Non-Isentropic Diesel Cycle (with variable specific heat)

[Funknatics]

Homework Statement

I must find temperature at each state, for starters.

This is a Diesel cycle with a non-isentropic compression process, but standard the rest of the way. The specific heat is variable with temperature. Fluid is air

Givens:
compression efficiency (\eta_{c}), initial temperature(T_{1}), qH, compression ratio r_{c} , curve fit for C_{v}(T)

Homework Equations

Non-isentropic with constant specific heat, k = 1.4
(1) T_{2} = T_{1} + \frac{T_{2s} - T_{1}}{\eta_{c}}

(2) \frac{T_{2s}}{T_{1}} = (\frac{V_{2}}{V_{1}})^{k-1}

(3) \eta_{c}=\frac{C_{v_{2s}}(T_{2s}-T_{1})}{C_{v_{2}}(T_{2}-T_{1})}}

(4) C_{v_{2s}} - C_{v} from average of T1 and T2s

The Attempt at a Solution

T_{2s} is an ideal assumption of the process assuming that it is isentropic. Using this, and the compression efficiency, I can find a temperature for state 2 through equation (1), assuming that the specific heat is constant, which is not true in this case.

I believe that equation (3) is the key to this, with a good assumption on the values of C_{v_{2}} and C_{v_{2s}}. One way I thought of doing this is to take the average of the temperatures and using the curve fit to find this specific heat. However, I can only do this for the ideal case as I don't have the final temperature.

This is where I am stuck.

Would it be valid to relate the compression efficiency to the change in temperature? If so, how? It would have to be a higher temperature because of the increase in entropy, and I would imagine it would look something along the lines of T1 + dT(1+ eta_c), or something (im not sure).

Any insight on this would be helpful. It frustrates me to have hit this road block right off the bat on this project.

Thanks in advance.

P.S. I got a bit lost in all the text and so if I left anything out, please let me know.

 

[Dr.D]

Would it be out of line to ask where this nonisentropic compression process comes from? Is this a home work problem, an experimentally observed result, or what? I have never heard of anyone wanting to model the compression stroke of a diesel engine as anything other than an adiabatic process, so I puzzled as to the motivation here.

 

[Funknatics]

It is an end of term thermodynamics project, and that was just given.

I used a weighted average Cp to model the process with most likely significant error, but within reason of the project. The emphasis on the project was to determine the trends of changing the compression ratio, heat input, and initial temperature so the exact numbers weren't as important.