Equations for Parabolic Refelctors

megamax42

Hello everyone, new to this forum, hoping you can give me a hand with finding formulas.

Background
I'm at Humboldt State University Studying Environmental Resources Engineering, currently taking the course Computational Methods 3. For our final project we need to solve an engineering problem using one or more numerical methods learned in class.

Topic Selection
For my final project I have chosen to optimize a CSP (concentrating solar power) farm. As seen below:



Approach
My approach to optimizing the farm is comparing the height of the 'collector' to the diameter of the dish and depth, as well as how many dishes are used.

An equation I was hoping to use was the beam intensity for light:

I = P/4[itex]\pi[/itex]r²

Variables explained:
I = light intensity
P = power
r = radius (or more accurately: distance)

I was also hoping to use the equation for the focal length of a parabolic reflector:

f = (D²)/16C

Variables explained:
D = dish diameter
f = focal distance
C = dish depth


And my idea was to use f (focal distance) from the focal length equation as the r (radius, or distance) in the light intensity equation. Then optimize D (dish diameter) and C (dish depth) to obtain a maximum I (light intensity).

Problem

I'm not sure if I'm on the right track with this approach. I can't help but feel like I'm missing a very important factor in this system of equations, I'm just not sure what it is.

Any help would be greatly appreciated. I am not looking for someone to give me the answers, just a push in the right direction.

Thanks!

megamax42

I found the error with my approach. The equations will lead to odd dimensions for the reflector, so I've decided to perform a polynomial interpolation on available prices for reflectors based off of the diameter, which is a variable found in the focal length equation. I can then minimize costs at the same time, hopefully leading to more reasonable values of D (dish diameter) and C (dish depth).