chapter 4 第四章 unsteady-state (transient) conduction 非稳态导热

Chapter 4 第四章

Unsteady-State (Transient) Conduction

非稳态导热

4-1 Introduction

1. Basic conception

Unsteady-state (or transient) conduction

Temperature distribution in a system varies with time

Steady-state conduction

Temperature distribution in a system does not varies with time

2. Example

The shell of steam turbine

Before start tf1=tw1=tw2=tf2

Admission of steam tf1

Inside of the shell

q1=h1(tf1-tw1)

At a particular time

h1A1(tf1-tw1)=h2A2(tw2-tf2)

Steady-state conduction

penetration time 穿透时间：Thermal layer 穿透深度

3. Problems to be solved

1). Temperature distribution at a given time

thermal stress ( 热应力 )

2). The time to reach a given temperature distribution

or steady-state

quenching ( 淬火过程 )

3). The heat transferred

3. Infinite plate subjected to sudden cooling of surfaces

Lxx i 20,0at 0, 0,0at 0 x

t

ax

t 12

2

Set1tti

ax

12

2

0,2at 0 Lx

xXx,

2

02 aCead

d

xCxCXdx

Xd sincos0 212

2

aexCxC2

)sincos( 21

00),0( 11

2

CeC a

02sin02sin),2(2

2 LLeCL a

,2,12

nL

n

1

]2/[

2sin),(

2

n

aLnn L

xneCx

inn

ni nC

L

xnCx

4

2sin)0,(

1

1

]2/[

2sin

14 2

n

aLn

i L

xne

n

Graphical form for calculation

t

ax

t 12

2

itxt 0,

0,00

xx

tx

ftthx

tLx

,

ftt Set

Math model

Third boundary condition no heat generation,

h=const. Tf =const.

2

2

xa

00, x

00

xx

hx x

Or

xCxCeCx sincos, 321

2

xBxAex sincos,2

From the first boundary condition

0,0

x

x 0cossin

2

xBxAe

x

Separating variables gives general solution

0 02

B B e xAex cos),(

2

Lx LAheLAe cossin22

LhL cossin

From the second boundary condition

L

Bi

L

hLhL

L

L

tancos

sin

h

LhLBi

/1

/

There are infinite number solutions to the equationThe final series form of the solution is

xeAx nn

nn cos,2

1

From initial condition

1

0 cos0,n

nn xAx

nnn

ni

nnn

nin LLL

LA

cossin

sin2

cossin

sin2

Lnn

nnn

nn

n

L

i

Lxe

x n

cossin

/cossin2

,

1

22

BiBinn tan

x

BiFofx

,,,

0

FoL

a 2

4-2 lumped-heat-capacity system

Generally t=f(x,y,z, )

• If internal conduction resistance 0 , uniform in temperature

• Temperature field t=f(x,y,z, ) reduces to t=f(), 0-D problem

• In fact, impossible for internal conduction resistance to be zero,

but if it is small enough , we believe t=f().

• This method is called lumped method or heat conduction with

negligible internal conduction resistance

( 集总参数法或忽略物体内部导热热阻的简化分析方法 )

1. Physical problem

k=const. Uniform T distribution Bi 0 , h=const. Find T=f()

2. Mathematical model

cz

t

y

t

x

ta

t

2

2

2

2

2

2

cd

dt

tthAq

ttV

hA

V

No boundary condition. Convection heat transfer taken as source item

Then

0at 0

tt

tthAd

dtVc

Vc

tthA

d

dt

tt

0at 0

Vc

hA

d

d

3. Solve

0 0d

cV

hAd

00ln

cV

hA

cV

hA

0

ln

cV

hA

e

0

Then

cVChA

RCRcV

hAthth

thth

11

BiFocV

hA

ee

00

BiFoAV

a

A

Vh

cV

A

A

hV

cV

hA

22

2

)/(

Characteristic dimension of solid (V/A=s)

)/( AVh

Bi 2/ AV

aFo

4. Heat transfer rate

fehAehAhAtthA cVhABiFo 00

Heat transferred from 0 to

000 00 cV

hAde

hA

cVhAdehAqd cV

hA

cV

hA

cV

hA

cVhAcV

hA

ecVecVecV

11 00

0

0

5. Biot number

resistance convection

resistance conduction internal

/1

/

h

LhLBi

•Bi L/λ has decisive effect, convection waits for conduction

tw t , become first kind of boundary condition

•Bi 0 L/λ is very little ， temperature tends towards uniform

•Bi = some certain value ， L/λ and 1/h play an important role

6. Applicability of lumped-capacity Analysis

1.0)/(

AVh

Bi


) thicknesshalf ( Plate LLA

AL

A

Vs

(radius) Cylinder rs

(radius) Sphere rs

•Plane wall BiV= Bi Cylinder BiV= Bi/2 Sphere BiV= Bi/3

•M= 1 (for plane wall), ½ (for cylinder) and 1/3 (for sohere)•Reason M= BiV /Bi

MAVh

BiV 1.0)/(

) thicknesshalf ( wallPlane LLA

AL

A

V

22Cylinder

2 R

Rl

lR

A

V

34

3/4Sphere

2

3 R

R

R

A

V


In our country

7. Time constantceeΘ cV

hA

0

constant timeis hA

cVc

36.8%

0

5%

cc3

The rate of temperature change

tan1

0

ced

d

c

2/,tan,0 c

1,0,0tan, Θc 02/,0 c c rate

4-3 Transient heat flow in a semi-infinite solid

1. Physical problem

i

i

ttx

ttx

tt

t

ax

t

,

,0

,0

1

0

2

2

i

i

x

x

ax

,

0,0

,0

12

2

Introduction of a new variable

mCx


0tt

ditermined be to & constants where mC

d

dm

d

dCxm m

1

d

dC

xxm

2

222

2

2

d

dCd

dCd

d

xxxxmm

Substitute into the eq.

2

2122

d

dac

d

dm m

To eliminate , set 2m +1=0, that is m = -1/2

Then 02

122

2

d

d

aCd

d

a

xCx

aC m

4 have we

41set

022

2

d

d

d

d

12ln2 C

d

dd

dddd

d

mCx

20 11

22

, CdeCeCd

d

0,0,0,0 2 Cx From initial condition

0,,0

0

11010

2

2

2

CCdeC

Then

a

xerfde

ax

4

2 4/

00

2

functionerror is 4 a

xerf

2 known when is it fig.2 From

%5error ,9953.0 ,00

TT

3. Two important parameters:

0,4 ax

xa

xo at ,,

16

2

4. Heat flow

erfx

ttAx

tA ix

0

aettA a

x

i2

12 20

a

ttA i )( 00

da

ttAd i

0

0

0 0

)( ittcA 02

qckq ,

5. Constant heat flux on semi-infinite solid

Governing eq. and initial condition are the sameBoundary condition

0for 0

0

xx

t

A

a

xerf

A

x

a

x

A

att i

21

4exp

/2 02

0

6. Energy pulse at surface

Instantaneous pulse of energy Q0/A

a

x

acAtt i 4

exp2

0

as allfor 0 xtt i

4-4 Convection boundary conditions

1.Convection boundary For the semi-infinite-solid problem, the boundary condition

0for )(0

0

x

x x

ttthA

The solution

ah

Xerfahhx

erfXtt

tt

i

i 1exp12

2

uret temperatenvironmen

solid of re temperatuinitial

2/

t

t

axX

i

2. Important case

1-D solids suddenly subjected to convection environment at T∞

Infinite plate

nnn

nn

n

L

i

Lxe

x n

cossin

/cossin2

,

1

22

x

BiFof ,,

0

,,r

rBiFof

i

0hr

Bi 20r

aFo

Infinite cylinder and sphere

• the nature of series: the first term is the largest, then the 2nd, 3rd …

when Fo is large enough the decrease is rapid.

• when Fo>0.2, the the error of the single-term approximation of the

series is less than <0.1% 。

111

11

cossin

cossin2

, 221

L

x

ex L

i

The ratio of (x,) to 0 is

L

xxx1

0

cos),0(

,,

It depends on the position and boundary condition, is independent of .

Initial condition has no effect on temperature distribution

Take infinite plate as an example

This region is called regular regime ( 正规热状况 )

or fully developed regime （充分发展阶段 )

0

0,

ii

x Fig. 4-13

ii cVTTcVQ 0

Fig. 4-7

Chart solution

3. Boit number and Fourier number

/hsBi

22 css

aFo

4. Applicability of the Heisler Charts

2.0Fo

4-5 Multidimensional systems

1. Physical problem: Infinite rectangular bar


cz

t

y

t

x

ta

t

2

2

2

2

2

2

TThy

tLy

y

ty

tthx

tLx

x

tx

tt

y

T

x

Ta

T

i

,

0,0

,

0,0

,0

2

1

2

2

2

2

Set dimensionless temperature difference

tt

ttΘ

ii

Substitute into eq.

0

,,,,

0,,

0,,

0,,

,,

10,,

2

1

2

0

0

1

2

2

2

2

Ly

y

x

Lx

y

yx

hLx

y

yx

x

yx

x

yx

hyL

yx

yxa

t

If equation

0

,,

0),(

10,

1

111

0

1

1

12

1

Lx

x

x

xt

hLt

x

xt

xtx

ta

t

Has a solution T1(x,), and

0

,,

0,

10,

,,

2

222

0

2

2

22

22

Ly

y

y

xt

hLt

y

yt

yt

y

yta

yt

has a solution of T2(y,)

,,then 21 ytxt

is the solution to the equation

Prove：

performing partial

differentiation of

1

22

1

, tt

tt

yx

x

tt

x

1

2x

tt

x

12

22

2

In a similar way

22

2

12

2

y

tt

y

3. Solution

000 2121

21

222

22

122

2

121

2

21

22

1

ttx

ta

tt

y

ta

tt

y

tt

x

tta

tt

tt

(x,y,)= t 1(x, ) t2 (y, ) satisfies the governing equation, and

1110,0,,, 21 ytxtyx also satisfies the initial condition.

00,,

,,,

20

12

0

yt

x

xtyt

x

yx

xx

11

,,,,

,,,, 1

22111LxLx x

xtyt

hytLt

x

yx

hyL

00,,

,, 21

112

1

ytx

xt

hLtyt

Lx

By the same way, it can be proved that (x,y,)= t 1(x, ) t2 (y, ) satisfy other two boundary conditions.

Substituting these relations in the governing equation

By the same way

* Constant 1st and 3rd kinds of boundary conditions

3 solidonintersecti

2 solidonintersecti

1 solidonintersecti

solidcombined

iiii

4. Heat transfer in multidimensional systems

201030102010total0

111Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

4-6 Transient numerical method

•

Space coordinate x 1N Space increment x

1. Mathematical models

2. Discretization of domain (node, grid)

3. Algebraic equations for all the nodes

4. Initial variable fields

5. Solving the algebraic equation

6. Analysis of the solutionTime coordinate 1I Time increment( 时间步长 )

T(p)n is the temperature of node (n,p)

0t

• Procedure is the same as steady-state

1-D, λ=const. problem

2

2

x

ta

t

2

)(1

)()(1

)()1( 2

x

ttta

tt pn

pn

pn

pn

pn

2

)1(1

)1()1(1

)()1( 2

x

ttta

tt pn

pn

pn

pn

pn

Implicit finite difference scheme ( 隐示格式 )

)(2

)(1

)(12

)1( 21 p

np

np

np

n tx

att

x

at

Governing equation

Forward difference

backward difference

Explicit finite difference scheme ( 显示格式 )

)1(2

)1(1

)1(12

)( 21

pn

pn

pn

pn t

x

att

x

at

)(2

)(1

)(12

)1( 21 p

np

np

np

n tx

att

x

at

Explicit formulation

If the time and space increment are chosen so that

22

a

xM

)(1

)(1

)1(

2

1 pn

pn

pn TTT

• If x△ and △ the rate that the solution proceeds the accuracy

• △ depends x△ , If M>2 the coefficient becomes negative,

the condition violates the 2nd law of thermodynamics.

• Stability problem, M<2 stable, M>2 instable

Boundary

)(,

)1(,)(

,

)(,

)(1,

)(,

)(1,

)(,

)(,1

2)(

22pnm

pnmp

nm

pnm

pnm

pnm

pnm

pnm

pnm

tty

xcttyh

y

ttx

y

ttx

x

tty

)(

,

2)(

1,)(

1,)(

,12)1(

, 4222 pnm

pnm

pnm

pnm

pnm t

xh

a

xtttt

xh

x

at

For uniform grid x= y

)(2

)(12

)1( 2222 pm

pm

pm T

k

xh

a

xTT

k

xh

x

aT

For 1-D problem

case D-2 for the )2/(2


kxh

kxh

a

x



kxh

kxh

a

x

Zero coefficient

To insure convergence

Set 2x

aFo

Fourier number in the numerical format

TBiFoTFoFoBiFoTT p

Mp

Mp

M 22221 )(1

)()1(

)()(1

)(1

)1( 21 pm

pm

pm

pm TFotTFoT

Infinite plate

Convection boundary node

Interior node

k

xhBi

Biot number in the numerical format

4-6 Thermal resistance and capacity formulation

Self-learning

chapter 4 第四章 unsteady-state (transient) conduction 非稳态导热

Documents