Heat Capacities 

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It should be obvious by now that to do energy balances you must be able to evaluate H for the fluid between the inlet and outlet conditions. To date, we have done this from tabulated data. We can also evaluate H from heat capacity data. Because we are always interested in the change in enthalpy between the inlet and outlet streams, we will need to calculate DH (or Dh, where lower case is used for the enthalpy per mole or per kg).

Constant volume heat capacity
Evaluation of DH for a constant volume process is done using the constant volume heat capacity, CV, to obtain DU and then the relationship between U and H to obtain DH. A short derivation is given here for completeness, but all that you are required to know is the highlighted (in red) item at the bottom of this section.



Derivation

                The first partial is defined as the constant volume heat capacity



Thus, we will use:

dU = CVdT      and        H = U + PV      (Evaluation of U and H when T and V are known)


Constant Pressure Heat Capacity

Evaluation of DH for a constant pressure process is done using the constant pressure heat capacity, CP, to obtain directly DH. A short derivation is given here for completeness, but again all that you are required to know is the highlighted (in red) item at the bottom of this section.




Thus, we will use:

dH = CPdT    (for gases)        and       dH = CPdT  + VdP      (for liquids and solids)
(Used when the variables T and P are known for the process)

Finding DH

To find DH for a change in temperature, we must integrate heat capacity data. For example, to find the DH for the change in temperature for an ideal gas from T1 to T2, we would integrate the above equation between the limits of T1 to T2:

In order to evaluate this integral, we must know something about the temperature dependence of CP. There are several different ways of getting data to do the integral. These are illustrated below.

1. Integrate T-dependent CP data(most accurate)
The temperature dependence CP is often tabulated as a function of temperature. For example, Table B.2 in the text lists CP = a + bT + cT2 + dT3 and provides the constants a, b, c, and d for a number of different fluids. To find the enthalpy difference between two streams, one at T1 and the other at T2, one would use:

 The DIPPR database uses this polynomial form for liquids and solids. The form used in the DIPPR database for ideal gases is:


This equation can be integrated numerically in Mathcad, for example, or you can use the analytical form of the integrated equation given by:
 

DH = ADT + BC[coth(C/T2) - coth(C/T1)] - DE[tanh(E/T2) - tanh(E/T1)]


2. Use constant values (often accurate enough for liquids and solids)
The heat capacity of a liquid or solid is less temperature dependent than that of a gas. Therefore many databases will list a constant for the heat capacity of solids and liquids. In this case, the enthalpy difference is easy to obtain:


The temperature-dependent DIPPR correlations are preferred when accuracy is important.

3. Graphical
If one has experimental heat capacity data rather than an explicit temperature-dependent functionality, one could either graph the data as a function of temperature and graphically integrate as shown below, or fit the data in Excel or Mathcad to a polynomial and then analytically integrate.


 

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