Question

. A Carnot heat pump is to be used to heat a house and maintain it at 22 °C in winter. When the outdoor temperature remains at 3 °C, the house is estimated to lose heat at a rate of 76,000 kJ/h. If the heat pump consumes 9 kW of power, how long does it need to run in a single day to keep the temperature constant inside the house?

Answer 1

The Carnot heat pump must work 3.624 hours per day to keep the temperature constant inside the house.

In this question we must apply first law of thermodynamics and features of Carnot heat pump to determine how much time the system must be on each day to keep the temperature constant inside the house.

The net heat daily loss of the house ($Q_{losses}$) is:

$Q_{losses} = \left(76000\,\frac{kJ}{h}\right)\cdot (24\,h)$

$Q_{losses} = 1.824\times 10^{6}\,kJ$

In order to keep the house warm, this heat must be equal to heat losses ($Q_{H}$):

$Q_{H} = Q_{losses}$ (1)

Besides, the Coefficient of Performance for a Carnot heat pump ($COP_{HP}$) is:

$COP_{HP} = \frac{T_{H}}{T_{H}-T_{L}}$ (2)

Where:

• $T_{L}$ - Temperature of the cold reservoir (Outdoors), measured in Kelvin.

• $T_{H}$ - Temperature of the hot reservoir (House), measured in Kelvin.

Given that $T_{L} = 276.15\,K$ and $T_{H} = 295.15\,K$, the Coefficient of Performance is:

$COP_{HP} = \frac{295.15\,K}{295.15\,K-276.15\,K}$

$COP_{HP} = 15.534$

For a real heat machine, the Coefficient of Performance is determined by the following expression:

$COP_{HP} = \frac{Q_{H}}{W}$ (3)

Where:

• $Q_{H}$ - Heat received by the house, measured in kilojoules.

• $W$ - Work consumed by the Carnot heat pump, measured in kilojoules.

The daily work consumed is now cleared in the previous expression:

$W = \frac{Q_{H}}{COP_{HP}}$ (3b)

$W = \frac{1.824\times 10^{6}\,kJ}{15.534}$

$W = 117419.853\,kJ$

The working time is calculated by dividing this result by input power. That is:

$\Delta t = \frac{W}{\dot W}$ (4)

$\Delta t = \left(\frac{117419.853\,kJ}{9\,kW} \right)\cdot \left(\frac{1}{3600}\,\frac{h}{s}\right)$

$\Delta t = 3.624\,h$

The Carnot heat pump must work 3.624 hours per day to keep the temperature constant inside the house.

To learn more on Carnot heat pump, we kindly invite to check this verified question: https://brainly.com/question/14019449