Question 6.2: Develop the property relations appropriate to the incompress...
Develop the property relations appropriate to the incompressible fluid, a model fluid for which both β and κ are zero (Sec. 3.2). This is an idealization employed in fluid mechanics.
Equations (6.27) and (6.28) written for an incompressible fluid become
d H=C_P dT +(1-\beta T) V d P (6.27)
d S=C_{p} \frac{d T}{T}-\beta V d P (6.28)
d H=C_P d T+V d P (A)
d S=C_P \frac{d T}{T}
The enthalpy of an incompressible fluid is therefore a function of both temperature and pressure, whereas the entropy is a function of temperature only, independent of P. With κ = β = 0, Eq. (6.29) shows that the internal energy is also a function of temperature only, and is therefore given by the equation, dU = CV dT. Equation (6.13), the criterion of exactness, applied to Eq. (A), yields:
\left(\frac{\partial U}{\partial P}\right)_T=(-\beta T+\kappa P) V (6.29)
\left(\frac{\partial M}{\partial y}\right)_x \left(\frac{\partial N}{\partial x}\right)_y (6.13)
\left(\frac{\partial C_P}{\partial P}\right)_T=\left(\frac{\partial V}{\partial T}\right)_P
However, the definition of β, given by Eq. (3.3), shows that the derivative on the right equals βV, which is zero for an incompressible fluid.
\beta \equiv \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P (3.3)
This implies that C_P is a function of temperature only, independent of P. Although H for an incompressible fluid depends upon pressure, CP does not.
The relation of C_P to C_V for an incompressible fluid is of interest. For a given change of state, Eqs. (6.28) and (6.36) must give the same value for dS; they are therefore equated. The resulting expression, after rearrangement, is:
d S=\frac{C_V}{T} d T+\frac{\beta}{\kappa} d V (6.36)
\left(C_P-C_V\right) d T=\beta T V d P+\frac{\beta T}{\kappa} d V
Upon restriction to constant V, this reduces to:
C_P-C_V=\beta T V\left(\frac{\partial P}{\partial T}\right)_V
Elimination of the derivative by Eq. (6.34) yields:
\left(\frac{\partial P}{\partial T}\right)_V=\frac{\beta} {\kappa} (6.34)
C_P-C_V=\beta T V\left(\frac{\beta}{\kappa}\right) (B)
Because β = 0, the right side of this equation is zero, provided that the indeterminate ratio β/κ is finite. This ratio is indeed finite for real fluids, and a contrary presumption for the model fluid would be irrational. Thus the definition of the incompressible fluid presumes this ratio is finite, and we conclude for such a fluid that the heat capacities at constant V and at constant P are identical:
C_P=C_V=C