ANALYSIS OF ISOTHERMALS AND HEAT FLOW LINES USING GeoGebra

1. ANALYSIS OF ISOTHERMALS AND
HEAT FLOW LINES USING
GEOGEBRA/MATLAB
Submitted by:
Saumya Tiwari
Dheeraj Mangal
Praman Satya
Shubham Sharma
Somesh Sharma

2. COMPLEX POTENTIAL
 We are now going to investigate the properties of a complex function the real and imaginary
part of which are conjugate functions. In particular we define the complex potential
 In the complex (Argand-Gauss) plane every point is associated with a complex number

3.  In general we can then write;
The fact that Cauchy-Riemann conditions hold for both and Ψ or equivalently
that these functions are conjugate, is a necessary and sufficient condition for the
function f to be analytic.
Now, if the function f is analytic, this implies that it is also differentiable,
meaning that the limit
ᶲ

4.  is finite and independent of the direction of
If then we pose it follows that

5. LAB EXPERIMENT AND MATERIAL USED:-
 Aluminum foil as metal for measuring the conduction with in it.
 Temperature at the both the ends is 0 C and 40 C.
 For const. temperature we use heating plate on as source of heat and ice bath on the other side
which work as sink for 0 C.
 Dimension of the foil and the plotted graph (28*22) cm.
 We find the isotherms length wise.
 Digital as well as analog thermometer for measuring the temperature.
 Marker for marking the temperature.

6. RESULT OF CHEMISTRY LAB EXPERIMENT AND
ROUGH SKETCH OF ISOTHERM

7. DETERMINATION OF GRAPH EQUATION USING
GEOGABRA:-
Here the lines in black signifies isotherms and the lines in red signifies the heat
lines associated with them equations of isotherm respectively.

8. PRACTICAL DATA
 Equation of isotherm Equation of heat flow
 a: 0.02x +5.84y = 43.82 j: -5.84x + 0.02y = -39.21
 b: y = 5.64 h: x = 2.56
 c: 0.12x + 4.28y = 20.8 i: -4.28x + 0.12y = -24
 d: 0.24x + 3.34y = 10.47 g: -3.34x + 0.24y = -12.51
 e: 0.08x +4.72y = 7.07 f: -4.72x + 0.08y = -4.98

9. THEORITICAL PROOF OF HARMONICITY OF THE FUNCTION AND
ITS HARMONIC FUNCTION:-
 FOLLOWING IS THE THEORITICAL CALCULATION WHICH WE GET BY TAKING
EQUATION (d) AS EXAMPLE AND PROFFING THAT IT IS HARMONIC OR NOT AND ITS
HARMONIC CONJUGATE

11. PHYSICAL INTERPRETATION:-
 When we applied constant temperature on the both end of aluminum with
temperature difference of 40’ C then we see that heat due to this
temperature difference we observe that as we move from higher temp. to
lower temperature i.e. 0 C. we also observe that the isothermal lines are
observed parallel to the source i.e. horizontal and the heat flows
perpendicular to these isotherms

12. PROPERTIES OF THIS FUNCTION:-
 This function (si) is harmonic.
 It’s Analytical along all complex plain i.e. it is an entire function.
 It has two component real and imaginary where real part show the isotherm
and imaginary part show the heat lines flow associated with it

13. THANK YOU