Concept explainers
In the method of separation of variables (Section 4.2)for two-dimensional, steady-state conduction, the separationconstant
Whether a negative or zero value of
Answer to Problem 4.1P
The boundary condition 4 results in a trivial solution so a negative or zero value of
Explanation of Solution
Formula Used:
The temperature distribution equation is given by,
The two dimensional steady state conduction equations is given by,
Calculation:
Substitute Equation (I) in Equation (II).
Divide Equation (III) by
Use separation constant
Case 1:
Assume
Substitute
From the above equation, the auxiliary equation is written as,
Here, the roots for the auxiliary equation are unequal and real.
The general solution for the auxiliary equation is given by,
Substitute
From the above equation, the auxiliary equation is written as,
The general solution for the auxiliary equation is given by,
Substitute
Apply boundary condition 1: at
Apply boundary condition 2: at
A trivial solutions results as the constant
Case 2:
Assume
Substitute
Integrate the above obtained equation.
Integrate the above equation.
Substitute
Apply boundary condition 2: at
Apply boundary condition 1: at
Apply boundary condition 3: at
Apply boundary condition 4: at
Conclusion:
The boundary condition 4 results in a trivial solution so a negative or zero value of
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Chapter 4 Solutions
Introduction to Heat Transfer
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