An air conditioner is used to keep the interior of a house...

An air conditioner is used to keep the interior of a house at a temperature of $21^{\circ} \mathrm{C}$ while the outside temperature is $32^{\circ} \mathrm{C}$ If heat leaks into the house at the rate of $11 \mathrm{kW},$ and the air conditioner has the efficiency of a Carnot engine, what is the mechanical power required to keep the house cool?

An air conditioner is used to keep the interior of a house at a temperature of $21^{\circ} \mathrm{C}$ while the outside temperature is $32^{\circ} \mathrm{C}$ If heat
leaks into the house at the rate of $11 \mathrm{kW},$ and the air conditioner
has the efficiency of a Carnot engine, what is the mechanical
power required to keep the house cool?

Key Concepts

Heat Transfer

Heat transfer is the process by which thermal energy moves from one body or region to another due to a temperature difference. In the context of refrigeration, understanding and quantifying the rate at which heat enters a system is essential, as it determines the amount of work required by the cooling mechanism to maintain the desired temperature.

Thermodynamic Temperature Scale

In thermodynamics, using an absolute temperature scale such as Kelvin is critical because performance calculations, such as those for the Carnot cycle or COP, depend on absolute differences in temperature. Converting temperatures from Celsius to Kelvin ensures that these relationships are accurately represented in the formulas.

Carnot Cycle

The Carnot cycle is an idealized thermodynamic cycle that represents the maximum possible efficiency a heat engine or refrigerator can achieve when operating between two thermal reservoirs. It serves as a benchmark, establishing limits on how much useful work can be extracted or required for cooling, based solely on the temperatures of the hot and cold sources.

Coefficient of Performance (COP)

The Coefficient of Performance is a metric used to gauge the effectiveness of refrigeration systems or heat pumps. It is defined as the ratio of the heat removed from the cold reservoir to the work input required. For an ideal, reversible system based on the Carnot cycle, the COP provides the upper limit on performance, and it is expressed in terms of absolute temperatures of the reservoirs.

Transcript

00:01 Hello everyone, i'm doing problem 59 from chapter 18.

00:03 It says an air conditioner used to keep the interior of a house at a temperature of 21 degrees celsius, while the outside temperature is 32 degrees celsius.

00:11 If heat leaks into the house at the rate of 11 kilowatts and the air conditioner has the efficiency of a car no engine, what is the mechanical power required to keep the house cool? so the first thing we should realize about this problem is that what it's giving us with the 11 kilowatts is, as it says, the heat that leaks into the house.

00:32 The rate at which the heat leaks in the house.

00:34 However, this is also the rate at which we need to extract the heat from the house if we're to keep the house at a constant temperature.

00:44 Okay, so that 11 kilowatts is actually just q sub c.

00:51 Okay, and then we know that qc is equal to 11 kilowatts.

00:56 We also know, because this is a carnot engine, that, or i should say a, it has the efficiency of a car new engine.

01:08 Then this is 1 minus tc over th.

01:16 Now that doesn't help us too much or it doesn't seem to because we don't know what the efficiency is and this also doesn't involve things that we want.

01:27 Because what we want is the power, but if we think about a different definition of efficiency, so we can make this set this equal to w over qh, which is a more broad definition of efficiency.

01:47 Okay, so now we're getting some things that we need because we have both tc and thh and we want to know w...

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