CBE 150A – Transport Spring Semester 2014 Non-Steady State Conduction.

CBE 150A – Transport Spring Semester 2014

Non-Steady State Conduction


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


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


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


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:


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


Neglect all but first term (for Fo greater than 0.1) and get:

bs

asT TT

TTst

2

28

ln21


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


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


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.


A sphere – heat transfer at boundary function of convective rate

Rr

sf r

TkTTh

A

q

Ts

Tf


Biot number ( Bi) = convection / conduction

k

hsBi

k

hrB mi

Flat plate

Cylinder and sphere


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


Conductive / convective mechanism plot

Figure 10.7

Change with time of the average temperature of a slab with external

convective resistance.


Conductive / convective mechanism plot

Figure 10.8

Change with time of the average temperature of a sphere with external

convective resistance.


Semi-infinite Solid Solid

t

xZ

dZeTT

TT Z

as

s

2

2 2

Ts

T at time t and position x

x

Solid


Semi-infinite Solid

Ts

T at time t

x

Solid

Graphical solution to preceding equation


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)


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 Non-Steady State Conduction.

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Transcript of CBE 150A – Transport Spring Semester 2014 Non-Steady State Conduction.