Given: Compressor speed is 1500rpm. The compressor has a bore of 3.125 in, a stroke of 3.26in and a connecting rod crank ratio to 3.5. Inlet pressure is atmospheric 14.7 psia, peak cylinder pressure is 132psia, and mean active pressure mep is 26psia. Flow is 8.9 cfm at mep, given 1.6 hp.Determine the force-time function within the compressor’s cylinder and the torque-time function acting on the input shaft on the compressor during any cycle.Assumption: The piston weight is 1 lb; the connecting rod weights 2 lb with its center of mass at 1/3 point from the big end. The crankshaft weights 5.4 lb including a counterbalance that optimally overbalances it to minimize the shaking force. The exponent for the gas law equation is k = 1.13

Question

Given: Compressor speed is 1500rpm. The compressor has a bore of 3.125 in, a stroke of 3.26in and a connecting rod crank ratio to 3.5. Inlet pressure is atmospheric 14.7 psia, peak cylinder pressure is 132psia, and mean active pressure mep is 26psia. Flow is 8.9 cfm at mep, given 1.6 hp.Determine the force-time function within the compressor’s cylinder and the torque-time function acting on the input shaft on the compressor during any cycle.Assumption: The piston weight is 1 lb; the connecting rod weights 2 lb with its center of mass at 1/3 point from the big end. The crankshaft weights 5.4 lb including a counterbalance that optimally overbalances it to minimize the shaking force. The exponent for the gas law equation is k = 1.13

Answer 1

To determine the force-time function within the compressor's cylinder and the torque-time function acting on the input shaft, we need to follow a step-by-step process. Here's how you can approach this problem:

Step 1: Calculate the displacement volume (Vd):
Vd = (π/4) × Bore^2 × Stroke
= (π/4) × (3.125)^2 × (3.26)

Step 2: Calculate the swept volume (Vs):
Vs = Vd × (1 + 1/(CR - 1))
= Vd × (1 + 1/(3.5 - 1))

Step 3: Calculate the clearance volume (Vc):
Vc = Vs / CR
= Vs / 3.5

Step 4: Calculate the compression ratio (CR):
CR = (Vs + Vc) / Vc

Step 5: Calculate the mass of the gas in cylinder (m):
m = (P × Vd) / (R × T)
Here, P is the peak cylinder pressure (132 psia), Vd is the displacement volume calculated in Step 1, R is the specific gas constant, and T is the absolute temperature.

Step 6: Calculate the mass of the gas in clearance volume (mc):
mc = (P × Vc) / (R × T)
Here, P is the inlet pressure (14.7 psia), Vc is the clearance volume calculated in Step 3, R is the specific gas constant, and T is the absolute temperature.

Step 7: Calculate the work done on the gas during the compression stroke:
Wc = m × (k / (k - 1)) × R × T × (1 - (Vc / Vs)^((k - 1) / k))
Here, k is the exponent for the gas law equation (1.13).

Step 8: Calculate the work done on the gas during the expansion stroke:
We = Wc / (CR^(k - 1))

Step 9: Calculate the net work done on the gas during the cycle:
Wnet = We - Wc

Step 10: Calculate the force acting on the piston:
F = Wnet / (Stroke × 12)

Step 11: Calculate the force on the connecting rod due to piston weight:
Fp = Piston Weight × (Stroke × 12 + (Stroke × 12) / 2)

Step 12: Calculate the force on the connecting rod due to connecting rod weight:
Fcr = Connecting Rod Weight × (Stroke × 12) / 2

Step 13: Calculate the force on the connecting rod due to crankshaft weight:
Fcs = Crankshaft Weight × (Stroke × 12) / 2

Step 14: Calculate the total force on the connecting rod:
Ftotal = Fp + Fcr + Fcs

Step 15: Calculate the torque on the input shaft:
Torque = Ftotal × (Bore / 2) × (3.14159 / 4)

By following these steps and plugging in the values you provided, you should be able to determine the force-time function within the compressor's cylinder and the torque-time function acting on the input shaft during any cycle.