Change of variables for this derivative in a heat transfer equation

In summary, the conversation discusses a question about understanding how to get from equation 6.5 to 6.5a. The person asking the question provides relevant information and explains the purpose of the equation. Another person responds with a suggestion and a solution to the question.
  • #1
rudy
45
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TL;DR Summary
How to perform change of variables for an equation in my textbook. I cannot reproduce their result.
Hello-

In the attached screenshot from my textbook, I am trying to understand how they get from equation 6.5 to 6.5a. I have attached my attempt to solve it, but I am stuck evaluating the left side. I do not see how to get their result.

Relevant information:
k, T_w, T_inf, h and L are all constants.
T (no subscript) and y are the only variables.

Any suggestions are appreciated. If it helps this equation is for heat transfer from a wall to the atmosphere. The idea is that at y = 0 (or on the surface of a plate) the heat transfer by conduction and the heat transfer by convection are equal. This part I understand, I just don't understand their steps.

Thanks for any help,

-rudy
 

Attachments

  • Textbook.png
    Textbook.png
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  • My attempt.jpg
    My attempt.jpg
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  • #2
Hi,

With ##\displaystyle {\partial (y/L)\over \partial y}= 1/L## your (6) is their (6a) !
 
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  • #3
Oh snap, thanks BvU. I thought it might be something obvious.
 
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1. What is the purpose of using a change of variables in a heat transfer equation?

The purpose of using a change of variables in a heat transfer equation is to simplify the equation and make it easier to solve. By changing the variables, we can often eliminate complex terms and reduce the equation to a more manageable form.

2. How do you determine which variables to change in a heat transfer equation?

The variables to change in a heat transfer equation are typically chosen based on the physical properties of the system being studied. For example, if the system involves heat transfer through a cylindrical object, it may be beneficial to change to polar coordinates to simplify the equation.

3. Can a change of variables affect the accuracy of the solution to a heat transfer equation?

Yes, a change of variables can potentially affect the accuracy of the solution to a heat transfer equation. It is important to carefully consider the chosen variables and ensure that the resulting equation accurately represents the physical system.

4. Are there any limitations to using a change of variables in a heat transfer equation?

There can be limitations to using a change of variables in a heat transfer equation. For example, some systems may not have a suitable set of variables that can be changed to simplify the equation. Additionally, certain boundary conditions may not be easily expressed in terms of the new variables.

5. How do you know if a change of variables is necessary in a heat transfer equation?

A change of variables is not always necessary in a heat transfer equation. It is typically used when the original equation is too complex to solve or when the new variables can better represent the physical system. It is important to carefully analyze the equation and consider the benefits and limitations of a change of variables before making a decision.

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