Post on 05-Jun-2020
MAE 310 course notes – Fall 2011 Copyrighted by R. D. Gould
Chapter 4 – Page 12
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
MAE 310 course notes – Fall 2011 Copyrighted by R. D. Gould
Chapter 4 – Page 13
HEAT TRANSFER FUNDAMENTALS
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
MAE 310 Muller Lec. 15 - 2
MAE 310 course notes – Fall 2011 Copyrighted by R. D. Gould
Chapter 4 – Page 14
HEAT TRANSFER FUNDAMENTALS
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
MAE 310 Muller Lec. 15 - 2
MAE 310 course notes – Fall 2011 Copyrighted by R. D. Gould
Chapter 4 – Page 15
HEAT TRANSFER FUNDAMENTALS
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
MAE 310 Muller Lec. 15 - 2
MAE 310 course notes – Fall 2011 Copyrighted by R. D. Gould
Chapter 4 – Page 16
HEAT TRANSFER FUNDAMENTALS
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:
MAE 310 Muller Lec. 15 - 2
MAE 310 course notes – Fall 2011 Copyrighted by R. D. Gould
Chapter 4 – Page 17
HEAT TRANSFER FUNDAMENTALS
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]
MAE 310 Muller Lec. 15 - 2
MAE 310 course notes – Fall 2011 Copyrighted by R. D. Gould
Chapter 4 – Page 18
HEAT TRANSFER FUNDAMENTALS
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]
MAE 310 Muller Lec. 15 - 2
MAE 310 course notes – Fall 2011 Copyrighted by R. D. Gould
Chapter 4 – Page 19
HEAT TRANSFER FUNDAMENTALS
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]
MAE 310 Muller Lec. 15 - 2
MAE 310 course notes – Fall 2011 Copyrighted by R. D. Gould
Chapter 4 – Page 20
HEAT TRANSFER FUNDAMENTALS
MAE 310 Muller Lec. 15 - 2
MAE 310 course notes – Fall 2011 Copyrighted by R. D. Gould
Chapter 4 – Page 21
HEAT TRANSFER FUNDAMENTALS
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 Muller Lec. 15 - 2