Improved lumped models for asymmetric cooling of a long slab by heat convection

Int. Comm. Heat Mass Transfer, Vol. 28, No. 7, pp. 973-983, 2001

Copyright © 200l Elsevier Science Led Printed in the USA. All rights reserved 0735-1933/0 I/S-see fronl matter

Pergamon

PIh S0735-1933(01)00301-3

IMPROVED

LUMPED

MODELS

FOR ASYMMETRIC

COOLING

OF A LONG SLAB BY HEAT CONVECTION

Jian Su Nuclear Engineering Department, C O P P E Universidade Federal do Rio de Janeiro CP 68509, Rio de Janeiro, 21945-970, Brazil [email protected]

(Communicated by J.P. Hartnett and W.J. Minkowycz)

ABSTRACT Unsteady cooling of a long slab by asymmetric heat convection is analyzed within the framework of lumped parameter model. As the application of classical lumped model is limited to values of Biot numbers less than 0.1075, we propose improved lumped models that can be applied in transient heat conduction with larger values of Blot numbers. The proposed lumped models are obtained through two point Hermite approximations for integrals. Closed form analytical solutions are given for the lumped models. It is shown by comparison with a reference finite difference solution of the original distributed parameter model that the proposed higher otder lumped model ( H l . 1 / H o , o approximation) yields significant improvement of average temperature prediction over the classical lumped model. © 2001 Elsevier Science Ltd

Introduction

Analysis of asymmetric cooling of a long slab by heat convection is important for metallurgical applications which involve heat treatment of steel strips. This problem has not been studied within the framework of a lumped model until a recent work by Alhama and Campo[1]. The question raised by Alhama and Campo[1] is the following: under what circumstances can the unsteady cooling of a long slab by asymmetric heat convection be treated with a lumped model? Theiz comparative study has led to the following conclusion: the simple lumped model is qualified to handle the general distributed model without incurring temperature errors that exceed 5~: as long as the combination of the Biot numbers are confined to the area circumscribed by a curvilinear rectangle that has its upper right vertex at 0.1075 for both Biot numbers. For combinations of Biot numbers that fall outside the boundaries of the curvilinear rectangle, the simple lumped model fails and the general distributed model must be used.

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J. Su

Vol. 28, No. 7

Limitation of the Biot numbers of asymmetric cooling to less than 0.1075 is inherent of the classical lumped model as that adopted by Alhama and Campo[1] in which moderate to low temperature gradients within the region are assumed.

However, Biot numbers might be much

higher in practical engineering problems. In other words, the moderate to low temperature gradient assumption is not reasonable in such applications, thus more accurate approach shouht be adopted. Cotta and Mikhailov[2] proposed a systematic formalism to provide improved lumped parameter fornmlations for steady and transient heat conduction problems based on Hermite approximation foi integrals that define average temperatures and heat fluxes. Regis et al.[3] have shown that for transient heat conduction in a nuclear fuel rod, good agreement between improved lumped model and original distributed model is obtained for Biot numbers up to 20. Su and Corral4] have presented a higher order lumped parameter formulation for simplified light water reactor (LWR) thermohydraulic analysis.

In this work, we propose improved lumped parameter models for asymmetric cooling of a long siab by heat convection that are applicable for higher Biot numbers than that considered by Alhama and Campo. The proposed lumped models are obtained through two point Hermite approximations for integrals[5, 6}. Close form analytical solutions are obtained for the lumped models. It is shown by comparison with a reference finite difference solution of the original distributed parameter model that the proposed higher order lumped model yields significant improvement of average temperature prediction over the classical lumped model.

Problem Formulation Consider unsteady cooling of a long slab of thickness 2L, initially at a uniform temperature T,. At t - 0, the slab is suddenly exposed to cooling fluids at both sides, with different heat transfer coefficients hi and h2, as illustrated in Fig. 1. The fluids at two sides of the slab may be different. but with same constant temperature To~. It is assumed that thermophysical properties of the solid material and of the fluids are constant. The mathematical formulation of the problem is given by

OT

02T

0~- =~0~TgT, 2'

in

-L
for

t>0,

(la)

-L
at

t=O,

(lb)

with initial and boundary conditions taken as

T(x,O)=~,

in

kOT=h.I(T-T~),

at

z=-L,

for

t>O,

(ic)

Voh 28, No. 7

COOLING OF A LONG SLAB

975

h 2,Too

hi, Too

FIG. 1 Asymmetric cooling of a l o n g slab

k OT = h 2 ( T - T o o ) ,

at

x=L,

for

t >0,

(ld)

where T is the temperature, t the time, x the spatial coordinate, a the thermal diffusivity of the slab, and k the thermal conductivity of the slab. Problem (1) can now be rewritten in dimensionless form as

O0 0~-

020 OX 2'

in

-1
0(X,0) = 1 ,

in

-1
O0 -BilO, OX

00 -Bi20, OX

at

at

< 1,

for

~->0,

(2a)

1,

at

T=0,

(2b)

X =-1,

for

T > 0,

(2c)

z=L,

for

t>0,

(2d)

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J. Su

Vol. 28, No. 7

where the dimensionless parameters are defined by

7' - T~

0 -

z

T~-T~' hlL k '

Bil -

c~t

X=~,

and

Lumped

~=~,

huL Bi2 = ~ .

Models

The spatially averaged dimensionless temperature is defined by

1/_ 1

O,v(~) =

~. -~ o(x, r)gX.

(3)

We operate Eq.(2a) by' (1/2).[!ldX, using the definition of average ternperatme, Eq (31. w<' get

dO~(r)

d~

1 O0

- ~(~lx<-

O0

~lx=-~).

(41)

Now. the boundary conditions Eqs.(2c, 2d) are used, we get

g0=~(~) dT

--

( B i l O ( - - 1 , T) +

Bi20(1, T)).

(5)

Equation (5) is an equivalent integro-differential formulation of the mathematical model, Eq. (2). with no approximation involved. Supposing that the temperature gradient, is suffqciently smooth over the whole spatial solution domain, the classical lumped system analysis (CLSA) is based on the assumption that the boundmy temperatures can be reasonably well approximated by the average temperature, as

0(-1,r)~0a~(r)

and

0(1,r) ~ 0 ~ ( r ) .

which leads to the simple lumped model,

dOav(r)

d~-

-

1

.

~(B,~I + Bi2)0o.(~-),

(6)

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COOLING OF A LONG SLAB

977

to be solved with the initial condition for the average temperature,

Oar(O) = 1.

(7)

The analytical solution of Eqs.(6, 7) is given by

1

0ov(~-) = exp (-~(B~:I + B~:2)7).

(s)

We now seek improved lumped models, in an attempt to offer enhancement to the approximation approach of classical lumped model. The basic idea is to provide a better relation between the boundary temperatures and the average temperature, which is to be developed from Hermite-type approximations for integrals that define the average temperature and the heat flux. We first employ the plain trapezoidal rule in both integrals for average temperature and heat flux (Ho,o/Ho,o approximation), in the form

~(o(-i,7) + 0(1,7)),

0o~(7) ~

_1 •

1

°°(x'7)a= 0(1,7) - 0 ( - 1 , 7 ) ~ A

(9)

00

00

~ (E21x=_l + ~1=1).

(io)

Using boundary conditions Eqs.(2c, 2d), Eq(10) becomes

(11)

0(1,T)--0(--1,T) ~BilO(-1,T)-Bi20(1,T).

Then we further improve the lumped model by employing two-side corrected trapezoidal rule in integral for average temperature, in the form

Oar(m) ~- 0(-1,~-)+

~

1 00

0(1,r)-Fg~-~x=

1 00i

~ -- 6 a N

!X=

,"

while keeping the plain trapezoidal rule in integral for heat flux (H1j/H0,0 approximation). Using boundary conditions Eqs.(2c, 2d), Eq.(12) becomes

(13) In the first improved lumped model (Ho,o/Ho,o), Eqs.(9, 11) form a system of two linear algebraic equations for two unknowns, 0(-1, T) and 0(1, T) that is solved to provide the relations

978

J. Su

Vol. 28, No. 7

between boundary t.emperatLtres and the average t.emperatme. These relations are then used 1o close t.tle ordinary differential equation (,5) for the average temperature, in tee from

dO,~. dr

Bi: + Bia + 2BilBi2,, "z + g~,t t Dz2

~,

to be solved with the initial condition, Eq.(7). The attalytical solution of Eqs.(14, 7) is giver by

0~v(r)=exp(

Bit+Biz+2BilBi9 2+Bi~+Bi2

) r .

(15)

In the second improved lumped model (HL:/Ho.o), E q s . ( l l , 13) form the s3.'stem of two lim.m algebraic equations for 0 ( - 1 , r) and 0(1, r). Similarly, the obtained relations are used to close the ordinary differential equation (5) for the averaged temperatme, in the form

dO=,___2,= dr

3(Bi: + Bi~ + 2BhBi~)iB )#,:,/rL, j 2(3 + 2Bil + 2B42 + Bil ,2

(16)

The analytical solution of Eqs.(16, 7) is given by

(]~(r) = exp (

3(Bi~ + Bi,2 + 2Bi~Bi~) 2(3 + 2Bi: + 2Bi~ + Bi~Bi2) )

1,5i

R e s u l t s and D i s c u s s i o n The analytical solutions of classical and improved lumped models, Eqs.(8, 15,

17),are

shown in

tabular and graphical forms in comparison with a reference finite difference solution of the original partial differential equation (2). Diffferent vMues of the Blot numbers of asymmetric cooling are chosen so as to assess the range of application of the lumped models. In Table 1, it is presented a comparison of the dimensionless average temperature obtaim, d by lumped models and the reference finite difference solution of the original distributed pa:ameter model at different values of time, for three combinations of values of Blot numbers,

bbr the

case of B'i: : 0.1 and Bi2 = 0.2, all lumped models, the classical and iutproved, give average temperatures with less than lV7c error up to r = 1.0, which could b~ accepted as sufficiently accurate approximations for most engineering applications. The proposed higher order lumped model (Hj,~/Ho,o approximation) gives less than 0.1% error up to r = 2.(k As values of the Blot numbers are increased, the improvement offered by the proposed models becomes more evident. For larger values of Bil and B%, the solution of the classical lumped model is in considerable errol

Vol. 28, No. 7

COOLING OF A LONG SLAB

979

TABLE 1 Comparison of lumped models against finite difference solution for average temperature O~v(r) at different values of time r

FD Solution

Bil = 0.1

B i 2 = 0.2

0.01

0.998519

CLAS

Ho,o/Ho,o

Hi. 1/Ho,o

0.998501

0.998523 (0.000%) 0.985326

0.998592 (0.007%) 0.986010 (0.0442,)

(-o.oo2%) 0.i0

0.985573

0,985112

0.931195

(-0.047%) 0.927743

-0.025%)

0.50

0.928752

0.931982

(-0.371%)

-0.262%)

(0.085%)

1.00 2.00

0,867814 0.753855

B~l = 0.1

Bi2 = 1.0

0.01

0.994856

0.10 0.50

0.954913 0.816019

1.00

0.679429

2,00

0.474357

0.860708

0,862581

0,868590

(-0.819%)

(-0.603%)

(00897,)

0.740818 (-1.729%)

0.744046

0.754449

(-1.301%)

(0.079~)

O.994515

0.995815

0.996328

(-0.034%)

(0.096t~)

(0.148c~)

0.946485

0,958932

0.963876

(-0.883%)

(0.421%)

(0.939~)

0.759572

0.810846

0.831967

(-6.917%)

(-0.634%)

(1.954~)

0.576950

0.657471

0.692169

(-15083%) 0.332871

B i 2 = 10.0

0.01

0.967533

0.10

0.822846

0.479099

(-8.873%)

(1.000%)

0.946485

0.976436

0.986802

(-2.175%)

(0.920%)

(1.992~)

0.576950

0.787839

0.875590

(-4.254%) 0.303520 (-38.476%) 0.092124

(6.,Ii0%) 0.514641 (4.318%) 0.264855

-65.730%)

( - 1.474%)

(-29.884%)

0.50

0.493338

0.063928 (-87.042%)

100

0.268818

2.00

0.080205

( 1 875¢~ )

0.432268

(-29.827%) Bi 1 = 1.0

(-3.232%)

0.004087 (-98.480%) 0.000017 (-99,979%)

0.008487

0070148

-89.419%)

(-12.539%)

980

J. Su

Vol. 28, No. 7

os (16

Oav

FD Soh

~ "~

Classical H0,0/H0,0

0002

. . . . 5,. /.q. .

I-f1,1/t-I0,0 ]

0.0

r

71.0

--

0.5

-~ 1.5

r--

2.0

T

FIG. 2 Comparison of dimensionless average temperature for B i ~ = 0.2 and Bi2 = 0.5

10 ...~.

FD Solution

08

\" ~ :%-~/_~..<.. 06

\,

,o, oa4o,o

~

)(-

H,,a400

Say 0.4

0.2

I --

00 00

I ---~

05

I

10

r~__

15

2t

T

FIG. 3 Comparison of dimensionless average temperature for B i l = 0.2 and B i 2 = 2.0.

10

r~,

0 8

i/~'~'~

.

.

.

.

. FD Solutiorl

--

0.4 i

0.0 ~ 00

Classical

'(ii I 0.5

I

'

I

F

I

10 T

15

20

FIG. 4 Comparison of dimensionless average temperature for B i l - 1.0 and B i 2 - 5.0.

V o l . 28, N o . 7

C O O L I N G OF A L O N G S L A B

981

1.0 08

FD Solution Classical

~

H0,0/I-I0,0 HI,l/H0,0

•

06~ Oc~v 04

,..,~

oo ~ -

~)~

-~-~V--

00

-~:-

05

I ....

'

I

1.0 T

~"

15

20

FIG. 5

Comparison of dimensionless average temperature for B~l

2.0 and Bi,e = 20.0.

10 FD Solution 08

(~av

~

- -- --

Classical

i( \,x~x.x

........ -"U

H00fl-100 H l i 1/H0',0

06

0.2 N ,, 00

....

E

00

05

10 T

I5

20

FIG. 6 Comparison of dimensionless average t e m p e r a t u r e for Bia = 5.0 and Bi2 = 20.0.

10 ~,,

(:tar 06

FD Solution

i ~

04

~<

HI,I/H0,0

il

0.2 i~ oo

" ~

k~-~----T 00

05

10 T

15

20

FIG. 7 C o m p a r i s o n o f d i m e n s i o n l e s s a v e r a g e t e m p e r a t u r e for

Bit

= 10.0 a n d

Bi2

= 20.0.

982

J. Su

Vol. 28, No. 7

compared to the reference finite difference solution. The improved lumped models

Ho,o/Ho,o approximations)

(Hl,1/Ho,o and

show a different behaviour, as the curves of average t e m p e r a t m e versus

time cross the curve obtained by the distributed model. The location of the interception point is a function of the Blot numbers. The

Ho,o/Ho,o formulation

is, as expected, globally less accurate

than the Hx.~/H0,0 formulation, but with better agreement for lower values of time. As shown in Figs.2 to 7, the higher order lumped model

(Hl.1/Ho,o approximation)

presents

good agreement with the reference finite difference soltuion for values of Blot number a~ high a~20.0, and the classical lumped model already deviates from the reference solution at By2 - 20.

Conclusion We present improved lumped models

(Hl,l/Ho,o

and

Hl,:/Ho,o

approximations) for unsteady

cooling of a long slab by asymmetric heat convection on the two sides. Based on comparison with a reference finite difference solution of the original distributed parameter model, we conclude that significant improvement has been achieved by the proposed higher order lumped model

(Hid/Ho,o)

over the classical lumped model. The exact range of application of the improved lumped models should be determined, however, by more elaborated comparative study as that was carried out by Alhama and Campo[1].

Acknowledgements The author acknowledges the Brazilian National Science Council (CNPq) by the financial support through grants No 523565/96-8 and No 462585/00-0 ( C N P q / C T P E T R O ) .

Nomenclature

Bil Bi2

Blot number relative of heat transfer at left surface

hi

convective heat transfer coefficient at left surface

h2

convective heat transfer coefficient at right surface

Blot number relative of heat transfer at right surface

k

thermal conductivity of the slab

L

half-thickness of the slab

t

time

T

temperature

T,

initial temperature of the slab

T~

temperature of fluid

:r

coordinate

X

dimensionless coordinate

Vol. 28, No. 7

COOLING OF A LONG SLAB

983

Greek symbols c~

thermal diffusivity

0

dimensionless temperature

p

density

r

dimensionless time

Subscmpts 0

initial

av

average

1

left surface

2

right surface

References 1. F. Alhama and A. Campo, The connection between the distributed and lumped models for asymmetric cooling of long slabs by heat convection, Int. Comm. Heat Mass Transfer. 28, 127 137, 2001. 2. R.M. Cotta and M.D. Mikhailov, Heat Conduction

Lumped Analysis, hltegral Tiansforms, Symbolic Computation, John Wiley & Sons, Chichester, 1997.

3. C.R. Regis, R.M. Cotta, and J. Su, Improved lumped analysis of transient heat conduction in a mtclear fuel rod, Int. Comm. Heat Mass Transfer, 27, 357 366, 2000. 4. J. Su and R.M. Cotta, Improved lumped parameter formulation for simplified LWR thermohydraulic analysis, Annals of Nuclear Energy, 28, 1019-1031, 2001. 5. J. Mennig, T. Auerbach, and W. Hi~lg, Two point Hermite approximations for the solution of linear initial value and boundary value problems, Comput. Methods. AppI. Mech. Engng., 39, 199 224, 198 3. 6. E.J. Corr6a and R.M. Cotta, Enhanced lumped-differential fornmlations of diffusion p~oblems,

AppI. Math. Modelling, 22, 137 152, 1998. Received July 2, 2001