Known A two-stage compressor with intercooling operates at steady state under specified conditions.
Find Determine the intercooler pressure for minimum total compressor work input, per unit of mass flowing.
Schematic and Given Data:
Engineering Model
1. The compressor stages and intercooler are analyzed as control
volumes at steady state.
2. The compression processes are isentropic.
3. There is no pressure drop for flow through the intercooler.
4. The temperature at the inlet to the second compressor stage
obeys T_{ d } \geq T_{1}
5. Kinetic and potential energy effects are negligible.
6. The working fluid is air modeled as an ideal gas.
7. The specific heat c_{p} and thus the specific heat ratio k are
constant.
Analysis The total compressor work input per unit of mass flow is
\frac{\dot{W}_{ c }}{\dot{m}}=\left(h_{ c }-h_{1}\right)+\left(h_{2}-h_{ d }\right)
Since c_{p} is constant,
\frac{\dot{W}_{ c }}{\dot{m}}=c_{p}\left(T_{ c }-T_{1}\right)+c_{p}\left(T_{2}-T_{ d }\right)
=c_{p} T_{1}\left(\frac{T_{ c }}{T_{1}}-1\right)+c_{p} T_{ d }\left(\frac{T_{2}}{T_{ d }}-1\right)
Since the compression processes are isentropic and the specific heat ratio k is constant, the pressure and temperature ratios across the compressor stages are related, respectively, by
\frac{T_{ c }}{T_{1}}=\left(\frac{p_{ i }}{p_{1}}\right)^{(k-1) / k} \quad \text { and } \quad \frac{T_{2}}{T_{ d }}=\left(\frac{p_{2}}{p_{ i }}\right)^{(k-1) / k}
Collecting results
\frac{\dot{W}_{ c }}{\dot{m}}=c_{p} T_{1}\left[\left(\frac{p_{ i }}{p_{1}}\right)^{(k-1) / k}-1\right]+c_{p} T_{ d }\left[\left(\frac{p_{2}}{p_{ i }}\right)^{(k-1) / k}-1\right]
Hence, for specified values of T_{1}, T_{ d }, p_{1}, p_{2}, \text { and } c_{p}, the value of the total compressor work input varies with the intercooler pressure only. To determine the pressure p_{ i } that minimizes the total work, form the derivative
\frac{\partial\left(\dot{W}_{c} / \dot{m}\right)}{\partial p_{ i }}=c_{p} T_{1} \frac{\partial}{\partial p_{ i }}\left(\frac{p_{ i }}{p_{1}}\right)^{(k-1) / k}+c_{p} T_{d} \frac{\partial}{\partial p_{ i }}\left(\frac{p_{2}}{p_{ i }}\right)^{(k- 1 ) / k}
=c_{p} T_{1}\left(\frac{k-1}{k}\right)\left[\left(\frac{p_{ i }}{p_{1}}\right)^{-1 / k}\left(\frac{1}{p_{1}}\right)-\frac{T_{ d }}{T_{1}}\left(\frac{p_{2}}{p_{ i }}\right)^{-1 / k}\left(\frac{p_{2}}{p_{ i }^{2}}\right)\right]
=c_{p} T_{1}\left(\frac{k-1}{k}\right) \frac{1}{p_{ i }}\left[\left(\frac{p_{ i }}{p_{1}}\right)^{(k-1) / k}-\frac{T_{ d }}{T_{1}}\left(\frac{p_{2}}{p_{ i }}\right)^{(k-1) / k}\right]
Setting the partial derivative to zero, we get
\frac{p_{ i }}{p_{1}}=\left(\frac{p_{2}}{p_{ i }}\right)\left(\frac{T_{ d }}{T_{1}}\right)^{k /(k-1)} (a)
Alternatively,
1 p_{ i }=\sqrt{p_{1} p_{2}\left(\frac{T_{ d }}{T_{1}}\right)^{k /(k-1)}} (b)
By checking the sign of the second derivative, it can be verified that the total compressor work is a minimum.
1 Observe that when T_{ d }=T_{1}, p_{ i }=\sqrt{p_{1} p_{2}}.
Skills Developed
Ability to…
• complete the detailed derivation of a thermodynamic expression.
• use calculus to minimize a function.
Quick Quiz
If p_{1}=1 \text { bar, } p_{2}=12 \text { bar, } T_{ d }=T_{1}=300 K, and k = 1.4, determine the intercooler pressure for minimum total compressor work, in bar, and the accompanying temperature at the exit of each compressor stage, in K. Ans. 3.46 bar, 428 K.
