MAE 310 course notes – Fall 2011 Copyrighted by R. D ... · MAE 310 course notes – Fall 2011...

MAE 310 course notes – Fall 2011 Copyrighted by R. D. Gould

Chapter 4 – 2

HEAT TRANSFER FUNDAMENTALS

IV-2. 1-D Transient Heat Transfer with Spatial Effect (use when Bi > 0.1)

The transient heat conduction problem for several simple shapes (constant k, no internal heat generation) subject to boundary conditions of practical importance have been computed. Analytic (infinite series) and graphical solutions are presented.

Geometries we will consider:

1. a long plane wall

2. a long solid cylinder

3. a sphere

all initially at a uniform temperature at t = 0 and with convection to a medium with fixed temperature at the exposed surface.

Other solutions are available.

MAE 310 Muller Lec. 15 - 2


Chapter 4 – Page 13


Plane Wall

T, h T, h fluid flow fluid flow

k dTdx

hT hTx L

k dTdx

hT hTx L

x=-L x=0 x=L

This problem is symmetric both geometrically and thermally.

T, h fluid flow

dTdx x

0

0

k dTdx

hT hTx L

x=0 x=L

Governing equation (1-D HCE)

tTCq

xTk

x p





Assume: , k = constant

Recall thermal diffusivitypC

k

Governing equation:

2

21T

xTt

0 < x < L, t > 0

Left B.C. Tx 0 at x = 0, t > 0

Right B.C. k Tx

hT hT

at x = L, t > 0

Initial Condition T Ti for t = 0 in 0 x L

Note: There are 8 independent variables

x, t, L, k, h, , , Ti T

0q





We can minimize the number of independent variables by defining non-dimensional parameters.

TT

TtxTtxi

,,* dimensionless temp.

x xL

* dimensionless space coor.

Bi hLk

Biot number:

resistanceconvresistancecond

.

.

Fo t tL

* 2 dimensionless time (Fourier number)

The non-dimensional equations are:

2

2

*

*

*

*x t 0 < x* < 1, t* > 0

*

*x 0 at x* = 0, t* > 0

*

**

xBi 0 at x* = 1, t*> 0

* = 1 at t* = 0, 0 x* 1

Note:There are only 3 independent variables in the non-dimensional formulation: x*, t*, Bi





Fourier number:

CW,Lvolume

instorageheatofrateCW,LvolumeinL

acrossconductionheatofrate

3

321

2*

3

tLC

L

p

Lk

Ltt

Fo =

Large Fourier # deeper heat penetration into a solid over a given time

tTxXtx ,*Assuming that , and using separation of variables gives:





Exact Solution for Plane wall

TT

TtxTxFoCtxin

nnn,cosexp,

1 position

*

infotime

2*

nn

nnC

2sin2sin4

Binn tan

2LtFo

Lxx *where and

and the eigenvalues are the positive roots of the transcendental equation

Table 5.1 (p. 301) gives the first root to this equation (App. B.3 gives the first 4 roots)

[Eq. 5.42a]

[Eq. 5.42b]

[Eq. 5.42c]





Approximate (or one term) Solution for Plane wall

TT

TtxTxFoCtxi

,cosexp, *1

211

*

11

11 2sin2

sin4

C

Bi11 tan

2LtFo

Lxx *where and

and the eigenvalue is the positive root of the transcendental equation

Table 5.1 (p. 301) gives the first root to this equation

If the Fo > 0.2 the infinite series converges such that one term is sufficient

[Eq. 5.43a]





Approximate (one term) Solution for Plane wall – continued

TTTT

TTTtTFoCt

i

o

io

,0exp,0 211

*

The total energy transferred up to any time t is given by:

The non-dimensional centerline temperature (x* =0) is given by:

*

1

1sin1 o

oQQ

[Eq. 5.44]

[Eq. 5.49]





Example 4.2Given: Consider a 304 stainless steel plane wall having the following

properties and given thermal conditions.

5 cm

T Ci 200 , 3mkg7900

T C 70 , s

m10178.42

6

CmW680 2

h , Ckg

kJ515.0

pC

CmW17

k

Find: The temperature at a distance 1.25 cm from faces 1 minute after the plate has been exposed to the convective environment. Also determine how much energy has been removed per unit area from the plate during this time? Rework this problem for an aluminum slab ( = 2700, k = 213, Cp = .9, = 8.765 10-5).


MAE 310 course notes – Fall 2011 Copyrighted by R. D ... · MAE 310 course notes – Fall 2011...

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