Question

The volumetric efficiency of a reciprocating compressor has the clearance ratio 'k' and the pressure ratio p2/p1 is calculated by using the following equation

• A. \( 1 + k + k (p_2/p_1)^{1/n} \)
• B. \( 1 - k + k (p_2/p_1)^{1/n} \)
• C. \( 1 + k - k (p_2/p_1)^{1/n} \)
• D. \( 1 + k - k (p_2/p_1)^n \)

Hint:

Volumetric efficiency of a reciprocating compressor is affected by the clearance volume and pressure ratio. The formula \( \eta_v = 1 + k - k(p_2/p_1)^{1/n} \) shows that a larger clearance ratio \( k \) and a higher pressure ratio \( p_2/p_1 \) lead to lower volumetric efficiency.

Answer 1

The Correct Option is C

Step 1: Define Volumetric Efficiency and Clearance Ratio for a Reciprocating Compressor.
Volumetric efficiency \( \eta_v \) is a measure of the effectiveness of a compressor in drawing in fresh air. It is defined as the ratio of the actual volume of free air drawn into the cylinder during the suction stroke to the piston displacement (swept volume). \[ \eta_v = \frac{\text{Actual volume of free air admitted}}{\text{Swept volume}} \] Clearance ratio \( k \) is the ratio of clearance volume (\( V_c \)) to swept volume (\( V_s \)): \[ k = \frac{V_c}{V_s} \]
Step 2: Derive the expression for volumetric efficiency assuming polytropic compression.
Consider the P-V diagram of a reciprocating compressor cycle. The total volume at the beginning of compression is \( V_1 = V_s + V_c \).
At the end of compression, the volume is \( V_2 \).
At the end of discharge, the volume is \( V_c \).
During the expansion of the gas trapped in the clearance volume from \( p_2 \) to \( p_1 \), the volume changes from \( V_c \) to \( V_4 \). The volume of fresh air drawn in is \( V_1 - V_4 \).
The volumetric efficiency is \( \eta_v = \frac{V_1 - V_4}{V_s} \).
For polytropic compression and expansion processes (with index \( n \)): For expansion of clearance volume: \( p_2 V_c^n = p_1 V_4^n \) \[ \frac{V_4}{V_c} = \left(\frac{p_2}{p_1}\right)^{1/n} \] \[ V_4 = V_c \left(\frac{p_2}{p_1}\right)^{1/n} \] Substitute this into the volumetric efficiency expression: \[ \eta_v = \frac{(V_s + V_c) - V_c \left(\frac{p_2}{p_1}\right)^{1/n}}{V_s} \] Divide each term in the numerator by \( V_s \): \[ \eta_v = \frac{V_s}{V_s} + \frac{V_c}{V_s} - \frac{V_c}{V_s} \left(\frac{p_2}{p_1}\right)^{1/n} \] Substitute \( k = \frac{V_c}{V_s} \): \[ \eta_v = 1 + k - k \left(\frac{p_2}{p_1}\right)^{1/n} \] This is the standard equation for the volumetric efficiency of a reciprocating compressor. The final answer is $\boxed{\text{3}}$.