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

1. Jul 14, 2006

### 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 !

2. Jul 14, 2006

### Astronuc

Staff Emeritus
Please provide the differential equation for the PK model.

3. Jul 14, 2006

### unknownreference

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

#### Attached Files:

• ###### PRKE Analytical solution.doc
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4. Jul 16, 2006

### Astronuc

Staff Emeritus
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 [Broken]
See bottom of page 4 for discussion of Runge-Kutta method.

Last edited by a moderator: May 2, 2017
5. Jul 16, 2006

### 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 !