Calculating the Cost of Heating with a Less Efficient Heat Pump

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SUMMARY

This discussion focuses on calculating the cost of heating a home using a heat pump with a coefficient of performance (COP) that is 75% of the theoretical maximum. The heat pump operates with a heat exchanger temperature of Tc=2°C and maintains baseboard radiators at Th=47°C. The cost to heat the house with ideal electric heaters is $1000, which serves as a baseline for comparison. Participants emphasize the importance of understanding the relationships between Qh (heat output) and Win (input work) in the context of thermodynamic efficiency.

PREREQUISITES
  • Understanding of thermodynamics principles, specifically the Coefficient of Performance (COP).
  • Familiarity with heat transfer concepts, including heat exchangers.
  • Basic knowledge of energy equations, particularly U=W+Q.
  • Ability to convert temperature units, specifically Celsius to Kelvin.
NEXT STEPS
  • Research the thermodynamic principles behind Coefficient of Performance (COP) in heat pumps.
  • Learn how to calculate heat output (Qh) and input work (Win) for different heating systems.
  • Explore the implications of efficiency losses in heating systems and their cost impact.
  • Study the differences between ideal electric heaters and heat pumps in terms of energy consumption and performance.
USEFUL FOR

Students studying thermodynamics, HVAC professionals, and homeowners interested in understanding the cost-effectiveness of heating systems.

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Homework Statement


Assume that you heat your home with a heat pump whose heat exchanger is at Tc=2∘C, and which maintains the baseboard radiators at Th=47∘C. If it would cost $1000 to heat the house for one winter with ideal electric heaters (which have a coefficient of performance of 1), how much would it cost if the actual coefficient of performance of the heat pump were 75% of that allowed by thermodynamics?

Homework Equations


Th/(Th-Tc)=K
U=W+Q
K=Qh/Win

The Attempt at a Solution


i converted to kelvin and then plugged the temperatures into the equation and set it equal to Qh/Win but i am unsure how to proceed. any help is greatly appreciated.
 
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It's best to type out your working so we can see exactly what you've done.

I assume 'K' is COP (Coefficient of performance), did you account for the given inefficiency here?

What does Qh represent? what are it's units? Is it the same for both the heater and the heat pump? How can you use that information?

What does W_in represent? what are it's units? Is it the same for both the heater and the heat pump? How can you use that information?
 

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