Modified heat equation steady state

In summary, the steady state equation for the given heat equation and boundary conditions is Ut=Uxx-4(U-T), with a particular solution of c1e2x+c2e-2x-T and constants obtained through variation of parameters. Alternatively, the general solution can be expressed as X(x) = Asinh(x)+Bsinh(T-x) + T, which simplifies the calculation of constants.
  • #1
dp182
22
0

Homework Statement


determine the steady state equation for the given heat equation and boundary conditions


Homework Equations


Ut=Uxx-4(U-T)
U(0,T)=T U(4,T)=0 U(x,0)=f(x)


The Attempt at a Solution


I put Ut=0
so 0=UInf''-4(Uinf-T)
then once I tried to integrate I ended up with a Uinf that's impossible to isolate is there a specific method used for this kind of heat equation?
 
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  • #2
The steady state is usually when the system is time independent, so:
[tex]
\frac{\partial U}{\partial t}=0
[/tex]
So this will turn your equation into and ODE, which I see you have already done and your ODE is:
[tex]
\frac{\partial^{2}U}{\partial x^{2}}-4U+4T=0
[/tex]
I am not too sure what T is, is it a constant? A function?
 
Last edited:
  • #3
dp182 said:

Homework Statement


determine the steady state equation for the given heat equation and boundary conditions


Homework Equations


Ut=Uxx-4(U-T)
U(0,T)=T U(4,T)=0 U(x,0)=f(x)


The Attempt at a Solution


I put Ut=0
so 0=UInf''-4(Uinf-T)
then once I tried to integrate I ended up with a Uinf that's impossible to isolate is there a specific method used for this kind of heat equation?

What do you mean by "impossible to isolate". Calling what you call Uinf u(x) for simplicity the equation you have written is just

u''(x) - 4u(x) = -4T
u(0) = u(T) = 0

A second order constant coefficient ODE.
 
  • #4
I believe T is constant. so I tried using variation of parameters to solve (Uinf=X) X''-4X=-4T
and came up with a particular solution of
c1e2x+c2e-2x-T then solved for the constants using boundary conditions. is this method right as all the example I have seen involve just integrating it twice.
 
  • #5
dp182 said:
I believe T is constant. so I tried using variation of parameters to solve (Uinf=X) X''-4X=-4T
and came up with a particular solution of
c1e2x+c2e-2x-T then solved for the constants using boundary conditions. is this method right as all the example I have seen involve just integrating it twice.

That should work, but I'm guessing it gets a bit messy solving for the constants. Here's a little trick you might like to learn. You probably know that instead of using the pair {e2x,e-2x} for the general solution of the homogeneous equation, you could have used {sinh(2x),cosh(2x)}, which simplifies things a bit when you calculate X(0) = 0. But an even better choice is the pair {sinh(x),sinh(T-x}. Try evaluating the constants when you write your solution

X(x) = Asinh(x)+Bsinh(T-x) + T

and you will see what I mean.
 

1. What is the modified heat equation steady state?

The modified heat equation steady state is a mathematical equation used to model the steady-state temperature distribution in a material with a heat source. It takes into account factors such as thermal conductivity, heat generation, and boundary conditions to determine the temperature distribution at a specific point in time.

2. How does the modified heat equation differ from the traditional heat equation?

The modified heat equation includes an additional term that accounts for heat generation within the material. This term is usually represented by a source function and can be used to model situations where heat is being generated or absorbed by the material.

3. What are the applications of the modified heat equation steady state?

The modified heat equation steady state has various applications in fields such as engineering, physics, and materials science. It can be used to analyze and design heat transfer systems, predict temperature distributions in materials, and understand the behavior of thermal processes.

4. How is the steady-state temperature distribution calculated using the modified heat equation?

The steady-state temperature distribution can be calculated by solving the modified heat equation using numerical or analytical methods. The equation can be solved for a specific material and boundary conditions, and the resulting temperature distribution can be plotted to visualize the behavior of the system.

5. What are the limitations of the modified heat equation steady state?

The modified heat equation steady state assumes a steady-state condition, which means that the temperature distribution remains constant over time. This may not be applicable in situations where the heat source or boundary conditions are changing. Additionally, the equation does not take into account factors such as convection or radiation, which can also affect the temperature distribution.

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