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Gas expands through an adiabatic turbine. The inlet stream flows at 100 bar and 600 K, while the outlet stream is at 20 bar and 445 K. The constant-pressure molar heat capacity of the gas is $30.0 + 0.02T$, where the molar heat capacity is in J/(mol K) and the temperature is in K. The process is at steady state.

Assume the gas is non-ideal and follows the following equation:

$$p=\frac{RT}{V_\mathrm m-b}+\frac{ap^2}{T\left(V_\mathrm m-b\right)}\tag{1}$$

with $\displaystyle a=0.001\ \frac{\mathrm{m^3\ K}}{\mathrm{bar\ mol}}$

and $\displaystyle b=0.00008\ \frac{\mathrm{m^3}}{\mathrm{mol}}$

For this gas the differential molar enthalpy change is:

$$\mathrm dH_\mathrm m=C_{\mathrm m,p}\mathrm dT+\left[V_\mathrm m-T\left(\frac{\partial V_\mathrm m}{\partial T}\right)_p\right]\mathrm dp\tag{2}$$

Assume that the ideal expression for the molar heat capacity given is still valid, but that it is only true as $p$ goes to 0.

And the question is; what is the shaft work of the turbine?

This question is from a thermodynamics course and to solve it students were sort of guided through a solution. First substitute equation (1) into equation (2) and find a general expression for the differential molar enthalpy change of the non-ideal gas. Then draw a $pT$ diagram of an appropriate path to evaluate the enthalpy change of the process, and just integrate along the path to find the answer.

I still can't figure out how this was supposed to be done.

Thanks to anyone who bothered to read/think about this problem as it is kind of long.

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You are aware that, from the open system version of the first law of thermodynamics, the shaft work per unit mass must be equal to the change in enthalpy per unit mass, correct? So you need to determine the change in enthalpy per unit mass between the gas entering the turbine and the gas exiting the turbine. Do we agree so far?

We know that enthalpy is a function of state. So we need to figure out a convenient path between state 1 at 100 bar, 600 K, and state 2 at 20 bar ,445 K such that the equation provided can easily be applied to determine the change in enthalpy. Consider the following 3 part path:

  1. Hold the temperature at 600 K and lower the pressure to say 1 bar, where the gas behaves as an ideal gas. Use the equation provided with dT set to 0 to evaluate $\Delta h$ for this step

  2. Hold the pressure constant at 1 atm, and use the ideal gas heat capacity to calculate $\Delta h$ for cooling the gas down from 600K to 445K.

  3. Hold the temperature constant at 445K and raise the pressure from 1 bar to 20 bars. Use the equation provided with dT set to 0 to evaluate $\Delta h$ for this step.

The sum of the enthalpy changes for these three steps adds up to the enthalpy change for the overall change.

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