This document discusses heat transfer and the heat equation. It provides an introduction to heat and the heat equation. It then discusses one-dimensional and three-dimensional heat equations. It shows stable and unstable solution graphs of the heat equation from MATLAB. Applications discussed include particle diffusion, Brownian motion, and the Schrodinger equation. Future trends discussed applying the heat equation to option pricing models and pressure diffusion. It then discusses heat transformers, how they work by circulating fluid through a mini-channel to cool electronics, and concludes discussing applications to biology.

1. Assignment of numerical method
Topic- Heat transformer(heat equation )
 SUBMITTED TO:-
Mrs.Hemlata saxsena
H.O.D of mathmetics
 SUBMITTED BY:-
 VINOD KUMAR (k11543)
 Himanshu
chauhan(k11579)
 BRANCH-
MECHANICAL

2. TABLE OF CONTENTS
Introduction
Heat equation
Types of heat equation dimensional
 types of graph
 applications
Recent trends/future scope
Heat transform
Working of heat transfrom
Conclusions

3. Introduction
 The purpose of our
presentation is to what is the
heat equation
 that we are able to model the
flow of heat through an
object.
 But before we begin our
discussion of the
mathematics of the heat
 equation, we must first
determine what is meant by
the termheat?

4. heat equation
 The heat equation arises in
the modeling of a number
of phenomena and is often
used in financial
mathematics in the
modeling of options. The
famous Black-Scholes
option pricing model's
differential equation can
be transformed into the
heat equation allowing
relatively easy solutions.
More generally in any coordinate
system:

5. One dimensional equation
The heat equation is derived from Fourier's
law and conservation of energy (Cannon 1984). By
Fourier's law. A uniform homogenous rod is made of
infinite number of molecules. They are interconnected by
cohesive force.

6. 0),,0(),,(f2
2






tLxtx
x
u
a
t
u
),0(),()0,( 0 Lxxuxu 
0),(),(),(),0( 21  ttgtLutgtu
for a function:
RLu  ),0[],0[:

7. The domain of the function:
RLu  ),0[],0[:
L x
t
)(),0( 1 tgtu  )(),( 2 tgtLu 
0
boundary
value
boundary
value
initial
value
)()0,( 0 xuxu 

8. Three dimensional heat equation
 In the special cases of wave propagation of heat in
an isotropic and homogeneous medium in a 3-
dimensional space, this equation is’

9. heat equation graph
After mat lab program receive the two type
graph
(1) stable solution of heat equation
(2) unstable solution of heat equation

10. Stable Solution of Heat Equation

11. unstable solution of heat equation

12. Application
Particle diffusion
Brownian motion:
Schrödinger equation for a free particle:
Thermal diffusivity in polymers:

13. Future trades
 heat equation the future trades are famous Black–S
uchholes option pricing model's differential
equation can be transformed into the heat equation
allowing relatively easy solutions from a familiar body
of mathematics.
Many of the extensions to the simple option models do
not have closed form solutions and thus must be
solved numerically to obtain a modeled option price.
The equation describing pressure diffusion in a porous
medium is identical in form with the heat equation.

14. Heat transform
The increased circuit density on today’s computer
chips is reaching the heat dissipation limit of air
cooling technology. The direct liquid cooling of chips is
being considered as a viable alternative. A broad
range of industries is driving the development of
compact, advanced cooling technology. One of the
most important of these applications is the removal of
high heat fluxes in microelectronic circuitry.

15. working of heat transform
Fluid enters the loop from a reservoir through a filter
and is continuously circulated by a peristaltic
pump. Constant temperature bath is at upstream of
the setup.
which controls the inlet flow temperature and
maintained constant temperature of fluid and it
goes to mini channel
Aluminum is selected as substrate because of
relatively high thermal conductivity (204.2w/mk) and
low mass density with high strength.

16. conclusion
 The heat equation equation also has immense
amounts of use in Biology where it is know as the if-
fusion equation and models the diffusion of a
substance through a system. We have only examined
the case where heat flows in one direction through a
thin rod.