How Does Temperature Difference Affect Air Conditioning Costs?

In summary, the cost of air conditioning in a building on a hot summer day is roughly proportional to the square of the temperature difference between the inside and outside, under the condition that heat enters entirely by conduction. This can be shown by considering the ideal COP and its relation to the temperature difference, the work input and its relationship to COP, and the amount of heat that must be removed by the air conditioner in relation to the temperature difference. Therefore, the cost of electricity is proportional to the square of the temperature difference.
  • #1
tralblaz
12
0

Homework Statement



Under many conditions, the rate at which heat enters as air conditioned building on a hot summer day is proportional to the difference in the temperature between the inside and outside, Th-Tc. (If the heat enters entirley by conduction,this statement will certainly be true. Radiation from the direct sunlight would be an exception.) Show that, under these condtions, the cost of air condtioning should be roughly proportional to the square of the temperature difference.

Homework Equations


use what is known about COP (coefficent of performance)

COP=Q/W

The Attempt at a Solution


I am thoroughly lost in how to approach this problem. I know that the COP is involved, but I have no idea where or how to start.

Could someone please get me started? Thanks
 
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  • #2
these are give away hints... but anyway.

1. consider the ideal COP, what is its relation to the temperature difference?

2. consider the work input (relating to cost of electricity) and its relationship between COP.

3. consider heat entering the house... if the air conditioner is "combating" this heat flow, how much heat must it take out (in relation to difference of temperature)?

4. relate 1, 2 and 3. So, what is cost of electricity proportional to?
 

FAQ: How Does Temperature Difference Affect Air Conditioning Costs?

What is a Carnot engine?

A Carnot engine is a theoretical heat engine that operates on the principles of thermodynamics. It consists of four main components: a hot reservoir, a cold reservoir, a working substance, and a mechanism to convert heat energy into mechanical work.

How does a Carnot engine work in air conditioning?

In air conditioning, a Carnot engine works by using the temperature difference between the hot outside air and the cold inside air to transfer heat from the inside to the outside. This is done by compressing and expanding a working fluid, such as a refrigerant, to absorb and release heat.

What are the advantages of using a Carnot engine in air conditioning?

One advantage of using a Carnot engine in air conditioning is its high efficiency. It is able to achieve the maximum possible efficiency for a heat engine, making it more energy-efficient compared to other types of engines. It also has a simpler design and requires less maintenance.

Are there any limitations to using a Carnot engine in air conditioning?

One limitation of using a Carnot engine in air conditioning is that it requires a large temperature difference between the hot and cold reservoirs to operate effectively. This can be difficult to achieve in certain environments, making it less practical for some applications.

How does the efficiency of a Carnot engine compare to other types of engines?

The efficiency of a Carnot engine is higher than that of other types of engines, such as Otto and Diesel engines. This is because it operates on the principle of maximum efficiency, while other engines have inherent losses due to their design and processes. However, the practical efficiency of a Carnot engine may be lower due to limitations in real-world applications.

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