Relevance to Thermodynamics

4. Relevance to Thermodynamics

We will now discuss the relevance of exact differentials to thermodynamic analyses.

a. Work and Heat Both work transfer W and heat transfer Q are path dependent, while properties such as P and T are path independent. If a gas that is initially at state 1 (cf. Figure 13 ) corresponding to the conditions T = 500 K, v 1 =2m 3 kg 1 –1 (i.e., P 1 = 71.8 kPa from Figure 13 ) is isothermally expanded (process AC) to v 2 =6m 3 kg –1 , following which heat addition occurs at fixed vol- ume so that the gas temperature rises to 1000 K, the pressure at state 2, i.e., P 2 , is found to be

47.8 kPa. The same end state may be achieved by first adding heat at constant volume (process AD) to raise the temperature to 1000 K, and then expanding isothermally to v 2 = 6 m 3 kg –1 . The final pressure following the latter process will still be 47.8 kPa.

In a closed system containing an ideal gas, the incremental work δW = P dV (this will

be discussed more thoroughly in Chapter 2, with the total work W for a process being given by the area under the corresponding P–v curve (e.g., Figure 12 ). For the path ACB the work W ACB is the area under the curve ACB, while for a process along ADB ( Figure 12 ), W ADB is the area

under that curve. It is apparent that W ACB ≠W ADB even though the initial and final pressures, temperatures and volumes (all of which are properties) are the same. Therefore, we can deter-

mine v for given values of T and P, but not the work done, since it is path dependent. So “v” could be tabulated at given T and P as in Steam and R 134a Tables ( Tables A-4 and A-5 ) but work cannot be tabulated . The differentials of path dependent quantities are inexact differen-

tials (e.g., δW, δQ etc.), and their cyclic integrals ∫ δQ ≠ 0 and ∫ δW ≠ 0. In general, the heat

transfer between any two states 1 and 2,

∫δQ ≠ Q 2 –Q 1 .

b. Integral Over a Closed Path (Thermodynamic Cycle) Over a cycle for which the final and initial states are identical

∫ (30) dT = ∫ dP = ∫ du == ... 0.

In general, for a process occurring between two distinct states 1 and 2, the property change

T dT =T 2 –T 1 1, P dP =P 2 –P 1 , etc.

The internal energy can be expressed as an exact differential by the relation du = T ds – Pdv, i.e., u = u(s,v), M(s,v) = T, and N (s,v) = –P. The exact differential criterion ( ∂T/∂v) s =

–( ∂P/∂s) v for this case is also referred to as a Maxwell relation, details of which are given in Chapter 7. In general, all system properties, e.g., T, P, V, v, u, U, etc., are path independent

and point functions, and, therefore, form exact differentials. Consider the exact differential form du = c v,0 dT + (a/T)v 2 dv where “a” and c v0 are constants. If the internal energy difference is to be determined between states A and B (cf. Figure 12 ) either of paths ACB (isothermal conditions along AC, and constant volume along CB), or ADB (v constant along AD and T

T=500

T=500

T=1000 K

D P=71.8

P=23.9

P=47.8 bar

v=2

v=6

v=6 m/kg 3

A path ACB

P=47.8 v

C P=71.8

path ADB

Figure 13: Illustration of path dependent work. A gas is ex- panded to state B using paths ACB or ADB.

unchanged along DB) can be used to integrate the expression, with the difference (u B –u A ) be- ing the same regardless of path. The Appendix contains several relations between irrotational scalar fields that are useful in fluid mechanics, criteria for exact differentials, and thermodynamic properties.