Question

The clearance is provided in the reciprocating compressor with clearance ratio of `k'. If the pressure ratio is \(\frac{P_2}{P_1}\), then the volumetric efficiency in terms of clearance ratio k, is

• A. \(1 + k - k\left(\frac{P_2}{P_1}\right)^{(1/n)} \)
• B. \(1 - k + k\left(\frac{P_2}{P_1}\right)^{(1/n)} \)
• C. \(1 - k - k\left(\frac{P_2}{P_1}\right)^{(1/n)} \)
• D. \(1 + k + k\left(\frac{P_2}{P_1}\right)^{(1/n)} \)

Hint:

Memory trick: As the pressure ratio $\frac{P_2}{P_1}$ increases, the trapped high-pressure gas expands more, taking up more space and reducing the volume of fresh air that can be drawn in. Therefore, the volumetric efficiency must drop as pressure ratio increases, meaning the formula must have a negative sign before the pressure ratio term: $1 + k - k(\dots)$.

Answer 1

The Correct Option is A

Concept: In a reciprocating compressor, clearance volume ($V_c$) is the small space left at the top of the cylinder head to prevent mechanical impact between the piston and the valves. The clearance ratio ($k$) is defined relative to the swept stroke volume ($V_s$): \[ k = \frac{V_c}{V_s} \] Volumetric efficiency ($\eta_v$) measures the effectiveness of a compressor in drawing in fresh gas. It is defined as the ratio of the actual volume of fresh air sucked into the cylinder during intake ($V_1 - V_4$) to the theoretical swept stroke volume ($V_s$): \[ \eta_v = \frac{V_1 - V_4}{V_s} \] During the delivery stroke, high-pressure gas remains in the clearance volume at pressure $P_2$. Before the intake valve can open, this trapped gas must expand polytropically down to the intake pressure $P_1$ according to the relationship $P_2 V_c^n = P_1 V_4^n$.

Step 1: Express expanded volume \(V_4\) in terms of clearance parameters.
From the polytropic expansion relation: \[ \frac{V_4}{V_c} = \left(\frac{P_2}{P_1}\right)^{1/n} \quad \Rightarrow \quad V_4 = V_c \left(\frac{P_2}{P_1}\right)^{1/n} \]

Step 2: Expand the expression for volumetric efficiency.
Substitute total volume definitions, noting that the maximum volume at Bottom Dead Center is $V_1 = V_s + V_c$: \[ \eta_v = \frac{(V_s + V_c) - V_4}{V_s} = \frac{V_s + V_c - V_c\left(\frac{P_2}{P_1}\right)^{1/n}}{V_s} \] Separating the fraction term-by-term yields: \[ \eta_v = \frac{V_s}{V_s} + \frac{V_c}{V_s} - \frac{V_c}{V_s}\left(\frac{P_2}{P_1}\right)^{1/n} \]

Step 3: Substitute the clearance ratio \(k = \frac{V_c}{V_s}\).
Replacing $\frac{V_c}{V_s}$ with $k$: \[ \eta_v = 1 + k - k\left(\frac{P_2}{P_1}\right)^{1/n} \] This derived expression matches Option (A).