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The ratio of the clearance volume to the displacement volume of a R12 reciprocating compressor is 0.05. Specific volume at inlet and outlet of compressor are 0.04 and 0.02 m3/kg respectively. Volumetric efficiency of the compressor is:
- a)95.0%
- b)47.5%
- c)38.0%
- d)19.0%
Correct answer is option 'A'. Can you explain this answer?
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The ratio of the clearance volume to the displacement volume of a R12 ...
To find the volumetric efficiency of the compressor, we need to understand the definitions of clearance volume, displacement volume, and volumetric efficiency.
- The clearance volume (Vc) is the volume remaining in the cylinder at the end of the suction stroke when the piston is at top dead center (TDC). It represents the space that is not swept by the piston during the compression stroke.
- The displacement volume (Vd) is the volume swept by the piston during the compression stroke from bottom dead center (BDC) to top dead center (TDC).
- The volumetric efficiency (ηv) of a compressor is the ratio of the actual volume of air delivered to the displacement volume. It gives an indication of how well the compressor fills the cylinder with air.
Now, let's solve the problem step by step.
Step 1: Calculate the clearance volume (Vc) and displacement volume (Vd).
Given: the ratio of clearance volume to displacement volume = 0.05
Let the displacement volume be Vd.
Vc = 0.05 * Vd
Step 2: Calculate the actual volume of air delivered.
The actual volume of air delivered (Vactual) is the sum of the displacement volume (Vd) and the clearance volume (Vc).
Vactual = Vd + Vc
Step 3: Calculate the volumetric efficiency.
The volumetric efficiency (ηv) is the ratio of the actual volume of air delivered (Vactual) to the displacement volume (Vd).
ηv = Vactual / Vd
Step 4: Substitute the values.
Given: Specific volume at inlet = 0.04 m3/kg
Specific volume at outlet = 0.02 m3/kg
The specific volume is the reciprocal of density. So, the density at the inlet (ρinlet) = 1 / 0.04 = 25 kg/m3
The density at the outlet (ρoutlet) = 1 / 0.02 = 50 kg/m3
We know that the displacement volume (Vd) is equal to the mass flow rate (m) divided by the density (ρ).
Vd = m / ρ
Since the mass flow rate is the same at both the inlet and outlet, we can write:
Vd = (m / ρinlet) = (m / ρoutlet)
Step 5: Simplify the equation.
Vactual = Vd + Vc
Vactual = (m / ρinlet) + (m / ρoutlet)
Vactual = m * (1 / ρinlet + 1 / ρoutlet)
ηv = Vactual / Vd
ηv = m * (1 / ρinlet + 1 / ρoutlet) / (m / ρinlet)
ηv = (1 / ρinlet + 1 / ρoutlet) / (1 / ρinlet)
Step 6: Substitute the values and calculate.
ηv = (1 / 25 + 1 / 50) / (1 / 25)
ηv = (2 / 50 + 1 / 50) / (1 / 25)
ηv = 3 / 50 * 25
ηv = 3 / 2
ηv = 1.5
The
- The clearance volume (Vc) is the volume remaining in the cylinder at the end of the suction stroke when the piston is at top dead center (TDC). It represents the space that is not swept by the piston during the compression stroke.
- The displacement volume (Vd) is the volume swept by the piston during the compression stroke from bottom dead center (BDC) to top dead center (TDC).
- The volumetric efficiency (ηv) of a compressor is the ratio of the actual volume of air delivered to the displacement volume. It gives an indication of how well the compressor fills the cylinder with air.
Now, let's solve the problem step by step.
Step 1: Calculate the clearance volume (Vc) and displacement volume (Vd).
Given: the ratio of clearance volume to displacement volume = 0.05
Let the displacement volume be Vd.
Vc = 0.05 * Vd
Step 2: Calculate the actual volume of air delivered.
The actual volume of air delivered (Vactual) is the sum of the displacement volume (Vd) and the clearance volume (Vc).
Vactual = Vd + Vc
Step 3: Calculate the volumetric efficiency.
The volumetric efficiency (ηv) is the ratio of the actual volume of air delivered (Vactual) to the displacement volume (Vd).
ηv = Vactual / Vd
Step 4: Substitute the values.
Given: Specific volume at inlet = 0.04 m3/kg
Specific volume at outlet = 0.02 m3/kg
The specific volume is the reciprocal of density. So, the density at the inlet (ρinlet) = 1 / 0.04 = 25 kg/m3
The density at the outlet (ρoutlet) = 1 / 0.02 = 50 kg/m3
We know that the displacement volume (Vd) is equal to the mass flow rate (m) divided by the density (ρ).
Vd = m / ρ
Since the mass flow rate is the same at both the inlet and outlet, we can write:
Vd = (m / ρinlet) = (m / ρoutlet)
Step 5: Simplify the equation.
Vactual = Vd + Vc
Vactual = (m / ρinlet) + (m / ρoutlet)
Vactual = m * (1 / ρinlet + 1 / ρoutlet)
ηv = Vactual / Vd
ηv = m * (1 / ρinlet + 1 / ρoutlet) / (m / ρinlet)
ηv = (1 / ρinlet + 1 / ρoutlet) / (1 / ρinlet)
Step 6: Substitute the values and calculate.
ηv = (1 / 25 + 1 / 50) / (1 / 25)
ηv = (2 / 50 + 1 / 50) / (1 / 25)
ηv = 3 / 50 * 25
ηv = 3 / 2
ηv = 1.5
The
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The ratio of the clearance volume to the displacement volume of a R12 reciprocating compressor is 0.05. Specific volume at inlet and outlet of compressor are 0.04 and 0.02 m3/kg respectively. Volumetric efficiency of the compressor is:a)95.0% b)47.5% c)38.0% d)19.0%Correct answer is option 'A'. Can you explain this answer?
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