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COURSE: MATHEMATICAL MODELLING OF MECHANICAL SYSTEMS
DEVELOPMENT OF EMPIRICAL MODELS FROM EXPERIMENTAL DATA PRACTICE QUESTION
An empirical model simply means model based on real world data collected from
measurements, experiments or observations.
PRACTICE QUESTION
An experiment of heat transfer measurement produces a data for the relationship between
Nusselt number Nu , Reynolds number Re and the Prandtl number Pr. The Nusselt number is
found to vary with the Reynolds number and the Prandtl number. Develop an empirical
model for the data given in the table below.
Exp No. Pr Re Nu
SOLUTION
The above set of data is a multiple input (Re, Pr) single output (Nu) model data. The
relationship between them can be expressed in the form of a power law equation.
๐
๐
Note: This equation will be called a linear equation if b and c equal 1 but that can only be
known after we solve for the variables a, b and c.
The goal here is to solve a set of linear equations that will give values for a, b and c. the best
way to get rid of the power is to take the natural logarithm of both sides.
๐
๐
ln Nu = ln (a โ Pr
b
โ Re
c
ln Nu = ln a + ln(Pr
b
) + ln(Re
c
ln Nu = ln a + b ln(Pr) + c ln(Re)
To simplify the solution we introduce variables to represent the logarithms.
Let: Z=ln Nu, A=ln a, X=ln Pr and Y=ln Re
We have 3 unknowns which means 3 equations are needed.
The first equation is gotten from the addition of the equation above for all the 6 data
obtained. I.e.
ฮฃZ =
Z
1
+ Z
2
+ Z
3
+ Z
4
+ Z
5
+ Z
6
ฮฃZ = A + bX1 + cY1 + A + bX2 + cY2 + A + bX3 + cY3 + A + bX4 + cY4 + A + bX
โด ๐๐ = ๐๐ + ๐๐๐ฐ + ๐๐๐ โฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ .. I
The second and third equation can be found by finding an equation for ฮฃXZ, ฮฃYZ.
๐
. II
๐
โฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ .. III
A NEW TABLE CORRESPONDING TO THE VARIABLES INTRODUCED WILL BE PRODUCED IN
ORDER TO OBTAIN VALUES FOR THE NEW VARIABLES.
Now we can input values for the variables in I, II and III.
The final equations become:
๐. ๐๔
๐๐๐๐ = ๐๐ + ๐. ๔
๔ฐ๐๔
๐๔ ๐ + ๐๔ . ๔๐๐๐๔
๔ ๐ โฆโฆโฆโฆโฆโฆโฆโฆโฆ .. โฆโฆโฆ .. I
๐. ๔ฐ๔
๔๐๔๐ = ๐. ๔
๔ฐ๐๔
๐๔ ๐ + ๔. ๐๐๐๐๔
๐ ๐ + ๐๐. ๔ ๐๔
๔
๔๐ ๐ .. โฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ. II
Exp
No.
Pr Re Nu X=ln Pr Y=ln Re Z= ln Nu XY XZ YZ X
2
Y
2
1
1.46 4000 4.2 0.378436 8.294050 1.435085 3.138771 0.543088 11.
4
2
1.42 3500 3.54 0.350657 8.160518 1.264127 2.861542 0.443275 10.
0
3
1.37 3250 3.12 0.314811 8.086410 1.137833 2.545689 0.358202 9.
6
4
1.2 3900 2.7 0.182322 8.268732 0.993252 1.507568 0.181091 8.
1
5
1.18 3450 2.58 0.165514 8.146130 0.947789 1.348302 0.156873 7.
5
6
1.21 3250 2.41 0.190620 8.086410 0.879627 1.541434 0.167675 7.
6
ฮฃ 1.
49.
0
6.657712 12.943306 1.850204 54.
0.
2
400.
