Calculating Work and State Variables for an ideal Stirling Engine

[WalkTex]

Homework Statement

Consider the ideal Stirling cycle working between a maximum temperature Th and min temp Tc, and a minimum volume V1 and a maximum volume V2. Suppose that the working gas of the cycle is 0.1 mol of an ideal gas with cv = 5R/2.

A) what are the heat flows to the cycle during each leg? Be sure to give the sign. For which legs is the heat flow positive?

B) What work is done by the cycle during each leg?

Homework Equations

W = Integral from Vi to Vf of PdV
W = QH-QC

The Attempt at a Solution

My attempt at a solution is given in the pdf below. My main problem was identifying some of these quantities without defined values for Vi, Vf, Qh, Qc or mass. I simply assumed some values that were gave in part C, but it was unclear to me if this is what was to be done, or if there was a way to calculate it without these values. For instance, in leg ii of the cycle, no mass is given for know data. How may I then calculate heat? Thanks in advance.

 

[DrClaude]

For part (b), do not use any numerical values. Just derive the formulas for the work.

I don't understand why you write $W=Q_H - Q_C$ for individual legs. During the cycle, the working substance is imagined to be in contact with either the hot or the cold reservoir one at a time. I would use $\Delta U = W + Q$, with $U$ the energy of the working substance.

 

[WalkTex]

For part (b), do not use any numerical values. Just derive the formulas for the work.

I don't understand why you write $W=Q_H - Q_C$ for individual legs. During the cycle, the working substance is imagined to be in contact with either the hot or the cold reservoir one at a time. I would use $\Delta U = W + Q$, with $U$ the energy of the working substance.

Thank you for the tip. For part A, does one not need the mass of the working substance to solve for heat?

 

[DrClaude]

For part A, does one not need the mass of the working substance to solve for heat?

No. Until you reach the point where you have to calculate actual values, such as efficiency, you should keep everything in generic terms.

I looked at your solution, and first I must say that your notation is confusing. For the volumes, you use $V_\mathrm{min}$ and $V_\mathrm{max}$ and $V_\mathrm{i}$ and $V_\mathrm{f}$, while the problem states $V_1$ and $V_2$. You should be also careful with the sign convention. When using $\Delta U = Q + W$, $W$ is in terms of the work done on the working substance, while for an engine, you usually want the work produced by the working substance to be positive. You get $W$ correct at then end, but because you dropped a minus sign along the way.

The efficiency you get is indeed low. You should check your calculation of $Q_\mathrm{pos}$ again (don't confuse $C_V$ and $c_V$...).

 

[Nickie]

Thank you for providing your solution for the ideal Stirling engine problem. Your approach is correct and you have correctly identified the work and heat flows for each leg of the cycle. However, I would like to clarify a few points in your solution.

Firstly, to calculate the heat flow for each leg, you can use the equation Q = mcΔT, where m is the mass of the working gas (given as 0.1 mol in the problem), c is the specific heat capacity of the working gas (given as 5R/2), and ΔT is the change in temperature during each leg. You can use the ideal gas law (PV = nRT) to calculate the temperature at each point in the cycle.

Secondly, in leg ii of the cycle, you can calculate the heat flow by using the equation Q = mcΔT, where ΔT is the difference between the temperature at point B and point C. The mass and specific heat capacity of the working gas remain the same throughout the cycle.

Lastly, in part C of the problem, you are given the values for the maximum and minimum temperatures and volumes, which you can use to calculate the specific work done during each leg of the cycle. The specific work is given by the equation W = (P2V2 - P1V1)/(γ-1), where P and V are the pressure and volume at each point in the cycle, and γ is the ratio of specific heats (for an ideal gas, γ = cp/cv = 5/3).

In conclusion, your solution is correct, but you can use the equations mentioned above to calculate the heat flow and specific work without assuming values for the variables. You can also refer to thermodynamic tables to find the values of heat and work for different temperatures and volumes. I hope this helps clarify your doubts. Keep up the good work!