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20.11. 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 time period deposits \(1.44 \times 10^{5} \mathrm{J}\) of heat into the outside air (a) What is the power consumption of the unit in watts? (b) What is the energy efficiency rating of the unit?

Short Answer

Expert verified
Power consumption: 767 W; EER: 7.67.

Step by step solution

01

Understand the problem components

The air-conditioner absorbs heat from the room and deposits more heat outside. The difference between these values gives the power consumed by the unit. We need to find the power consumption in watts and the energy efficiency rating (EER).
02

Calculate the Power Consumption in Watts

Power consumption is given by the difference in heat deposited and heat absorbed divided by the time in seconds. The heat absorbed is \(9.80 \times 10^{4} \mathrm{J/min}\) and the heat deposited is \(1.44 \times 10^{5} \mathrm{J/min}\). First, convert minutes to seconds: 1 minute = 60 seconds.\[P = \frac{(1.44 \times 10^{5} - 9.80 \times 10^{4}) \mathrm{J}}{60 \text{ s}} = \frac{4.6 \times 10^{4} \mathrm{J}}{60 \text{ s}} \approx 766.67 \mathrm{W}\]
03

Define Energy Efficiency Rating (EER)

EER is a measure of how efficiently an air conditioner cools and is calculated as the ratio of heat removed from the room in J per time in hours to the energy consumed in watts. The EER will be \[EER = \frac{9.80 \times 10^{4} \mathrm{J/min} \times 60 \text{ min/hr}}{766.67 \, \mathrm{W}} = \frac{5.88 \times 10^{6} \, \mathrm{J/hr}}{766.67 \, \mathrm{W}} \approx 7.67\]
04

Prepare the final results

Using the calculations above, the power consumption of the unit is approximately 767 watts. The energy efficiency rating (EER) of the unit is approximately 7.67.

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Key Concepts

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

Heat Transfer
Heat transfer is a fundamental concept in thermodynamics, describing how energy moves from one system to another. In the context of an air-conditioner, this is the process by which heat is absorbed from the indoor air and deposited outside.

There are three primary modes of heat transfer: conduction, convection, and radiation.
  • **Conduction** happens in solids where vibrating particles transfer energy to neighboring particles.
  • **Convection** occurs in fluids; currents carry heat away, as seen with the flow of air through an air-conditioning unit.
  • **Radiation** involves energy transfer via electromagnetic waves. No physical medium is necessary, allowing energy to travel through space.
The air-conditioner uses these mechanisms primarily convection to transport heat efficiently. Understanding heat transfer is essential for optimizing system performance and ensuring environmental comfort.
Energy Efficiency Rating
The Energy Efficiency Rating (EER) quantifies how effectively an air conditioning unit converts electrical energy into cooling. It's a ratio of the cooling capacity (how much it can cool) to the power consumed (electrical energy used). The higher the EER, the more efficient the appliance.

EER can be calculated using the formula: \[ EER = \frac{\text{Cooling capacity (BTU/hr)}}{\text{Power consumption (Watts)}} \]A higher EER means that the unit uses less electricity to perform the same amount of work, reducing energy consumption and cutting electricity bills.

Environmental sustainability is increasingly important. A high EER is beneficial as it leads not only to cost savings but also minimizes your carbon footprint. Therefore, when purchasing an air conditioner, looking for a high EER can guide you to more sustainable and economical choices.
Power Consumption
Power consumption refers to the amount of electrical energy used by a device, measured in watts. Calculating this for an air conditioner means determining how much energy it requires to function over a given time. This involves three main components: input energy, time, and efficiency.

In this context, we find it by taking the difference between the heat deposited outside and the heat absorbed from the room, divided by time in seconds. Using formula: \[ P = \frac{(Q_{ ext{out}} - Q_{ ext{in}})}{t} \] where \(Q_{ ext{out}}\) is the energy deposited outside and \(Q_{ ext{in}}\) is the energy absorbed.

Understanding this helps users be aware of devices that consume higher energy, which can impact their utility bills. Modern devices often aim for lower power consumption to be cost-effective and eco-friendly.
Physics Education
Physics education provides the foundational understanding of principles like thermodynamics and heat transfer that are essential in everyday appliances like air-conditioners. It encourages critical thinking and problem-solving skills necessary for tackling real-world issues.

By studying physics, students learn to calculate and understand complex problems involving energy transformations and efficiency. This includes:
  • Interpreting formulas and applying them to calculate metrics such as EER and power consumption.
  • Exploring the laws of thermodynamics to understand energy conservation and transfer.
  • Visualizing abstract concepts through practical examples, making complex theories relatable.
The skills gained from physics education are indispensable in various technical fields, leading to innovations in energy-efficient appliances and sustainable solutions. Engaging deeply with physics can empower students to contribute thoughtfully to advancement and technological sustainability.

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Most popular questions from this chapter

20.38. You are designing a Carnot engine that has 2 \(\mathrm{mol}\) of \(\mathrm{CO}_{2}\) as its working substance; the gas may be treated as ideal. The gas is to have a maximum temperature of \(527^{\circ} \mathrm{C}\) and a maximum pressure of 5.00 \(\mathrm{atm}\) . With a heat input of 400 \(\mathrm{J}\) per cycle, you want 300 \(\mathrm{J}\) of useful work (a) Find the temperature of the cold reservoir. (b) For how many cycles must this engine run to melt completely a 10.0 kg block of ice originally at \(0.0^{\circ} \mathrm{C}\) , using only the beat rejected by the engine?

A diesel engine performs 2200 \(\mathrm{J}\) of mechanical work and discards 4300 \(\mathrm{J}\) of heat each cycle. (a) How much heat must be supplied to the engine in each cycle? (b) What is the thermal efficiency of the engine?

20.42. Heat Pump. A heat pump is a heat engine run in reverse. In winter it pumps heat from the cold air outside into the warmer air inside the building, maintaining the building at a comfortable temperature. In summer it pumps heat from the cooler air inside the building to the warmer air outside, acting as an air conditioner. (a) If the outside temperature in winter is \(-5.0^{\circ} \mathrm{C}\) and the inside temperature is \(17.0^{\circ} \mathrm{C}\) , how many joules of heat will the heat pump deliver to the inside for each joule of electrical energy used to run the unit, assuming an ideal Carnot cycle? ( b) Suppose you have the option of using electrical resistance heating rather than a heat pump. How much electrical energy would you need in order to deliver the same amount of heat to the inside of the house as in part (a)? Consider a Carnot heat pump delivering heat to the inside of a house to maintain it at \(68^{\circ} \mathrm{F}\) . Show that the beat pump delivers less heat for each joule of electrical energy used to operate the unit as the outside temperature decreases. Notice that this behavior is opposite to the dependence of the efficiency of a Carnot heat engine on the difference in the reservoir temperatures. Explain why this is so.

20.13. A Camot 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?

20.39. A Carnot engine whose low-temperature reservoir is at \(-90.0^{\circ} \mathrm{C}\) has an efficiency of 40.0\(\%\) . An engineer is assigned the problem of increasing this to 45.0\(\%\) . (a) By how many Celsius degrees must the temperature of the high-temperature reservoir be increased if the temperature of the low-temperature reservoir remains constant? (b) By how many Celsius degrees must the temperature of the low-temperature reservoir be decreased if the temperature of the high-temperature reservoir remains constant?

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