The connection between the distributed and lumped models for asymmetric cooling of long slabs by heat convection

Int. Comm. Heat Mass Transfez Vol. 28, No. 1, pp. 127-137. 2001 Copyright 0 2001 Elsevier Science Ltd Printed in the USA. All rights reserved 073%1933/01/$-see front matter

Peqamon

PII:SO735-1933(01)00220-2

THE CONNECTION BETWEEN THE DISTBIBUTED AND LUMPED MODELS FOR ASYMMETBIC COOLING OF LONG SLAB!3 BY HEAT CONVECTION

FmneiscoAlhama

Dpto. de Fisica Aplicada E.T.S. hgehems Induhales UniversidadPolithica de Chtqeaa 30203 Cartageaq Murcia SPAIN Alltotliocampo of Eqheering Idaho State University Pocatello, ID 83209

cow

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

ABSTRACT The unste&_xolii

ofa long slabby symmhc heatconvectionis controlkdby a single Biot numberBi=hLlkappiiedatbathsurhces. Itiskntnvnthatthethermalresponseofthelongslab maybeanaylpAwithalumpedmodtlwithasmallerrorwbeneverBi
Introduction

Three possibilities may be conteanplatedfor the cooling of a long slab by heat convection in metallurgicalapplications involving heat treatiagofsteel strips. They are: (a) symmetriccooling to the same fluid, (b) asymmehc coolii to the same fluid or (c) asymmetriccooling to two differentfluids. In case (a), the heat exchange occurs by way of an interplayof two thermalresistances,oae by heatanhction inside the slab, and the other by heat convection between the slab surfaces and the fluid [ 1,2]). Then, if the internal 127

Vol. 28, No. 1

F. Alhama and A. Camp0

128

resistance is negligible when compared to the sur&ce-fluid resistance, the Biot number Bi = hL/k becomes very small. The spatial variations of temperature inside the slab are virtually imperceptible.

lke5Wfmentsare

equivalent to writing if

Bi a O(1)

-

T(t) = T,(t)

(1)

where T(t) and TL(t) designate the mean temperatures computed f&n the distributed and the lumped models, respectively.

Therefore, this ensuing criterion may be applied to long slabs which comply with the following

conditions: (1) are very thin, (2) are constructed 6om materials with large thermal conductivities, or (3) are exposed to fluids with weak mean convective coefiicients.

Conversely, when the cooling of a long slab occurs in an asymmetric fashion as in cases (b) and (c), two Biot numbers Bi, = L&/k (for the left surface) and Bi2 = LhJk (for the right surlhce) manage the bilateral heatexchangebetweentheslabandthefluid(s).

Underthesep remises, there is an interplay of three resistances,

one by heat conduction inside the slab, and the other two by heat convection behveen the exposed slab surfhces and the fluid(s). A literature search has not revealed any work so far’bn the topic of asymmetric cooling of slabs within the fkunework of a lumped model. Certainly, it may be of interest to metallurgical engineers to f%xlthe relative magnitudes of Bi, and Bil that justify the utilimtion of a quick lumped model for asymmetric cooling of a slab by heat convection.

This short paper seeks to answer this question.

Consider the unsteady cooling of a long&b

ofthkk~~

2L having a

UIIZOITIIinitial tanperatu~,

Ti,

throughout. Fromt=Oonward,theslabissuddenlyexposedtothesamefluidon~sidesortotwodiffereat fluids. In either case, the tempemtun

of the fluid(s) is constant, T,. Unequal cooling conditions are applied

atthetwoexposedsurfacesoftheslab,forinstance,surface coefficient h,, whereas sur&ce 2 is conne&d pressumed that the thermophysical temperature.

1 isassociatedwithameanheattransfer

to a mean heat transfer coefficient hz (see Fig. I). It is also

properties of the solid material and the fluid(s) are not influenced by

ASYMMETRIC

Vol. 28, No. 1

fluid h,,

COOLING OF LONG SLABS BY HEAT CONVECTION

1

fluid

129

2:

X

To,

h,

0

L

>

1 l-+4-+-l FIG. 1 Long slab with asymmehc cuovective

cooling

Distributed Model

The tempe&ure inside a long slab obeys the unidirectional heat conduction equation

dT

c3=T

at

h=

-=a-

(2)

The initial condition is

T=Ti,

’ t=o

kg =h,(T - T_),

-k$

=

h,(T-TJ,

(24

(W

x = -L

W)

x=L

The majority of the analyses of unsteady heat conduction of solid bodies have been traditionally condUctedwithtbedishibutedandthelumpedmodels,~~~variatioaalandintegralmodelSbaVe~Used to

a lesser degree [ 11. Tbere are two potential avewx

fix cumparisoa between the temperature solutions

furnished by the distributed aad the lumped models. One avuwe reties on the calculation of the mean

F. Alhama and A. Camp0

130

Vol. 28, No. 1

temperature T(t) (a global temperature that is common to the two models): (3)

where V designates the volume of the solid body. ‘Ihe other avenue contrasts local temperatures produced by the distributed mode4only, such as the tnsimumtanperatureT_~thetwominimumtemperaturesT(L,t) and T&t) at the two e-xposexlswfices.

Inasmuch as for symmethc cooling of a slab of thickness 2L the chacteristic

havedeemeditappropriatetoretainthesamefigureL~rthegeoeralcaseof~ccoding.

length is L = V/A we Anadequate

set of dimensionless variables for T, x, and t may be

fJ=-

T - T,

r=-

T, - T_’

r

(4)

“-z

L%l’

where the respective scales are (Ti - T,), L*/a and L. Accordingly, the heat conduction equation (1) in dimensionless form becomes ae _=.E!? &

(5)

ax2

Also, tbe initial condition

is

5=0

fl = 1,

aidthebamdaryconditionsare

_?!=Bi ax

I’

-?!=Bi ax

x=

0

l’

0

-1

x=

I

Vol. 28, No. 1

ASYMMETRIC

COOLING OF LONG SLABS BY HEAT CONVECTION

131

(6)

Transfomution of theDistributed Model into a Lumued Model

The next step in the present compamtive study is geared toward the systematic conversion of the general distributed model, eq. (5), along with tbe boundary conditions, eqs. (5b) and (5c), into a particular lumpedmodel. The most importantsteps ofthe mahmatical derivationhave been includedin this sub-section.

AtthebegGng,theheatconductionequation(5)mayberwraqedas

ae ___=a ae al

(1

(7)

axax

IntegratingwithrespecttothedimensionlessspacevariableXbetweentbeLimitsX=-IandX=

1,leadsto

the integralform

I I?!!,= al -1

,o ‘a

-I

E

ax

(8)

a

Later,by virtue of the Leibnitz theorem(see Chant and Hilbert [3]), tbe LHS of this equation is reordered topennitthedi&e&ahontooccurpriortotheintegratian.

Becausetbeupperandlowerlimitsoftheintegral

are numbers,this c4nnmutativeoperationyields (9)

Here, it may be real&d that the term to be d&whed

in the LHS

of eq. (9) is prxisely the dimerkonless

meantemperaturedefinedby eq. (6). Also, the iuteghon of the RHS of eq. (9) can be completedimmediately to give

(10)

Vol. 28, No. 1

F. Alhama and A. Campo

132

of eq. (10) are replaced by their equivalent temperature relations. This operation leads to ae

-

=

-Bi2O&T) - Bi,e(-1,~)

(11)

ar This

equation reveals the presence ofmixed temperatures;

in other words, the LHS involves the dimensionless

mean temperature, G(r), whereas the two dimensionless surface temfwatums

O&r) and 0(-l ,r) are embodied

into the RI-IS. To resolve this discrepancy, the lirst level of the lumped model proposes a quasi temperature uniformity inside. the slab; that is G(r) = O(-1,~) J O&r) at all tunes. Clearly, since G depends on r only, this conversion procedure culminates with the ordinary ditXerential equation of first order:

dr

= -Bi2 0, - Bi, 0,

(12)

where the dimensionless lumped temperature is designated by Or_within the context of a lumped model. Then, the initial condition in eq. (5) is rewritten accordingly as

8, = 1,

r=O

(13)

lheanalyticalsolutionofeqs.(12)and(13)isreadilygivenby f3, = exp( -Bi2 - Bi,) ‘F

(14)

For the common case of symmetric cooling (Bi, = Bir = Bi) this type of exponential solution is of remarkable importance in the analysis of unsteady heat conduction [ 1,2]. With increasing time, evaluation of the singleterm exponential equation (14) yields very small numbers.

At the opposite. end, the single-term exponential

maximum of 1 at r = 0, retrieving the initial condition, eq. @a), perfectly. Boussinesq

equation(14)reachesa

designated the single-term exponential relation of eq. (14) as a fundamental solution (see Kondrat’ev [4]). Conversely, Kondrat’ev [4] himselfprekred

to rename the exponential equation (14) as the regular condition;

this condition implies some sort of intermediam condition tbat bridges the gap between an initial conditinn (e, = l)andthefinal~ySfafecoadition(8,=0)forlongtimes. case involving asymmek

cooling of a slab.

Finally, the original physical variables afkting resulting in the equation

-

=

Thesameargumentscanbeexter&dtothe

the lumped problem may be retrieved in eq. (14),

ASYMMETRIC

Vol. 28, No. 1

COOLING OF LONG SLABS BY HEAT CONVECTION

133

For completeness, the analog thermal network capable of simulating the lumped model for the asymmetric convective cooling of a slab is displayed in Fig. 2.

s Ti

r”__ -

-

c-p,

_

$

difkential dcpcmht

1

x2=R2

network

it may be real&d that a distributed model governed by the partial

heat eooduetion equation (5) io two kkpendmt variable 0 (temperature)

R

n1= ’

FIG. 2 Analog resistan~itao~

From a mathematical shndpoiit,

’

has been success~Uy

variables (space variable X and time r) with the reduced to a lumped model. This new model is

wpresented by a first-order, ordhy

ditlkeotial kat equation (12) where the depcahnt

temperature, k, and the hdepehnt

variable is the time, z.

variable is the lumped

Now, the question that needs to be answered is the following: what are the satisfaetoty combinations of Bi, and Bi, that cause a near overlapping bctweea the mean temperature distribution e(z) determined by the distributed model and the hunpedtemperature

distributioa C+(T)obtained by the lumped model? This issue will

beexamioedinaforhomingsectioo.

Discussion of Results

Numerical temperature fields in the slab, 0(X,r), have been calculated w&be Network Method for unnbiions

Resistance-Capaeitaace

of small values of Bi, and Bi, that lie in the vieioity of zero, i.e., both Bi,

Vol. 28, No. 1

F. Alhama and A. Campo

134

and Bi2 << O(1). The method exploits the analogy between elect& circuit theory and heat surv9ed

conduction

theory

by Pas&his and Baker [S] back in 1942. The main fbature of this physical analogy is the use of

discrete intervals of the space variable and real, continuous tune as the independent variable. In those days, it was necessary to build an equivalent network and determine the desii

voltage and current variations by

actual measurements in an apparatus called a thermal analyzer. The thermal analyzer was just a network composed of resistances and capacitors selected to duplicate a thermal system. With the advent of fast digital computers tbe numerical calculations can be done very rapidly and etliciently with the computer code PSPICE [6,7]. This computer code was designed to simulate unlimited circuit networks.

For each temperature field obtained with PSPICE, three local temperature variations are pre-selected; they are: (a) the maximum temperature, O_(s) in the interval (-1 < X < 1). (b) the tcrnperature at the left sur&cc, 8(-1,~) and (c) the temperature at the right surface, O&r).

The last two surface temperatures are

connected to relative or absolute minima. Inside the slab, the location of the maximum temperature

moves

progressively from the leg sur&cc x = -L with an unposed Bi, = 0 toward the mid-plane x = 0 when Bi, and Bi, are equal.

One option for the impkmentation of the lumped criterion for symmetric cooling of a slab aflkted by only one Bi = LTinc, stipulates that the surfacato-maximurn 5 % [ 1,2].

This

same

criteria

when

extended

to the case

by the two inequalities [O_(t) - e(-1,~)]/9_(r)

temperature

ratios

have

to be liiti,

to less than

of asymmetric coohng of a slab, may be expressed

< 0.05 and [O_(T) s(l,s)]/O_(r)

< 0.05.

Those subsets of combinations of Bi, and Biz which are suitable for the lumped slab with arymmetric

convective cooling are plotted in Fig. 3 forming a curvilinear rectangle OABC. The trends observed in the curvilinear rectangle may be explained as follows. l’be vertkal stmight side OA corresponds to a lumped shah with an insulated left surfke (Bi, = 0) and a convective right surthce (Bi2 5 0.0537)‘. In contra!& the upper right vertex B is associated with a lumped slab having two symmetric convective surfaces, so that Bi, = Bi, = 0.1075. The central portion of the curvikar

rectangle spanning from OA to B horiaontally and 6om the

curved side OC to the curved side AB vertically represents a lumped slab with aqnmetk

convective surfaces

(Bi, # BiJ under study here. As an example, take a tixed value of Bi, at 0.678; then the reading from Fig. 3 indicates that the acceptable range of Bi, that complies with the Lumped criterion is confined to the

i&Nd

0.0145 C Bit < 0.0970.

’

This extreme may be envisioned as a long slab of thickness featuring a single Biot number, Bi = Lli/k = 0.1075

L = 2L with unilateral cooliig

Vol. 28, No. 1

ASYMMETRIC

.../.

7..

+ ._j

..i..

i

0.04

__.............................

0.02

0

i

.._....._.............

~

0

0.02

135

COOLING OF LONG SLABS BY HEAT CONVECTION

..i

j..

4 ;-:;/ 0.04

--I 0.08

0.1

0.08

0.12

Bi, FIG. 3 Region of validity of the lumped slab with asymmetric comxtive

cooling

Finally, a curve fit of the gathemi Biot number data for the two bounding lines that f?ame the curvilinear redaogle in Fig. 3 &livers two predictive equations. Here, Bi, is the dependent variable and Bi, is the independent variable. The predictive equation for the upper bound of Bi, is:

Bi2Lb= 0.054

+ 0.229Bilo””

W)

and the other predictive equation for the lower bowl of Bi, is:

0

for

-0.009

Bi, < 0.05 + 219.949Bi,“su

fov Bi, > 0.05

(17)

F. Alhama and A. Camp0

136

Vol. 28, No. 1

Conclusions

For asymmehc

wrwxti~cooling

of long slabs influenced by two different Biot numbers, Bi, and Bi,,

the present comparative study has ledto the folknving statement: the simple lumped m&l

is qualified to handle

the general distributed model without incurring temperature errors that exceed 5%. This Hatementistrueas long as the combiions

of Bi, aad Bi, are wnfined to the anx circumsc ribed by the curvilinear rectangle in

Fig. 3. For combinations of Bi, and Bi, that fAl outside the boundaries of the curvilinear rectangle, the simple lumped model fails and the general distributed model must to be used.

Nomenclature Bi

Biot number for symmetric cooliig. Li&

Bi,

Bid number at left surf&e for asymmetric cooling, L&/k

Bi,

Biot number at right surface for asymmetric coohag, L&h

c,

specific heat capacity

6

mean convective coefficient at left surfh

I;,

mean convective coeflicient at right surf&e

k

thermal cuaductivity

2L

thickness of slab with bilateral cooling

L

thickness of slab with unilateral cooling

t

time

T

temperature

7

mean temperature, eq. (3)

V

volume of slab

X

space variable

X

dimensionless x, eq. (4)

Greek Letters a

thermal dit&sivity

9

dimensionless T, eq. (4)

P

density

r

dimensionless t or Fourier number, eq. (4)

SubscriDts L

lumped

max

maximum

Vol. 28, No. 1

1,2

ASYMMETRIC

COOLING OF LONG SLABS BY HEAT CONVECTION

137

left and right surfaces

Rcferencq

1.

V. Arpaci, Conduction Heat Transfer, Addison-Wesley,

2.

A.V. Luikov, Atwlyrical Heat DQjWon Theory, Academic, New York, 1968.

3.

R. Coutant and D. Hilbert, Methods ofMathematical Physics, Interscience. New York, 1953.

4.

Y.V. Kondrat’ev, Unsteady State Heat Transfer, Iliffe Books, London, United Kingdom, 1966.

5.

V. Pas&is

6.

L.W. Nagel. SPICEZ: A computer program to simulate semiconductor circuits, Memo No. UCB/ERL M520, Electronic Research Laboratory, University of Caliirnia, Berkeley, CA, 1975.

7.

PSPICE, Version 6.0, 1994. ‘kxosim

Reading, MA, 1966.

and H.D. Baker, Trans. ASME 64, 105-I 12, 1942.

Corp., 20 Fairbanks St., Irvine, CA, 92718.

Received October 16, 2000