Question

3. (20 points) Sandler 6.18 The Clausius equation of state is P(V – b) = RT where b is a constant. (a) Show that for this volumetric equation of state Cp(P,T) = Cy(P,T) +R Cp(P,T) = CP(T) Cy(V,T) = Ci(T) (b) For a certain process the pressure of a gas must be reduced from an initial pressure P, to the final pressure P2. The gas obeys the Clausius equation of state, and the pressure reduction is to be accomplished by passing the gas through a flow constriction, such as a pressure-reducing valve, or by passing it through a small gas turbine (which we can assume to be both reversible and adiabatic). Obtain expressions for the final gas temperature in each of these cases in terms of the initial state and the properties of the gas.

Answer 1

Answer (a)

The Clausius equation for specific heat difference is given as:

$\small P(V - b ) = RT$

The general expression for specific heat difference is given as:

$\small C_{P}- C_{V} =-T(\frac{\partial V}{\partial T})_{p}(\frac{\partial P}{\partial V})_{T}$

   $\small = -T(\frac{R}{P})(\frac{- P}{V-b})$

  $\small = (\frac{TR}{P(V-b)})\times R$

TR TR)) x

$\small C_{P}- C_{V} = R$

$\small C_{P}(P,T)= C_{V}(P,T) + R$

The specific heat capacity at constant pressure is given as :

$\small C_{P}(P,T)= C_{P}(\frac{\partial V}{\partial T})_{P} + P(\frac{\partial V}{\partial T})_{P}$

  $\small = C_{P}(\frac{\partial V}{\partial T})_{P} + 0$

$\small C_{P}(P,T)= C_{P}(T)$

The specific heat capacity at constant volume :

$\small C_{V}(V,T)= C_{V}(\frac{\partial V}{\partial T})_{V} + (\frac{\partial T}{\partial T})_{V}$

$\small = C_{V}(\frac{\partial V}{\partial T})_{V} + 0$

Therefore, $\small C_{V}(V,T)= C_{V}(T)$

Answer (b)

The ideal gas equation is given by :

$\small P(V - b ) = RT$

Case 1 : The pressure is reducing when pressure is reducing a small gas turbine

$\small h_{1} + \frac{V_{1}^{2}}{2g} + g.z + Q = h_{2} + \frac{V_{2}^{2}}{2g} + g.z + W$

$\small u_{1} + P_{1}.V_{1} + \frac{V_{1}^{2}}{2g} + g.z + 0 = u_{2} + P_{2}.V_{2} + \frac{V_{2}^{2}}{2g} + g.z + mC_{p}.(T_{2} - T_{1})$

Solving above and determining T2 , gives :

$\small T_{2} =T_{1} + \frac{u_{1} + P_{1}.V_{1} + \frac{V_{1}^{2}}{2g} - u_{2} - P_{2}.V_{2} - \frac{V_{2}^{2}}{2g} }{mC_{p}}$

Case 2 : The pressure is reducing when we use pressure reducing value

$\small \frac{P_{1}.(V_{1} - b )}{R.T_{1}} = \frac{P_{2}.(V_{2} - b )}{R.T_{2}}$

  $\small T_{2} = \frac{P_{2}.T_{1}(V_{2} -b )}{P_{1}.(V_{1} - b)}$

Hence, above equation is our required solution.