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1. We let T denote the temperature in the insulation at the radius r. What expression denotes the temperature gradient (temperature change per unit length) in the direction of radius r? 2. The law of heat conduction, also known as Fourier's law, states that the rate of heat transfer (flow) through a material is proportional to the temperature gradient and to the area through which the heat flows. Following this description, create an equation that describes the amount of heat Q flowing out of the pipe walls as a product of thermal conductivity, surface area, and the temperature gradient from #1.

1. We let T denote the temperature in the insulation at the radius r. What expression denotes the temperature gradient (temperature change per unit length) in the direction of radius r? 2. The law of heat conduction, also known as Fourier's law, states that the rate of heat transfer (flow) through a material is proportional to the temperature gradient and to the area through which the heat flows. Following this description, create an equation that describes the amount of heat Q flowing out of the pipe walls as a product of thermal conductivity, surface area, and the temperature gradient from #1.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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I need help with this assigment. I just need help with questions 4, 5, and 6, pls.

Consider a typical cross section of a thick pipe (see figure) with the given thermal conductivity k. We assume the following: (1) all heat flow through the insulation is radial (in the direction of the radius), (2) for the unit length of pipe, all the heat passing into the pipe wall through its inner surface will eventually pass into the air through its outer surface, and (3) the same amount of constant heat loss per time Q will pass through every cylindrical area between r, and r. T(T) To 1. We let T denote the temperature in the insulation at the radius r. What expression denotes the temperatre gradient (temperature change per unit length) in the direction of radius r? 2. The law of heat conduction, also known as Fourier's law, states that the rate of heat transfer (flow) through a material is proportional to the temperature gradient and to the area through which the heat flows. Following this description, create an equation that describes the amount ofheat Q flowing out of the pipe walls as a product of thermal conductivity, surface area, and the temperature gradient from #1. 3. Rearrange your equation in #2 to obtain a differential equation for T(r), then derive the general solution for T(r) using separation of variables.
Transcribed Image Text:Consider a typical cross section of a thick pipe (see figure) with the given thermal conductivity k. We assume the following: (1) all heat flow through the insulation is radial (in the direction of the radius), (2) for the unit length of pipe, all the heat passing into the pipe wall through its inner surface will eventually pass into the air through its outer surface, and (3) the same amount of constant heat loss per time Q will pass through every cylindrical area between r, and r. T(T) To 1. We let T denote the temperature in the insulation at the radius r. What expression denotes the temperatre gradient (temperature change per unit length) in the direction of radius r? 2. The law of heat conduction, also known as Fourier's law, states that the rate of heat transfer (flow) through a material is proportional to the temperature gradient and to the area through which the heat flows. Following this description, create an equation that describes the amount ofheat Q flowing out of the pipe walls as a product of thermal conductivity, surface area, and the temperature gradient from #1. 3. Rearrange your equation in #2 to obtain a differential equation for T(r), then derive the general solution for T(r) using separation of variables.
4. Use the initial condition T(ro) = To and the general solution in #3 to derive the particular solution for T(r). Your answer should have no unknown constants. 5. The figure also shows a boundary condition, T(r,) = T,. Plug this boundary condition into the solution in #4 to derive an expression for the constant Q as a function of ro, r1, To,T1. 6. Combine the results from #4 and #5 to obtain an equation for T (r) that is no longer a function of Q. This is the temperature T at a given radius r through the pipe wall, which only depends on rɔ, r1, To, T1.
Transcribed Image Text:4. Use the initial condition T(ro) = To and the general solution in #3 to derive the particular solution for T(r). Your answer should have no unknown constants. 5. The figure also shows a boundary condition, T(r,) = T,. Plug this boundary condition into the solution in #4 to derive an expression for the constant Q as a function of ro, r1, To,T1. 6. Combine the results from #4 and #5 to obtain an equation for T (r) that is no longer a function of Q. This is the temperature T at a given radius r through the pipe wall, which only depends on rɔ, r1, To, T1.
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