Problem 6.ART TATConsider the cylindrical pipe shown in the figure. Heat is being generated in the pipe wall at a rateof R in units of W/m³, and the thermal conductivity of the wall is k, the heat transfer coefficientsat the inside and outside of the pipe are h; and ho, respectively.At a given location along the pipe axis, the temperature of the fluid flowing inside of the pipe is Tiand that at the outside is Toi.ii.Find the expression for the temperature distribution in the pipe wall.Find the total heat transfer rate from the pipe to both fluids.

Given: Length of the pipe is L Heat generation in the pipe wall is R W/m3 There is a radius in the…

Answered: Problem 6. ' i. ii. V 1 R2 1 s F…

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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Problem 6.
ART TAT
Consider the cylindrical pipe shown in the figure. Heat is being generated in the pipe wall at a rate
of R in units of W/m³, and the thermal conductivity of the wall is k, the heat transfer coefficients
at the inside and outside of the pipe are h; and ho, respectively.
At a given location along the pipe axis, the temperature of the fluid flowing inside of the pipe is Ti
and that at the outside is To
i.
ii.
Find the expression for the temperature distribution in the pipe wall.
Find the total heat transfer rate from the pipe to both fluids.

Expert Solution

Step 1

Given:

Length of the pipe is L

Heat generation in the pipe wall is R W/m³

There is a radius in the pipe say R' where the heat transfer rate will be zero due to the both side heat transfer with heat generation.

Let the temperature is T' at the interface at radius R' which is the maximum temperature inside the pipe wall.

Because the heat generated within the pipe wall from radius R' to R₂ will be:

$\to q_{2} = (\frac{π}{4} ({R_{2}}^{2} - R'^{2}) L) \times R$

And this generated heat will be transferred to outside the pipe wall.

So the heat transfer rate to ouside the wall will be:

$\to h_{o} (2 {πR}_{2} L) (T_{s} - T_{o}) = q_{2} \to h_{o} (2 {πR}_{2} L) (T_{s} - T_{o}) = (\frac{π}{4} ({R_{2}}^{2} - R'^{2}) L) \times R \to T_{s} = T_{o} + \frac{({R_{2}}^{2} - R'^{2}) L \times R}{8 h_{o} R_{2}} . . . . . . . . . . . . . . . . . . . . . . . . . . e q u a t i o n 1$

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