3.1 Concept of Internal Energy
We have noted in chapter 1 that the two most common modes of energy exchanged by a thermodynamic system and its surroundings are work and heat. The interconvertibility between these two forms of energy was first demonstrated by the British scientist James P. Joule during 1840s by a series of carefully executed experiments. The experimental setup he used is shown as a schematic in fig. 3.1.  Known quantities of a set of fluids (water, oil, and mercury) were placed in an
Fig. 3.1 Schematic of Joule’s Experimental System
insulated, rigid vessel and stirred by means of rotating shaft provided with vanes. The amounts of work done on the fluid by the stirrer were measured in terms that needed to lower or raise a weight, and the resultant change in the temperature of the fluid was recorded. The key observation made by Joule was that for each fluid a fixed amount of work was required per unit mass for every degree of temperature rise caused by the rotating paddle wheel. Further, the experiments showed that the temperature of the fluid could be restored to its initial value by the transfer of heat by bringing it in contact with a cooler object. These experimental findings demonstrated for the first time that inter-convertibility exists between work and heat, and therefore the latter was also a form of energy.  
Joule’s observation also provided the basis for postulation of the concept of internal energy (introduced briefly in section 1.3). Since work are heat are distinctly different forms of energy how is it possible to convert one into another? The question can be answered if one assumes that although these two types of energies are distinct in transit across a thermodynamic system boundary, they must eventually be stored within a thermodynamic system in a common form. That common form is the so-called internal energy. As we have already discussed in section 1.3, such a form of energy can only repose at the microscopic level of atoms and molecules, essentially in the form of translational, vibrational and rotational energies. To this may be added the potential energy of intermolecular interactions (as introduced in section 2.2).  On a sub-molecular scale energy is associated with the electrons and nuclei of atoms, and with bond energy resulting from the forces holding atoms together as molecules. With these considerations one is in a position to rationalize the observation that while a system may receive energy in the form of work done on it, it may part with it also in the form of heat to another body or surroundings and be restored to its state prior to receipt of work. This is possible as in the interim between these two processes all energy may be stored in the form of internal energy.

As may be evident from the foregoing discussion, the addition of heat or work from an external source can lead to enhancement of the microscopic form of systemic (internal) energy. As also noted in chapter 1 the terminology “internal” is applied mainly to distinguish it from the mechanical potential and kinetic energies that a thermodynamic system may also possess by virtue of its position and velocity with respect to a datum. The latter two may then be thought of as “external” forms of energy.

It is important to note that like other intensive, macroscopic variables such as pressure, temperature, mass or volume, internal energy is a state variable as it is wholly dependent on the energy states that its atoms / molecules. Thus any change in the (say, specific) internal energy due to a process would only depend on the initial and final states, and not on the path followed during the change. Thus as for changes in P, V or T, one may write: 


However, unlike P, V, T or mass, U isnot a directly measurable property. Besides, in common with potential and kinetic energies, no absolute values of internal energy are possible. However, this is not of particular significance as in thermodynamic processes one is always interested in changes in energies rather than their absolute values.
3.2 The First Law of Thermodynamics
The empirical conclusion that heat and internal energy belong to the general category of energies, help extending the law of conservation of mechanical energy, which states that potential and kinetic energies are fully inter-convertible. As already discussed in chapter 1, a thermodynamic system may possess any other forms of energy such as surface energy, electrical energy, and magnetic energy, etc. Thus one may arrive at an extended postulate that all forms are energies are inter-convertible. This constitutes the basis of the First Law of Thermodynamics, which may be stated as follows:

Energy can neither be destroyed nor created, when it disappears in one form it must re-appear at the same time in other forms
It must be said that there is no formal proof of the first law (or indeed of other laws of thermodynamics) is possible, but that no evidence have been found to date that violates the principle enunciated by it.

For any thermodynamic process, in general one needs to account for changes occurring both within a system as well as its surroundings. Since the two together forms the “universe” in thermodynamic terms, the application of the first law to a process leads to the following mathematical form:



Where finite change occurring during the thermodynamic process


Application of the First Law to Closed Systems
In general, a thermodynamic system in its most complex form may be multi-component as well as multiphase in nature, and may contain species which react chemically with each other. Thermodynamic analysis tends to focus dominantly on the energy changes occurring within such a thermodynamic system due to change of state (or vice versa), and therefore it is often convenient to formulate the first law specifically for the system in question. Here we focus on closed systems, i.e., one that does not allow transfer of mass across its boundary. As already pointed out work and heat may enter or leave such a system across its boundary (to and fro with respect to the surrounding) and also be stored in the common form of internal energy. Since in a system may also possess potential and kinetic energies, one may reframe the first law as follows.

If the energy transfer across the system boundary takes place only the form of work and heat:



The above equation may also be written in a differential form:
(3.1)
If there is no change in potential and kinetic energies for the system or it is negligible – as is usually true for most thermodynamic systems of practical interest – the above equation reduces to: 
(3.2)
One of the great strengths of the mathematical statement of the first law as codified by eqn. 3.2 is that it equates a state variable (U) with two path variables (Q, W). As a differentiator we use the symbol δ to indicate infinitesimal work and heat transfer (as opposed to used state variables).  The last equation potentially allows the calculation of work and heat energies required for a process, by simply computing the change in internal energy. As we shall see later (chapters. 4 & 5) changes in internal energy can be conveniently expressed as functions of changes in state properties such as T, P and V.
In the above equation the term δW represents any form of work transfer to or from the system. In many situations of practical interest the thermodynamic work for closed systems is typically the PdV work (eqn. 1.6). Hence in such cases one may reframe eqn. 3.2 as follows: 

In keeping with the definition of work above, we adopt the following convention : 

   
The process of change in a thermodynamic system may occur under various types of constraints, which are enlisted below:
  • Constant pressure (isobaric)
  • Constant volume (isochoric)
  • Constant temperature (isothermal)
  • Without heat transfer (adiabatic)
The mathematical treatment of each of these processes is presented below.
For a constant pressure process (fig. 3.2), we may write:


The term H is termed enthalpy. It follows that like UH is also a state variable. On integrating the differential form of the equation above one obtains for the process:
Or: 
(3.4)
Fig. 3.2 Schematic of an Isochoric Path
On the other hand if the process occurs under isochoric (const. V) conditions (shown in fig. 3.3) the first law leads to: 
Fig. 3.3 Schematic of an Isochoric Path





3.3 Application of the First Law to Open Systems
While the last section addressed processes occurring in closed systems, the wider application of the first law involves formulating the energy balance differently in order to accommodate the fact that most thermodynamic systems, i.e., equipments, in continuous process plants are essentially open systems: they allow mass transfer across their boundaries (i.e., through inlet and outlet).  Examples include pumps, compressors, reactors, distillation columns, heat exchangers etc. Since such open systems admit both material and energy transfer across their boundaries the thermodynamic analysis necessarily involves both mass and energy balances to be carried out together. Also such systems may in general operate under both steady (during normal plant operation) and unsteady states (say during startup and shutdown). As we will see the former state is a limiting case of the more general situation of unsteady state behavior.
Mass Balance for Open Systems:
For generality consider an open system with which has multiple inlets (1, 2) and outlets (3, 4). The volume enclosed by the physical boundary is the control volume (CV). The general mass balance equation for such a system may be written as:  
Fig. 3.6 Schematic of an open system

(3.27)
The mass flow rate   is given by:
(3.28)
Where,    
The first term on the left side of the eqn. (3.27) denotes sum of all flow rates over all inlets, while the second term corresponds to the summation over all outlet flow rates. For the system shown in fig. 3.5 the eqn. 3.27 may be written as:
(3.28)
The last equation may be reframed in a general way as follows: 
(3.29)
Where the symbol  
Equation 3.29 may be react as:   


The above equation simplifies under steady flow conditions as the accumulation term for the control volume, i.e., 

In such a case for the simplest case of a system with one inlet and outlet (say: 1 and 2 respectively), which typically represents the majority of process plant equipments, the mass balance equation reduces to:
(3.31)
Energy Balance for Open Systems: 
Consider the schematic of an open system as shown in fig. 3.7. For simplicity we assume one 
Fig. 3.7 Schematic of an open system showing flow and energy interactions

inlet and one exit ports to the control volume. The thermodynamic states at the inlet i and exit e are defined by the P, V, Tu (average fluid velocity across the cross section of the port), and Z, the height of the port above a datum plane. A fluid element (consider an unit mole or mass) enters the CV carrying internal energy, kinetic and potential energies at the inlet conditions (PiTi, with molar volume as Vi) and leaves values of these energies at the exit state conditions (PeTe, with molar volume as Ve). Thus the total specific energy of the fluid at the two ports corresponds to the sum of specific internal, potential and kinetic energies, given by:  In addition, the CV exchanges heat with the surroundings at the rate , and say a total work (in one or more forms) at the rate of 

In the schematic we, however, have shown a specific work form, shaft work, that is delivered to or bythe system by means of rotatory motion of a paddle wheel which, as we will see later in the section, is implicated in many typical process plant units.  As with material balance one may write a total energy balance equation for the control volume as follows:

Or:
                                                               
(3.32)
The general total work term should include all forms of work. We draw the reader’s attention to the fact that the total work interaction also should include that needed to push fluid into the CV as well as that implicated in pushing it out of CV.  The fluid state at the inlet or exit is characterized by a set of state properties, U, V, H, etc. Consider a unit mass (or mole) of fluid entering the CV. This fluid element obviously needs to be “pushed” by another that follows it so as to make the formed enter the CV. In essence a fluid element of (specific) volume V is pushed into the CV at a pressure P. This is akin to a P-V form of work (as in the case of a piston-in-a-cylinder system) that is done on the CV and so may be quantified as – PiViThe same considerations apply at the exit in which case in pushing out a similar fluid element at exit conditionsi.e., – PeVeThus, eqn. 3.32 may be rewritten as follows:

                                  
Where, 
(3.33)
The term represents sum of all other forms of work associated with the process occurring within the CV. This residual work term may include the shaft work, P-V work resulting from expansion or contraction of the CV, electrical work, and so on. As the last two work terms on the left side of the eqn. 3.33 are associated with the flow streams we may rewrite the equation as follows: 
(3.34)
On rearranging: 

Or:
(3.35)
 (3.36)
It may be noted that eqn. 3.34 assumes that the CV is fixed in space and therefore no overall potential of kinetic energy terms depicting these mechanical energies for the control volume is included. This, of course, is valid for all process plant applications.  In addition, For many cases of practical importance (though not all) in a chemical plant the kinetic and potential energy changes between the inlet and exit streams may not be significant, whence the last equation may be simplified as: an equipment may be neglected; hence: 

Further, for the special case where the only shaft work is involved, the above equation may be simplified to: 
(3.38)
For steady state applications the eqn. 3.34 reduces to:
(3.39)
Further, if the kinetic and potential energy changes associated with the flow streams are insignificant, it follows that:        
(3.40)
Since under steady state  is constant we may write:
ΔH = Q + W   (3.41)
Where and W denote the work and heat interactions per unit mass of mole of fluid flowing through the system flowing through system. Once again if Ws is the only form of work interaction between the system and the surrounding then: 
ΔH = Q + Ws    (3.42)
Examples of process plant units to which eqn. 3.42 applies are: pumps, compressors, turbines, fans, blowers, etc. In all cases a rotatory part is used exchange work between the system and surrounding.

(3.3)