(Fig. 29a)

Drag: We have already introduced this term in the previous lecture. There are two types of drag: form drag and wall or shear drag. The former is because of the fluid–pressure on the solid–surface and acts perpendicular to the surface wall. Drag is because of shear–forces and acts parallel to the surface.




(Fig. 29b)

 A horizontal plate parallel to fluid flow will experience drag only because of wall shear.



form drag

wall drag
 
 (already defined in previous lectures) 
 (Stoke’s Law for Rep < 1) 
 
 in general depends upon Reynolds number, but is also dependent on the shape and orientation of the particle with respect to the flow. Charts are available to determine drag coefficient.



(Fig. 29b)


 Terminal or settling velocity of a single particle in fluid may be calculated as




 If the fluid moves–up with velocity , velocity of the particle with respect to a stationary observer,
, so that the relative velocity or drag remains the same.

 depends on which cannot be directly calculated because the velocity (settling) is not known. Therefore, iteration is required to determine . In the previous lecture, we tookup such an example. However, there is a criterion to check if the settling is in the Stoke’s regime (creeping flow) or in Newton’s region (high flow when inertial effects are important), which is calculated independent of velocity:
Criteria for settling 

y (Stoke’s regime)




 (As an exercise, substitute in the expression for terminal velocity with , and use the criterion to obtain the above expression, for the Stoke’s regime)

Hindered settling If there are particles in the fluid, then the settling of a single particle will be influenced by the presence of the neighboring particles. In such case, the settling velocity is larger than that of a single–particle. 

 where is the volume–fractions (not bed–porosity) of fine–suspension of particles in fluid, 


The viscosity of a suspension is also affected by the presence of the dispensed phase and should be accordingly used in the calculation of Reynolds number: 

= viscosity of suspension; = viscosity of pure fluid
Flow through packed bed (continued)

Pressure–drop calculation:



Notes 
 Actual or real packed beds are randomly packed with irregular size particles
 The flow–path of a fluid though the packed bed is tortuous.

 For the theoretical analysis to calculate pressure–drop, actual flow channels are replaced with parallel cylindrical conduits of constant cross–section. Particles are assumed to be of the same size and shape having constant sphericity, .
 Pressure–drop occurs due to inertial and viscous effects. At high Reynolds number, inertial effects prevail, whereas the viscous effects are important at low Reynolds number. In general,
For a packed–bed 

 (Propose) 
(Recall: wall shearstress in tubular laminar flow, 
 
Similarly, pressure–drop at high Reynolds number, . Therefore, Pressuredrop in packed beds is related to pressure–drop due to viscous and inertial effects via two empirical constants, .

 (multiply both numerator and denominator by L) 
 , S_{O} = cross sectional area of packed–bed 
 


 
 

Therefore,


 
 
Substituting,


Or, 

Setting (based on experimental data) 
We obtain 

 Ergun's equation

One also defines as the friction factor for a packed–bed 

Therefore,


If

(Blake – Plummer equation) 
It is pointed out that the Ergun's equation is applied to calculate pressure drop across packed bed consisting of small size particles

Also, note that is to be interpreted as energy loss due to drag or friction per unit mass of the fluid , so that the term can be substituted in the general mechanical energy balance equation, consisting of KE, gravitational and shaftwork heads.


where frictional–pressure drop

 

Fluidization

 When a liquid or gas is passed at a relatively small velocity though a bed of solid particles, the particles do not move. Fluid moves through the voids between the particles; pressure–drop is calculated by Ergun's equation. If the flow rate is steadily increased, pressure–drop (or drag) increases. Eventually, particles tend to move and bed expands a little. A stage is reached when the pressure–drop balances the weight of solid particles and buoyancy. Now, the bed apparently seems to be boiling. Particles–movement increases; yet they do not leave the bed. Such bed is termed as ‘fluidized bed'.
 Mechanistically, the frictional force (drag) between particles and fluid just counterbalances the weight of the particles; the vertical component of compressive forces between particles disappear and equates the effective weight.
 A simple experiment can be carried out to observe the movement of particles packed within a glass or Perspex made column. The height of the bed and the pressure–drop across the bed can be measured with accuracy:



(fig. 32a)
Fixed bed  Bed expands a little  Incipiently or minimum fluidized bed 
(No movement of particles)  Particles are unlocked and the bed expands from an height of  
Bed continues to expand
(for liquid only) till there is a carryover of solid particles 






(Fig. 32c)

Veryoften a hysteresis is observed, if the velocity is gradually decreased: 


(Fig. 32d)
 Fluidized–bed has fluid–like behavior. It will appear boiling at the surface, with the particles moving up and down in the bed, especially on top of the surface.
 The minimum fluidization porosity, or the porosity at the minimum fluidization condition is particle–size and type–specific. Some examples are:



Size



Sharp sand, 
 


Adsorption carbon 
0.72

0.69


FischerTropsch catalyst 

0.58


   
 For liquid, the state of fluidization past the minimum fluidization stage is called homogeneous/ smooth/particulate/non–bubbling fluidized bed, as the bed expands smoothly. At higher velocity, there is a carryover of particles. Slurry flow ensues.
 For gases, the particulate or homogeneous fluidization occurs only for small (fine) particles. For large particles, bubbles are formed. At even higher velocity, vigorous fluidization occurs, with turbulent motion of solid clusters and bubbles. Such state is called “Fast Fluidized Bed”. There may be carryover/entrainment of particles with the outgoing gas.

Minimum fluidization velocity

The minimum fluidization velocity can be calculated by equating the pressure–drop across the fixed packed–bed, calculated from Ergun's equation to that from the expression for fluidized bed under particulate (smooth) conditions.
Let us calculate the pressuredrop from the 2 ^{nd} expression:
Under fluidization conditions, pressure–drop equals effective weight of solid, as intraparticle forces disappear and solids float in the bed exhibiting ‘liquid–like ‘behavior. For a fluidized bed of length of L and bedporosity of ,

Weight of solidparticles–buoyancy

 
Or

 

, etc.

where

Rcall:



(Fig. 33a)
At the minimum fluidization condition:
Apply Ergun's equation for ‘fixed–bed' at minimum fluidization condition or at the incipience of fluidization:
, where superficial average velocity at minimum fluidization state

Equate:


The aboveequation is quadratic on (minimum fluidization velocity) and may be written in the following form:

, where

  

For small particles


For large particles
To avoid or reduce carryover of particles form the fluidized bed, keep the gas velocity between . Recall 
Terminal velocity, for low Reynolds number and, 
for high Reynolds number
With the expressions for and known for small (viscous–flow) and large (inertial flow) particles or Reynolds number, one can take the ratio of and :
For small
For spherical particles, and assuming

Therefore, a bed that fluidizes at 1cm/s could preferably be operated with velocities < 50 cm/s, with few particles carried out or entrained with the exit gas.
For large
Or,
Therefore, operating safety margin in a bed of coarse particles is smaller and there is a disadvantage for the use of coarse particles in a fluidized bed.
However, make a note that the operating particle size is also decided by the other factors such as grinding cost, pressuredrop, heat and masstransfer aspects.
Filtration

 Removal of solids from fluid (gas or liquid) by a filtering medium on which solid particles are deposited.
 For filtration, external force is applied to a (gas or liquid + solid) mixture to make it flow through the medium.
 Filtration, when applied to gas cleaning, usually refers to the removal of fine particles like dust from air or flue gas. In such case, a polymeric fiber or cloth is wrapped over a pretreated metallic cylinder, capable of capturing micron size particles, including soot and flyash.
 Very large size ceramic based filters for high temperature applications are also commercially available.

In this and the next lectures we will confine our discussion to liquid – solid filtration .

 The liquid–solid filtration is often called “cake–filtration”, because the separation of solids from the slurry by the filtering medium is effective during the initial stages of filtration. Later, the ‘cakes' or deposits collected over the medium act as the filter. Therefore, cake thickness increases during filtration and the resistance (hydraulic) offered by the cake–material is larger than that by the filtering medium.
 There are two types of operation:
 Constantpressure
 Constant filtering rate
In the 1^{st} case, filtering rate varies with time, whereas in the 2^{nd} case, pressure–drop increases with time.

 For ideal cake filtration, cake should be stable and large porosity. There are two common types of filters:
 The plate and frame press
 Rotarydrum filter

The plate and frame press filter

 Consists of series of plates and frames sandwitched alternatively; cakes are builtup inside the frame–clamber. Cloth, filtering medium, is supported on a corrugated material. There are slurry and filtrate ports.



(Fig. 35a)
While designing the plate and frame press filter, dismantling and re–assembling times, removal of cake from each frame, and other operations such as washing and drying of cakes should also be taken into consideration.

Rotary Filter



(Fig. 35b)


See the schematic above. The portion of the cylinder (rotary drum) submerged in the trough is subjected to vacuum. A layer of solids builds upon the drum as the liquid is drained through cloth, slots, compartments, pipe to the tank, which collects the filtered water.
In the washing/drying zone; vacuum is removed; cakes are removed by scrapping it off with a knife, doctor blade. The process is continuous whereas the plate and frame press filter is a batch process.

surface area and volume of the cakeparticles (solids of the slurry), respectively


(Important to note is the time–change of pressure and cake–thickness)


superficial velocity of filtrate
Principles of filtration (continued )

Case 1: Constant Pressuredrop Filtration


differential mass of the cake

Substituting,


Assuming, incompressible cake (Const )

(pressure–drop) through cake

, where, total mass of cake.

upstreampressure of filter–media
Define,
= property of cake
Pressure–drop through filter medium
hydraulic resistance of filter medium
Now ‘C' as the mass of the particles deposited in the filter per unit volume of the filtrate,
It can be shown that
where,
If turns out that
Replacing in the expression for
This is the working equation for cake filtration.
Case 1: constant–pressure filtration


(Fig. 36a)



Therefore, (one can calculate form the initial filtrationdata when resistance due
to cake = 0)
One can also write,
= constant (known)



(Fig. 36b)
on integration
The above expression can be integrated to develop an expression for the amount of cake formed over time ‘t' or the production rate of cake for the rotary drum filter:




(Fig. 36c)

A = Total area of filtration
Case 2: Constant Rate Filtration 


(Fig. 36d)



Or

(neglecting )
here, v is constant.
varies linearly with time.
(Such operation is difficult to run, i.e, keeping volumetric flow rate constant)
Agitation of liquids 
 The unit operation is used to prepare liquid–mixture by bringing in contact two liquids in a mechanically agitated vessel or container.
 Agitation refers to the induced motion of liquid in some defined may, usually in circulatory pattern and is achieved by some mechanical device.

Why agitation? 
 Dispenses a liquid which is immiscible with the other liquid by forming an emulsion or suspension of few drops.
 Suspends relatively lighter solid particles
 Promotes heat transfer between the liquid in the think or container and a coil or jacket surrounding the container
 Blends miscible liquids



(Fig. 40a)
 The equipment consists of a tank with an insulated jacket, baffles, shaft with motor, impeller, and other accessories such as thermometer and dip leg.

The role of baffles is to remove stratification in the radial direction and improve mixing,




 Typical configurationdimensions are:

 Two types of impellers:
 Radial flow impellers (flow is induced in radial or tangential directions)
 Axial flow impellers (currents are parallel to the axis of impeller shaft)

 Two types of geometrical configurations:



(Fig. 40c)
Flow patterns in agitated vessels 
There are three principal currents in the vessel during agitation: (a) radial (perpendicular to the shaft) (b) tangential (tangential to the circular path) (c) longitudinal (parallel to the shaft)
 Radia!
 Longitudinal
 Swirling



(Fig. 40d)

Notes: 
 Tangential component induces vortex and swirling, which in turn create stratification responsible for non–uniform mixing. In such case fluid particles are followed by another fluid particle.
 At relatively higher rpm, the center of vortex may reach impeller and air may be sucked in. This may not be desirable.
 Swirling can be minimized by placing the shaft slightly away from the center of the vessel, or by putting baffles. In the latter–configuration, tangential streamlines will also be reduced.
Power requirement

Dimensional analysis is used to determine the power requirement. Variables are
Relatively larger viscous fluid requires high power for mixing. Similarly, high density fluid–mixture also require large power for mixing:

From Buckingham theorem, no of independent dimensionless groups can be formed. For (6+m) variables, there will be (3 + m) groups:

 Power number, ,
 Reynolds number, , where is the tangential velocity of the tip of the impeller or
 Froude number

The other groups are
(Power number is analogous to friction factor and equals drag force on an unit area of impeller per KE of unitfluidvolume )

Or,
(Here, Reynolds number is based on peripheral speed and diameter of impeller)
Graphical results are available for different types of impellers to calculate power number:




(Slop is 1 on log–log plot for )

 As in the case of tubular flow flow, viscous effects are predominant and density of fluid is not important at low Reynolds number.

(Tables are available to calculate )

Or

 At high Reynolds number , power number is independent of the Reynolds number and viscosity is not important. Flow is fully turbulent.

Or
(Tables are available to calculate P)
Cyclone (Centrifugal settler)

 The equipment separates solid particles from a gas (eg. dust laden flue gas), based on the principle of centrifugal force, which is much stronger than gravitational force. Cyclone works relatively more efficiently at high gas flow rates.
 The equipment requires large flow rates/velocity to create a swirling movement inside the column. Cyclone, as such, does not have moving parts but may require a blower upstream to impart KE to the gas laden with particles.

 

(Fig. 41a)

(Fig. 41b)











The real trajectory of gas and particles is difficult to analyze. The particles laden gas enters the cyclone from the sideway (see top view) at a high flow rate and moves downward in a swirling/ spiral path.
Solid particles are thrown outward radially due to centrifugal force. They strike the walls of cyclone and settle down. Gas, on the other hand, will move radially inward, then upward through the least hydrodynamically resistance – path to the exit.
Gas moving in spiral reaches the apex of the cone, then moves upward in a smaller spiral
( ) path to the exit at the top, as the opening at the bottom is filled with solid particles. For the gas, the least resistance – path is the exit at the top. For the particles, the least resistance path is the exit at the bottom.
Mechanistically, if the centrifugal force acting on the particles is larger than the drag (inward) by the gas, the particles will strike the walls and settle down, else they will move inward alongwith the gas. At a radius r, where these two forces are equal, particle will rotate in equilibrium and move downward till they hit the slant walls and are collected. Gas on the other hand has a very high upward flow rate at the center, typically in the corediameter of . Any particle in the zone will be carried upward.
















Form , the theoretical cut–diameter, d _{p} is determined from the settling velocity equation:
(Note that it is assumed that particles settle in Stoke's regime)










 











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