Modern Engineering Thermodynamics

378 CHAPTER 11: More Thermodynamic Relations
Note that following a path of constant T for the second integral in this equation is very logical since f depends
on only v and T and we are integrating over v.
For example, if we had an equation of state of the form pv = RT + αT 2 /v, then
and
then,
Equation (11.19) now gives
T
∂p
∂T
v
T
p = RT
v + αT2
v 2
∂p
= R ∂T v + 2αT
v
v 2
∂p
∂T
v
= RT
v + 2αT2
v 2
− p = RT
v + 2αT2 − RT
v 2 v + αT2
v 2 = αT2
v 2
Z T1
Z v1
u 1 − u 0 = c 0 v dT +
T 0
=
=
Z T1
v 0 =∞
αT 2
dv
v 2
c 0 v dT + αT2 − 1
v 1
v
T 0
∞
Z T1
c 0 v dT + αT 1 2
v
T 0 1
and if the zero pressure specific heat c 0 v is constant over the temperature range T 0 to T 1 , this equation reduces to
u 1 − u 0 = c 0 v ðT 1 − T 0 Þ − αT2 1
v 1
A similar equation can be easily developed for a second state, and we can then combine them to produce an
equation for the change in specific internal energy between these two states for this material as
u 2 − u 1 = c 0 v ðT 2 − T 1 Þ − α T2 2
v
− T2 1
2 v 1
and the reference state values have completely cancelled out.
11.7 CONSTRUCTING TABLES AND CHARTS
We are now able to use Eqs. (11.19), (11.22), (11.24), and (11.26) to construct thermodynamic tables and charts.
The construction of thermodynamic tables and charts like the ones in Thermodynamic Tables to accompany Modern
Engineering Thermodynamics require, first of all, that a great deal of accurate experimental p, v, T, andc v (or c p )databe
obtained. These data are reduced to mathematical equations through curve-fitting techniques. The resultant mathematical
equations are used to derive equations for u, h, ands using the thermodynamic property relations discussed
previously. One of the simplest methods for generating saturation and superheat tables is carried out as follows.
A. The following four data sets must be developed from appropriate experiments:
Data set 1. Saturation temperature and saturation pressure (T sat , p sat ).
Data set 2. Pressure, specific volume, and temperature in the superheated vapor region and along the
saturated vapor curve (p, v, T).
Data set 3. Saturated liquid specific volume (or density) and saturation temperature (v f , T sat ).
Data set 4. Low- (or zero) pressure constant volume specific heat, c 0 v and temperature T in the superheated
vapor region and along the saturated vapor curve. The superscript 0 is used to denote the fact that the c 0 v
values are measured at essentially zero pressure.
B. Once these four data sets have been obtained, a mathematical equation is curve fit to each of them to
obtain four mathematical equations of the form
Curve fit 1: p sat = p sat ðT sat Þ
(11.31a)