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A window air-conditioner unit absorbs \(9.80 \times 10^{4} \mathrm{~J}\) of heat per minute from the room being cooled and in the same period deposits \(1.44 \times 10^{5} \mathrm{~J}\) of heat into the outside air. What is the power consumption of the unit in watts?

Short Answer

Expert verified
The power consumption of the unit is approximately 766.67 watts.

Step by step solution

01

Understand the Energy Transfer

The air-conditioner absorbs \(9.80 \times 10^{4} \text{ J/min}\) from the room and deposits \(1.44 \times 10^{5} \text{ J/min}\) outside. The difference represents the energy consumed by the air-conditioner, which is the work done by the unit.
02

Calculate the Energy Consumed

The energy consumed by the air-conditioner per minute is given by the difference between the heat deposited outside and the heat absorbed from the room: \[E_{consumed} = 1.44 \times 10^{5} \text{ J/min} - 9.80 \times 10^{4} \text{ J/min}\] which equals \(4.6 \times 10^{4} \text{ J/min}\).
03

Convert Energy Consumption to Watts

Convert the energy consumed per minute into watts. Since 1 watt (W) is equivalent to 1 joule per second (J/s), we convert minutes to seconds: \[P = \frac{4.6 \times 10^{4} \text{ J/min}}{60 \text{ s/min}} \approx 766.67 \text{ W}\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Energy Transfer
Energy transfer involves the movement of energy from one place to another or from one form to another. In thermodynamics, this transfer can occur in various forms such as heat, work, or through the consumption of power.
In the case of a window air-conditioner, energy transfer happens in two primary ways:
  • The air-conditioner absorbs heat from the room to cool it down.
  • It deposits this heat, along with additional energy consumed in the process, to the outside environment.
Understanding energy transfer is crucial because it lays the foundation for analyzing systems like air-conditioners. This process helps us see how much energy is being conserved, used, or wasted, which is the cornerstone of efficiency calculations.
Power Calculation
Power calculation refers to determining how much energy is used over a particular period, usually expressed in watts.
To calculate power, first, find out the energy consumed in joules and the time over which it is consumed, typically in seconds. Use the formula:
\[ P = \frac{E}{t} \]
Where:- \( P \) is the power in watts (W),- \( E \) is the energy in joules (J),- \( t \) is the time in seconds (s).
For an air-conditioner, if it consumes energy per minute, as in our example, we convert the time from minutes to seconds (1 min = 60 s) to compute power. This conversion is necessary because power is generally measured concerning time in seconds, giving us the rate of energy consumption over time.
Work Done
In thermodynamics, work done is a measure of energy transfer that doesn't involve heat transfer. In the context of an air-conditioner, work is the additional energy required by the unit to move the heat from the inside to the outside.
To determine the work done by an air-conditioner, subtract the energy absorbed from the indoors from the energy deposited outside. This difference represents the effort or work required to operate the cooling system.
The formula used is:
\[ W = Q_{out} - Q_{in} \]
Where:
- \( W \) is the work done by the air-conditioner,- \( Q_{out} \) is the heat deposited outside,- \( Q_{in} \) is the heat absorbed from the cooled room.
This reveals how efficient the air-conditioner is by showing how much additional energy is used beyond simple heat transfer from inside to outside.
Air-Conditioner Efficiency
Air-conditioner efficiency is a measure of how well the unit performs its function of cooling compared to the energy it consumes. This efficiency is closely related to the concept of the coefficient of performance (COP) in thermodynamics.
To calculate the efficiency of an air-conditioner, consider both the energy input and the output. The higher the efficiency, the less energy is needed to achieve the desired cooling effect.
  • An air-conditioner with a high efficiency uses less energy to move the same amount of heat compared to an inefficient one.
  • Efficiency can often be improved by ensuring the unit is well-maintained and adequately sized for the space it is cooling.
In our previous example, understanding the work done and power calculation can help us determine the system's efficiency. This helps in optimizing energy use and minimizing operational costs in the long term.

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Most popular questions from this chapter

A Carnot engine whose high-temperature reservoir is at \(620 \mathrm{~K}\) takes in \(550 \mathrm{~J}\) of heat at this temperature in each cycle and gives up \(335 \mathrm{~J}\) to the low-temperature reservoir. (a) How much mechanical work does the engine perform during each cycle? (b) What is the temperature of the low-temperature reservoir? (c) What is the thermal efficiency of the cycle?

A gasoline engine has a power output of \(180 \mathrm{~kW}\) (about \(241 \mathrm{hp}\) ). Its thermal efficiency is \(28.0 \%\). (a) How much heat must be supplied to the engine per second? (b) How much heat is discarded by the engine per second?

The Kwik-Freez Appliance Co. wants you to design a food freezer that will keep the freezing compartment at \(-5.0^{\circ} \mathrm{C}\) and will operate in a room at \(20.0^{\circ} \mathrm{C}\). The freezer is to make \(5.00 \mathrm{~kg}\) of ice at \(0.0^{\circ} \mathrm{C}\), starting with water at \(20.0^{\circ} \mathrm{C}\). Find the least possible amount of electrical energy needed to make this ice and the smallest possible amount of heat expelled into the room.

A Carnot engine has an efficiency of \(59 \%\) and performs \(2.5 \times 10^{4} \mathrm{~J}\) of work in each cycle. (a) How much heat does the engine extract from its heat source in each cycle? (b) Suppose the engine exhausts heat at room temperature \(\left(20.0^{\circ} \mathrm{C}\right) .\) What is the temperature of its heat source?

A refrigerator has a coefficient of performance of \(K=2.0 .\) Each cycle, it absorbs \(3.40 \times 10^{4} \mathrm{~J}\) of heat from the cold reservoir. The refrigerator is driven by a Carnot engine that has an efficiency of \(e=0.5 .\) (a) How much mechanical energy is required each cycle to operate the refrigerator? (b) During each cycle, how much heat flows into the Carnot engine?

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