Seperation of variables - Product solutions for unsteady heat conduction

In summary, the conversation is discussing problem 12C.1 in the book Transport Phenomena by Bird, Stewart, and Lightfoot. The problem involves a block of material initially at uniform temperature T0 being suddenly exposed to temperature T1 at all surfaces. The suggested solution is T=X(x,t)Y(y,t)Z(z,t), but there is confusion about how to do a separation of variables for 3 dimensions. It is suggested to use T=X(x)Y(y)Z(z)W(t) instead, and it is mentioned that writing down equations and explaining the confusion will be necessary to make progress.
  • #1
Chard3000
2
0
Hey guys,

I was wondering about problem 12C.1 in Transport phenomena by Bird, Stewart and lightfoot.

The problem states that a block of material initially at uniform T0 is suddenely exposed to T1 at all surfaces.

Assume a solution of T=X(x,t)Y(y,t)Z(z,t)

any help with separation of variables of this type (3D)

thanks
 
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  • #2
Have you tried substituting that expression into the heat equation?
 
  • #3
I have tried to to... but I do not understand how to do a separation of variables for 3 dimensions.
 
  • #4
Did T = X(x,t)Y(y,t)Z(z,t) come from a suggestion in the problem or did you come up with that? T = X(x) Y(z) Z(z) W(t) is a better choice.

Either way, unless you start writing down some equations and explain what's confusing you, you won't get too far.
 

What is "Separation of Variables" in relation to unsteady heat conduction?

"Separation of Variables" is a mathematical technique used to solve partial differential equations, such as the heat conduction equation. It involves breaking down a complex equation into simpler parts, which can then be solved individually and combined to find the overall solution.

How does the "Separation of Variables" method work for unsteady heat conduction?

The "Separation of Variables" method for unsteady heat conduction involves separating the variables of time and space in the heat conduction equation. This results in a series of simpler equations, which can be solved using different techniques, such as Fourier series or Laplace transforms. The solutions from each equation are then combined to obtain the overall solution for the system.

What are the key assumptions made in the "Separation of Variables" method for unsteady heat conduction?

The "Separation of Variables" method for unsteady heat conduction makes several key assumptions, including that the material being studied is homogeneous and isotropic, the boundary conditions are constant, and the system is in steady-state. These assumptions allow for the use of simpler equations and techniques to solve for the temperature distribution.

What are the advantages of using the "Separation of Variables" method for unsteady heat conduction?

The "Separation of Variables" method for unsteady heat conduction has several advantages, including its ability to handle complex boundary conditions and easily incorporate changes in the geometry of the system. It also allows for the use of different mathematical techniques, making it a versatile approach for solving heat conduction problems.

What are the limitations of the "Separation of Variables" method for unsteady heat conduction?

While the "Separation of Variables" method is a powerful tool for solving unsteady heat conduction problems, it does have some limitations. It may not always be applicable to more complicated systems, and the resulting solutions may only be valid for certain ranges of time and space. Additionally, it may be challenging to find exact solutions, leading to the use of numerical methods instead.

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