First-Order Nonlinear ODE from transient heat transfer

In summary, the conversation is about a problem from a heat transfer book that led to a differential equation. The professor said there wouldn't be an analytical solution and suggested using an iterative method. However, the person tried Wolfram Alpha and found a solution in the form of a closed integral in terms of four roots of a polynomial. They were initially confused but were able to understand it with the help of someone else.
  • #1
mafra
10
0
A problem from a heat transfer book with conduction and radiation led me to a differential equation like this:

T'(t) = a - b*T(t) - c*T(t)^4

Although my professor said that there wouldn't be an analytical solution for this one and to get the answer by an iterative method I got curious and tried Wolfram Alpha. For my surprise it gave a solution, but I just can't understand the answer and neither the steps for solving it

Here it is:
http://www.wolframalpha.com/input/?i=T'(t)+=+a+-+b*T(t)+-+c*T(t)^4

Could anyone give me some light here?
 
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  • #2
The solution is shown in attachment.
WolframAlpha gives a closed form of the integral in terms of the four roots of the polynomial :
a - b*T - c*T^4 = c*(T-w1)*(T-w2)*(T-w3)*(T-w4)
 

Attachments

  • Integral.JPG
    Integral.JPG
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  • #3
Thanks mate! That's simpler than I thought. So problably he was referring to an analytical solution for T(t)
 

1. What is a first-order nonlinear ODE in the context of transient heat transfer?

A first-order nonlinear ODE, or ordinary differential equation, is a mathematical equation that describes the relationship between a function and its derivatives. In the context of transient heat transfer, it is used to model the change in temperature over time in a system with nonlinear heat transfer properties.

2. How is a first-order nonlinear ODE derived for transient heat transfer?

The first-order nonlinear ODE for transient heat transfer is derived from the fundamental laws of heat transfer, such as Fourier's law and the heat conservation equation. These laws are combined and simplified using mathematical techniques to create the ODE that describes the behavior of the system.

3. What are the key assumptions made in the first-order nonlinear ODE for transient heat transfer?

Some of the key assumptions made in this ODE include the assumption of one-dimensional heat transfer, constant material properties, and negligible internal heat generation. These assumptions may not hold true in all systems, but they allow for a simplified mathematical model that can still provide valuable insights.

4. How is a first-order nonlinear ODE solved for transient heat transfer?

The first-order nonlinear ODE for transient heat transfer can be solved using analytical or numerical methods. Analytical methods involve finding an exact solution to the ODE, while numerical methods use approximations and algorithms to find a numerical solution. The choice of method depends on the complexity of the ODE and the desired level of accuracy.

5. What are some practical applications of the first-order nonlinear ODE for transient heat transfer?

The first-order nonlinear ODE for transient heat transfer has many practical applications, including in the design and analysis of heating and cooling systems, thermal insulation, and temperature control systems. It can also be used to study the behavior of materials under different thermal conditions and to optimize energy efficiency in various industries.

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