Thermodynamics, two engine device

[zandbera]

Homework Statement

Suppose you build a two-engine device with the exhaust energy output from one heat engine supplying the input energy for a second heat engine. We say that the two engines are running in series. Let e₁ and e₂ represent the efficiencies of the two engines. (a) The overall efficiency of the two-engine device is defined as the total work output divided by the energy put into the first engine by heat. Show that the overall efficiency is given by:
e = e₁ + e₂ - e₁e₂

Homework Equations

e = W_eng / |Q_h| = 1 - |Q_c| / |Q_h|
e = 1 - T_c/T_h

The Attempt at a Solution

Since the first engines Qc is supplying the Qh for the second engine, I said Qc1 = Qh2, and then I tried using that equality in the first formula there but I don't think that's working out for me

 

[rl.bhat]

Write down the expression for e1 and e2.
From these expression find the ratios of Qc1/Qh1 and Qc2/Qh2.
If you multiply these ratios you, will get the overall efficiency.

 

[zandbera]

assuming that Qc1 = Qh2,

e1 = 1 - Qc1/Qh1
e2 = 1 - Qc2/Qc1

Qc1/Qh1 = e1 - 1
Qc2/Qc1 = e2 - 1

multiplying them, i get e1e2 - e1 - e2, which is -1 times the answer I am trying to show. why is this?

 

[rl.bhat]

assuming that Qc1 = Qh2,

e1 = 1 - Qc1/Qh1
e2 = 1 - Qc2/Qc1

Qc1/Qh1 = e1 - 1
Qc2/Qc1 = e2 - 1

multiplying them, i get e1e2 - e1 - e2, which is -1 times the answer I am trying to show. why is this?

Your simplification is wrong.
Qc1/Qh1 = e1 - 1
Qc2/Qc1 = e2 - 1
it should be
Qc1/Qh1 = 1 -e1
Qc2/Qh2 = 1 - e2

 

[zandbera]

thank you i got it now.
one question though: how do you know to multiply the ratios? as far as i understand, it gives you the efficiency for Qc2 / Qh1 so that would be like the energy input from the first engine to the energy output of the second engine so it would be the efficiency for the entire cycle?

 

[rl.bhat]

Work done in the first engine = Qh1 - Qc1 = Qh1*e1 = Qh1 - Qh2...(1)
Work done in the second engine = Qh2 - Qc2 = Qh2*e2...(2)
So the total work done = Qh1 - Qc2 = Qh1*e1 + Qh2*e2...(3)
From the equation (1) Qh2 = Qh1 - Qh1e1
Put this value in eq. (3) and find the efficiency.

 

[Andrew Mason]

thank you i got it now.
one question though: how do you know to multiply the ratios? as far as i understand, it gives you the efficiency for Qc2 / Qh1 so that would be like the energy input from the first engine to the energy output of the second engine so it would be the efficiency for the entire cycle?

Overall efficiency is:
$$\eta_T = 1 - \frac{Q_{c2}}{Q_{h1}}$$

But:

$$\frac{Q_{c2}}{Q_{h1}} = \frac{Q_{c2}}{Q_{c1}} \frac{Q_{c1}}{Q_{h1}}$$

And since:

$$\frac{Q_{c1}}{Q_{h1}} = 1 - \eta_1$$ and

$$\frac{Q_{c2}}{Q_{c1}} = \frac{Q_{c2}}{Q_{h2}} = 1 - \eta_2$$

So:

$$\eta_T = 1- (1-\eta_1)(1-\eta_2)$$AM