A reciprocating compressor is a device that compresses air by a back-and-forth straight-line motion, like a piston in a cylinder. Consider a reciprocating compressor running at 150 rpm. During a compression stroke, 1.30 mol of air is compressed. The initial temperature of the air is 390 K, the engine of the compressor is supplying 8.4 kW of power to compress the air, and heat is being removed at the rate of 1.9 kW. Calculate the temperature change per compression stroke.
The temperature change per compression stroke can be calculated using the equation:
ΔT = (8.4 kW - 1.9 kW) / (1.30 mol * 8.314 J/mol*K * 390 K)
ΔT = 0.0045 K/stroke
To calculate the temperature change per compression stroke, we need to use the first law of thermodynamics, which states that:
Change in internal energy (ΔU) = Heat added - Work done
Given the following values:
- Number of moles of air (n) = 1.30 mol
- Initial temperature (T1) = 390 K
- Power supplied (P) = 8.4 kW = 8400 W
- Heat removal rate (Q) = 1.9 kW = 1900 W
First, let's calculate the work done (W) using the power supplied:
Work done (W) = Power supplied (P) × Time (t)
Since we know the compressor runs at 150 rpm, the time (t) for one compression stroke can be calculated as:
t = 1 min / 150 = 0.0067 min = 0.0067 × 60 = 0.40 seconds
Now we can calculate the work done:
W = P × t = 8400 W × 0.4 s = 3360 J
Next, let's calculate the heat added (Q) using the heat removal rate:
Heat added (Q) = Heat removal rate (Q)
Now, we can calculate the change in internal energy (ΔU):
ΔU = Q - W = 1900 J - 3360 J = -1460 J
Finally, we can calculate the temperature change (ΔT) using the ideal gas law equation:
ΔU = nCv(ΔT)
Where:
- n is the number of moles of air
- Cv is the molar specific heat at constant volume for air (assume Cv = 20.8 J/(mol·K))
Rearranging the equation to solve for ΔT:
ΔT = ΔU / (nCv)
ΔT = (-1460 J) / (1.30 mol × 20.8 J/(mol·K))
ΔT = -55.96 K
Therefore, the temperature change per compression stroke is approximately -55.96 K.
To calculate the temperature change per compression stroke, we need to use the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system.
Mathematically, the equation can be written as:
ΔU = Q - W
where ΔU is the change in internal energy, Q is the heat added to the system, and W is the work done by the system.
In this case, we are given that the reciprocating compressor is running at 150 rpm, which means it completes 150 compression strokes in one minute (assuming the compressor operates at a constant speed throughout).
To calculate the temperature change per compression stroke, we need to determine the heat added and the work done per compression stroke.
First, let's calculate the heat added per compression stroke. We are given that the heat is being removed at a rate of 1.9 kW. Since power is the rate at which work is done, we can assume that the heat removed is equal to the work done:
Q = 1.9 kW
Next, let's calculate the work done per compression stroke. We are given that the engine of the compressor is supplying 8.4 kW of power. Power is defined as the rate at which work is done, so we can calculate the work done per compression stroke by dividing the power by the number of compression strokes per minute:
W = (8.4 kW) / 150 compression strokes = 0.056 kW or 56 W
Now that we have the heat added and the work done per compression stroke, we can use the first law of thermodynamics to find the change in internal energy:
ΔU = Q - W
ΔU = 1.9 kW - 0.056 kW = 1.844 kW or 1844 W
Finally, we can calculate the temperature change per compression stroke using the ideal gas law, which states that the change in temperature is directly proportional to the change in internal energy:
ΔU = nCvΔT
Rearranging the equation to solve for ΔT, where n is the number of moles, Cv is the molar specific heat at constant volume, and ΔT is the change in temperature:
ΔT = ΔU / (nCv)
We are given that 1.30 mol of air is compressed. The specific heat at constant volume for air is approximately 20.8 J/mol·K. Therefore, we can substitute these values into the equation:
ΔT = 1844 W / (1.30 mol * 20.8 J/mol·K)
Calculating this, we get:
ΔT ≈ 68.54 K
Therefore, the temperature change per compression stroke is approximately 68.54 K.