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Derive an expression for volumetric efficiency of a single stage single acting reciprocating air compressor with clearance and discuss at least four factors that affect volumetric efficiency.
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The volumetric efficiency of a compressor is the ratio of the actual free air delivered to the displacement of the compressor. The expression for a single stage reciprocating compressor can be derived as follows-

Let,

$\eta_v = \textit{Volumetric efficiency}\\ V_s = \textit{Swept volume}\\ V_a = \textit{Actual volume}\\ V_c = \textit{Clearance volume}$ The volumetric efficiency is given by the expression,

$\eta_v = \frac{v_1 - v_4}{v_1 - v_3} = \frac{v_a}{v_s}\\ \hspace{0.25cm}=\frac{v_s +v_c - v_4}{v_s}\\ \hspace{0.25cm}=\frac{v_s -(v_4 - v_c)}{v_s}$

But,

$P_3V_3^n = P_4V_4^n\\ P_2V_c^n = P_1V_4^n.....\textit{as}\hspace{0.25cm}p_3 = p_2\hspace{0.25cm}\&\hspace{0.25cm}p_1 = p_4\hspace{0.25cm}\&\hspace{0.25cm}V_3 =V_c\\ \frac{V_4}{V_c} = (\frac{p_2}{p_1})^2\\ \eta_v = \frac{V_s - V_c(\frac{V_4}{V_c} - 1)}{V_s}\\ \hspace{0.25cm}=\frac{V-s - V_c((\frac{p_2}{p_1})^\frac{1}{n} - 1)}{V_s}\\ \hspace{0.25cm}= 1 - \frac{V_c}{V_s}((\frac{p_2}{p_1})^\frac{1}{n} - 1)\\ \eta_v = 1 - \gamma((\frac{p_2}{p_1})^\frac{1}{n} - 1) .........where\hspace{0.25cm} \gamma = \frac{V_c}{V_s}$

The following factors affect volumetric efficiency:

• Increase in clearance volume causes a decrease in volumetric efficiency.

• Increase in compression ratio also decreases the volumetric efficiency.

• Piston ring leakage allows gas from compression chamber to escape the piston and enter the other end of cylinder. This reduces the volumetric efficiency.

• Valve slip leads to reversal of gas flow through the valves. This leads to volume loss at the inlet and outlet valves. Thus, decreasing the volumetric efficiency.