Heat engine and Heat pump in combination

[canadiansmith]

Question is attached

I have spent a great deal of time on this problem and I am hoping someone can help me out.
My attempt at problem.

first for the heat engine
QL/Qw1 = TL/Twaste
so efficiency = 1- Tl/Twaste
efficiency = 0.93808
and I know that efficiency = Wout/ Qw1


This is as far as i got for the heat engine...

For Heat pump
i found cop = TH/(TH-TL) = 4.23

 

[rude man]

Never mind efficiency. The problem doesn't ask for efficiency. It only wants you to compute dQh/dt.

ANYWAY ...

write the equations for the 1st and 2nd laws. That's 4 equations.

Then write the relationship between Qw1, Qw2 and 5 MW.

5 equations, 5 unknowns. Away you go. Solve for Qh.

(Note - the 1st and 2nd laws are usually written in terms of energy rather than power. Don't let that slow you down. Just pretend it's energy for 1 second, which of course is 1 J/s = 1 W.

 

[canadiansmith]

Never mind efficiency. The problem doesn't ask for efficiency. It only wants you to compute dQh/dt.

ANYWAY ...

write the equations for the 1st and 2nd laws. That's 4 equations.

Then write the relationship between Qw1, Qw2 and 5 MW.

5 equations, 5 unknowns. Away you go. Solve for Qh.

(Note - the 1st and 2nd laws are usually written in terms of energy rather than power. Don't let that slow you down. Just pretend it's energy for 1 second, which of course is 1 J/s = 1 W.

I am still a little confused. There are 2 equations for both first and second law?
i only have delta U = Q-W
and Q = integral of TodeltaS

 

[rude man]

I am still a little confused. There are 2 equations for both first and second law?
i only have delta U = Q-W
and Q = integral of TodeltaS

Yes, but you have two mechanisms, the heat engine and the heat pump. Each has its own
1st & 2nd law equations.

BTW dQ = TdS is not an expression of the second law. It's just a formula to calculate entropy. What you need is an expression reflective of the fact that entropy is unchanged in a reversible cycle.

As for the 1st law, you are right, but what can you say about Qw1, QL, Qw2, QH and W? Remember that U is a state function so over a cycle it doesn't change either, reversible or not.

 

[DeeNos]

I would like to offer some clarification and further explanation on the concepts of heat engines and heat pumps, as well as how they can be used together.

A heat engine is a device that converts heat energy into mechanical work, such as the movement of a piston or turbine. It operates in a cyclic process, taking in heat from a high temperature source (TL) and releasing some of that heat to a lower temperature sink (Twaste). The remaining heat is converted into work, which is represented by the equation QW1 = QL - Qwaste. The efficiency of a heat engine is given by the formula QL/QW1 = TL/Twaste, where TL is the temperature of the high temperature source and Twaste is the temperature of the low temperature sink. This efficiency is always less than 1, and the closer the temperatures are, the lower the efficiency will be.

On the other hand, a heat pump is a device that uses mechanical work to transfer heat from a low temperature source (TL) to a higher temperature sink (TH). It operates in a reverse cycle compared to a heat engine, taking in work (W) and releasing heat to the high temperature sink (TH). The coefficient of performance (COP) of a heat pump is defined as the ratio of heat transferred to the amount of work put in, and is given by COP = TH/(TH-TL). This value can be greater than 1, indicating that a heat pump can transfer more heat than the work put in.

Now, in terms of using a heat engine and heat pump in combination, one possible application is in a combined heat and power (CHP) system. In this system, a heat engine is used to generate electricity, while the waste heat from the engine is then used to power a heat pump and provide heating or cooling for a building. This allows for the efficient use of both the heat engine and heat pump, as the waste heat from the engine is not wasted but instead utilized for another purpose.

In terms of the given problem, it seems that the heat engine and heat pump are being used in conjunction to transfer heat from a low temperature source (TL) to a high temperature sink (TH). The efficiency of the heat engine is calculated to be 0.93808, meaning that about 94% of the heat is converted into work. The COP of the heat pump is calculated to be 4.23, indicating that for every