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

TABLE OF CONTENTS Introduction Heat equation Types of heat equation dimensional types of graph applications Recent trends/future scopeHeat transformWorking of heat transfrom Conclusions

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?

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:

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.

0),,0(),,(f2

2

tLxtx

xua

tu

),0(),()0,( 0 Lxxuxu

0),(),(),(),0( 21 ttgtLutgtu

for a function:

RLu ),0[],0[:

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

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’

heat equation graph After mat lab program receive the two type

graph (1) stable solution of heat equation (2) unstable solution of heat equation

Stable Solution of Heat Equation

unstable solution of heat equation

Application Particle diffusionBrownian motion:Schrödinger equation for a free particle:Thermal diffusivity in polymers:

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.

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.

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.

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.