The Stirling cycle is similar to the Otto cycle, except that the compression and

Question

The Stirling cycle is similar to the Otto cycle, except that the compression and expansion of the gas are done at constant temperature, not adiabatically as in the Otto cycle. The Stirling cycle is used in external combustion engines. (In fact, burning fuel is not necessary; any way of producing a temperature difference will do: solar, geothermal, ocean temperature gradient, etc.) Heat is supplied by a steady source outside the cylinder, instead of explosively inside the cylinder as in the Otto cycle, eliminating intake and exhaust valves, and making Stirling-cycle engines quieter than Otto-cycle engines. While small Stirling engines are used for a variety of purposes, Stirling engines for automobiles have not been successful because they are larger, heavier, and more expensive than conventional automobile engines. In the cycle, the working fluid goes through the following sequence of steps (the figure ): Compressed isothermally at temperature T_1 from the initial state a to state b, with a compression ratio r. Heated at constant volume to state c at temperature T_2. Expanded isothermally at T_2 to state d. Cooled at constant volume back to the initial state a. Assume that the working fluid is n moles of an ideal gas (for which C_V is independent of temperature). Calculate Q, W, and DeltaU for the process a rightarrow b. Express your answers in terms of the variables T_1, T_2, C_V, n, R, and r. Enter your answers separated by commas. Calculate Q, W, and DeltaU for the process b rightarrow c. Express your answers in terms of the variables T_1, T_2, C_V, n, R, and r. Enter your answers separated by commas. Calculate Q, W, and DeltaU for the process c rightarrow d. Express your answers in terms of the variables T_1, T_2, C_V, n, R, and r. Enter your answers separated by commas. Calculate Q, W, and DeltaU for the process d rightarrow a. Express your answers in terms of the variables , T_1, T_2, C_V, n, R, and r. Enter your answers separated by commas.

Explanation / Answer

Work done in Isothermal process

W = n R T ln (vf / vi)

and from First law of thermodynamics   

dQ = dU + dW

in isothermal process the temperature is constant so the change in internal enrgy is zero so

dQ = dW

part A)   from state a ----> b is

      W = 2.303 nRTlog( vf / vi )

in isothermal process Q = W in compression Vf is less than vi work is done on the gas .

and delta U is zero

Part B)   from b ---> c     temperature is T2, change in volume is zero so the work done is zero

                                  dQ = dU +dW

                                   W = 0, Q = 0, dU =0

part C)   from c ----> d

       Isothermal expansioin at T2 ,

           work done = 2.303 n RT log (vf / vi)               vf > vi work done by the gas ,

                               Q = W , delta U = 0

part D) from d -----> a   initial state

                      change in volume is zero so the work done is zero

                                  dQ = dU +dW

                                   W = 0, Q = 0, dU =0

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