# Worksheet: Heat Pumps and Refrigerators

In this worksheet, we will practice calculating the mechanical work required by heat pumps to transfer heat from low to high temperature reservoirs.

**Q7: **

In the heat exchanger shown in the accompanying diagram, a fluid enters the pipe at the point 1 at a temperature of . The fluid exits the pipe at the point 3 at a temperature of . Another fluid enters the exchanger’s inner pipe at the point 4. The second fluid enters this pipe at a temperature of and leaves the pipe at the point 2 at a temperature of . What is the logarithmic-mean temperature difference of the fluids?

**Q8: **

In the heat exchanger shown in the accompanying diagram, a fluid flowing into the
exchanger’s inner pipe has a specific heat capacity of
.
Fluid enters the pipe at the point 1 with a flow rate of
3.00 kg/s and at a
temperature of . The
fluid exits the pipe at the point 3 at a temperature of . Another fluid with a
specific heat capacity of enters the exchanger’s inner pipe at the point 4
with a flow rate of 0.500 kg/s.
The second fluid enters this pipe at a temperature of
and leaves the pipe at the point 2 at a temperature of . The area of contact between
the inner and outer pipes is 2.00 m^{2}. What is the overall heat transfer coefficient in the
heat exchanger?

**Q9: **

In the heat exchanger shown in the accompanying diagram, a fluid flowing into the exchanger’s outer pipe has a specific heat capacity of .Fluid enters the pipe at the point 2 with a flow rate of 3.00 kg/s and at a temperature of .The fluid exits the pipe at the point 4 at a temperature of .Another fluid with a specific heat capacity of enters the exchanger’s inner pipe at the point 1. The second fluid enters this pipe at a temperature of and leaves the pipe at the point 3, at a temperature of . What is the flow rate of the fluid that moves through the inner pipe?

**Q12: **

A Carnot refrigerator operates between the temperatures at its cold reservoir and at its hot reservoir. and vary, causing changes in the work required to cool the cold reservoir by 1.0 J.

Find the work required to cool the cold reservoir by 1.0 J for and .

Find the work required to cool the cold reservoir by 1.0 J for and .

Find the work required to cool the cold reservoir by 1.0 J for and .

Find the work required to cool the cold reservoir by 1.0 J for and .

**Q13: **

A motor with a power output of 430 W operates a Carnot refrigerator with a cold reservoir temperature of and a hot reservoir temperature of .

What is the rate of cooling of the refrigerator’s interior due to the work done by the refrigerator?

What is the rate of heating of the refrigerator’s exterior due to the work done by the refrigerator?

**Q17: **

A Carnot refrigerator heats the air around it, which is at a temperature of . Determine how much power the refrigerator requires to freeze 2.0 g of water per second if the water is at an initial temperature of . Use a value of for the specific heat capacity of water and use a value of 334 kJ/kg for the latent heat of fusion of ice.

**Q20: **

In the heat exchanger
shown in the accompanying diagram, a fluid flowing into the exchanger’s inner
pipe has a specific heat capacity of . Fluid enters the pipe
at the point 1 with a flow rate of 3.00 kg/s and at a temperature
of . The fluid exits the pipe at the point 3 at a temperature
of . Another fluid with a specific heat capacity of
enters the exchanger’s inner pipe at the point 4 with a flow rate
of 0.500 kg/s. The second fluid enters this pipe at a temperature of and
leaves the pipe at the point 2 at a temperature of .
The overall heat-transfer coefficient of the heat exchanger is 1,500 W/m^{2}⋅K. What is the area for heat transfer?