Heat pumps and coeffiecent of performance

[rmorelan]

Hi everyone, I was hoping to ask a thermodynamics problem, specificially one about heat pumps and coeffiecent of performance. I am wondering if I am missing something.

I am given that heat pump is heat a building. Inside temperature is to be maintained at 22 degree celcius, outside is -5.0 celcius. The coefficent of performance is 3.8, and the the pump delivers 7.6MJ of heat each hour to the building. I need to find out at what rate work needs to be applied to run the pump.

Homework Equations

Now I believe the equation of the the coeffiecient of performance (CP) is equal to:

cp = |Q|/W

Now this is where I am confused, as I don't understand what (if anything) I am supposed to do with the temperatures given. I know cp, and I know Q, so I can solve for W needed to run the pump can't I? I would need to convert the heat produced to be in SI in order to get Watts as a result of course.

thanks for your help!
rob

 

[Hootenanny]

If we treat the heat pump as a Carnot heat pump we can say that;

$$\frac{Q_2}{Q_1} = \frac{T_2}{T_2}$$

Hence;

$$\eta^{HP}_{C} = \frac{Q_1}{W} = \frac{Q_1}{Q_1-Q_2} = \frac{T_1}{T_1-T_2}$$

Where T₁ and T₂ are the temperatures of the hot and cold reservoirs respectively.

 

[m2003]

Hello Rob,

Yes, you are on the right track. The coefficient of performance (COP) of a heat pump is defined as the ratio of heat output (Q) to the work input (W). So, using the equation you provided, we can rearrange it to solve for W:

W = Q/CP

In this case, Q is given to be 7.6 MJ per hour. However, we need to convert this to SI units, which is Joules per second (J/s) or Watts (W). To do this, we divide 7.6 MJ by 3600 seconds (1 hour) to get 2111.11 J/s or 2111.11 W.

Next, we need to calculate the COP. This is given to be 3.8. So, we can plug this into the equation for COP and solve for W:

3.8 = 2111.11/W

W = 2111.11/3.8

W = 555.55 W

Therefore, the rate of work needed to run the heat pump is 555.55 Watts.

Hope this helps! Keep up the good work in thermodynamics.