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

Step 1: Fix a convenient scale for the volumes.
Since the clearance ratio \(k\) is a pure ratio, we can pick a convenient number for the swept volume without losing generality. Let the swept (stroke) volume be \(V_s = 1\) unit. Then, by the definition \(k = V_c/V_s\), the clearance volume is simply \(V_c = k\), and the total cylinder volume at the start of compression (bottom dead centre) is \(V_1 = V_s + V_c = 1 + k\).

Step 2: Expand the trapped gas back to suction pressure.
At the end of delivery, the clearance volume \(V_c = k\) is filled with gas at the high pressure \(P_2\). Before fresh gas can enter, this trapped gas must expand polytropically until its pressure drops to the suction pressure \(P_1\):\[ P_2 V_c^{\,n} = P_1 V_4^{\,n} \quad \Rightarrow \quad V_4 = V_c \left(\frac{P_2}{P_1}\right)^{1/n} = k\left(\frac{P_2}{P_1}\right)^{1/n} \]

Step 3: Find the fresh charge actually drawn in.
Fresh gas only starts entering once the piston has moved past \(V_4\), so the volume of fresh gas sucked in during the intake stroke is \(V_1 - V_4\). Since we set \(V_s = 1\), the volumetric efficiency is this fresh volume divided directly by the stroke volume:\[ \eta_v = \frac{V_1 - V_4}{V_s} = \frac{(1+k) - k\left(\frac{P_2}{P_1}\right)^{1/n}}{1} = 1 + k - k\left(\frac{P_2}{P_1}\right)^{1/n} \]\[ \boxed{\eta_v = 1 + k - k\left(\frac{P_2}{P_1}\right)^{1/n}} \]
This is exactly option (A).