Understanding a specific Chebyshev integral

Benit13

Math Team
Apr 2014
2,166
739
Glasgow
I have a situation where I'm trying to understand the physics of a cooling tower. Part of the solution for the model I'm working with involves an integral:

\(\displaystyle C = \int_{T_{low}}^{T_{high}} \frac{1}{\left(h' - h_a\right)} dT\)

where \(\displaystyle h'\) is a slowly varying, monotonically increasing function with temperature, T, and \(\displaystyle h_a\) is linearly increasing with a gradient \(\displaystyle m\). Both functions for \(\displaystyle h'\) and \(\displaystyle h_a\) can be evaluated for any known T and \(\displaystyle h' > h_a\) for all \(\displaystyle T_{low} \le T \le T_{high}\).

Now... according to a particular reference (I don't have a copy, but it's known as British Standard 4485... catchy!) the solution to this integral involves performing a Chebyshev approximation and then performing a rectangular integration on the pieces:

\(\displaystyle C = \frac{\Delta T}{4}\sum_{i=1}^{4} \frac{1}{\left(h'(T_i) - h_a(T_i)\right)}\)

\(\displaystyle \Delta T = T_{high} - T_{low}\)

where
\(\displaystyle T_1 = T_{high} - 0.1026728 \Delta T\)
\(\displaystyle T_2 = T_{high} - 0.4062038 \Delta T\)
\(\displaystyle T_3 = T_{low} + 0.4062038 \Delta T\)
\(\displaystyle T_4 = T_{low} + 0.1026728 \Delta T\)

I don't understand this approximation at all. Could someone please explain this to me (or point out some links or references that explain this outcome)?
 
Jun 2019
493
262
USA
Since no one's answered, let me say that I can't tell you specifically where those numbers came from, but I can maybe give you a general idea. If these were Taylor series-based integrals, I could probably derive it for you on the spot. With those, you are using Taylor series expansion to approximate derivatives (or integrals) of the function using only a finite footprint (such as the four points you have here).

For example:
\(\displaystyle f(x_1) = f(x_0) + f'(x_0)(x_1-x_0) + \frac{1}{2} f''(x_0)(x_1-x_0)^2 + O((x_1-x_0)^3)\)
\(\displaystyle f(x_2) = f(x_0) + f'(x_0)(x_2-x_0) + \frac{1}{2} f''(x_0)(x_2-x_0)^2 + O((x_2-x_0)^3)\)

Then by choosing a certain number of points and/or a certain desired accuracy (leading order of the dropped terms), you can combine these expansions to approximate any of the derivatives \(\displaystyle f^{(n)}(x_0)\) as a weighted average of neighbouring points \(\displaystyle c_1 f(x_1) + c_2 f(x_2) + ...\), e.g.,
\(\displaystyle f'(x_0) = \frac{f(x_2)-f(x_1)}{x_2-x_1} + O((x_2-x_1)^3), ~if~ x_2-x_0=x_0-x_1\) (central difference), or \(\displaystyle +O((x_2-x_1)^2)\) otherwise.
Definite integrals work similarly, though I'd have to sit down with one for a few minutes to figure it out (Runge-Kutta methods are developed this way, though).

Chebyshev approximations, if I'm not mistaken, approximate the function not as a superposition of Nth-order polynomials, but as polynomials in sines and cosines. You'd have to go googling for derivations to find the exact source of those coefficients, though.

Out of curiosity, are you sure British Standard 4485 is still the industrial standard? (Not that it would surprise me terribly.) Assuming your Rayleigh/Prandtl numbers aren't changing wildly, it's not expensive to calculate the convection coefficients for dozens or even hundreds of temperatures, and get a much better approximation of the integral than just relying on four points.
 

Benit13

Math Team
Apr 2014
2,166
739
Glasgow
Thanks for your reply. I've been researching Chebyshev approximation techniques and integrals now for several days and I wrote a Python script that manages to perform function approximations for any user-input function and solve the integral between two arbitrary limits. It's been a good exercise. Numerical recipes sort of came to the rescue in this regard.

I'm not certain whether BS4485 is *the* current standard used for cooling tower performance calculations, but it seems vestiges of it are still sitting in old codes for modelling cooling towers. It's adoption is probably due to the simplicity of the method for hand calculations in the 80s.

The closest I've found to finding out the origin of those particular numbers are these two characteristics:

1. The roots of Chebyshev polynomials of the first kind are:

\(\displaystyle x_j = \cos\left(\frac{\pi(2j-1)}{2N}\right)\)

where N is the order of the the polynomial and j = 1, ..., N. These are the points which should be used to evaluate integrals and consequently those numbers should probably be associated with cosines of some angle. However, it seems no combination of j or N can give the values specified in the table.

2. I found a table in a book on thermal heat transfer which provides weights and ordinate directions for a discretized ordinate method for radiative heat transfer (see attached file). The methods described in the book talk about approximation methods like Chebyshev approximation, but applied to radiative transfer problems. Those S4 numbers are the same ones! Unfortunately, I can't find any references that explain where those numbers come from and how they are calculated; the original reference in the table is hidden behind a paywall.
 

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