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Chapter 16: Problem 29

A well-insulated house of moderate size in a temperate climate requires an average heat input rate of \(20.0 \mathrm{~kW}\). If this heat is to be supplied by a solar collector with an average (night and day) energy input of \(300 \mathrm{~W} / \mathrm{m}^{2}\) and a collection efficiency of \(60.0 \%,\) what area of solar collector is required?

Short Answer

Expert verified
A solar collector area of 111.1 m虏 is required.

Step by step solution

01

Understand the Given Data

We have a house that requires an average heat input rate of 20.0 kW. The solar collector provides an energy input of 300 W/m虏, and the collection efficiency is 60%.
02

Determine Usable Energy Rate

First, calculate the usable energy rate from the solar collector by accounting for its efficiency. The usable energy is the input energy multiplied by the efficiency: \[ \text{Usable Energy Rate} = 300 \text{ W/m}^2 \times 0.60 = 180 \text{ W/m}^2. \]
03

Establish the Total Energy Requirement

Since the house requires an average heat input rate of 20.0 kW, we need to match this rate using the solar collectors. Ensure both rates (house requirement and collector output) are in the same units. Convert the house's energy requirement to watts since 1 kW = 1000 W: \[ 20.0 \text{ kW} = 20000 \text{ W}. \]
04

Calculate Required Solar Collector Area

To find the required area of the solar collector, divide the total energy requirement of the house by the usable energy output rate of the solar collector:\[ \text{Required Area} = \frac{20000 \text{ W}}{180 \text{ W/m}^2} = 111.11 \text{ m}^2. \]
05

Finalize the Calculation

Round the area to a practical value considering significant figures from the problem's data. The calculated area of 111.11 m虏 suggests that a solar collector around 111.1 m虏 is needed.

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Key Concepts

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

Heat Input Rate
The term "heat input rate" is crucial when understanding energy requirements of a building. It defines the average amount of heat energy a structure needs to maintain comfort. In our scenario, the house demands a heat input rate of \(20.0\, \text{kW}\). These requirements consider various factors:
  • Exterior weather conditions, which influence the house's ability to retain heat.
  • Insulation efficiency, affecting the rate at which heat escapes.
  • Size and layout of the house, impacting total heat distribution needs.
Ensuring we're providing the correct heat input rate is essential to maintain adequate temperature levels without excessive energy waste.
Energy Efficiency
Energy efficiency measures how well a system converts energy input into useful output, in this case, heat. The solar collector's efficiency is given as 60%. This means that 60% of the solar energy captured is converted into usable heat for the house. Improving energy efficiency reduces total energy costs and environmental impact. Several factors can enhance efficiency:
  • Innovative technology, such as special coatings that increase solar energy absorption.
  • Optimal orientation and tilting of collectors to maximize sunlight capture.
  • Routine maintenance to ensure components function correctly.
Understanding these elements helps ensure maximum efficacy when using solar technology for heating.
Solar Collector Area
The solar collector area is the physical space required to capture solar energy effectively. For our exercise, the usable energy rate of 180 \(\text{W/m}^2\) guides this calculation. We need to determine how much surface area is necessary to produce the required 20000 W. Calculating the required collector area involves:
  • Dividing the house's total energy requirement by the usable output per square meter.
With an efficiency-adjusted output of 180 \(\text{W/m}^2\), calculations render a solar collector area of roughly 111.1 \(\text{m}^2\). This extensive collection area ensures sufficient energy to meet the house's heat needs consistently.
Energy Conversion
Energy conversion is the process where sunlight is transformed into heat energy via solar collectors. This involves capturing solar radiation and converting it into usable heat output for a home. The solar collectors are designed to optimize this conversion, balancing factors such as:
  • Material properties, which affect conductivity and absorption of sunlight.
  • System loss minimization, ensuring least energy is lost in the conversion process.
In our example, despite an input energy of 300 \(\text{W/m}^2\) from sunlight, only 180 \(\text{W/m}^2\) is effectively utilized due to efficiency constraints. Thus, energy conversion directly influences both the design and operational efficiency of solar heating systems.

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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 certain nuclear power plant has a thermal efficiency \(e=0.25\). Its rate of heat input from the nuclear reactor is \(1300 \mathrm{MW}\). What would be the reduction in the rate of discarded heat if the plant's efficiency were increased to \(e=0.3 ?\)

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 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 \(10.0 \mathrm{~kg}\) block of ice originally at \(0.0^{\circ} \mathrm{C}\), using only the heat rejected by the engine?

Each cycle, a certain heat engine expels \(250 \mathrm{~J}\) of heat when you put in \(325 \mathrm{~J}\) of heat. Find the efficiency of this engine and the amount of work you get out of the \(325 \mathrm{~J}\) heat input.

A Carnot engine operates between two heat reservoirs at temperatures \(T_{\mathrm{H}}\) and \(T_{\mathrm{C}}\). An inventor proposes to increase the efficiency of the engine by increasing both \(T_{\mathrm{H}}\) and \(T_{\mathrm{C}}\) by a factor of \(2 .\) Will this plan work? Why or why not?
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