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The Heat Diffusion Equation (X, Y, Z) : Recap

The document discusses the heat diffusion equation in cylindrical and spherical coordinates and describes the boundary and initial conditions needed to solve the heat equation. These include radial, circumferential, axial, polar, and azimuthal conditions. It also provides an example problem on determining the rate of heat transfer and temperature change at boundaries for a medium, and discusses the three common types of boundary conditions: Dirichlet, Neumann, and third kind conditions.

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AMARTYA MONDAL
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32 views13 pages

The Heat Diffusion Equation (X, Y, Z) : Recap

The document discusses the heat diffusion equation in cylindrical and spherical coordinates and describes the boundary and initial conditions needed to solve the heat equation. These include radial, circumferential, axial, polar, and azimuthal conditions. It also provides an example problem on determining the rate of heat transfer and temperature change at boundaries for a medium, and discusses the three common types of boundary conditions: Dirichlet, Neumann, and third kind conditions.

Uploaded by

AMARTYA MONDAL
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Recap

Thursday, August 20, 2020 9:45 AM

The Heat Diffusion Equation (x,y,z)

Cylindrical Coordinates ( )

Boundary and Initial Conditions Page 1


Radial Circumferential Axial

Boundary and Initial Conditions Page 2


Spherical Coordinates (

Radial Polar Azimuthal

Boundary and Initial Conditions Page 3


Radial Polar Azimuthal

Boundary and Initial Conditions Page 4


Sample Problem
Thursday, August 20, 2020 9:45 AM

Determine the rate of heat transfer entering the wall (x = 0) and leaving the wall (x = 1 m).

Boundary and Initial Conditions Page 5


Determine the rate of change of energy storage in the wall.

Determine the time rate of temperature change at x = 0, 0.25, and 0.5 m.

Boundary and Initial Conditions Page 6


Boundary and Initial Conditions Page 7
Boundary and Initial Conditions
Thursday, August 20, 2020 9:45 AM

To determine the temperature distribution in a medium, it is necessary to


solve the appropriate form of the heat equation.
However, such a solution depends on the physical conditions existing at
the boundaries of the medium and, if the situation is time dependent, on
conditions existing in the medium at some initial time.
Because the heat equation is second order in the spatial coordinates,
two boundary conditions must be expressed for each coordinate needed to
describe the system.
Because the equation is first order in time, however, only one condition,
termed the initial condition, must be specified.

Three kinds of boundary conditions typically encountered in heat transfer


problems are summarized in the table below

1st kind- DIRICHLET CONDITION

2nd kind- NEUMANN CONDITION

3rd kind

Boundary and Initial Conditions Page 8


Sample Problem

Assumptions:

Boundary and Initial Conditions Page 9


How will the temperature vary with time @ x = 0 and x = L?

Sample Problem

Boundary and Initial Conditions Page 10


Sample Problem

Boundary and Initial Conditions Page 11


Boundary and Initial Conditions Page 12
Boundary and Initial Conditions Page 13

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