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Chapter 5

Transient Conduction Notes

5.2 Spatial Effects

If the Biot number Bi 0.1 temperature gradients within the solid is not negligible any more and

temperature depends on time and position.

The Infinite Plane Wall with Convection

Consider an infinite plane wall with constant thermal properties ,thickness 2L,and in effect

infinite in the other directions .At the two surfaces the wall exchanges heat by convection with a

surrounding fluid where fluid temperature T

and the convective heat transfer coefficient is h

on both sides. Consideration of symmetry reveals that we need to consider only the half of plate.

Temperature within the solid depends on the following physical parameters, and in particular

c

T T(x, t,T ,L ,k, ,h)

Here k

c

is the thermal diffusivity of the solid.

Let us define dimensionless temperature as follows

*

i i

T T

T T

Accordingly * must be in the range

*0 1 .

2

A dimensionless spatial coordinate may be defined as

* x

xL

Where c

L L is the half thickness of the plane wall and finally a dimensionless time may be

defined as

*

2

tt Fo

L

Dimensionless temperature takes the following form

* *f (x ,Fo,Bi)

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The Infinite Cylinder with Convection

Consider an infinite cylinder with constant thermal properties ,radius 0

r and in effect

infinite in the z direction .At surface the cylinder exchanges heat by convection with a

surrounding fluid.

The fluid temperature is T

and the convective heat transfer coefficient is h . See

Figure given above.

The dimensionless temperature distribution can be expressed as

* *

*

0

2

0

0

f (r , Fo, Bi)

rr

r

tFo

r

hrBi

k

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Sphere with Convective Boundary Condition

The dimensionless temperature distribution can be expressed as

* *

*

0

2

0

0

f (r , Fo, Bi)

rr

r

tFo

r

hrBi

k

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Equations for heat transfer and temperature distributions have been evaluated numerically by a

number of investigators for ranges of three parameters. The results are presented here.

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Semi-Infinite Solid

Case I:Semi-Infinite Solid with Constant Surface Temperature

A semi-infinite solid extends to infinity in all but one direction, it is characterized by a single

identifiable surface. Semi-infinite solid is confined to the domain 0 x which is initially See figure given below

If a sudden change of conditions is imposed at the surface, transient, one-dimensional conduction

process is confined to thin region near the surface into which temperature changes have

penetrated. Interior cools slowly. This model may be used to determine transient heat transfer

near the surface of the earth or to approximate the transient response of a finite solid, such as a

thick slab.

The solid is initially at a uniform temperature i

T and for t>0 the boundary surface at x=0 is

changed constant temperature s

T and is maintained at that temperature for t>0. At

ix , T T .

Mathematical Model:

2

2

T 1 T

x t

iT(x,0) T

sT(0,t) T

iT( ,t) T

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Heat transfer is given as

Error function is tabulated at the end of notes.

Dimensionless temperature

i

i

T(x, t) T

T T

is plotted as a function of dimensionless parameter

x

t

.See figure on next page

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Case II Semi-Infinite Solid with Constant Surface Heat Flux

The solid is initially at a uniform temperature i

T and for t>0 the boundary surface at x=0 is

subjected to a constant heat flux " 2

sq (W / m ) .Temperature distribution is given by

where erfc(w) is called complimentary error function and it is defined as

erfc(w) 1 erf (w)

Case III: Semi-Infinite Solid with Convection at the Surface

The solid is initially at a uniform temperature i

T and for t>0 the boundary surface at x=0 is

subjected to convection with a fluid temperature T

with a heat transfer coefficient h . The

temperature distribution is given as.

Dimensionless temperature

i

i

T(x, t) T

T T

is plotted as a function of dimensionless parameter

x

t

.See figure below.

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P(X)=infinite plane C(X)=infinite cylinder S(X)=semi infinite body

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