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.

Q1:

A refrigerator has a coefficient of performance of 3.0 and it requires an input of 2.0×10 J of work per cycle.

How much heat per cycle is removed from the cold reservoir?

How much heat per cycle is discarded to the hot reservoir?

Q2:

A refrigerator discards 80.0 J of heat per cycle and its coefficient of performance is 6.0.

How much heat does the refrigerator remove from its cold reservoir per cycle?

How much work must the refrigerator do per cycle?

Q3:

Calculate the maximum coefficient of performance for a vapor absorption refrigeration system operating with a generator temperature of 90C, an absorber temperature of 5C, and a condenser temperature of 30C. Answer to three-significant-figure precision.

Q4:

A refrigerator with a coefficient of performance of 3.00 is used to produce 5.25 g of ice per second from water at a temperature 0C. What is the minimum input power required by this refrigerator?

Q5:

A Carnot refrigerator operates between temperatures of 30C and 5C. What is its coefficient of performance?

Q6:

Calculate the Carnot coefficient of performance for a refrigeration system operating with a condenser temperature of 300 K and an evaporator temperature of 260 K.

Q7:

In the heat exchanger shown in the accompanying diagram, a fluid enters the pipe at the point 1 at a temperature of 10.0C. The fluid exits the pipe at the point 3 at a temperature of 15.0C. Another fluid enters the exchanger’s inner pipe at the point 4. The second fluid enters this pipe at a temperature of 90.0C and leaves the pipe at the point 2 at a temperature of 30.0C. 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 4.00/kJkgC. Fluid enters the pipe at the point 1 with a flow rate of 3.00 kg/s and at a temperature of 10.0C. The fluid exits the pipe at the point 3 at a temperature of 15.0C. Another fluid with a specific heat capacity of 2.00/kJkgC 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 90.0C and leaves the pipe at the point 2 at a temperature of 30.0C. The area of contact between the inner and outer pipes is 2.00 m2. 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 4/kJkgC.Fluid enters the pipe at the point 2 with a flow rate of 3.00 kg/s and at a temperature of 10C.The fluid exits the pipe at the point 4 at a temperature of 15C.Another fluid with a specific heat capacity of 2/kJkgC enters the exchanger’s inner pipe at the point 1. The second fluid enters this pipe at a temperature of 90C and leaves the pipe at the point 3, at a temperature of 30C. What is the flow rate of the fluid that moves through the inner pipe?

Q10:

During one cycle, a refrigerator removes 450 J from a cold reservoir and rejects 700 J to its hot reservoir.

What is the refrigerator’s coefficient of performance?

How much work per cycle does the refrigerator require to operate?

Q11:

A refrigerator has a coefficient of performance of 2.50. The refrigerator requires 230 J of work per cycle to operate.

How much heat per cycle does the refrigerator extract from the cold reservoir?

How much heat per cycle does the hot reservoir absorb?

Q12:

A Carnot refrigerator operates between the temperatures 𝑇 at its cold reservoir and 𝑇 at its hot reservoir. 𝑇 and 𝑇 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 𝑇=9.0C and 𝑇=25C.

Find the work required to cool the cold reservoir by 1.0 J for 𝑇=75C and 𝑇=25C.

Find the work required to cool the cold reservoir by 1.0 J for 𝑇=135C and 𝑇=25C.

Find the work required to cool the cold reservoir by 1.0 J for 𝑇=275C and 𝑇=25C.

Q13:

A motor with a power output of 430 W operates a Carnot refrigerator with a cold reservoir temperature of 6.0C and a hot reservoir temperature of 27C.

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?

Q14:

A Carnot heat pump in a house operates between temperatures of 3.0C and 18C. By how much does the pump heat the house for every 1.0 J of work that it performs?

Q15:

Find the minimum work that a refrigerator must be supplied with if it is to cool a freezer’s interior by 40 J per cycle. The freezer’s interior temperature is 8.0C and its exterior temperature is 23C.

Q16:

A Carnot engine operates between heat reservoirs at temperatures of 550 K and 350 K. The high-temperature reservoir is heated by 1,400 J per cycle. How much work per cycle does the engine perform?

Q17:

A Carnot refrigerator heats the air around it, which is at a temperature of 20C. 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 0.0C. Use a value of 4,184/JkgC for the specific heat capacity of water and use a value of 334 kJ/kg for the latent heat of fusion of ice.

Q18:

An engineer must design a refrigerator that does 300 J of work per cycle to cool a freezer’s interior at a temperature of 10C by 2,100 J per cycle. What is the maximum air temperature for which this condition can be met?

Q19:

A Carnot cycle is operating between a heat source at a temperature of 95C and a heat sink at a temperature of 27C in order to power a refrigerator that operates between temperatures of 12C and 27C. What power output must the Carnot cycle provide to allow the refrigerator to cool its interior by 13 J/s?

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 4.00/kJkgC. Fluid enters the pipe at the point 1 with a flow rate of 3.00 kg/s and at a temperature of 10C. The fluid exits the pipe at the point 3 at a temperature of 15.0C. Another fluid with a specific heat capacity of 2.00/kJkgC 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 90.0C and leaves the pipe at the point 2 at a temperature of 30.0C. The overall heat-transfer coefficient of the heat exchanger is 1,500 W/m2⋅K. What is the area for heat transfer?

Q21:

A refrigerator is kept in a room with temperature of 95.0F. The refrigerator heats its surroundings at a rate of 1.00 kW. What is the minimum power needed to keep the refrigerator interior at a temperature of 10.4F?

Q22:

What is the maximum coefficient of performance of a heat pump, whose temperatures of hot and cold reservoirs are 327C and 27C, respectively?

Q23:

A 300 W heat pump operates between the ground, the temperature of which is 0C, and the interior of a house at 22C. What is the maximum amount of heat per hour that the heat pump can supply to the house? Give your answer to three significant figures.

  • A 1 . 5 5 × 1 0 J
  • B 1 . 5 2 × 1 0 J
  • C 1 . 4 5 × 1 0 J
  • D 1 . 5 7 × 1 0 J
  • E 1 . 4 8 × 1 0 J

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