Deriving Efficiency of a Diesel Engine Formula

[CallMeShady]

Homework Statement

The Diesel cycle is an idealized representation of the process that occurs in a Diesel combustion engine, as shown in the graph below. Starting at point a, the Diesel cycle consists of an adiabatic compression, an isobaric expansion, and adiabatic expansion and an isochoric cooling. Find the efficiency in terms of only the compression ratio, r = V_a/V_b, γ (gamma), T_Hot and T_Cold. Show all work!

Homework Equations

e = 1 + Q_C/Q_H
γ (gamma) = c_p/c_v
dQ = dU + dW

For adiabatic expansion, TV^γ-1 = constant, if the number of moles is constant PV^γ = C where C is a constant.

Work for adiabatic process = W = (C/1-γ)(V₂^1-γ - V₁^1-γ)

The Attempt at a Solution

I am having problems starting up with this problem and writing it in terms of temperature. I know that temperature at T_a is equal to T_c which in turn is equal to T_H from what I know. But, how do I begin after setting up the problem like this? I am sorry for not providing a "thorough" attempt at a solution; I am having problems at the initial steps which is why I can't do much. Any light on this problem would be appreciated.

 

[haruspex]

I know that temperature at Ta is equal to Tc

How can that be? Surely Ta=T_cold.

 

[kuruman]

An easy way to organize your thoughts would be to construct a Table as shown below and fill in the blanks. Start by figuring out the temperatures at each point using the ideal gas law. For example, $T_a=\dfrac{p_a~rV}{nR}$. Then use $T_a(rV)^{\gamma -1}=T_b(V)^{\gamma -1}$ to find $T_b$ and so on for the remaining temperatures. Once you have all the temperatures, you can find the changes in internal energy using $\Delta U=nC_V\Delta T$. Then you can fill in the rest of the entries.

The first law takes the form $\Delta U = Q-W$ where $W$ is the work done by the gas. To keep myself honest, I would calculate the work and the heat entering the gas independently and verify that the first law holds across each row. I would also verify that the sums of the entries in each column obey $~~0=Q_{\text{Total}}-W_{\text{Total}}$.

Once you have completed the table, it's an easy matter to find the efficiency and answer any other question related to the cycle.

	ΔU​	Q​	W​
a → b
b → c
c → d
d → a
Total	​