**Heat Capacities **

**Back to last
section: Incompressible Flow**

**Continue to next section: Evaluation Along a
Chosen Path**

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 D*H* (or D*h*,
where lower case is used for the enthalpy *per* mole or per kg).

**Constant volume heat capacity**

Evaluation of D*H* for a constant
volume process is done using the constant volume heat capacity, *C*_{V},
to obtain D*U* and then the relationship
between *U* and *H* to obtain D*H.*
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__

- Recall that
*U*is a function of two variables:*U = U(T,V)*. We may therefore write (by taking the total differential of*U*):

The first partial is __defined__ as the constant volume heat capacity

- The
__constant volume heat capacity__is a small quantity of heat added isochorically divided by the corresponding temperature rise. Note that if the volume is constant, no*PV*work is done and*C*is also the change in internal energy with respect to_{V}*T*.

- ------
We will learn to evaluate this in ChE 373. For now we will take this as
zero. It is in fact identically 0 for an
and close to zero for liquids and solids.__ideal gas__

Thus, we will use:

**d U
= C_{V}dT and
H = U + PV (Evaluation of U and H
when T and V are known)**

**Constant Pressure Heat Capacity**

Evaluation of D*H* for a
constant pressure process is done using the constant pressure heat capacity, *C*_{P},
to obtain directly D*H.* 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.

- Like
*U*,*H*is a function of two independent variables for a pure fluid. We may therefore write*H = H(T,P)*which gives the total differential

- It can be shown that

- The
__constant pressure heat capacity__is a small quantity of heat added isobarically divided by the corresponding temperature rise. Note that if the pressure is constant,*C*can also be thought of as the change in enthalpy with respect to_{P}*T*.

- ------ We
will learn to evaluate this in CHE 373. For now we will take this as 0 for
gases and
*V*for solids and liquids. It is exactly zero for, and approximately equal to__ideal gases__*V*for solids and liquids.T

Thus, we will use:

**d H
= C_{P}dT (for gases)
and dH = C_{P}dT
+ VdP (for liquids and solids)**

To find D*H*
for a change in temperature, we must integrate heat capacity data. For example,
to find the D*H* for the change in
temperature for an ideal gas from *T*_{1} to *T*_{2},
we would integrate the above equation between the limits of *T*_{1}
to *T*_{2}:

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

**1. Integrate T-dependent C _{P}
data(most accurate)**

The temperature dependence

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:

D*H* = *A*D*T* + *BC*[coth(*C*/*T*_{2})
- coth(*C*/*T*_{1})] - *DE*[tanh(*E*/*T*_{2})
- tanh(*E*/*T*_{1})]

**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.