Compressor Design

Problem Statement

In a two-stage compressor, the working gas leaving the first stage of compression is cooled by passing through a heat exchanger. Then, it enters the second stage of compression to increase the efficiency. The question is: What should be the intermediate pressure $p_2$ at which intercooling is done to minimize the work input to the compressor?

Mathematical Model

For an ideal gas undergoing isentropic compression, the total work input to a two-stage compressor is given by:

$W=c_p{T}_1\left[{\left(\frac{p_2}{p_1}\right)}^{\frac{k-1}{k}}+{\left(\frac{p_3}{p_2}\right)}^{\frac{k-1}{k}}-2\right]$

Where:

• $c_p$ is the specific heat of the gas at constant pressure
• $T_1$ is the temperature at which the gas enters the compressor
• $p_1$ is the initial pressure
• $p_2$ is the intermediate pressure after the first stage
• $p_3$ is the final pressure after the second stage
• $k$ is the ratio of specific heat at constant pressure to that at constant volume of the gas

Optimisation Setup

Design Variable

• $p_2$ = intermediate pressure (scalar, 1-dimensional problem)

Objective Function

We want to minimize the work input:

$L(p_2):=W=c_p{T}_1\left[{\left(\frac{p_2}{p_1}\right)}^{\frac{k-1}{k}}+{\left(\frac{p_3}{p_2}\right)}^{\frac{k-1}{k}}-2\right]$

Solution Approach

To find the optimal intermediate pressure, we differentiate the work function with respect to $p_2$ and set it to zero:

$\frac{dW}{d{p}_2}=c_p{T}_1\left[\frac{k-1}{k}\cdot \frac{1}{p_1}\cdot {\left(\frac{p_2}{p_1}\right)}^{\frac{k-1}{k}-1}-\frac{k-1}{k}\cdot \frac{p_3}{p_2^2}\cdot {\left(\frac{p_3}{p_2}\right)}^{\frac{k-1}{k}-1}\right]=0$

Simplifying:

$\frac{1}{p_1}\cdot {\left(\frac{p_2}{p_1}\right)}^{\frac{k-1}{k}-1}=\frac{p_3}{p_2^2}\cdot {\left(\frac{p_3}{p_2}\right)}^{\frac{k-1}{k}-1}$

Further manipulation leads to:

${\left(\frac{p_2}{p_1}\right)}^{\frac{k-1}{k}}={\left(\frac{p_3}{p_2}\right)}^{\frac{k-1}{k}}$

Taking both sides to the power of $\frac{k}{k-1}$:

$\frac{p_2}{p_1}=\frac{p_3}{p_2}$

Solving for $p_2$:

$p_2^2=p_1{p}_3$

$p_2=\sqrt{p_1{p}_3}$

Therefore, the optimal intermediate pressure is the geometric mean of the initial and final pressures.

Verification of Minimum

To verify this is indeed a minimum, we can check the second derivative:

$\frac{d^{2}W}{d{p}_2^2}>0\text{ at }{p}_2=\sqrt{p_1{p}_3}$

The second derivative is positive at the critical point, confirming that we have found a minimum.

Physical Interpretation

The result $p_2=\sqrt{p_1{p}_3}$ has an important physical interpretation:

1. It means the pressure ratio is the same across both compression stages: $\frac{p_2}{p_1}=\frac{p_3}{p_2}$
2. This balanced pressure ratio distributes the compression work evenly across the stages
3. The intermediate cooling then becomes most effective, reducing the overall work requirement

Example Calculation

Suppose:

• Initial pressure $p_1=1$ bar
• Final pressure $p_3=9$ bar

The optimal intermediate pressure would be: $p_2=\sqrt{1\times 9}=3\text{ bar}$

With this optimal pressure, the compression ratios for both stages are equal: $\frac{p_2}{p_1}=\frac{p_3}{p_2}=3$

Extension to Multi-Stage Compression

For an $n$-stage compressor with intercooling between each stage, the optimal pressure ratios should be equal across all stages:

$\frac{p_2}{p_1}=\frac{p_3}{p_2}=\dots =\frac{p_{n+1}}{p_n}={\left(\frac{p_{n+1}}{p_1}\right)}^{1/n}$

Key Insights

1. This problem demonstrates the application of optimization to improve energy efficiency
2. The optimal solution has a simple and elegant form (geometric mean)
3. The result has a clear physical interpretation: equal pressure ratios minimize work
4. Multi-stage compression with intercooling can significantly reduce the work input compared to single-stage compression
5. This principle is widely applied in industrial compressors, refrigeration systems, and gas turbines