Heat Pump Power and COP: How Does Temperature Affect Efficiency?

In summary, the conversation discusses a demonstrational heat pump and its operation using heat QC and QH. The compressor adds work W which is paid for with the electric bill. The temperature in the hot reservoir is increased by 25.8 °C in 1616 seconds and the average power given to the hot reservoir is calculated to be 667 W. The temperature in the hot reservoir is measured over time and a straight line is fit to the data using the least square method. The coefficient of performance COPHP for the heat pump is calculated to be 4.3, but there may be some confusion about the units of the variable a, which should be in K/s for COP to be unitless.
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In a demonstrational heat pump, according to figure 1, heat QC = Qin is taken from a "cold reservoir" that contains 10 liter mix of water and glycol. Heat QH = Qout is dissipated to a “hot reservoir” that contains 10 liters of water. QC is the heat that you normally get "for free" from a suitable reservoir and QH is useful heat that we normally use for heating. The compressor adds the work W that we pay for with the electric bill.

figure1.jpg
Figure 1. A heat pump with the most important parts drawn. Compressor, expansion valve, cold and hot reservoir.

a) The temperature in the hot reservoir was increased with 25.8 °C in 1616 s. How big average power has been given to the hot reservoir?

b) The diagram in figure 2 shows how the temperature TH in the hot reservoir is varies with time t. The measurement started when the heat pump started.

figure2.jpg
Figure 2. The variation of temperature with time in the hot reservoir.

We fit a straight line to the measurement points with the least square method and get:
[tex]T_{H} = a \cdot t + b[/tex] where a = 0,0163 °C/s and b = 25,7 °C.

We assume that the power of the compressor is constantly 158 W during the time of measurement. Calculate the coefficient of performance COPHP for the heat pump when the temperature varies according to the above.
[tex]COP_{HP}(t)=\frac{Q_{H}}{W}=\frac{dQ_{H}(t)}{dt}/\frac{dW(t)}{dt}=\frac{P_{out}(t)}{P_{in}(t)}[/tex]
where Pout is the power that is transmitted to the hot reservoir and Pin is the electric power used by the compressor.

Hint:

[tex]P_{out}=\frac{dQ_{H}}{dt}=\frac{dQ_{H}}{dT}\cdot \frac{dT}{dt}=m\cdot c\cdot \frac{dT}{dt}[/tex]

So if we start with question a) to calculate P = Q/t we can use that Q=m*c*ΔT so for for the 10 liters of water in the hot reservoir we get P=10*4186*25.8/1616=668 W.
However it says the answer for a) is 667 W. Any idea why the difference?

For question b) we have Vf = Pout / Pin where Pout = m*c*dT/dt
dT/dt = a*t^-1 so Vf = m*c*a*t^-1 / Pin = 10*4186*0.0163 / 158 = 4.3

Here the number is correct but I wonder about something about the units. The variable a is said to have unit °C/s so when deriving with respect to time we should get °C/s^2 ? But in order for the answer to be correct the unit should rather be K/s in order for COP to be unitless? Please help how can this be cleared up?
 
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  • #2
T = a.t + b

∴ dT/dt = a

Effect said:
dT/dt = a*t^-1
 

FAQ: Heat Pump Power and COP: How Does Temperature Affect Efficiency?

What is a heat pump?

A heat pump is a device that transfers heat from one location to another. It works by using a small amount of energy to move heat from a cooler area to a warmer one, making the cooler space cooler and the warmer space warmer.

How is the power of a heat pump measured?

The power of a heat pump is typically measured in kilowatts (kW). This is a unit of power that represents the amount of energy needed to do work at a rate of 1,000 watts per second.

What is COP and how is it related to heat pump power?

COP stands for Coefficient of Performance and it is a measure of the efficiency of a heat pump. It is calculated by dividing the heat output of the heat pump by the energy input. The higher the COP, the more efficient the heat pump is at transferring heat.

What factors affect the power and COP of a heat pump?

The power and COP of a heat pump can be affected by a variety of factors including the temperature difference between the inside and outside air, the size and design of the heat pump, and the type of refrigerant used. It is important to properly size and maintain a heat pump to ensure optimal performance.

How can I improve the power and COP of my heat pump?

To improve the power and COP of a heat pump, it is important to properly maintain it by regularly changing filters and scheduling professional inspections. Additionally, using a programmable thermostat and sealing air leaks in your home can help reduce the workload on the heat pump and improve its efficiency.

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