Heat Pump Thermodynamics Question

[mcomputing]

I have a general question, it's not homework or anything, we are studying heat pumps and thermal efficiencies and COP. There's an example in my book that has a heat pump that is heating a house and it asks to find the minimum theoretical cost per day. Of course the solution and detailed steps are in my textbook.

However I was thinking what if I wanted to raise the temperature from any given T1 to T2 using a heat pump that gives me the COP and the heat loss or Q_loss and the mass of the house. I want to find how long it would take to raise the temperature from let's say 10 degrees Celsius to 22 degrees Celsius. I think I have an idea of how to go about solving such a problem.

Is the the process:

I have the energy balance equation which gives me sum(Q) = mC_v*delta(T)
I have a sum to be the Qin - Qloss = mC_v*delta(T). If I am given the Qloss and the mC_v and the Qin I can obviously solve for delta(T) and use that to find Tf if I am given a Ti.

Then I can use the COP equation of COP = Q.in/W.in to find Q.in. and Q.in would be equal to Qin/delta(t_sec).

Now my question is what if I wasn't given W.in for the heat pump, how would I be able to solve for it with my given information. Please advise. Thank you very much.

 

[Curl]

Then you will have some sort of differential equation. How good is your math?

 

[Andrew Mason]

Now my question is what if I wasn't given W.in for the heat pump, how would I be able to solve for it with my given information.

The minimum work required by the heat pump is a function of the temperatures between which it is operating. For a Carnot heat pump:

COP = Qh/W = Qh/(Qh-Qc) = Th/(Th-Tc) so

W= Qh(Th-Tc)/Th = Qh(1-Tc/Th)

AM

 

[mcomputing]

Thanks for all your help. I was able to figure this out by asking my professor. He basically told me that not being given W.in is beyond the scope of the class I am taking. I think I was over thinking the question and imagining all possibilities.

 

[khemist]

Thanks for all your help. I was able to figure this out by asking my professor. He basically told me that not being given W.in is beyond the scope of the class I am taking. I think I was over thinking the question and imagining all possibilities.

I hate it when teachers say this.

 

[Curl]

I hate it when teachers say this.

This is a real-world problem (which I thought of and solved myself a while back) but he's too lazy to explain it to the student.That's the human mentality:

"If it don't make my paycheck bigger, I don't give a flying f***."