Example of a Thermodynamics Problem that Cannot Be Solved with Only the Mass and Energy Balances^7 A compressor is a gas pumping device that takes in gas at low pressure and discharges it at a higher pressure. Since this process occurs quickly compared with heat transfer, it is usually assumed to

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Example of a Thermodynamics Problem that Cannot Be Solved with Only the Mass and Energy Balance[latex]s^{7}[/latex]

A compressor is a gas pumping device that takes in gas at low pressure and discharges it at a higher pressure. Since this process occurs quickly compared with heat transfer, it is usually assumed to be adiabatic; that is, there is no heat transfer to or from the gas during its compression. Assuming that the inlet to the compressor is air [which we will take to be an ideal gas with [latex]C_{ P }^{*}[/latex] = 29.3 J/(mol K)] at 1 bar and 290 K and that the discharge is at a pressure of 10 bar, estimate the temperature of the exit gas and the rate at which work is done on the gas (i.e., the power requirement) for a gas flow of 2.5 mol/s.

Accepted Answer

(Since there are two unknown thermodynamic quantities, the final temperature and the rate at which work is being done, we can anticipate that the mass and energy balances will not be sufficient to solve this problem.)
The system will be taken to be the gas contained in the compressor. The differential form of the molar mass and energy balances for this open system are

[latex]\begin{aligned}&\frac{d N}{d t}=\dot{N}_{1}+\dot{N}_{2} \&\frac{d U}{d t}=\dot{N}_{1} \underline{H}_{1}+\dot{N}_{2} \underline{H}_{2}+\dot{Q}+\dot{W}\end{aligned}[/latex]

where we have used the subscript 1 to indicate the flow stream into the compressor and 2 to indicate the flow stream out of the compressor.
Since the compressor operates continuously, the process may be assumed to be in a steady state,

[latex]\begin{aligned}\frac{d N}{d t} &=0 \quad ext { or } \quad \dot{N}_{1}=-\dot{N}_{2} \\frac{d U}{d t} &=0\end{aligned}[/latex]

that is, the time variations of the mass of the gas contained in the compressor and of the energy content of this gas are both zero. Also, [latex]\dot{Q}[/latex] = 0 since there is no heat transfer to the gas, and [latex]\dot{W}=\dot{W}_{s}[/latex] since the system boundaries (the compressor) are not changing with time. Thus we have

[latex]\dot{W}_{s}=\dot{N}_{1} \underline{H}_{2}-\dot{N}_{1} \underline{H}_{1}=\dot{N}_{1} C_{ P }^{*}\left(T_{2}-T_{1} ight)[/latex]

or

[latex]\underline{W}_{s}=C_{ P }^{*}\left(T_{2}-T_{1} ight)[/latex]

where [latex]\underline{W}_{s}=\dot{W}_{s} / \dot{N}_{1}[/latex] is the work done per mole of gas. Therefore, the power necessary to drive the compressor can be computed once the outlet temperature of the gas is known, or the outlet temperature can be determined if the power input is known.
We are at an impasse; we need more information before a solution can be obtained. It is clear by comparison with the previous examples why we cannot obtain a solution here. In the previous cases, the mass balance and the energy balance, together with the equation of state of the fluid and the problem statement, provided the information necessary to determine the final state of the system. However, here we have a situation where the energy balance contains two unknowns, the final temperature and [latex]\underline{W}_{s}[/latex]. Since neither is specified, we need additional information about the system or process before we can solve the problem. This additional information will be obtained using an additional balance equation developed in the next chapter.

Question 3.4-4: Example of a Thermodynamics Problem that Cannot Be Solved wi...

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