Is 'stacking' engines a viable way to maximize efficiency?

Question

Suppose we have $n$ carnot engines connected together labelled as $C_1,C_2...C_n$, now suppose we feed an energy of $E_1$ into $C_1$, now by the second law of thermodynamics, it must be that efficiency can't be $1$ , so some heat is dumped to environment, let's say we 'direct' the heat dump $E_2$ into $C_2$, and we keep doing this procedure till we reach the $n-1th$ engine. Now, the heat dump of this engine into n+1th engine must be very small in magnitude ( say we have enough engines such that this happens), then this if we look at the at the total efficiency of the process:

$$ e= \frac{Q_{in-sys} - Q_{out-sys} }{Q_{in-sys} }$$

The question I have is how to achieve the step where we direct the heat loss from the $ith$ engine to the $i+1th$ engine. If there was a way to do that I Think this would be a viable method for maximizing heat engnine output.

Answer 1

I will try to expand on Tiger guys answer, and more specifically on the example for the exhaust of a gas turbine.

A typical gas turbine has a maximum temperature of about $1000^oC$ and assuming that the output gases are at 500[C], so in theory you could get a maximum efficiency of $$e_1= 1- \frac{500+273}{1000+273}=60%$$

So there would have been about 40% energy available to recover.

Now the thing is that if the second engine goes from 500 to 23$[^oC]$, then

$$e_2= 1- \frac{23+273}{500+273}=38%$$

So in total you would have gained, 60% from the original energy and 0.38*0.4=15.2% from the second engine. So a total of about 75%.

Now the thing is that if you were to have a single engine starting from 1000 and ending at 23, you would theoretically get a maximum efficiency of 76.75%.

So multiple engines, don't necessarily have a better efficiency.

Of course, reality comes into play, and the maximum efficiency of about 70% in the above gas turbine example, is in real life less than 40% at best, even at the most carefully engineered and designed machines (due to losses and other not recoverable energy).

Answer 2

Capturing waste heat is a very viable way to increase the efficiency of a system, but it isn't a panacea.

Real world limits come into play. The 2nd Law says you'll never be able to capture all the waste heat, and as the heat loss gets less (and thus the temperature of the waste heat), the lower the efficiency of the machine that captures it.

There are lots of ways, for example, to capture exhaust heat of a gas turbine. But then capturing the waste heat from that process quickly becomes impractical. This might work as a thought experiment, but like trying to use a Stirling engine in the real world, barriers show up quickly.

Answer 3

I'd say it's a fairly common exploit to squeeze what extra you can out of your heat source. The Titanic had stacked engines. The center shaft was driven by a blow-down turbine fed from the exhausts from the port and starboard engines.

The P 51 had a Meredith effect radiator.