A clever chemical engineer has devised the thermally operated elevator shown at

Question

A clever chemical engineer has devised the thermally operated elevator shown at the bottom of this page. The elevator compartment is made to rise by electrically heating the air contained in the piston and cylinder drive mechanism, and the elevator is lowered by opening a valve at the side of the cylinder, allowing the air to escape slowly once the elevator compartment returns to the lower level, a small compressor removes the remaining air in the cylinder and replaces it with air at 20 degree celcius and a pressure sufficient to support the elevator compartment. The cycle can then be repeated. There is no heat transfer between the piston, cylinder, and the gas; the weight of the piston, elevator, and elevator contents is 4000 kg; the piston has an area of 2.5 m2; and the volume in the cylinder when the elevator is at its lowest level is 25 m3. There is no friction between the piston and the cylinder, and the air in the cylinder is assumed to be an ideal gas with Cp = 30 J/mol K.

a.) What is the pressure in the cylinder throughout the process?

b.) How much heat must be added to the air during the process of raising the elevator 3 m, and what is the final temperature of the gas?

c.) What fraction of the heat added is used in doing work? What fraction is used in raising the temperature of the gas?

d.) How many moles of air must be allowed to escape in order for the elevator to return to the lowest level?

Explanation / Answer

Given initial temperature,Ti = 200C
area of piston, A = 2.5 m2
volume contained in the cylinder V = 25 m3
Cp = 30 J/mol-K
mass of the piston, elevator, and elevator contents, m = 4000kg
a)weight of the piston, elevator, and elevator contents, w = 4000kg*9.81m/s2 = 39240 N {g=9.81m/s2}
Pressure inside the cylinder will always be equal to just hold theweight.
so, Pi = W/A = 39240/2.5 = 15696 N/m2
assuming process as reversible process, pressure will always remainthesame              {to hold the weight and No jerk in the motion of the elevator}
so, Pressure inside the cylinder throughout the process =15696 N/m2 .................................ans(a)
b)number of moles of air inside, n = PV/RT =(15696*25)/(8.314*293.15) = 161
work done by gas to raise the elevator by 3 m = mgh = 4000kg*9.81m/s2*3m = 117720 J
Now applying 1st law of thermodynamics, Q = U+W
or, nCpdT = nCvdT +W
or, n(Cp-Cv)dT = W
but, we know that, Cp-Cv =R = 8.314 J/mol-K
so, 161*8.314*dT = 117720
or, dT = Tf - Ti = 87.945
or, Tf = 20+87.945 = 107.945 0C
and heat added to the system, Q = nCpdT = 161*30*87.945
         = 424774.35 J = 424.77kJ  ..........................................................ans(b)
c) fraction of heat used in doing work = W/Q =117720/424774.35
                                                        = 0.2771 = 27.71%
so fraction used in raising the temperature = 1-0.2771 = 0.7229=72.2 % ..............................ans(c)
d)Let n' is the moles allowed to escape
assuming outside temperature = Ti = 200C,
applying conservation of energy
(n-n')CvTf' + n'CvTi = nCvTf + mgh, whereTf' is the temperature of the system afterreturn to initial state
Now, n' = PV/RTf' or,Tf' = PV/n'R, subtituting in the aboveequation
Cv[(n-n')*PV/n'R + n'Ti -nTf] = mgh
Now, Cv = Cp-R = 30-8.314 = 21.686 J/mol-K
so, [(161-n')*15696*25/(n'*8.314) + n'*293.15 - 161*381.095] =117720/21.686
or, [7598797.21/n' - 47197.49 + 293.15n' - 61356.29] = 5428.38
or, 7598797.21/n' + 293.15n' = 5428.38+47197.49+ 61356.29=113982.16
or, 25921.19/n' + n' = 388.81
or, n'2-388.81n' +25921.19 =0
solving the quadratic equation gives us n' = 85.445 and 303.364
n' should be less than o equal to n (n=161)
so, no of moles allowed to escape, n' = 85.44 .............................................................................ans(d)
e) It should not be considered as the good idea as only 27.17% ofthe heat supplied is used in actual work. also there is work ofcompressor involved we does not come in the calculationsabove. ........................................ans(e)

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