Carnot Heat Pump: Solving |Q_c|/|Q_h|=T_c/T_h

[frankwilson]

A Carnot engine uses hot and cold reservoirs that have temperatures of 1684 and 842 K, respectively. The input heat for this engine is |QH|. The work delivered by the engine is used to operate a Carnot heat pump. The pump removes heat from the 842-K reservoir and puts it into a hot reservoir at a temperature T`. The amount of heat removed from the 842-K reservoir is also |QH|. Find the temperature T`.


|Q_c|/|Q_h|=T_c/T_h
|Q_h|=|W| + |Q_c|


I'm having trouble visualizing this problem. Are there two separate engines? Doing a little rearranging, I was able to get down to |W| = 1/2 |Q_h|. I figure that since there are no values for either heat value or work that they cancel out. I'm just not sure how to proceed. I worked it one way and got my final T` to be 1684 K, but I don't feel too confident about it. Anyone out there know where I should start or if I'm even on the right track?

 

[Andrew Mason]

A Carnot engine uses hot and cold reservoirs that have temperatures of 1684 and 842 K, respectively. The input heat for this engine is |QH|. The work delivered by the engine is used to operate a Carnot heat pump. The pump removes heat from the 842-K reservoir and puts it into a hot reservoir at a temperature T`. The amount of heat removed from the 842-K reservoir is also |QH|. Find the temperature T`.|Q_c|/|Q_h|=T_c/T_h
|Q_h|=|W| + |Q_c|I'm having trouble visualizing this problem. Are there two separate engines?

There are two Carnot devices. One is a heat engine, the other is a heat pump.

Doing a little rearranging, I was able to get down to |W| = 1/2 |Q_h|. I figure that since there are no values for either heat value or work that they cancel out. I'm just not sure how to proceed. I worked it one way and got my final T` to be 1684 K, but I don't feel too confident about it. Anyone out there know where I should start or if I'm even on the right track?

Write out the equation for the COP of the heat pump as a function of Tc and T': COP = W/Qc

Since W = Qh/2 and Qc = Qh, that leaves you with an equation with only one unknown: T'.

AM

 

[Vespa71]

Or for simplicity maybe just Tc/Th=Qc/Qh, then rearrange and substitute. By the way, AM, COP-figures for consumer-info usually are larger than 1, calculated Qc/W for coolers, aren't they? Anyway, no need to bring in Cop or efficiency as long as we know enough T's and Q's.

 

[Andrew Mason]

Or for simplicity maybe just Tc/Th=Qc/Qh, then rearrange and substitute. By the way, AM, COP-figures for consumer-info usually are larger than 1, calculated Qc/W for coolers, aren't they?

Yes - a slip there. COP = output/input = heat removed/work input = Qc/W. Thanks.

AM

 

[rwisz]

I understand your confusion and concerns about this problem. Let me provide some clarification and guidance to help you solve it.

First, it is important to understand that a Carnot heat pump is essentially a reversed Carnot engine. In a Carnot engine, heat is taken from a hot reservoir and converted into work, while in a Carnot heat pump, work is used to transfer heat from a cold reservoir to a hot reservoir.

In this problem, we are given the temperatures of the hot and cold reservoirs, 1684 K and 842 K respectively, and the amount of heat removed from the cold reservoir, |Q_c| = |Q_h|. We are asked to find the temperature T` of the hot reservoir, which is the same as the temperature of the hot reservoir in the Carnot engine.

To solve this problem, we can use the equation |Q_c|/|Q_h|=T_c/T_h, where |Q_c| is the heat removed from the cold reservoir, and |Q_h| is the heat added to the hot reservoir. We know that |Q_c| = |Q_h|, so we can substitute this into the equation and simplify to get T_c/T_h = 1.

Now, we also know that |Q_h| = |W| + |Q_c|, where |W| is the work delivered by the engine. Since we are dealing with a Carnot engine, we know that the work delivered is equal to half of the heat added, so we can write |W| = 1/2 |Q_h|. Substituting this into the equation above, we get T_c/T_h = 1/2.

Finally, we can substitute the temperatures of the hot and cold reservoirs into the equation and solve for T`:

T_c/T_h = 1/2
(842 K)/(1684 K) = 1/2
T` = (1/2)(1684 K) = 842 K

So, the temperature of the hot reservoir in the Carnot engine and Carnot heat pump is 842 K. I hope this helps clarify the problem for you. If you have any further questions, please do not hesitate to ask.