responsible for motion of the fluid, and it subsequently evaporated when it rises to the surface. This regime is called the interfacial evaporation regime.
Section QS: The section QS is composed of section QR and section RS. In QR section, bubbles begin to form on the surface of the wire and are dissipated in the liquid after detaching from the heating surface. If the excess temperature ( further increases, bubbles form rapidly on the surface of the heating wire, and released from it, rise to the surface of the liquid, and are discharged into the top of the water surface (fig 6.3). This particular phenomenon is shown in section RS. Near the point S, the vapour bubbles rise as columns and bigger bubbles are formed. The vapour bubbles break and coalesce thus an intense motion of the liquid occurs which inturn increases the heat transfer coefficient or heat transfer flux to the liquid from the heating wire. The section QS is known as nucleate boiling.
Fig. 6.3: (a) Formation of tiny bubbles, and (b) Grown up bubbles
Section ST: At the beginning of the section ST or at the end of the section , the maximum number of bubbles are generated from the heating surface. The bubbles almost occupy the full surface of the heating wire. Therefore, the agitation becomes highest as they discharge from the surface. Thus, maximum heat transfer coefficient is obtained at point S. However, once the population of the bubbles reaches to maximum, the nearby bubbles coalesce and eventually a film of vapour forms on the heating surface. This layer is highly unstable and it forms momentarily and breaks. This is known as transition boiling (nucleate to film). In this situation the vapour film (unstable) imparts a thermal resistance and thus the heat transfer coefficient reduces rapidly.
Section TU: If the excess temperature is further increased, the coalesced bubbles form so rapidly that they blanket the heating surface (stable vapour film) and prevent the inflow of fresh liquid from taking their place. The heat conducts only by the conduction through this stable vapour film. As a result the flux of heat transfer decreases continuously and reaches a minimum at point U. All the resistance to the heat transfer is imposed by this layer stable layer of vapour film.
Section UV: At very high excess temperature the heat transfer is facilitated by the radiation through the vapour film and thus the heat transfer coefficient start increasing. Infact the excess temperature in this regime is so high that the heating wire may get melted. This situation is known as boiling crises. The combine regime of ST, TU, and UV is known as film boiling regime.
At this stage it would be interesting to know the Leidenfrost phenomenon, which was observed by Leidenfrost in 1756. When water droplets fall on a very hot surface they dance and jump on the hot surface and reduces in size and eventually the droplets disappear. The mechanism is related to the film boiling of the water droplets. When water droplet drops on to the very hot surface, a film of vapour forms immediately between the droplet and the hot surface. The vapour film generated provide and upthrust to the droplet. Therefore, the droplet moves up and when again the droplet comes in the contact of the hot surface, the vapour generated out of the water droplet and the phenomenon continues till it disappears.
The effectiveness of nucleate boiling depends primarily on the ease with which bubbles form and free themselves from the heating surface. The important factor in controlling the rate of bubble detachment is the interfacial tension between the liquid and the heating surface. If this interfacial tension is large the bubbles tends to spread along the surface and blocked the heat transfer area, rather than leaving the surface, to make room for other bubbles. The heat transfer coefficient obtained during the nucleation boiling is sensitive to the nature of the liquid, the type and condition of the heating surface, the composition and purity of the liquid, agitation, temperature and pressure.
Fact: Film boiling is not normally desired in commercial equipment because the heat transfer rate is low for such a large temperature drop.
6.2.1 Nucleation boiling
Rohsenow correlation may be used for calculating pool boiling heat transfer
where,
C_{sf} and n are the constants and depend on the liquid and heating surface combination for boiling operation, for example,
All the properties are to be evaluated at film temperature.
6.2.2 Maximum heat flux
Maximum heat flux corresponding to the point S in the fig.6.2 can be found by Leinhard correlation,
The notations are same as for eq.6.2.
6.2.3 Film boiling
where, k_{v} is the thermal conductivity of the vapour, µ_{v} is the viscosity of the vapour, d is the characteristic length (tube diameter or height of the vertical plate), other notations are same as for eq. 6.2.
If the surface temperature is high enough to consider the contribution of radiative heat transfer, the total heat transfer coefficient may be calculation by,
where, h_{r} is the radiative heat transfer coefficient and is given in eq.6.4.
Upto this section, we have discussed about the boiling phenomenon where the liquid phase changes to vapour phase. In the subsequent sections, we will study the opposite phenomena of boiling that is condensation, where the vapour phase changes to the liquid phase.
6.3 Heat transfer during condensation
Condensation of vapours on the surfaces cooler than the condensing temperature of the vapour is an important phenomenon in chemical process industries like boiling phenomenon. It is quite clear that in condensation the phase changes from vapour to liquid. Consider a vertical flat plate which is exposed to a condensable vapour. If the temperature of the plate is below the saturation temperature of the vapour, condensate will form on the surface and flows down the plate due to gravity. It is to be noted that a liquid at its boiling point is a saturated liquid and the vapour in equilibrium with the saturated liquid is saturated vapour. A liquid or vapour above the saturation temperature is called superheated. If the noncondensable gases will present in the vapour the rate of condensation of the vapour will reduce significantly.
Condensation may be of two types, film condensation and dropwise condensation. If the liquid (condensate) wets the surface, a smooth film is formed and the process is called film type condensation. In this process, the surface is blocked by the film, which grows in thickness as it moves down the plate. A temperature gradient exists in the film and the film represents thermal resistance in the heat transfer. The latent heat is transferred through the wall to the cooling fluid on the other side of the wall. However, if the liquid does not wet the system, drops are formed on the surface in some random fashion. This process is called dropwise condensation. Some of the surface will always be free from the condensate drops (for a reasonable time period).
Now, with the help of the above discussion one can easily understand that the condensate film offers significant heat transfer resistance as compared to dropwise condensation. In dropwise condensation the surface is not fully covered by the liquid and exposed to the vapour for the condensation. Therefore, the heat transfer coefficient will be higher for dropwise condensation. Thus the dropwise condensation is preferred over the film condensation. However, the dropwise condensation is not practically easy to achieve. We have to put some coating on the surface or we have to add some additive to the vapour to have dropwise condensation. Practically, these techniques for dropwise condensation are not easy for the sustained dropwise condensation. Because of these reasons, in many instances we assume film condensation because the film condensation sustained on the surface and it is comparatively easy to quantify and analyse.
6.4 Film condensation on a vertical flat plate
Figure 6.4 shows a vertical wall very long in zdirection. The wall is exposed to a condensable vapour. The condensate film is assumed to be fully developed laminar flow with zero interfacial shear and constant liquid properties. It is also assumed that the vapour is saturated and the heat transfer through the condensate film occurs by condensation only and the temperature profile is assumed to be linear.
Fig. 6.4: Condensation of film in laminar flow
The wall temperature is maintained at temperature T_{w} and the vapour temperature at the edge of the film is the saturation temperature T_{v}. The condensate film thickness is represented by δ_{x}, a function of x. A fluid element of thickness dx was assumed with a unit width in the zdirection.
The force balance on the element provides,
F_{1} = F_{2}  F_{3}
where, shear force is the viscosity of the condensate (liquid). In the subsequent sections of this module, the subscripts l and v will represent liquid and vapour phase.
Thus,
On integrating for the following boundary condition, u = 0 at y = 0; no slip condition.
Equation 6.6 shows the velocity profile in the condensate falling film.
The corresponding mass flow rate of the condensate for dy thickness and unit width of the film,
where dy is the length of the volume element at y distance. The rate of condensation for dx.1 (over element surface) area exposed to the vapour can be obtained from the rate of heat transfer through this area.
The rate of heat transfer
The thermal conductivity of the liquid is represented by k_{l}. The above rate of heat transfer is due to the latent heat of condensation of the vapour. Thus,
The specific latent heat of condensation is represented by λ. On solving eqs.6.7 and 6.8, for boundary layer conditions (x = 0; δ_{x} = 0)
The eq. 6.9 gives the local condensate film thickness at any location x. If h is the film heat transfer coefficient for the condensate film, heat flux through the film at any location is,
The local Nusselt number will be,
We can also calculate the average heat transfer coefficient along the length of the surface,
In eq. 6.10, the liquid properties can be taken at the mean film temperature The equation 6.10 is applicable for Pr > 0.5 and ≤ 1.0
It can also be understood that at any location on the plate the liquid film temperature changes from T_{v} to T_{w}. It indicates that apart from latent heat some amount of sensible heat will also be removed. Thus, to take this into account and to further improve the accuracy of Nusselt’s equation (eq. 6.10), a modified latent heat term can be used in place of λ. The term J_{a} is called the Jacob number as is defined by eq. 6.11. All the properties are to be evaluated at film temperature.
In the previous discussion we have not discussed about the ripples or turbulent condition of the condensate film as it grows while coming down from the vertical wall. The previous discussion was applicable only when the flow in the condensate film was 1D and the velocity profile was half parabolic all along the length of the wall. However, if the rate of condensation is high or the height of the condensing wall is more, the thickness of the condensate film neither remains small nor the flow remains laminar.
The nature of the flow is determined by the film Reynolds number (Re_{f}). The local average liquid velocity in the film can be obtained by eq. 6.6.
Now, the Re_{f} can be calculated by,
where D is the hydraulic diameter of the condensate film. The hydraulic diameter can be calculated by the flow area (δ_{x}.1) and wetted perimeter (unit breadth, thus 1). It has been found that, if
Case 1: Re_{f} ≤ 30; the film remains laminar and the free surface of the film remains wave free.
Case 2: 30 < Re_{f} < 1600; the film remains laminar but the waves and ripples appear on the surface. Case 3: Re_{f} ≥ 1600; the film becomes turbulent and surface becomes wavy.
The corresponding average heat transfer coefficient can be calculated by the following correlation,
The Nusselt number in case1 is defined as Modified Nusselt number or condensation number (Co).
The above relations may also be used for condensation inside or outside of a vertical tube if the tube diameter is very large in comparison to condensate film thickness. Moreover, the relations are valid for the tilted surfaces also. If the surface make an angle “θ” from the vertical plane the “g” will be replaced by “g.cosθ” in the above equations
Radiation Heat Transfer
In the previous chapters it has been observed that the heat transfer studies were based on the fact that the temperature of a body, a portion of a body, which is hotter than its surroundings, tends to decrease with time. The decrease in temperature indicates a flow of energy from the body. In all the previous chapters, limitation was that a physical medium was necessary for the transport of the energy from the high temperature source to the low temperature sink. The heat transport was related to conduction and convection and the rate of heat transport was proportional to the temperature difference between the source and the sink.
Now, if we observe the heat transfer from the Sun to the earth atmosphere, we can understand that there is no medium exists between the source (the Sun) and the sink (earth atmosphere). However, still the heat transfer takes place, which is entirely a different energy transfer mechanism takes place and it is called thermal radiation.
Thermal radiation is referred when a body is heated or exhibits the loss of energy by radiation. However, more general form “radiation energy” is used to cover all the other forms. The emission of other form of radiant energy may be caused when a body is excited by oscillating electrical current, electronic bombardment, chemical reaction etc. Moreover, when radiation energy strikes a body and is absorbed, it may manifest itself in the form of thermal internal energy, a chemical reaction, an electromotive force, etc. depending on the nature of the incident radiation and the substance of which the body is composed.
In this chapter, we will concentrate on thermal radiation (emission or absorption) that on radiation produced by or while produces thermal excitation of a body.
There are many theories available in literature which explains the transport of energy by radiation. However, a dual theory is generally accepted which enables to explain the radiant energy in the characterisation of a wave motion (electromagnetic wave motion) and discontinuous emission (discrete packets or quanta of energy).
An electromagnetic wave propagates at the speed of light (3×10^{8} m/s). It is characterised by its wavelength λ or its frequency ν related by
Emission of radiation is not continuous, but occurs only in the form of discrete quanta. Each quantum has energy
where, = 6.6246×10^{34} J.s, is known as Planck’s constant.
Table 7.1 shows the electromagnetic radiation covering the entire spectrum of wavelength
Table 7.1: Electromagnetic radiation for entire spectrum of wavelength
It is to be noted that the above band is in approximate values and do not have any sharp boundary.
7.1 Basic definition pertaining to radiation
Before we further study about the radiation it would be better to get familiarised with the basic terminology and properties of the radiant energy and how to characterise it.
As observed in the table 7.1 that the thermal radiation is defined between wavelength of about 1×10^{1} and 1×10^{2} μm of the electromagnetic radiation. If the thermal radiation is emitted by a surface, which is divided into its spectrum over the wavelength band, it would be found that the radiation is not equally distributed over all wavelength. Similarly, radiation incident on a system, reflected by a system, absorbed by a system, etc. may be wavelength dependent. The dependence on the wavelength is generally different from case to case, system to system, etc. The wavelength dependency of any radiative quantity or surface property will be referred to as a spectral dependency. The radiation quantity may be monochromatic (applicable at a single wavelength) or total (applicable at entire thermal radiation spectrum). It is to be noted that radiation quantity may be dependent on the direction and wavelength both but we will not consider any directional dependency. This chapter will not consider directional effect and the emissive power will always used to be (hemispherical) summed overall direction in the hemisphere above the surface.
7.1.1 Emissive power
It is the emitted thermal radiation leaving a system per unit time, per unit area of surface. The total emissive power of a surface is all the emitted energy, summed over all the direction and all wavelengths, and is usually denoted as E. The total emissive power is found to be dependent upon the temperature of the emitting surface, the subsystem which this system is composed, and the nature of the surface structure or texture.
The monochromatic emissive power E_{λ}, is defined as the rate, per unit area, at which the surface emits thermal radiation at a particular wavelength λ. Thus the total and monochromatic hemispherical emissive power are related by
and the functional dependency of E_{λ} on λ must be known to evaluate E.
7.1.2 Radiosity
It is the term used to indicate all the radiation leaving a surface, per unit time and unit area.
where, J and J_{λ} are the total and monochromatic radiosity.
The radiosity includes reflected energy as well as original emission whereas emissive power consists of only original emission leaving the system. The emissive power does not include any energy leaving a system that is the result of the reflection of any incident radiation.
7.1.3 Irradiation
It is the term used to denote the rate, per unit area, at which thermal radiation is incident upon a surface (from all the directions). The irradiative incident upon a surface is the result of emission and reflection from other surfaces and may thus be spectrally dependent.
where, G and G_{λ} are the total and monochromatic irradiation.
Reflection from a surface may be of two types specular or diffusive as shown in fig.7.1.
Fig. 7.1: (a) Specular, and (b) diffusive radiation
Thus,
7.1.4 Absorptivity, reflectivity, and transmitting
The emissive power, radiosity, and irradiation of a surface are interrelated by the reflective, absorptive, and transmissive properties of the system. When thermal radiation is incident on a surface, a part of the radiation may be reflected by the surface, a part may be absorbed by the surface and a part may be transmitted through the surface as shown in fig.7.2. These fractions of reflected, absorbed, and transmitted energy are interpreted as system properties called reflectivity, absorptivity, and transmissivity, respectively.
Fig. 7.2: Reflection, absorption and transmitted energy
Thus using energy conservation,
where, are total reflectivity, total absorptivity, and total transmissivity. The subscript λ indicates the monochromatic property.
In general the monochromatic and total surface properties are dependent on the system composition, its roughness, and on its temperature.
Monochromatic properties are dependent on the wavelength of the incident radiation, and the total properties are dependent on the spectral distribution of the incident energy.
Most gases have high transmissivity, i.e. (like air at atmospheric pressure). However, some other gases (water vapour, CO_{2} etc.) may be highly absorptive to thermal radiation, at least at certain wavelength.
Most solids encountered in engineering practice are opaque to thermal radiation Thus for thermally opaque solid surfaces,
Another important property of the surface of a substance is its ability to emit radiation. Emission and radiation have different concept. Reflection may occur only when the surface receives radiation whereas emission always occurs if the temperature of the surface is above the absolute zero. Emissivity of the surface is a measure of how good it is an emitter.
7.2 Blackbody radiation
In order to evaluate the radiation characteristics and properties of a real surface it is useful to define an ideal surface such as the perfect blackbody. The perfect blackbody is defined as one which absorbs all incident radiation regardless of the spectral distribution or directional characteristic of the incident radiation.
A blackbody is black because it does not reflect any radiation. The only radiation leaving a blackbody surface is original emission since a blackbody absorbs all incident radiation. The emissive power of a blackbody is represented by , and depends on the surface temperature only.
Fig. 7.3: Example of a near perfect blackbody
It is possible to produce a near perfect blackbody as shown in fig.7.3.
Figure 7.2 shows a cavity with a small opening. The body is at isothermal state, where a ray of incident radiation enters through the opening will undergo a number of internal reflections. A portion of the radiation absorbed at each internal reflection and a very little of the incident beam ever find the way out through the small hole. Thus, the radiation found to be evacuating from the hole will appear to that coming from a nearly perfect blackbody.
7.2.1 Planck’s law
A surface emits radiation of different wavelengths at a given temperature (theoretically zero to infinite wavelengths). At a fixed wavelength, the surface radiates more energy as the temperature increases. Monochromatic emissive power of a blackbody is given by eq.7.10.
Equation 7.10 is known as Planck’s law. Figure 7.4 shows the representative plot for Planck’s distribution.
Fig. 7.4: Representative plot for Planck’s distribution
7.2.2 Wien’s law
Figure 7.4 shows that as the temperature increases the peaks of the curve also increases and it shift towards the shorter wavelength. It can be easily found out that the wavelength corresponding to the peak of the plot (λ_{max}) is inversely proportional to the temperature of the blackbody (Wein’s law) as shown in eq. 7.11.
Now with the Wien’s law or Wien’s displacement law, it can be understood if we heat a body, initially the emitted radiation does not have any colour. As the temperature rises the λ of the radiation reach the visible spectrum and we can able to see the red colour being height λ (for red colour). Further increase in temperature shows the white colour indicating all the colours in the light.
7.2.3 The StefanBoltzmann law for blackbody
Josef Stefan based on experimental facts suggested that the total emissive power of a blackbody is proportional to the fourth power of the absolute temperature. Later, Ludwig Boltzmann derived the same using classical thermodynamics. Thus the eq. 7.12 is known as StefanBoltzmann law,
where, E_{b} is the emissive power of a blackbody, T is absolute temperature, and σ (= 5.67 X 10^{8}W/m^{2}/K^{4}) is the StefanBoltzmann constant.
The StefanBoltzmann law for the emissive power gives the total energy emitted by a blackbody defined by eq.7.3.
7.2.4 Special characteristic of blackbody radiation
It has been shown that the irradiation field in an isothermal cavity is equal to E_{b}. Moreover, the irradiation was same for all planes of any orientation within the cavity. It may then be shown that the intensity of the blackbody radiation, I_{b}, is uniform. Thus, blackbody radiation is defined as,
where, is the total intensity of the radiation and is called the spectral radiation intensity of the blackbody.
7.2.5 Kirchhoff’s law
Consider an enclosure as shown in fig.7.2 and a body is placed inside the enclosure. The radiant heat flux (q) is incident onto the body and allowed to come into temperature equilibrium. The rate of energy absorbed at equilibrium by the body must be equal to the energy emitted.
where, E is the emissive power of the body, is absorptivity of the of the body at equilibrium temperature, and A is the area of the body.
Now consider the body is replaced by a blackbody i.e. E → E_{b} and= 1, the equation 7.14 becomes
Dividing eq. 7.14 by eq.7.15,
At this point we may define emissivity, which is a measure of how good the body is an emitter as compared to blackbody. Thus the emissivity can be written as the ratio of the emissive power to that of a blackbody,
On comparing eq.7.16 and eq.7.17, we get
Equation 7.18 is the Kirchhoff’s law, which states that the emissivity of a body which is in thermal equilibrium with its surrounding is equal to its absorptivity of the body. It should be noted that the source temperature is equal to the temperature of the irradiated surface. However, in practical purposes it is assumed that emissivity and absorptivity of a system are equal even if it is not in thermal equilibrium with the surrounding. The reason being the absorptivity of most real surfaces is relatively insensitive to temperature and wavelength. This particular assumption leads to the concept of grey body. The emissivity is considered to be independent of the wavelength of radiation for grey body.
7.3 Grey body
If grey body is defined as a substance whose monochromatic emissivity and absorptivity are independent of wavelength. A comparative study of grey body and blackbody is shown in the table 7.2.
Table7.2: Comparison of grey and blackbody
Illustration 7.1
The surface of a blackbody is at 500 K temperature. Obtain the total emissive power, the wavelength of the maximum monochromatic emissive power.
Solution 7.1
Using eq. 7.12, the total emissive power can be calculated,
E_{b} = σT^{4}
where, σ (= 5.67 X 10^{8} W/m^{2}/K^{4}) is the StefanBoltzmann constant. Thus at 500 K,
E_{b} = (5.67 X 10^{8})(5000^{4}) W/m^{2}
E_{b} = 354.75 W/m^{2}
The wavelength of the maximum monochromatic emissive power can be obtained from the Wien’s law (eq. 7.11),
λ_{max}T = 2898
In the previous chapters it has been observed that the heat transfer studies were based on the fact that the temperature of a body, a portion of a body, which is hotter than its surroundings, tends to decrease with time. The decrease in temperature indicates a flow of energy from the body. In all the previous chapters, limitation was that a physical medium was necessary for the transport of the energy from the high temperature source to the low temperature sink. The heat transport was related to conduction and convection and the rate of heat transport was proportional to the temperature difference between the source and the sink.
Now, if we observe the heat transfer from the Sun to the earth atmosphere, we can understand that there is no medium exists between the source (the Sun) and the sink (earth atmosphere). However, still the heat transfer takes place, which is entirely a different energy transfer mechanism takes place and it is called thermal radiation.
Thermal radiation is referred when a body is heated or exhibits the loss of energy by radiation. However, more general form “radiation energy” is used to cover all the other forms. The emission of other form of radiant energy may be caused when a body is excited by oscillating electrical current, electronic bombardment, chemical reaction etc. Moreover, when radiation energy strikes a body and is absorbed, it may manifest itself in the form of thermal internal energy, a chemical reaction, an electromotive force, etc. depending on the nature of the incident radiation and the substance of which the body is composed.
In this chapter, we will concentrate on thermal radiation (emission or absorption) that on radiation produced by or while produces thermal excitation of a body.
There are many theories available in literature which explains the transport of energy by radiation. However, a dual theory is generally accepted which enables to explain the radiant energy in the characterisation of a wave motion (electromagnetic wave motion) and discontinuous emission (discrete packets or quanta of energy).
An electromagnetic wave propagates at the speed of light (3×10^{8} m/s). It is characterised by its wavelength λ or its frequency ν related by
Emission of radiation is not continuous, but occurs only in the form of discrete quanta. Each quantum has energy
where, = 6.6246×10^{34} J.s, is known as Planck’s constant.
Table 7.1 shows the electromagnetic radiation covering the entire spectrum of wavelength
Table 7.1: Electromagnetic radiation for entire spectrum of wavelength
It is to be noted that the above band is in approximate values and do not have any sharp boundary.
7.1 Basic definition pertaining to radiation
Before we further study about the radiation it would be better to get familiarised with the basic terminology and properties of the radiant energy and how to characterise it.
As observed in the table 7.1 that the thermal radiation is defined between wavelength of about 1×10^{1} and 1×10^{2} μm of the electromagnetic radiation. If the thermal radiation is emitted by a surface, which is divided into its spectrum over the wavelength band, it would be found that the radiation is not equally distributed over all wavelength. Similarly, radiation incident on a system, reflected by a system, absorbed by a system, etc. may be wavelength dependent. The dependence on the wavelength is generally different from case to case, system to system, etc. The wavelength dependency of any radiative quantity or surface property will be referred to as a spectral dependency. The radiation quantity may be monochromatic (applicable at a single wavelength) or total (applicable at entire thermal radiation spectrum). It is to be noted that radiation quantity may be dependent on the direction and wavelength both but we will not consider any directional dependency. This chapter will not consider directional effect and the emissive power will always used to be (hemispherical) summed overall direction in the hemisphere above the surface.
7.1.1 Emissive power
It is the emitted thermal radiation leaving a system per unit time, per unit area of surface. The total emissive power of a surface is all the emitted energy, summed over all the direction and all wavelengths, and is usually denoted as E. The total emissive power is found to be dependent upon the temperature of the emitting surface, the subsystem which this system is composed, and the nature of the surface structure or texture.
The monochromatic emissive power E_{λ}, is defined as the rate, per unit area, at which the surface emits thermal radiation at a particular wavelength λ. Thus the total and monochromatic hemispherical emissive power are related by
and the functional dependency of E_{λ} on λ must be known to evaluate E.
7.1.2 Radiosity
It is the term used to indicate all the radiation leaving a surface, per unit time and unit area.
where, J and J_{λ} are the total and monochromatic radiosity.
The radiosity includes reflected energy as well as original emission whereas emissive power consists of only original emission leaving the system. The emissive power does not include any energy leaving a system that is the result of the reflection of any incident radiation.
7.1.3 Irradiation
It is the term used to denote the rate, per unit area, at which thermal radiation is incident upon a surface (from all the directions). The irradiative incident upon a surface is the result of emission and reflection from other surfaces and may thus be spectrally dependent.
where, G and G_{λ} are the total and monochromatic irradiation.
Reflection from a surface may be of two types specular or diffusive as shown in fig.7.1.
Fig. 7.1: (a) Specular, and (b) diffusive radiation
Thus,
7.1.4 Absorptivity, reflectivity, and transmitting
The emissive power, radiosity, and irradiation of a surface are interrelated by the reflective, absorptive, and transmissive properties of the system. When thermal radiation is incident on a surface, a part of the radiation may be reflected by the surface, a part may be absorbed by the surface and a part may be transmitted through the surface as shown in fig.7.2. These fractions of reflected, absorbed, and transmitted energy are interpreted as system properties called reflectivity, absorptivity, and transmissivity, respectively.
Fig. 7.2: Reflection, absorption and transmitted energy
Thus using energy conservation,
where, are total reflectivity, total absorptivity, and total transmissivity. The subscript λ indicates the monochromatic property.
In general the monochromatic and total surface properties are dependent on the system composition, its roughness, and on its temperature.
Monochromatic properties are dependent on the wavelength of the incident radiation, and the total properties are dependent on the spectral distribution of the incident energy.
Most gases have high transmissivity, i.e. (like air at atmospheric pressure). However, some other gases (water vapour, CO_{2} etc.) may be highly absorptive to thermal radiation, at least at certain wavelength.
Most solids encountered in engineering practice are opaque to thermal radiation Thus for thermally opaque solid surfaces,
Another important property of the surface of a substance is its ability to emit radiation. Emission and radiation have different concept. Reflection may occur only when the surface receives radiation whereas emission always occurs if the temperature of the surface is above the absolute zero. Emissivity of the surface is a measure of how good it is an emitter.
7.2 Blackbody radiation
In order to evaluate the radiation characteristics and properties of a real surface it is useful to define an ideal surface such as the perfect blackbody. The perfect blackbody is defined as one which absorbs all incident radiation regardless of the spectral distribution or directional characteristic of the incident radiation.
A blackbody is black because it does not reflect any radiation. The only radiation leaving a blackbody surface is original emission since a blackbody absorbs all incident radiation. The emissive power of a blackbody is represented by , and depends on the surface temperature only.
Fig. 7.3: Example of a near perfect blackbody
It is possible to produce a near perfect blackbody as shown in fig.7.3.
Figure 7.2 shows a cavity with a small opening. The body is at isothermal state, where a ray of incident radiation enters through the opening will undergo a number of internal reflections. A portion of the radiation absorbed at each internal reflection and a very little of the incident beam ever find the way out through the small hole. Thus, the radiation found to be evacuating from the hole will appear to that coming from a nearly perfect blackbody.
7.2.1 Planck’s law
A surface emits radiation of different wavelengths at a given temperature (theoretically zero to infinite wavelengths). At a fixed wavelength, the surface radiates more energy as the temperature increases. Monochromatic emissive power of a blackbody is given by eq.7.10.
Equation 7.10 is known as Planck’s law. Figure 7.4 shows the representative plot for Planck’s distribution.
Fig. 7.4: Representative plot for Planck’s distribution
7.2.2 Wien’s law
Figure 7.4 shows that as the temperature increases the peaks of the curve also increases and it shift towards the shorter wavelength. It can be easily found out that the wavelength corresponding to the peak of the plot (λ_{max}) is inversely proportional to the temperature of the blackbody (Wein’s law) as shown in eq. 7.11.
Now with the Wien’s law or Wien’s displacement law, it can be understood if we heat a body, initially the emitted radiation does not have any colour. As the temperature rises the λ of the radiation reach the visible spectrum and we can able to see the red colour being height λ (for red colour). Further increase in temperature shows the white colour indicating all the colours in the light.
7.2.3 The StefanBoltzmann law for blackbody
Josef Stefan based on experimental facts suggested that the total emissive power of a blackbody is proportional to the fourth power of the absolute temperature. Later, Ludwig Boltzmann derived the same using classical thermodynamics. Thus the eq. 7.12 is known as StefanBoltzmann law,
where, E_{b} is the emissive power of a blackbody, T is absolute temperature, and σ (= 5.67 X 10^{8}W/m^{2}/K^{4}) is the StefanBoltzmann constant.
The StefanBoltzmann law for the emissive power gives the total energy emitted by a blackbody defined by eq.7.3.
7.2.4 Special characteristic of blackbody radiation
It has been shown that the irradiation field in an isothermal cavity is equal to E_{b}. Moreover, the irradiation was same for all planes of any orientation within the cavity. It may then be shown that the intensity of the blackbody radiation, I_{b}, is uniform. Thus, blackbody radiation is defined as,
where, is the total intensity of the radiation and is called the spectral radiation intensity of the blackbody.
7.2.5 Kirchhoff’s law
Consider an enclosure as shown in fig.7.2 and a body is placed inside the enclosure. The radiant heat flux (q) is incident onto the body and allowed to come into temperature equilibrium. The rate of energy absorbed at equilibrium by the body must be equal to the energy emitted.
where, E is the emissive power of the body, is absorptivity of the of the body at equilibrium temperature, and A is the area of the body.
Now consider the body is replaced by a blackbody i.e. E → E_{b} and= 1, the equation 7.14 becomes
Dividing eq. 7.14 by eq.7.15,
At this point we may define emissivity, which is a measure of how good the body is an emitter as compared to blackbody. Thus the emissivity can be written as the ratio of the emissive power to that of a blackbody,
On comparing eq.7.16 and eq.7.17, we get
Equation 7.18 is the Kirchhoff’s law, which states that the emissivity of a body which is in thermal equilibrium with its surrounding is equal to its absorptivity of the body. It should be noted that the source temperature is equal to the temperature of the irradiated surface. However, in practical purposes it is assumed that emissivity and absorptivity of a system are equal even if it is not in thermal equilibrium with the surrounding. The reason being the absorptivity of most real surfaces is relatively insensitive to temperature and wavelength. This particular assumption leads to the concept of grey body. The emissivity is considered to be independent of the wavelength of radiation for grey body.
7.3 Grey body
If grey body is defined as a substance whose monochromatic emissivity and absorptivity are independent of wavelength. A comparative study of grey body and blackbody is shown in the table 7.2.
Table7.2: Comparison of grey and blackbody
Illustration 7.1
The surface of a blackbody is at 500 K temperature. Obtain the total emissive power, the wavelength of the maximum monochromatic emissive power.
Solution 7.1
Using eq. 7.12, the total emissive power can be calculated,
E_{b} = σT^{4}
where, σ (= 5.67 X 10^{8} W/m^{2}/K^{4}) is the StefanBoltzmann constant. Thus at 500 K,
E_{b} = (5.67 X 10^{8})(5000^{4}) W/m^{2}
E_{b} = 354.75 W/m^{2}
The wavelength of the maximum monochromatic emissive power can be obtained from the Wien’s law (eq. 7.11),
λ_{max}T = 2898
7.4 Radiative heat exchanger between surfaces
Till now we have discussed fundamental aspects of various definitions and laws. Now we will study the heat exchange between two or more surfaces which is of practical importance. The two surfaces which are not in direct contact, exchanges the heat due to radiation phenomena. The factors those determine the rate of heat exchange between two bodies are the temperature of the individual surfaces, their emissivities, as well as how well one surface can see the other surface. The last factor is known as view factor, shape factor, angle factor or configuration factor.
7.4.1 View factor
In this section we would like to find the energy exchange between two black surfaces having area A_{1} and A_{2}, respectively, and they are at different temperature and have arbitrary shape and orientation with respect to each other. In order to find the radiative heat exchange between the bodies we have to first define the view factor as
Thus the energy leaving surface 1 and arriving at surface 2 is E_{b1}A_{1}F_{12} and the energy leaving surface 2 and arriving at surface 1 is E_{b2}A_{2}F_{21}. All the incident radiation will be absorbed by the blackbody and the net energy exchange will be,
Q = E_{b1}A_{1}F_{12}  E_{b2}A_{2}F_{21}
At thermal equilibrium between the surfaces Q_{12} = 0 and E_{b1} = E_{b2}, thus
0 = E_{b1} (A_{1}F_{12}  A_{2}F_{21})
Equation 7.19 is known as reciprocating relation, and it can be applied in general way for any blackbody surfaces.
Though the relation is valid for blackbody it may be applied to any surface as long as diffuse radiation is involved.
7.4.1.1 Relation between view factors
In this section we will develop some useful relation of view factor considering fig. 7.5
Fig. 7.5: Exchange of energy between area A1 and A2 (A is area of blackbody)
View factor for radiation from A_{1} to the combined area A_{2},
and using the reciprocating relations for surface 1 and 4,
Using eq. 7.21 and 7.22,
Thus the unknown view factor F_{14} can be estimated if the view factors F_{12} and F_{13}, as well as their areas are (A_{1}, A_{2}) known.
Now, consider a flat plate (for eg.) which is emitting the radiation, it can be understood that the radiation of the flat plat cannot fall on its own surface (partly or fully). Such kind or surfaces are termed as “not able to see itself”. In such situations,
F_{11} = F_{22} = F_{33} = F_{44} = 0
However, if the surface can see itself like concave curved surfaces, which may thus see themselves and then the shape factor will not be zero in those cases.
Another property of the shape factor is that when the surface is enclosed, then the following relation holds,
where, F_{ij} is the fraction of the total energy leaving surface i which arrives at surface j.
In case of Nwalled enclosure, some of the view factors may be evaluated from the knowledge of the rest and the total N^{2} view factors may be represented in square matrix form shown below,
7.5 Heat exchange between non blackbodies
Evaluation of radiative heat transfer between black surfaces is relatively easy because in case of blackbody all the radiant energy which strikes the surface is absorbed. However, finding view factor is slightly complex, but once it can be done, finding heat exchange between the black bodies is quite easy.
When non blackbodies are involved the heat transfer process becomes very complex because all the energy striking on to the surface does not get absorbed. A part of this striking energy reflected back to another heat transfer surface, and part may be reflected out from the system entirely. Now, one can imagine that this radiant energy can be reflected back and forth between the heat transfer surfaces many times.
In this section, we will assume that all surfaces are in the analysis are diffuse and uniform in temperature and that the reflective and emissive properties are constant over all surfaces.
Fig. 7.6: (a) Surface energy balance for opaque surface (b) equivalent electrical circuit
It is also assumed that the radiosity and irradiation are uniform over each surface. As we have already discussed that the radiosity is the sum of the energy emitted and the energy reflected when no energy is transmitted (as opaque body), or
where, is the emissivity and E_{b} is the blackbody emissive power. Because the transmissivity is zero due to opaque surface and absorptivity of the body (grey) will be equal to its emissivity by Kirchhoff’s law.
Thus, eq.7.24 becomes
The net energy leaving the surface is the difference between the radiosity and the irradiance (fig.7.6a),
The eq.7.26 can be analogous to the electrical circuit as shown in fig.7.6(b). The numerator of the eq.7.26 is equivalent to the potential difference, denominator is equivalent to the surface resistance to radiative heat, and left part is equivalent to the current in the circuit.
In the above discussion we have considered only one surface. Now we will analyse the exchange of radiant energy by two surfaces, A_{1} and A_{2}, as shown in the fig.7.7a.
Fig. 7.7: (a) Energy exchange between two surfaces, (b) equivalent circuit diagram
The radiation which leaves surface 1, the amount that reaches surface 2 is
J_{1}A_{1}F_{12}
Similarly, the radiation which leaves system 2, the amount that reaches surface 1 is
J_{2}A_{2}F_{21}
The net energy transfer between the surfaces,
Reciprocity theorem states that
It also resembles an electrical circuit shown in fig.7.7b. The difference between eq.7.26 and 7.27 is that in eq.7.27 the denominator term is space resistance instead of surface resistance.
Now, to know, the net energy exchange between the two surfaces we need to add both the surface resistances along with the overall potential as shown in the fig.7.8. Here the surfaces see each other and nothing else.
Fig. 7.8: Radiative nature for two surfaces which can see each other nothing else
Evaporators
Evaporation is the vaporization of a liquid. Chemical process industries, in general, use evaporator for the vaporization of a solvent from a solution. We have already discussed the heat transfer for boiling liquids in early chapter. However the evaporation is so important operation in chemical process industry that it is considered an individual operation. In this chapter we will focus on the evaporation with an objective to concentrate a solution consisting of a nonvolatile solute and a volatile solvent. If we continue the evaporation process, the residual mater will be solid, which is known as drying. However, our aim is not to dry but to concentrate the solution, moreover, we will also not deal with the crystallization, in which the evaporation leads to formation of crystal in the solution. It is suggested that reader should learn the difference between evaporator, drying and crystallization.
As we will deal with the solution for the evaporation process, a few of the facts must be known about the solution properties.
9.1 Solution properties
Knowledge of solution properties is important for the design of the equipment for evaporation. Some of the important properties of the solution are given below,
9.1.1 Concentration
Initially, the solution may be quite dilute and the properties of the solution may be taken as the properties of solvent. As the concentration increases, the solution becomes viscous and heat transfer resistance increases. The crystal may grow on the heating coil or on the heating surface. The boiling points of the solution also rise considerably. Solid or solute contact increases and the boiling temperature of the concentrated solution became higher than that of the solvent as the same pressure (i.e. elevation in boiling point).
9.1.2 Foaming
Many of the materials like organic substance may foam during vaporization. If the foam is stable, it may come out along the vapor known as entrainment. Heat transfer coefficient changes abruptly for such systems.
9.1.3 Degradation due to high temperature
The products of many chemical, food, pharmaceutical industries etc. are very temperature sensitive and they may get damaged during evaporation. Thus special case or technique is required for concentrating such solution.
9.1.4 Scaling
Many solution have tendency to deposit the scale on the heating surface, which may increase the heat transfer resistance. These scales produce extra thermal resistance of significant value. Therefore, scaling in the equipment should not be ignored thus descaling becomes an important and routine matter.
9.1.5 Equipment material
The material of the equipment must be chosen considering the solution properties so that the solution should neither be contaminated nor react with the equipment material.
9.2 Evaporator
Equipment, in which evaporation is performed, is known as evaporator. The evaporators used in chemical process industries are heated by steam and have tubular surface. The solution is circulated in the tube and the tubes are heated by steam. In general the steam is the saturated steam and thus it condenses on the outer tube surface in order to heat the tube. The circulation of the solution in the tube have reasonable velocity in order to increase the heat transfer coefficient and remove of scales on the inner surface of the tube. The steam heated tubular evaporators may be classified as natural and forced circulation evaporators.
9.2.1 Natural circulation evaporator
In this category the main evaporators are,
As the name indicates, the circulation of the solution is natural and the density difference derives it. The solution gets heat up and partially vaporized as it flows up the tubes. The heated liquid flows up because of the density difference. Vaporliquid disengagement occurs above the tube. Thick liquor comes down from this down comer and withdrawn from the bottom. The naturalcirculation evaporators may be used if the solution is quite dilute. In the dilute solution the natural circulation will be at sufficient speed. It may also be used when the solution does not have suspended solid particles. As the solution stays in the tube for larger time, the solution should not be heat sensitive.
The Calandria type or shorttube evaporators have short tubes as compared to the long tube evaporators. The shorttube evaporation uses circulation and solution flows many times in the evaporators. However, in case of the long tube evaporator the flow is once through.
9.2.2 Forced circulation evaporator
Natural circulation evaporators have many limitations (as mentioned earlier) through they are economical as compared to forced circulation evaporator. A forced circulation evaporator has a tubular exchanger for heating the solution without boiling. The superheated solution flashes in the chamber, where the solution gets concentrated. In forced circulation evaporator horizontal or vertical both type of design is in practice. The forced circulation evaporators are used for handling viscous or heat sensitive solution.
9.2.3 Falling film evaporator
Highly heat sensitive materials are processed in falling film evaporators. They are generally oncethrough evaporator, in which the liquid enters at the top, flows downstream inside the heater tubes as a film and leaves from the bottom. The tubes are heated by condensing steam over the tube. As the liquid flows down, the water evaporates and the liquid gets concentrated. To have a film inside of the tube, the tube diameter is kept high whereas the height low to keep the residence time low for the flowing liquid. Therefore, these evaporators, with noncirculation and short resistance time, handle heat sensitive material, which are very difficult to process by other method. The main problem in falling film evaporator is the distribution of the liquid uniformly as a thin film inside the tube.
9.3 Performance of steam heated tubular evaporators
The performance of a steam heated tubular evaporator is evaluated by the capacity and the economy.
9.3.1 Capacity and economy
Capacity is defined as the no of kilograms of water vaporized per hour. Economy is the number of kg of water vaporized per kg of steam fed to the unit. Steam consumption is very important to know, and can be estimated by the ratio of capacity divided by the economy. That is the steam consumption (in kg/h) is
Steam Consumption = Capacity / Economy
9.3.2 Single and multiple effect evaporators
In single effect evaporator, as shown in fig. 9.1, the steam is fed to the evaporator which condenses on the tube surface and the heat is transferred to the solution. The saturated vapor comes out from the evaporator and this vapor either may be vented out or condensed. The concentrated solution is taken out from the evaporator.
Now we can see if we want the further concentrate, the solution has to be sent into another similar evaporator which will have the fresh steam to provide the necessary heat.
It may be noted that in this process the fresh steam is required for the second evaporator and at the same time the vapor is not utilized. Therefore it can be said the single effect evaporator does not utilized the steam efficiently. The economy of the single effect evaporator is thus less than one. Moreover, the other reason for low economy is that in many of the cases the feed temperature remains below the boiling temperature of the solution. Therefore, a part of the heat is utilized to raise the feed temperature to its boiling point.
Fig.9.1: Single effect evaporator
In order to increase the economy we may consider the arrangement of the two evaporators as shown in the fig. 9.2.
The figure 9.2 shows that the two evaporators are connected in series. The saturated vapor coming out from the evaporator1 is used as steam in the second evaporator. Partially concentrated solution works as a feed to the second evaporator. This arrangement is known as double effect evaporator in forward feed scheme. A few of the important point that we have to note for this scheme is that the vapour leaving evaporator2 is at the boiling temperature of the liquid leaving the first effect. In order to transfer this heat from the condensing vapor from the evaporator1 to the boiling liquid in evaporator2, the liquid in evaporator2 must boil at a temperature considerable less than the condensation temperature of the vaporization, in order to ensure reasonable driving force for heat transfer. A method of achieving this is to maintain a suitable lower pressure in the second effect so that the liquid boils at a lower temperature. Therefore, if the evaporator1 operates at atmospheric pressure, the evaporator2 should be operated at same suitable vacuum.
Fig.9.2: Double effect evaporator with forward feed scheme
The benefit of the use of multiple effect evaporators is that in this arrangement multiple reuse of heat supplied to the first effect is possible and results in improved steam economy.
9.3.3 Boiling point elevation
The evaporators produce concentrated solution having substantially higher boiling point than that of the solvent (of the solution) at the prevailing pressure. The increase in boiling point over that of water is known as boiling point elevation (BPE) of the solution. As the concentration increases the boiling point of the solution also increases. Therefore, in order to get the real temperature difference (or driving force) between the steam temperature and the solution temperature, the BPE must be subtracted from the temperature drop. The BPE may be predicted from the steam table (in case water is a solvent).
An empirical rule known as Dühring rule is suitable for estimating the BPE of strong solution. The Dühring rule states that the boiling point of a given solution is a linear function of the boiling point of the pure water at the same pressure. Therefore, if the boiling point of the solution is plotted against that of the water at the same pressure, a straight line results. Different lines are obtained at different concentrations. The fig. 9.3 shows representative Dühring plots for a solution (nonvolatile solute in water).
Fig.9.3: Representative Dühring lines for a system (nonvolatile solute in water) mole fraction of solute in the solution (a) 0.1 (b) 0.2 (c) 0.25 (d) 0.39 (e) 0.35 (f) 0.45 (g) 0.5 (h) 0.6 (i) 0.7
The fig.9.3 helps to find out the boiling point of solution at moderate pressure. For example if a solution having ‘x’ mole fractions of solute have a pressure over it such that water boils at T° C, by reading up from the xaxis at T °C to the line for the x mole fraction solution and then moving horizontally to the yaxis, the boiling point of the solution can be found at that pressure.
9.4 Temperature profile in an evaporator
Let us consider the case of longtube vertical evaporator heated by steam. After boiling and flashing of the superheated liquid, the disengagement of the vapor and liquid occur in vapor space of the evaporator and the recycled liquid flows down the external pipe. A part of this concentrated liquid is withdrawn as a product and the remaining part get mixed with a feed and again enter the evaporator tube. If T_{BP} is the boiling of the liquid in the evaporator as the prevailing pressure, then the temperature of the liquid in the tube will be T_{BP}. The temperature of the recycled stream entering the tubes will then also be T_{BP}, if the feed is sufficiently hot. Now, we will imagine how the temperature is changing in the tube. Let us see that when the liquid flows up in the tube, its temperature rises because at the bottom of the tube the pressure is higher (vapor chamber pressure + hydrostatic pressure + frictional loss) as compared to the top of the tube. Therefore, a liquid starts boiling at a level when its temperature rises to its saturation temperature at the pressure at that point. After the boiling in between the tube, as liquid goes up in the tube, the local temperature drops because of the reduction in the local pressure. It may also be mentioned that as the liquid moves up it gets concentrated and thus the boiling point of the solution also increases as the liquid traversed up in the tube. The liquid temperature profile in the tube is shown in the fig.9.4 for low (plot i) and high (plot ii) liquid velocity. The liquid temperature in the tubes increases up to certain height and then the temperature decreases due to the loss of superheat. At higher velocity the temperature raise is less and the liquid boils near the top of the tube. The plot (iii) shows the shell side temperature profile where steam is heating the tube. As can be seen, the slightly superheated steam enters the shell and soon the temperature of the steam losses its sensible heat and then condenses on the tubes and provide the latent heat of condensation (at temperature T_{steam}) to the tube and before boiling from the shell may get slightly subcooled. The plot (iv) is the boiling temperature of the water (T_{w}) at the pressure in the vapor chamber. Thus, the BPE=T_{BP}T_{w}and the true temperature during force is the difference between the plot (iii) and the plot (i) or (ii).
It can be understood with the help of the discussion and fig.9.4 that the temperature changes all along the length of the tube. Thus, the real temperature driving force will be the difference in steam temperature and liquid temperature always the high. However, it is practically not easy to determine the temperature profile in the tube. Therefore, the driving force can be taken as (T_{steam}  T_{BP}) for the design purpose.
9.5 Heat transfer coefficient
The correlation used in the boiling and condensation may be used here. If the evaporator operates at very high liquid velocity so that the boiling occurs at the top end of the tube, the following correlation (eq. 9.1) may be used,
where, D is the inner diameter of the tube, k is the thermal conductivity of the liquid or solution.
Fig.9.4: Temperature profiles in an evaporator
Fouling is a concern in the evaporator; therefore the following equation (eq.9.2) may be used for the overall heat transfer coefficient with time,
where, t is the time for where the evaporator is the operation, α is a constant for a particular liquid, U_{dirty} and U_{clean} all the overall heat transfer coefficient of the dirty and clean evaporator.
9.6 Method of feeding: Multiple effect evaporators
The fig.9.5, 9.6, 9.7, and 9.8 show the four different feeding arrangement of feed to the evaporators. In the fig.9.5 the liquid feed is pumped into the first effect and the partially concentrated solution is sent to the second effect and so on. The heating steam is also sent through the first effect to another effect. This particular strategy is known as forward feed. In the forward feed the concentration of the liquid increases from first effect to the subsequent effects till the last effect. It may be noted that the first effect is that in which the fresh steam is fed, whereas the vapour generated in the first effect is fed to the next evaporator (connected in series with the first effect) is known as second effect and so on.
The forward feed requires a pump for feeding dilute solution to the first effect. The first effect is generally at atmospheric pressure and the subsequent effects are in decreasing pressure. Thus, the liquid may move without the pump from one effect to another effect in the direction of decreasing pressure. However, to take out the concentrated liquid from the last effect may need a pump.
The backward feed arrangement is very common arrangement. A tripleeffect evaporator in backward arrangement is shown in the fig.9.6. In this arrangement the dilute liquid is fed to the last effect and then pumped through the successive effects to the first effect. The method requires additional pumps (generally one pump in between two effects) as shown in the fig. 9.6. Backward feed is advantageous and gives higher capacity than the forward feed when the concentrated liquid is viscous, because the viscous fluid is at higher temperature being in the first effect. However, this arrangement provides lower economy as compared to forward feed arrangement.
The combination of forwardfeed and backwardfeed is known as mixed feed arrangement. In mixed feed the dilute liquid enters in between effects, flows in forward feed to the end of the effect and then pumped back to the first effect for final concentration. Figure 9.7 shows triple effect mixed feed arrangement. This mixed feed arrangement eliminates the need of a few of the pumps. Moreover, it still passes the most concentrated liquid through the first effect, which is having higher temperature among all the effect (being at highest pressure compared to other effects).
Another common evaporator arrangements, which is more common in crystallization is parallel feed where feed is admitted individually to all the effects. Figure 9.7 shows such arrangement.
Fig.9.5: Forward feed arrangement in tripleeffect evaporator (dotted line: recycle stream)
Fig.9.6: Backward feed arrangement in tripleeffect evaporator (dotted line: recycle stream)
Fig.9.7: Mixed feed arrangement in tripleeffect evaporator (dotted line: recycle stream)
Fig.9.8: Parallel feed arrangement in tripleeffect evaporator
9.7 Enthalpy Balance
9.7.1 Single effect evaporator The latent heat of condensation of the steam is transferred to the boiling solution through the heating surface in order to vaporize the water. Thus, two enthalpy balances are required one for the liquid and another for the steam.
The following assumptions are required, in order to make the enthalpy balance,
The enthalpy balance for the steam side is,
Where,
Enthalpy balance for the liquid side is (eq.9.3),
Where,
The enthalpy balance at steam side and liquid side will be same in the absence of any heat loss (eq.9.4). Thus,
The area of heat transfer A can be calculated from
When ΔT = (T_{b}  T_{c});
T_{b} = Saturated temperature of steam in the shell T_{s} = Boiling point of the solution at the prevailing pressure U_{D} = Overall coefficient (dirty)
9.7.2 Effect of heat of dilution
Most of the solutions when mixed or diluted at constant temperature do not give significant heat effect. It is generally true for organic solutions (like sugar). However, many of the inorganic solutions (like sulfuric acid, potassium hydroxide, calcium carbonate etc.) evolve significant heat on dilution. Therefore, an equivalent amount of heat is required (in addition to the latent heat of vaporization), when dilute solutions of these inorganic chemicals are concentrated. Enthalpyconcentration diagram are helpful in order to find the enthalpy of the solution at different concentration of these chemicals in the solution.
9.7.3 Multiple effect evaporators
The steam goes into Ieffect and heat the solution by the latent heat of condensation. If the heat required to boil the feed is negligible, it follows that practically all this heat
must appear as latent heat in the vapor that leaves the Ieffect and enter into IIeffect as steam. The temperature of the condensate leaving the IIeffect will be very near the temperature T_{1} of the vapors from the boiling liquid in the Ieffect. Thus, in steady state operation all the heat that was expanded in creating vapor in the Ieffect must be given by when this same vapor condenses in the IIeffect and so on.
The heat delivered into the IIeffect will be,
The
Similarly, for IIIeffect
It can be seen (eq. 9.5) that the temperature drops in a multiple effect evaporator is approximately inversely proportional to the heattransfer coefficient.
The total available temperature drop will be given by eq.9.6,
where,
T_{s} : Steam temp. (Ieffect); T_{v3} : Vapor temperature leaving IIIeffect BPE : boiling point elevation in the solution in various effects 
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