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ME 435 Thermal Energy Systems
Lecture #13
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Curve Fitting
As mentioned last time, one example of the equations that we will need for a compressor simulation are
the power and capacity as functions of the saturated evaporating and saturated condensing temperatures.
Power = f1 (SET,SCT)
Capacity = f2 (SET,SCT)
We saw last time that equations like this are 3-D and represent surfaces. Performance curves are 2-D
representations (“topological maps”) of these surfaces.
Fitting techniques can be used to develop the functional dependence from manufacturer’s tabular data.
The two techniques we will be discussing are:
1) Exact fitting: (this lecture) where the curve is forced to hit each and every data point
2) Method of least squares: (next lecture) where the curve is a “best fit” but doesn’t hit every
point
Many times, components can be modeled using simple polynomials. A typical 2-D polynomial can be
written as:
y(x) = a0 + a1 x + a2 x2 + a3 x3 + …
The ai are the coefficients for the polynomial: these are what we will be after when we do a curve fit!
In the case of 3-D surfaces, there are two independent variables involved, let’s say x and z. A typical
form for y(x,z) would be:
y(x,z) = a0 + a1 x + a2 z + a3 x z +a4 x2 + a5 z2 + a6 x2 z + a7 x z2 + a8 x2 z2 + …
(Note: this is only one possible form…we will discuss other possible forms later!)
Our basic task, once we have the (y,x,z) data will be to
1) Determine the functional form for the y(x,z) equation
2) Determine the ai coefficients
3) CHECK THE RESULT!!!
3-D Curve Fitting : Exact Fitting
The idea behind exact fitting is to select a set of data points and determine an equation that passes
through these points exactly! The set of points may be all of the data points, or a subset of the data for
the performance parameter we are considering. When a data set of subset is fit exactly, an equation is
written for each data point. This results in a set of n equations with n unknowns. These unknowns are
the ai coefficients.
Consider the following example. In this example, an equation is desired for the outlet water temperature
from a cooling tower as a function of the ambient wet bulb temperature (WBT) and the range R. Note:
R is defined as the difference between the inlet and outlet temperatures of the water passing through the
cooling tower. The manufacturer’s data for a specific cooling tower are given in the following table:
Outlet Water Temperature (°C)
WBT (°C)
R (°C)
20
23
26
10
25.9
27.5
29.4
16
27.0
28.4
30.2
22
28.4
29.6
31.3
The form of the equation that is desired is:
Outlet temperature = t = f (R,WBT)
In this table, there are 9 data points. This is all that we have. We are interested in developing an
equation that fits all 9 data points exactly. Using the exact fitting ides, we need to write 9 (!) equations
with 9 unknowns: a1 through a9.
Step 1: write the polynomial.
If we assume that the polynomial model we mentioned on the previous page is sufficient to represent the
data, then we must use a 9-term, 3-D function. To keep life simple, we will use the lowest terms of the
polynomial that we possibly can!
t = a1 + a2 R + a3 WBT + a4 R WBT + a5 R2 + a6 WBT2
+ a7 R2 WBT + a8 R WBT2 + a9 R2 WBT2
Step 2: substitute the data points into the equation to form the 9 equations (leaving the ai as the
unknowns…)
a1 + 10 a2 + 20 a3 + 200 a4 + 100 a5 + 400 a6 + 2000 a7 + 4000 a8 + 40000 a9 = 25.9
a1 + 10 a2 + 23 a3 + 230 a4 + 100 a5 + 529 a6 + 2300 a7 + 5290 a8 + 52900 a9 = 27.5
a1 + 10 a2 + 26 a3 + 260 a4 + 100 a5 + 676 a6 + 2600 a7 + 6760 a8 + 67600 a9 = 29.4
a1 + 16 a2 + 20 a3 + 320 a4 + 256 a5 + 400 a6 + 5120 a7 + 6400 a8 +102400 a9 = 27.0
a1 + 16 a2 + 23 a3 + 368 a4 + 256 a5 + 529 a6 + 5888 a7 + 8464 a8 +135424 a9 = 28.4
a1 + 16 a2 + 26 a3 + 416 a4 + 256 a5 + 676 a6 + 6656 a7 +10816 a8 +173056 a9 = 30.2
a1 + 22 a2 + 20 a3 + 440 a4 + 484 a5 + 400 a6 + 9680 a7 + 8800 a8 +193600 a9 = 28.4
a1 + 22 a2 + 23 a3 + 506 a4 + 484 a5 + 529 a6 +11132 a7 +11638 a8 +256036 a9 = .29.6
a1 + ?? a2 + ?? a3 + ??? a4 + ???? a5 + ???? a6 + ???? a7 + ???? a8 + ?????? a9 = ??.?
Group Exercise #1:
Fill in the question marks in the last equation above. You have 10 min.
The 9 equations above can be solved for the ai using a variety of numerical techniques (or software
packages. Whatever technique you use, you are solving the following matrix equation:
[A] [a] = [b]
[a] = [A]-1 [b]
A solution may be obtained from a matrix operation:
where [A]-1 is the inverse of the [A] matrix.
1
1
1
1
1
1
1
1
?
10
10
10
16
16
16
22
22
??
20
23
26
20
23
26
20
23
??
200
230
260
320
368
416
440
506
???
100
100
100
256
256
256
484
484
????
400
529
676
400
529
676
400
529
????
2000
2300
2600
5120
5888
6656
9680
11132
????
4000
5290
6760
6400
8464
10816
8800
11638
????
40000
52900
67600
102400
135424
173056
193600
256036
??????
[A]
a1
a2
a3
a4
a5
a6
a7
a8.
a9
=
25.9
27.5
29.4
27.0
28.4
30.2
28.4
29.6
?.?
[a]
=
[b]
Of course this is one of those jobs for which a computer or calculator is best suited. I suggest you see if
your calculator will do matrix inverses. (It will probably look as if you are “dividing” the vector [b] by
the matrix [A] to get the vector [a], although technically that is not what you are doing.)
You may also use a spreadsheet or EES to get your solution.
I will hand out a solution in class.
The final step (as always) is to check your answer! Remember that this equation is supposed to fit
every data point exactly!
Group Exercise #2:
Check the solution that I hand out in class to see whether or not it is indeed correct.
The equation for what is known as the percent deviation is:
 x(data )  x(calc) 
% dev  100
.
x(data )


What is the percent deviation for each of the 9 manufacturer’s data points?
Next we will examine how to determine what the functional form of a set of data points should look like.