Chapter 4 Unsteady state conduction.pdf

Sheu Long Jye Dep. of Mechanical Engineering

Chung Hua University [email protected]

Chapter 4 Unsteady state conduction (Text: J. P. Holman, Heat Transfer, 8th ed., McGraw Hill, NY)

Consider 1-D unsteady state conduction problem in a slab (Fig. 4-1)

2

2

0

1 , 0 , 0

0 , 0

0 0, 0

( ) 0 , 0

T q T x L tx k t

T T in x L tT at x txTk h T T at x L tx

α

∞

∂ ∂+ = ≤ ≤ >

∂ ∂= ≤ ≤ =

∂= = >

∂∂

+ − = = >∂

There are totally 7 parameters: 0, , , , , ,k q L T h and Tα ∞ Nondimensinoalization,

02

02 2

2 2 2

20 0

2

2

2

20

,

1

( ) ( )

.( )

T TxletL T T

T T q TL k t

qL L Tk T T T T t

G

t qLwhere Fourier No G dimensionaless heat sourceL k T T

ξ θ

θξ α

θξ α

θ θξ τ

ατ

∞

∞

∞

∞ ∞

∞

−= =

−

− ∂ ∂+ =

∂ ∂

∂ ∂+ =

∂ − − ∂

∂ ∂+ =

∂ ∂

= = = =−

With initial and boundary conditions

1 0 1, 0

0 0, 0

0 0, 0.

.

in

at

Bi at

hLwhere Bi Biot Nok

θ ξ τθ ξ τξθ θ ξ τξ

= ≤ ≤ =∂

= = >∂∂

+ = = >∂

= =

There are only 2 paramters: G and Bi. The physical meanings of Fo and Bi:

2 3

2 3 3

( / ) ./

t k L L rate of heat conduction across L in L FoL cL t rate of heat storage in Lα

ρ= = =

The larger the Fourier No. is, the deeper is the penetration of heat inot a solid over a given period of time.



/1/

/

hL L kA conduction resistanceBik hA convection resistanceh heat transfer coefficient at surface of a solid

k L internal conductance of solid across L

= = =

= =

Biot number is the ratio of the heat transfer coefficient to unit conductance of a solid over the characteristic length. For solids in the shape of a slab, long cylinder, or a sphere with no internal heat generation, 1-D, transient temperature distribution within solid may be considered uniform if

0.1 5%,ss

hL VBi with error Lk A

= < < =

If we analyze systems which are considered uniform in temperature at any instant during the transient conduction, this type of analysis is called lumped heat capacity system. Consider a hot ball immersed in a cool run of water and assume that lumped heat capacity method might be used. The convection heat loss form the body is evidenced as a decrease in internal energy of the body (Fig. 3-2)

0

0

( ) , (0)

, .hA ttcV

dThA T T cV T Tdt

T T cVe e time constantT T hA

ρ τ

ρ

ρτ

∞

− −∞

∞

− = − =

−= = = =

−

When the internal resistance of the body is significient, the temperature may not be assumed uniform and heat equation for 1-D slab problem is

2

2

1T Tx tα

∂ ∂=

∂ ∂

Consider the heat equation

2

2

0

1 , 0 , 0

( ,0) , 0 , 0(0, ) , 0

i

T T x tx t

subject to IC and BCsT x T x tT t T t

α∂ ∂

= < < ∞ >∂ ∂

= < < ∞ =

= >

Solving by Laplace transform defined as



0

[ ] ( ) , [ '] (0)stL f f t e dt L f sf f∞ −= = −∫ .

The Laplace transform of the governing equations gives

2

2

0

/0

1 10

0

(0, )

, /

[ ] ( ) , [ ]2 2

2( ) 1 ( ) 1 , ( 1)

qx qxi

s xi i

s x

i i

x

dT s TTdx

TT ss

TT Ae Be q ss

The BCs givesT T TT es s

x e xT L T T T T erfc L erfcst t

where erfc x erf x e d Table A

α

α

η

α α

α

α α

ηπ

−

−

−

− −

−

− = −

=

= + + =

−= +

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟= = + − =⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎜ ⎟

⎝ ⎠

= − = − −∫

∵

2

0

0

0

0 4

( ) 1 ( )2 2

( )2

"

i

i

i

xi t

T T x xerfc erfT T t tT T xerfT T t

T qq kx A

T Tk et

α

α α

α

πα

−

−= = −

−−

=−

∂= − =

∂

−=

If heat flux boundary condition is applied,



2

2

"0

0

/

"0

0

1 1/ 2 4

" "1 1/ 20 04

[ ] 2( )2

2[ ] ( )2

x

s xi

x

s x xt

xt

i

Tk qxTT Aes

qTkx s

e t xL e xerfcs ts

q q xt xT L T T e erfck k t

α

αα

α

απ α

α

απ α

=

−

=

−−−

−−

∂− =

∂

= +

∂− =

∂

⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟= − ⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠

⎛ ⎞= = + − ⎜ ⎟⎝ ⎠

∵

If convective boundary condition is applied, the solution is

2

00

00

1

( )

( )( ) ,( ) ( / )

1 1[ ]( ) 2 2

2

xx

qxi

xx

iqxi i

qxx t

h xi k

i

Tk h T TxTT Aes

hTTk hTx s

h T Th T T T kA T es kq h s s q h k

e x xL erfc e erfc ts q t t

T T xerfc eT T t

β αβ β αβ β βα α

α

∞ ==

−

∞

==

∞−∞

−− − +

− +

∞

∂− = −

∂

= +

∂− = −

∂

−−= = +

+ +

⎛ ⎞⎛ ⎞ ⎛ ⎞= − +⎜ ⎟⎜ ⎟ ⎜ ⎟+ ⎝ ⎠ ⎝ ⎠⎝ ⎠

− ⎛ ⎞= −⎜ ⎟− ⎝ ⎠

∵

2

2

2

,2

h tk

i

i

x h terfckt

T T x h tfT T kt

α αα

αα∞

⎛ ⎞+⎜ ⎟⎜ ⎟

⎝ ⎠⎛ ⎞−

= ⎜ ⎟⎜ ⎟− ⎝ ⎠

Laplace transform on infinite slab.



2

2

0

1 , , 0

( ,0) ,

0

( )

i

x

T T L x L tx t

subject to IC and BCsT x T L x L

Tx

Tk h T T at x Lx

α

=

∞

∂ ∂= − < < >

∂ ∂

= − < <

∂=

∂∂

− = − =∂

2

2

0

1 , , 0

( ,0) ,

0

i

x

let T T

L x L tx t

subject to IC and BCsx L x L

x

k h at x Lx

θ

θ θα

θ θθ

θ θ

∞

=

= −

∂ ∂= − < < >

∂ ∂

= − < <

∂=

∂∂

− = =∂

By Laplace transform

2

2

22

2

( )

cosh sinh ,

cosh( sinh cosh )

1 .....2 2 2

i

i

i i

h h tL xk k

i

qx s

sA qx B qx qs

h qxs s kq qL h qL

L x L x h L xerfc erfc e erfc tkt t t

α

θθ θ

θθα

θ θθ

θ θ αα α α

− +

∂− = −

∂

= + + =

= −+

⎧ ⎫⎡ ⎤− + −⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎪ ⎪= − + + + +⎨ ⎬⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦⎪ ⎪⎩ ⎭

The solutions of 1-D transient heat conduction in infinite slab, cylinder, and sphere are summarized and figured by Heilser chart. (Fig. 4-5 to 4-16) Multidimensional system Consider an infinite rectangular bar (Fig.)



2 2

1 1 2 22 2

12

1 11 12

22

2 22 22

1 2

1

1 , , , 0 (*)

( , )

1

( , )

1

(*)

L x L L z L tx z t

If x t satisfies

in L x Lx t

and x t satisfies

in L z Lz t

Then satisfiesif at the surface the medium subjects to

k h at xx

θ θ θα

θ

θ θα

θ

θ θα

θ θ θ

θ θ

∂ ∂ ∂+ = − < < − < < >

∂ ∂ ∂

∂ ∂= − < <

∂ ∂

∂ ∂= − < <

∂ ∂=

∂− =

∂ 1

2 2

,L and

k h at z Lzθ θ

=

∂− = =

∂

θ can be obtained as

1 22

1 1 11 1 12

22 2 2

2 2 22

,

1 ,

1 ,

where

k h at x Lx t x

k h at z Lz t z

θ θ θ

θ θ θ θα

θ θ θ θα

=

∂ ∂ ∂= − = =

∂ ∂ ∂∂ ∂ ∂

= − = =∂ ∂ ∂

Heat transfer in multidimensional systems for 2-D system:

0 0 0 01 2 1

1total

q q q qq q q q

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + −⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

For 3-D

0 0 0 0 0 0 01 2 1 3 1 2

1 1 1total

q q q q q q qq q q q q q q

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + − + − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Examples: Transient Finite Difference Method Consider a 2-D body within which the heat flow is governed by



2 2

2 2

21, 1, ,

2 2

2, 1 , 1 ,

2 2

1, ,

( )

2,

( )2

,( )

m n m n m n

m n m n m n

p pm n m n

T T Tk cx y t

T T TTx x

T T TTy y

T TTt t

ρ

+ −

+ −

+

∂ ∂ ∂+ =

∂ ∂ ∂+ −∂

≈∂ Δ

+ −∂≈

∂ Δ

−∂≈

∂ Δ

If the nodal temperature derivatives are evaluated at time p:

11, 1, , , 1 , 1 , , ,

2 2

1, 1, 1, , 1 , 1 ,

2

2 2 1( ) ( )

( ) (1 4 )

( )

p p p p p p p pm n m n m n m n m n m n m n m n

p p p p p pm n m n m n m n m n m n

T T T T T T T Tx y t

if x yT Fo T T T T Fo T

twhere Fox

α

α

++ − + −

++ − + −

+ − + − −+ =

Δ Δ ΔΔ = Δ

= + + + + −

Δ=

Δ

If the system is 1-D in x (Fig.)

11 1( ) (1 2 )p p p p

m m m mT Fo T T Fo T++ −= + + −

The above expression are called explicit formulations Stability criterion: For 1-D system: 1/ 2Fo ≤ . For 2-D system: 1/ 4Fo ≤ . Example: (Fig.) Δx and Δt are such that

1

1.121.1 1.1(2 2) (1 2 ) 2.1 2.2 0.21 2( 2 )2 2

p ndm

Fo

T break Law+

=

= + + − = − <i i

The boundary nodes: Consider the transient energy balance at node (m,n) on a boundary surface, (Fig)



11, , , 1 , , 1 , , ,

,

1, 1, , 1 , 1 ,

2

( )2 2 2

1{2 2 [ 2 4] }

,( )

p p p p p p p pm n m n m n m n m n m n m n m np

m n

p p p p pm n m n m n m n m n

T T T T T T T Tx x xk y k k h y T T c yx y y t

if x y

T Fo BiT T T T Bi TFo

t h xFo Bix k

ρ

α

+− + −

∞

+∞ − − +

− − − −Δ Δ ΔΔ + + + Δ − = Δ

Δ Δ Δ ΔΔ = Δ

= + + + + − −

Δ Δ= =

Δ

Corresponding 1-D relation is

11

1{2 2 [ 2 2] }p p pm m mT Fo BiT T Bi T

Fo+

∞ −= + + − −

Stability criterion:

For 1-D system: 1 ,1 2 2 02

Fo Fo BiFo≤ − − ≥ .

For 2-D system: 1 ,1 4 2 04

Fo Fo BiFo≤ − − ≥ .

Implicit formulations:

1 1 1 1 1 1 11, 1, , , 1 , 1 , , ,

2 2

1 1 1 1 1, 1, 1, , 1 , 1 ,

1

2 2 1( ) ( )

(1 4 ) ( )

p p p p p p p pm n m n m n m n m n m n m n m n

p p p p p pm n m n m n m n m n m n

T T T T T T T Tx y t

if x yFo T Fo T T T T T

linear equations systemAX B X A B

α

+ + + + + + ++ − + −

+ + + + ++ − + −

−

+ − + − −+ =

Δ Δ ΔΔ = Δ

− − + + + =

= ⎯⎯→ =

Chapter 4 Unsteady state conduction.pdf

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