Can an analytical solution for PRKE be found with ramp reactivity ?

[unknownreference]

Hi there,

Before i state what my actual question is, let me give some background on some work i have done on this. I have been trying to solve the 0-D PRKE transient problem for a step and a ramp input numerically using couple of high precision methods.

1) Using MATLAB's inbuilt ode23s solver.
2) Using a custom written adaptive solver

My problem is that i want to find the order of accuracy or maybe even just the exact error in my adaptive method in comparison to the actual true solution. Without the exact analytical solution, i would have to assume that the solution from MATLAB is exact and compute the error. But i am reluctant to do that because i want to test few embedded numerical methods with higher orders of accuracy than matlab's 2-3 method.

So now, i did derive the analytical solution for PRKE with a step input (constant reactivity) and tested my method with the solution i computed analytically. It perfectly matches !

But when the reactivity is a function of time (Ramp or say parabolic change), i end up with an integro-differential form which does require approximating the integral numerically, which in turn will introduce truncation errors in the calculation. Now is there any way to get around this ? OR have i completely missed out some other alternative ?

I would appreciate any and all help that someone can provide. Thanks in advance !

 

[Astronuc]

Please provide the differential equation for the PK model.

 

[unknownreference]

Please provide the differential equation for the PK model.

Please see my attached document file for the equations and for the question in more detail. Please let me know if something is confusing !

 

[Astronuc]

I think an analytical solution might be possible, and could be found using Laplace transforms.

For a ramp, $\rho(t)\,=\,\rho_o\,+\,\gamma\,t$, but $\rho_o$ could be zero.

The problem in the attachment shows one precursor (or an effective precursor) concentration. Is this correct?

Numerically, one can use the Runge-Kutta method for solving first order differential equations.

Interesting paper - "The reactor point-kinetics equations: semi-analytical methods versus numerical methods"
http://sab.sscc.ru/imacs2005/papers/T2-I-72-0937.pdf
See bottom of page 4 for discussion of Runge-Kutta method.

 

[unknownreference]

Astronuc,

Thanks for the reply.

Yes. I have shown only one precursor group in the document because i just wanted to simplify the system. For the analytical solution though, i do not think it makes any difference but just couple more equations to solve to obtain the coefficients.

I have been trying to use Laplace transforms for the PRKE but since my math with transforms is a bit rusty, it is taking me a while. But it is good to know that i am going in the right direction !

And thanks for the great reference document. I am sure it will help me out further in my current research.

I will try to derive the analytical solution and if not, i might have to go with the numerical solution with a very high accurate numerical scheme. I will let you know what i stumble onto along the way and post some interesting observations when i have my result.

Thanks again !