CBE 150A – Transport Spring Semester 2014
Non-Steady State Conduction
CBE 150A – Transport Spring Semester 2014
Goals:
By the end of today’s lecture, you should be able to: define the mechanisms for non-steady state conduction determine the time required to transfer heat to and from:
flat plates cylinders spheres
describe the difference between constant surface
combined convection and conduction.
Non-steady State Conduction
CBE 150A – Transport Spring Semester 2014
Outline:
I. Conduction for a constant boundary surface temperature• Flat plate• Cylinder• Sphere
II. Conduction for a rate based boundary temperature• Flat plate• Cylinder• Sphere
CBE 150A – Transport Spring Semester 2014
Heat Flow Heat Flow
Ts Ts
dx
Infinitely long solid slab (no end effects)(constant surface temperature)
dtt
TAdxcdxdt
x
TkA p
2
2
Heat Balance
2s
CBE 150A – Transport Spring Semester 2014
2
22
x
T
x
T
c
k
t
T
p
Where: = thermal diffusivity = k/cp
Divide by cp A dx dt to yield:
00)3
)2
0)1
xatx
T
tallforsxatTT
xallfortatTT
s
a
Boundary Conditions:
CBE 150A – Transport Spring Semester 2014
Integrated Solution:
.......
25
1
9
18111 259
2ooo FaFaFa
as
bs eeeTT
TT
Where: Ts = constant average temperature of surfaceTa = initial temperature of slabTb = average temperature of the slab at time tFo = Fourier number = tT/s2
= thermal diffusivity = k/cp
tT time for heatings = one-half slab thicknessa1 = (/2)2
CBE 150A – Transport Spring Semester 2014
Neglect all but first term (for Fo greater than 0.1) and get:
bs
asT TT
TTst
2
28
ln21
CBE 150A – Transport Spring Semester 2014
For infinitely long (no end effects) cylinder:
ooo FFF
as
bs eeeTT
TT 9.745.3078.5 0534.0131.0692.0
bs
asmT TT
TTrt
692.0ln
78.5
2
Where: Fo = tT / rm2
CBE 150A – Transport Spring Semester 2014
For a sphere:
ooo FFF
as
bs eeeTT
TT 8.885.3987.9 0676.0152.0608.0
bs
asmT TT
TTrt
608.0ln
87.9
2
Where: Fo = tT / rm2
CBE 150A – Transport Spring Semester 2014
Constant surface temperature plot
Figure 10.5
Average temperatures during unsteady-state heating or cooling of a large slab, and infinitely long cylinder, or a sphere.
CBE 150A – Transport Spring Semester 2014
A sphere – heat transfer at boundary function of convective rate
Rr
sf r
TkTTh
A
q
Ts
Tf
CBE 150A – Transport Spring Semester 2014
Biot number ( Bi) = convection / conduction
k
hsBi
k
hrB mi
Flat plate
Cylinder and sphere
CBE 150A – Transport Spring Semester 2014
For a sphere at low Biot number:
Assuming an effective internal coefficient and an overall heat-transfer coefficient
bfmb
mp TTrUdt
Tdrc
23 4
3
4
coefficentconductionconvectioncoupledEmpiricalk
r
hUm /
5
11
mpbf
b
rc
dtU
TT
Td
3
mpaf
bf
rc
tU
TT
TT
3
ln
CBE 150A – Transport Spring Semester 2014
Conductive / convective mechanism plot
Figure 10.7
Change with time of the average temperature of a slab with external
convective resistance.
CBE 150A – Transport Spring Semester 2014
Conductive / convective mechanism plot
Figure 10.8
Change with time of the average temperature of a sphere with external
convective resistance.
CBE 150A – Transport Spring Semester 2014
Semi-infinite Solid Solid
t
xZ
dZeTT
TT Z
as
s
2
2 2
Ts
T at time t and position x
x
Solid
CBE 150A – Transport Spring Semester 2014
Semi-infinite Solid
Ts
T at time t
x
Solid
Graphical solution to preceding equation
CBE 150A – Transport Spring Semester 2014
Problem Solution Matrix
Problem StatementSteady State
CalculateU,T,Q,A
Non-Steady State
Constant Surface (Ts)or
Convective Film (Tf)TAvg or T Position
ResourcesFig. 11.1.2
Fig. 10.5ResourcesFig. 11.1.3
Ts
Tf
CalculateUo or ho
Resources Fig. (b-g)
TAvg
Fig. 10.7Fig. 10.8
Eqn. 10.32
T Position
TAvg
T Position
TAvg or T Position
Geometry(sphere, slab, cylinder)
CBE 150A – Transport Spring Semester 2014
Ten Minute Problem - The Thanksgiving Turducken
I am cooking a 20 lb turducken (a turkey - stuffed with a duck - stuffed with a chicken – stuffed with stuffing – see photo below) for Thanksgiving dinner. How long will it take to cook ???
Initial temperature (T) of turducken on my kitchen counter = 70 FT oven = 350 FT of stuffing for a "done" turducken = 165 FExternal heat transfer coefficient for my Magic Chef natural circulating oven = 0.40 BTU / hr ft2 F
Assume the turducken is a fat thing that approximates a spherical geometry.Volume = 4/3 r3
Surface area = 4 r2
Effective density of turducken = 65 lb/ ft3
Effective heat capacity of turducken = 0.83 BTU / lb FThermal conductivity of turducken = 0.35 BTU / ft hr F
CBE 150A – Transport Spring Semester 2014
CBE 150A – Transport Spring Semester 2014
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