--- title: "Effect of clearance and volumetric efficiency" subject: "Applied Thermodynamics" chapter: 2 key: "semester_3/applied_thermodynamics:clearance-volumetric-efficiency-derivation" source: https://www.bctengineer.com/q/semester_3/applied_thermodynamics/clearance-volumetric-efficiency-derivation site: BCT Engineering --- # Effect of clearance and volumetric efficiency Applied Thermodynamics · Chapter 2: Air Compressors - Asked in: year not given, 6 marks - Video: [APPLIED THERMODYNAMICS Reciprocating Compressor Effect of Clearance and Efficiencies](https://www.youtube.com/watch?v=RD-4CerEycM) — BR Mech Academy - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/applied_thermodynamics/clearance-volumetric-efficiency-derivation - Whole chapter: https://www.bctengineer.com/md/semester_3/applied_thermodynamics/chapter-2 ## Question What is clearance volume in a reciprocating compressor? Explain its effect on the compressor, and derive an expression for the volumetric efficiency in terms of the clearance ratio and the pressure ratio. ## Short answer Clearance volume Vc is the cylinder volume at the end of delivery; clearance ratio C = Vc/Vs. Re‑expansion of trapped air reduces fresh intake. Volumetric efficiency ηv = Va/Vs = 1 + C − C (p2/p1)^(1/n); higher clearance or pressure ratio lowers ηv. ## Answer **Clearance volume** $V_c$ is the volume left in the cylinder (including the valve ports) when the piston is at the end of the delivery stroke. It is provided so that the piston does not strike the cylinder head and to leave room for the valves. The clearance ratio is $C = V_c/V_s$, where $V_s$ is the swept volume. #### Effect of clearance - The trapped air re-expands during the suction stroke, so suction starts only after the piston has moved some distance. - The volume of fresh air drawn in is less than the swept volume, so the capacity falls. - The work per kg of air delivered is not changed (the compression and re-expansion work of the clearance air cancel for the same index), but the machine must be larger for a given output. - A greater pressure ratio and a larger clearance reduce the volume drawn in; at a high enough ratio no air is delivered at all. #### Derivation of volumetric efficiency ```text p p2 | 3+--+2 | | \ | \ \ | \ \ p1 | 4+---\--+1 +-----+-+----+--+---- V Vc V4 V1 <---Va---> ``` The swept volume is $V_s = V_1 - V_c$, so $V_1 = V_c + V_s$. The effective suction volume (air drawn in) is $V_a = V_1 - V_4$, where $V_4$ is the volume of the re-expanded clearance air. The clearance air expands polytropically from $p_2$ to $p_1$: $p_2V_c^{\,n} = p_1V_4^{\,n}$, so $$ V_4 = V_c\left(\frac{p_2}{p_1}\right)^{1/n} $$ $$ \begin{aligned} V_a &= V_s + V_c - V_c\left(\frac{p_2}{p_1}\right)^{1/n}\\ \eta_v &= \frac{V_a}{V_s} = 1 + \frac{V_c}{V_s} - \frac{V_c}{V_s}\left(\frac{p_2}{p_1}\right)^{1/n} \end{aligned} $$ $$ \boxed{\eta_v = 1 + C - C\left(\frac{p_2}{p_1}\right)^{1/n}} $$ Hence $\eta_v$ decreases when the clearance ratio $C$ or the pressure ratio $p_2/p_1$ increases. This is one reason for using multistage compression at high pressure ratios. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the effect of clearance volume on a compressor? A. The fresh air drawn in is less than the swept volume, so capacity falls B. It increases the work per kg of air C. It increases the delivery temperature only D. It raises the swept volume Answer: A. Trapped air re-expands and occupies part of the stroke, reducing air drawn in. (Key point: Effect on capacity) 2. How does clearance affect work per kg of air delivered? A. It doubles B. It is unchanged, since compression and re-expansion work cancel for the same index C. It decreases to zero D. It increases with clearance ratio Answer: B. Re-expansion returns work, so work per kg is not changed; the machine just becomes larger. (Key point: Work per kg) 3. The volumetric efficiency is A. 1 - C (p2/p1)^(n-1) B. 1 + C + C (p2/p1)^(1/n) C. 1 + C - C (p2/p1)^(1/n) D. C (p2/p1)^(1/n) - 1 Answer: C. Va = Vs + Vc - Vc (p2/p1)^(1/n); dividing by Vs gives the expression. (Key point: Volumetric efficiency) 4. If the pressure ratio increases, the volumetric efficiency A. increases B. stays constant C. becomes unity D. decreases, which is one reason for multistage compression Answer: D. A higher ratio expands the clearance air more, so less fresh air is drawn. (Key point: Trend) ## Related in Applied Thermodynamics - [Work of single-stage compressor without clearance](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_thermodynamics%3Asingle-stage-compressor-work-derivation) — Applied Thermodynamics, chapter 2 - [Components of reciprocating air compressor](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_thermodynamics%3Areciprocating-compressor-components) — Applied Thermodynamics, chapter 2 - [Cylinder output with clearance numerical](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_thermodynamics%3Acompressor-volumetric-efficiency-numerical) — Applied Thermodynamics, chapter 2 - [Actual VCR cycle with compressor efficiency](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_thermodynamics%3Avcr-actual-compressor-efficiency-numerical) — Applied Thermodynamics, chapter 3 - [Two-stage compressor with perfect intercooling](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_thermodynamics%3Atwo-stage-compressor-intercooling-numerical) — Applied Thermodynamics, chapter 2 - [Single-stage compressor power numerical](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_thermodynamics%3Asingle-stage-compressor-numerical) — Applied Thermodynamics, chapter 2 ## Related in other subjects - [Ideal Brayton: efficiency and net power, pressure ratio 10](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Abrayton-5m3s-pr10-1400k) — Fundamental of Thermodynamics & Heat Transfer, chapter 6 - [Brayton cycle efficiency and optimum pressure ratio](https://www.bctengineer.com/question.md?key=semester_7%2Fturbo_machines%3Abrayton-cycle-efficiency-optimum-pressure-ratio) — Turbo Machines, chapter 3 - [Ideal gas turbine 100 kW: pressure ratio, work, efficiency](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Aideal-gas-turbine-100kw-pressure-ratio) — Fundamental of Thermodynamics & Heat Transfer, chapter 6 - [Otto cycle efficiency and mean effective pressure](https://www.bctengineer.com/question.md?key=semester_6%2Finternal_combustion_engines%3Aotto-cycle-efficiency-numerical) — Internal Combustion Engines, chapter 1