An electric kitchen range has a total wall area of 1.401.40 m2 and is insulated with a layer of fiberglass 4.004.00 cm thick. The inside surface of the fiberglass has a temperature of 175175°C, and its outside surface is at 35.035.0°C. The fiberglass has a thermal conductivity of 0.040 W/m⋅K0.040\;W/m\cdot K. What is the heat current through the insulation, assuming it may be treated as a flat slab with an area of 1.401.40 m2 ?

Ch 17: Temperature and Heat

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 17: Temperature and Heat

Problem 58a

Chapter 17, Problem 58a

An electric kitchen range has a total wall area of $1.40$ m² and is insulated with a layer of fiberglass $4.00$ cm thick. The inside surface of the fiberglass has a temperature of $175$ °C, and its outside surface is at $35.0$ °C. The fiberglass has a thermal conductivity of $0.040 W / m \cdot K$ . What is the heat current through the insulation, assuming it may be treated as a flat slab with an area of $1.40$ m² ?

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Identify the formula for heat current through a flat slab, which is given by Fourier's law of heat conduction: $ Q = \frac{k \cdot A \cdot (T_{inside} - T_{outside})}{d} $, where $ Q $ is the heat current, $ k $ is the thermal conductivity, $ A $ is the area, $ T_{inside} $ and $ T_{outside} $ are the temperatures of the inside and outside surfaces, and $ d $ is the thickness of the slab.

Substitute the given values into the formula: $ k = 0.040 \text{ W/m K} $, $ A = 1.40 \text{ m}^2 $, $ T_{inside} = 175°C $, $ T_{outside} = 35.0°C $, and $ d = 0.04 \text{ m} $.

Calculate the temperature difference $ \Delta T = T_{inside} - T_{outside} = 175°C - 35°C $.

Plug the values into the formula: $ Q = \frac{0.040 \cdot 1.40 \cdot (175 - 35)}{0.04} $.

Simplify the expression to find the heat current $ Q $. This will give you the rate at which heat is transferred through the fiberglass insulation.

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Key Concepts

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

Thermal Conductivity

Thermal conductivity is a material property that indicates how well a material can conduct heat. It is defined as the amount of heat that passes through a material with a given thickness, area, and temperature difference. In this problem, the fiberglass's thermal conductivity is crucial for calculating the heat current through the insulation.

Heat Current

Heat current, also known as heat flow rate, is the rate at which heat energy transfers through a material. It is calculated using the formula Q = kA(T1 - T2)/d, where k is the thermal conductivity, A is the area, T1 and T2 are the temperatures on either side of the material, and d is the thickness. Understanding this concept is essential for solving the problem of heat transfer through the fiberglass.

Temperature Gradient

The temperature gradient is the change in temperature across a material per unit distance. It is a driving factor for heat transfer, as heat flows from regions of higher temperature to lower temperature. In this scenario, the temperature difference between the inside and outside surfaces of the fiberglass creates a gradient that influences the heat current calculation.

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Textbook Question

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