Fundamentals of Heat and Mass Transfer
7th Edition
ISBN: 9780470501979
Author: Frank P. Incropera, David P. DeWitt, Theodore L. Bergman, Adrienne S. Lavine
Publisher: Wiley, John & Sons, Incorporated

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Textbook Question
Chapter 3, Problem 3.1P

Consider the plane wall of Figure 3.1, separating hot and cold fluids at temperatures T , 1 and T , 2 , respectively. Using surface energy balances as boundary conditions at x = 0 and x = L (see Equation 2.34). obtain the temperature distribution within the wall and the heat flux in terms of T , 1 , T , 2 , h 1 , h 2 , k , and L.

Expert Solution & Answer
To determine

The temperature distribution within the wall and heat flux.

## Answer to Problem 3.1P

The expression of temperature distribution is T(x)=h1h2kh1+kh2+h1h2Lx+kh1T,1+kh2T,2+h2Lh1T,1kh1+kh2+h1h2L and heat flux is q=h1kh21h2+1h1+Lk(T,1T,2) .

### Explanation of Solution

Given:

The temperature of hot fluid is T,1

The temperature of cold fluid is T,2

Formula Used:

The expression of one dimensional heat equation is given by,

ρcpdTdt=ddx(kdTdx)+q˙

The expression of convection heat rate is given by,

qconv=h(TsT)

Calculation:

Assume steady state, constant material properties with no heat generation.

The heat equation is calculated as,

ddx(kdTdx)=0

Integrate the above equation once.

kdTdx=C1C1=qk

Integrate above equation again.

T(x)=C1x+C2

Apply boundary condition at x=0 and x=L .

At x=0 , q=k( dT dx)x=0=h1(T ,1T s,1)

At x=L , q=k( dT dx)x=L=h2(T s,2T ,2)

The surface temperature at x=0 is given by,

Ts,1=C10+C2

The surface temperature at x=L is given by,

Ts,2=C1L+C2 .

Solve all the equation to find unknown.

C1=h1h2kh1+kh2+h1h2L

C2=kh1T,1+kh2T,2+h2Lh1T,1kh1+kh2+h1h2L

q=h1kh2kh1+kh2+h1h2L(T,1T,2)

The temperature distribution is calculated as,

T(x)=h1h2kh1+kh2+h1h2Lx+kh1T,1+kh2T,2+h2Lh1T,1kh1+kh2+h1h2L

The heat flux is calculated as,

q=h1kh21h2+1h1+Lk(T,1T,2)

Conclusion:

Therefore, the expression of temperature distribution is T(x)=h1h2kh1+kh2+h1h2Lx+kh1T,1+kh2T,2+h2Lh1T,1kh1+kh2+h1h2L and heat flux is q=h1kh21h2+1h1+Lk(T,1T,2) .

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