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The best COP hp Of a heat pump for home use

A heat pump used to warm a home must employ a cycle that produces a working fluid at temperatures greater than typical indoor temperature so that heat transfer to the inside can take place. Similarly, it must produce a working fluid at temperatures that are colder than the outdoor temperature so that heat transfer occurs from outside. Its hot and cold reservoir temperatures therefore cannot be too close, placing a limit on its COP hp size 12{ ital "COP" rSub { size 8{"hp"} } } {} . (See [link] .) What is the best coefficient of performance possible for such a heat pump, if it has a hot reservoir temperature of 45 . 0 º C size 12{"45" "." 0°C} {} and a cold reservoir temperature of 15 . 0 º C size 12{-"15" "." 0°C} {} ?

Strategy

A Carnot engine reversed will give the best possible performance as a heat pump. As noted above, COP hp = 1 / Eff size 12{ ital "COP" rSub { size 8{"hp"} } =1/ ital "Eff"} {} , so that we need to first calculate the Carnot efficiency to solve this problem.

Solution

Carnot efficiency in terms of absolute temperature is given by :

Eff C = 1 T c T h . size 12{ ital "Eff" rSub { size 8{C} } =1 - { {T rSub { size 8{c} } } over {T rSub { size 8{h} } } } } {}

The temperatures in kelvins are T h = 318 K size 12{T rSub { size 8{h} } ="318"" K"} {} and T c = 258 K size 12{T rSub { size 8{c} } ="258"" K"} {} , so that

Eff C = 1 258 K 318 K = 0 . 1887 . size 12{ ital "Eff" rSub { size 8{C} } =1 - { {"258"" K"} over {"318 K"} } =0 "." "1887"} {}

Thus, from the discussion above,

COP hp = 1 Eff = 1 0 . 1887 = 5 . 30 , size 12{ ital "COP" rSub { size 8{"hp"} } = { {1} over { ital "Eff"} } = { {1} over {0 "." "1887"} } =5 "." "30",} {}

or

COP hp = Q h W = 5 . 30 , size 12{ ital "COP" rSub { size 8{"hp"} } = { {Q rSub { size 8{h} } } over {W} } =5 "." "30",} {}

so that

Q h = 5.30 W . size 12{Q rSub { size 8{h} } =5 "." "30"" W" "." } {}

Discussion

This result means that the heat transfer by the heat pump is 5.30 times as much as the work put into it. It would cost 5.30 times as much for the same heat transfer by an electric room heater as it does for that produced by this heat pump. This is not a violation of conservation of energy. Cold ambient air provides 4.3 J per 1 J of work from the electrical outlet.

The figure shows a schematic diagram of a heat pump. The hot and cold reservoirs are shown as two rectangular boxes attached to a vertical rectangular wall. The hot reservoir is at temperature T sub c equals negative fifteen degrees Celsius and the hot reservoir is at a temperature T sub h equals forty five degrees Celsius. Work W is shown to enter from an electrical outlet. Heat Q sub c is shown to enter the cold reservoir at an outside air temperature of negative five degrees Celsius and Q sub h is shown to leave the hot reservoir at an inside air temperature of twenty degrees Celsius.
Heat transfer from the outside to the inside, along with work done to run the pump, takes place in the heat pump of the example above . Note that the cold temperature produced by the heat pump is lower than the outside temperature, so that heat transfer into the working fluid occurs. The pump’s compressor produces a temperature greater than the indoor temperature in order for heat transfer into the house to occur.

Real heat pumps do not perform quite as well as the ideal one in the previous example; their values of COP hp size 12{ ital "COP" rSub { size 8{"hp"} } } {} range from about 2 to 4. This range means that the heat transfer Q h size 12{Q rSub { size 8{h} } } {} from the heat pumps is 2 to 4 times as great as the work W size 12{W} {} put into them. Their economical feasibility is still limited, however, since W size 12{W} {} is usually supplied by electrical energy that costs more per joule than heat transfer by burning fuels like natural gas. Furthermore, the initial cost of a heat pump is greater than that of many furnaces, so that a heat pump must last longer for its cost to be recovered. Heat pumps are most likely to be economically superior where winter temperatures are mild, electricity is relatively cheap, and other fuels are relatively expensive. Also, since they can cool as well as heat a space, they have advantages where cooling in summer months is also desired. Thus some of the best locations for heat pumps are in warm summer climates with cool winters. [link] shows a heat pump, called a “ reverse cycle” or “ split-system cooler” in some countries.

A residential heat pump.
In hot weather, heat transfer occurs from air inside the room to air outside, cooling the room. In cool weather, heat transfer occurs from air outside to air inside, warming the room. This switching is achieved by reversing the direction of flow of the working fluid.

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Source:  OpenStax, College physics ii. OpenStax CNX. Nov 29, 2012 Download for free at http://legacy.cnx.org/content/col11458/1.2
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