How to solve a equation of this kind

T' = a*(T^4 - r^4) + b*(T^4 - s^4) + P*(1 - η(T))

The above equation is driving me nuts... the 'η' is a function of T(Temperature) the efficiency and initial value of T is known.
Say at t = 0 T is 298

I need to find the temperature of the system after a given time say 60s...

Ok, i'm assuming that T' means dT/dt, and that a, b, P, r and s are not functions of t.

If it were the case, this is a separable DE, and can be solved by the standard way.

The only problem is you will get t as a function of T, instead of T as a function of t as wished, but i'm afraid this is the closest you can get of an analytical solution.

However, if you need only numbers, i strongly suggest integrating it numerically, as it is a much easier way to get the answer you need.

epenguin
Homework Helper
Gold Member
How to solve a equation of this kind

T' = a*(T^4 - r^4) + b*(T^4 - s^4) + P*(1 - η(T))

The above equation is driving me nuts... the 'η' is a function of T(Temperature) the efficiency and initial value of T is known.
Say at t = 0 T is 298

I need to find the temperature of the system after a given time say 60s...

My amateur answer :uhh: is I would have thought:
you can solve

T' = a*(T^4 - r^4)

It is a 'straightforward' integration - I think I have done it buried in one of my early posts here somewhere. Standardish anyway. Edit: now I remember one way was to express (T4 - r 4) as (T2 - r2)(T2 - r2) then you have a simple partial fractions problem and then standard integrals.

You can integrate each term. Then any linear combination of those integrals plus and arbitrary constant determined by the physics should be a solution of the overall equation.

Then you have two problems.

One is whether this is the most general solution. On a quick look it seemed to me it is but anyway get a solution first.

The other is that last term. As we don't know the form of η we can only write for the integral of the last term P*(x - ∫ η(T) dT ) . Then if η is given as data integrate that graphically or numerically. But I would guess that this is a function varying much more lowly than the fourth powers you have in your other terms so is a relatively small correction, and that it is adequately modelled by a simple expression perhaps empirical, e.g. linear or quadratic.

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Hi all,

Thanks for taking time and answering my question. I know to solve for T^4 terms. Thats not the problem.
And I don't require analytical solution. I am interested only in numbers. I think you would have realized that the equation is an energy balance equation. 'η' is the efficiency of the system which is a function of 'T' the temperature of the system. If η were a constant, I would simply integrate it. But, η(T) will vary as T increases. When power 'P' i.e. heat is incident, T will rise... will the rise be determined only by material constants? No...

Rise in T will also be affected by the behavior of η(T). η is a complex program.
My intention is this...
I have written a program for η... I want to call it in the energy balance equation and pass 'T' as a parameter into η. As η decreases with T... Rise in T will be more dramatic at higher temperatures...