The engine has an efficiency e1. The engine takes input heat Qh from a hot reservoir and delivers work W1. The heat rejected by this engine is used as input heat for a second engine, which has an efficiency e2 and delivers work W2. The overall efficiency of the two-engine device is the total wor kdfelivered (W1 + W2) divided by the input heat Qh. Find an expression for the overall efficiency in terms of e1 and e2.
To find the expression for the overall efficiency in terms of e1 and e2, let's break down the problem step by step.
First, we need to find the work delivered by each engine. The work delivered by the first engine, W1, can be calculated using the equation:
W1 = Qh * e1
This equation gives us the work done by the first engine using the input heat Qh and its efficiency e1.
Next, we need to find the work delivered by the second engine, W2. The heat rejected by the first engine is used as the input heat for the second engine. Therefore, the input heat for the second engine is equal to the heat rejected by the first engine.
The heat rejected by the first engine can be calculated using the equation:
Qr = Qh - W1
Substituting the value of W1, we get:
Qr = Qh - (Qh * e1)
Simplifying, we have:
Qr = Qh * (1 - e1)
Now, we can calculate the work delivered by the second engine, W2, using its efficiency e2:
W2 = Qr * e2
Substituting the value of Qr, we get:
W2 = (Qh * (1 - e1)) * e2
Finally, we can find the total work delivered by both engines:
W_total = W1 + W2 = Qh * e1 + (Qh * (1 - e1)) * e2
The overall efficiency of the two-engine device is given by the ratio of the total work delivered to the input heat:
Efficiency = (W1 + W2) / Qh = (Qh * e1 + (Qh * (1 - e1)) * e2) / Qh
Simplifying, we get:
Efficiency = e1 + (1 - e1) * e2
Therefore, the expression for the overall efficiency in terms of e1 and e2 is:
Efficiency = e1 + (1 - e1) * e2