Heat Pump Q: Cost of Winter Heat w/ 75% CoP

In summary, the conversation discusses the calculation of the cost to heat a house using a heat pump with an actual coefficient of performance of 75% of the ideal value. The solution involves converting temperatures to Kelvin and equating the coefficient of performance with the ratio of Qh (heat output) to Win (input work), resulting in a cost of $187.5 to heat the house for one winter.
  • #1
ewnair
3
0

Homework Statement



Assume that you heat your home with a heat pump whose heat exchanger is at Tc = 2degrees celcius and which maintains the baseboard radiators at Th = 47 degrees celcuiys. 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



I understand that K = Th/(Th-Tc)

The Attempt at a Solution



I converted the temperature to kelvin and solved for the ideal coefficient which was 7.11. 75% of that is 5.33. Equated this to Qh/Win and assumed that Qh is the same for both actual and ideal pump. I multiplied 1000 with 7.11/5.33 and got an answer of $1330. However the answer is incorrect. Can anyone help please?
 
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  • #2
Once you have figured out that the heat pump's actual COP is 5.33, the 7.11 figure should not enter into the calculation again.

... 75% of that is 5.33. Equated this to Qh/Win and assumed that Qh is the same for both actual and ideal pump.

Yes, absolutely. So what is Win, using this equation?
 
  • #3
I got $1330 which isn't correct. i equated the 2 equations and plugged in Win = 1000 and K = 7.11 to find out Qh. Then i used this value of Qh and K= 5.33 to get Win which is $1330 but the answer is wrong.
 
  • #4
K is not 7.11, it is 5.33.

ewnair said:
I converted the temperature to kelvin and solved for the ideal coefficient which was 7.11. 75% of that is 5.33. Equated this to Qh/Win ...

Yes, 5.33 should be equated with Qh/Win. Then solve the equation for Win.
 
  • #5
Redbelly98 said:
K is not 7.11, it is 5.33.
Yes, 5.33 should be equated with Qh/Win. Then solve the equation for Win.

The Problem states that K=1 (which is kind of misleading). So if we have a actual engine with K=5.33, our work should be less.

So, 1=Qh/(1000) and
5.33=Qh/(dollars we pay)

you will get $187.5 as your answer (you might think that is weird because an "actual" engine is more efficient than a carnot engine)
 

Q: How does the cost of using a heat pump for winter heating compare to other heating options?

The cost of using a heat pump for winter heating is generally lower than other options, such as electric furnaces or baseboard heaters. This is because heat pumps transfer heat from the outside air instead of generating it, making them more energy efficient.

Q: What is the average cost of using a heat pump for winter heating?

The average cost of using a heat pump for winter heating will vary depending on factors such as climate, size of the heat pump, and energy prices. However, on average, homeowners can expect to pay around $1,500 to $2,500 per year for heating with a heat pump.

Q: How does the Coefficient of Performance (COP) affect the cost of winter heating with a heat pump?

The Coefficient of Performance (COP) is a measure of a heat pump's efficiency. The higher the COP, the more efficient the heat pump is at transferring heat. A heat pump with a COP of 75% will use 75% less energy to produce the same amount of heat as an electric furnace, resulting in lower heating costs.

Q: Are there any additional costs associated with using a heat pump for winter heating?

There may be some additional costs associated with using a heat pump for winter heating, such as installation and maintenance costs. However, these costs are typically offset by the energy savings and lower operating costs of a heat pump compared to other heating options.

Q: Can I save even more money on winter heating costs with a heat pump?

Yes, there are ways to save even more money on winter heating costs with a heat pump. Some tips include setting the thermostat to a lower temperature, properly maintaining the heat pump, and ensuring your home is well-insulated to minimize heat loss. Additionally, some areas may offer incentives or rebates for using energy-efficient heating options like heat pumps.

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