A 5.0-m-diameter garden pond is 30 cm deep. Solar energy is incident on the pond at an average rate of 400 W/m2. If the water absorbs all the solar energy and does not exchange energy with its surroundings, how many hours will it take to warm from 15°C to 25°C?

Ch 19: Work, Heat, and the First Law of Thermodynamics

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 19: Work, Heat, and the First Law of Thermodynamics

Problem 38

Chapter 19, Problem 38

A 5.0-m-diameter garden pond is 30 cm deep. Solar energy is incident on the pond at an average rate of 400 W/m². If the water absorbs all the solar energy and does not exchange energy with its surroundings, how many hours will it take to warm from 15°C to 25°C?

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Calculate the volume of the pond. The pond is cylindrical, so its volume is given by the formula: \( V = \pi r^2 h \), where \( r \) is the radius (half the diameter) and \( h \) is the depth. Convert the depth from cm to meters before substituting.

Determine the mass of the water in the pond. Use the relationship \( m = \rho V \), where \( \rho \) is the density of water (approximately \( 1000 \ \text{kg/m}^3 \)) and \( V \) is the volume calculated in the previous step.

Calculate the energy required to raise the temperature of the water from 15°C to 25°C. Use the formula \( Q = mc\Delta T \), where \( m \) is the mass of the water, \( c \) is the specific heat capacity of water (\( 4186 \ \text{J/kg°C} \)), and \( \Delta T \) is the temperature change (\( 25 - 15 = 10 \ \text{°C} \)).

Determine the rate at which energy is absorbed by the pond. The power absorbed is given by \( P = I A \), where \( I \) is the solar energy incident rate (\( 400 \ \text{W/m}^2 \)) and \( A \) is the surface area of the pond (\( A = \pi r^2 \)).

Calculate the time required to absorb the energy \( Q \) at the rate \( P \). Use the formula \( t = \frac{Q}{P} \). Convert the time from seconds to hours by dividing by 3600.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Energy Absorption

Energy absorption refers to the process by which an object takes in energy from its surroundings. In this context, the pond absorbs solar energy at a rate of 400 W/m², which is the power per unit area received from sunlight. This energy is crucial for raising the temperature of the water in the pond.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius. For water, this value is approximately 4.18 J/g°C. Understanding this concept is essential for calculating how much energy is needed to increase the pond's water temperature from 15°C to 25°C.

Heat Transfer and Time Calculation

Heat transfer involves the movement of thermal energy from one object to another, which in this case is the energy absorbed by the pond water. To determine how long it will take to warm the water, one must calculate the total energy required to achieve the desired temperature change and then relate this to the rate of energy absorption from the solar radiation.

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