An air conditioner is a device used to cool the inside of a...

An air conditioner is a device used to cool the inside of a home. It is, in essence, a refrigerator in which mechanical work is done and heat removed from the (cooler) inside and rejected to the (warmer) outside. A home air conditioner operating on a reversible Carnot cycle between the inside, absolute temperature $T_2$, and the outside, absolute temperature $T_1>T_2$, consumes $P$ joules $/ \mathrm{sec}$ from the power lines when operating continuously. (a) In one second, the air conditioner absorbs $Q_2$ joules from the house and rejects $Q_1$ joules outdoors. Develop a formula for the efficiency ratio $Q_2 / P$ in terms of $T_1$ and $T_2$. (b) Heat leakage into the house follows Newton's law $Q=A\left(T_1-T_2\right)$. Develop a formula for $T_2$ in terms of $T_1, P$, and $A$ for continuous operation of the air conditioner under constant outside temperature $T_1$ and uniform (in space) inside temperature $T_2$. (c) The air conditioner is controlled by the usual on-off thermostat and it is observed that when the thermostat set at $20^{\circ} \mathrm{C}$ and an outside temperature at $30^{\circ}$, it operates $30 \%$ of the time. Find the highest outside temperature, in ${ }^{\circ} \mathrm{C}$, for which it can maintain $20^{\circ} \mathrm{C}$ inside (use $-273^{\circ} \mathrm{C}$ for absolute zero). (d) In the winter, the cycle is reversed and the device becomes a heat pump which absorbs heat from outside and rejects heat into the house. Find the lowest outside temperature in ${ }^{\circ} \mathrm{C}$ for which it can maintain $20^{\circ} \mathrm{C}$ inside. FIGURE CAN'T COPY.

An air conditioner is a device used to cool the inside of a home. It is, in essence, a refrigerator in which mechanical work is done and heat removed from the (cooler) inside and rejected to the (warmer) outside.

A home air conditioner operating on a reversible Carnot cycle between the inside, absolute temperature $T_2$, and the outside, absolute temperature $T_1>T_2$, consumes $P$ joules $/ \mathrm{sec}$ from the power lines when operating continuously.
(a) In one second, the air conditioner absorbs $Q_2$ joules from the house and rejects $Q_1$ joules outdoors. Develop a formula for the efficiency ratio $Q_2 / P$ in terms of $T_1$ and $T_2$.
(b) Heat leakage into the house follows Newton's law $Q=A\left(T_1-T_2\right)$. Develop a formula for $T_2$ in terms of $T_1, P$, and $A$ for continuous operation of the air conditioner under constant outside temperature $T_1$ and uniform (in space) inside temperature $T_2$.
(c) The air conditioner is controlled by the usual on-off thermostat and it is observed that when the thermostat set at $20^{\circ} \mathrm{C}$ and an outside temperature at $30^{\circ}$, it operates $30 \%$ of the time. Find the highest outside temperature, in ${ }^{\circ} \mathrm{C}$, for which it can maintain $20^{\circ} \mathrm{C}$ inside (use $-273^{\circ} \mathrm{C}$ for absolute zero).
(d) In the winter, the cycle is reversed and the device becomes a heat pump which absorbs heat from outside and rejects heat into the house. Find the lowest outside temperature in ${ }^{\circ} \mathrm{C}$ for which it can maintain $20^{\circ} \mathrm{C}$ inside.
FIGURE CAN'T COPY.

Key Concepts

Thermostatic Control

Thermostatic control refers to the method by which a system automatically regulates its temperature by switching the operating device (like a heater or air conditioner) on and off. This control mechanism is essential for maintaining a desired indoor temperature despite external variations and is commonly used in household heating and cooling systems.

Steady-State Energy Balance

Steady-state energy balance involves equating the energy being supplied, removed, and lost in a system so that its overall internal energy remains constant over time. It is a critical concept in thermodynamics for analyzing systems like refrigerators and heat pumps, where energy inputs and losses must balance for the system to maintain a constant interior temperature.

Heat Pump Operation

Heat pump operation is the process in which a device transfers heat from a colder external environment to a warmer interior space by reversing the refrigeration cycle. This application of thermodynamic principles allows the same system to provide heating rather than cooling, with its performance again limited by the ideal efficiency of a Carnot cycle.

Coefficient of Performance (COP)

The Coefficient of Performance (COP) is a measure of the effectiveness of refrigeration and heat pump systems. It is defined as the ratio of the cooling or heating provided to the work input, and for an ideal reversible cycle its value depends solely on the temperatures of the reservoirs. The COP helps in assessing how efficiently a device uses energy to transfer heat.

Reversible Carnot Cycle

The reversible Carnot cycle is an idealized thermodynamic cycle that provides the maximum possible efficiency that any heat engine or refrigerator operating between two temperature reservoirs can achieve. It serves as a theoretical benchmark for the performance of practical devices, emphasizing the dependence of efficiency on the absolute temperatures of the hot and cold reservoirs.

Heat Transfer and Newton's Law of Cooling

Newton's law of cooling describes the rate of heat transfer between a system and its surroundings as proportional to the temperature difference between them. This principle is fundamental in evaluating energy losses or gains in thermal systems and is widely used in developing models for steady-state conditions in heating and cooling applications.

Transcript

00:01 So here's a problem 92 in chapter 19, and this problem talks about the air conditioner involves a refrigerant.

00:08 Part a, what is the sign of q for the expansion? so for the expansion process, it is the evaporation process.

00:22 And evaporation process is an endothermic process.

00:30 And for endothermic process, delta h is positive.

00:36 So even though in the system does not occur at a constant pressure.

00:41 So q is not exactly equal to delta h, but they still share the same signs.

00:47 So the sign of the q will also be positive.

00:52 Part b is asking what is the sign of q for compression process? so because compression is the reverse process of expansion, the q would be negative.

01:05 Part c, in a central air conditioner system, one chamber is inside the home and the other one is outside.

01:15 So this is asking you which is inside the house and which is outside.

01:21 So we know that the expansion chamber, cue of the system will increase, so the cue of the surrounding will decrease, causing the surrounding temperature.

01:41 To decrease.

01:45 So in this case we put the expansion chamber inside the house.

01:51 So the air around this chamber will be cooled.

01:56 And the compression, we have q of the system decrease, so the surrounding will be heated.

02:08 The surrounding temperature will increase...

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