- Email: [email protected]
Boundary layer approach to heat and mass transfer in porous bodies of cylindrical geometry
IN. I, En@g Sci., 1976. Vol. 14, pp. !+75-!?%I. Pnpmoo
h.
PriUed iu Goal
fkih
BOUNDARY LAYER APPROACH TO HEAT AND MASS TRANSFER IN POROUS BODIES OF CYLINDRICAL GEOMETRY K. N. RAI and R. N. PANDEY Applied Mathematics Section, Institute of Technology, Banaras Hindu University,
Varanasi-221005.
India
Ahstraet-Integral balance method is applied to determine the transfer potentisds of heat and nmsa inporous bodies of simple geometry. The solutions thus obtained are compared with the exact solutions of the linear problems. Further, the method is utilized in developing solutions for a noo-linear probkm also. Criterial influence over the transfer potentials is shown graphically. The advantage of the present method is to provide quick approximate results to non-linear problems involving temperature and moisture dependent physical properties of the medium. 1. INTRODUCTION
A PROCESS engineer is often faced with the problems involving simultaneous heat and mass transfer in porous media. For instance, the dehydration of food-stuff, drying of wood by infrared rays, drying of diatomic slabs, mineral wool and autoclave concrete, which are of immense industrial importance, involve simultaneous transfer of heat and mass during their processing. Usually the shape of the porous matrix in which the transfer of heat and mass takes place is quite complicated and cannot be easily described by means of well defined boundaries; it is therefore proper to consider them in simple geometrical configurations like cylinder, sphere and plate for the analytaical solution. In general, the differential equations governing the coupled phenomena of heat and mass transfer in porous media are complicated in nature. Kumar and Narang[ l] have developed a technique which is an extension of Goodman’s[2] technique to obtain the solutions of certain problems in coupled phenomena of heat and mass transfer in an infinite porous plate of finite thickness. Here we have used a similar technique to obtain approximate solutions of heat and mass transfer potentials in an infinite cylindrical porous body with Merent types of boundary conditions and constant initial distribution of potentials. In this technique, it is assumed that a thermal boundary layer and a mass boundary layer exist, whose thicknesses grow with time. The thicknesses of these layers are specified by the surfaces where there is no heat and mass transfer and the temperature and concentration have their uniform initial vaiues. Therefore, as long as the thicknesses of these layers are less than the radius of the cylinder, it behaves as an infinite medium, because the boundary conditions at the axis of the cylinder do not matter. At the transition times-when the thicknesses of two layers are separately equal to the radius of the cylinder-the boundary conditions prescribed on the surface of the cylinder come into play. Therefore, for the solution, these problems are split up into two phases-one valid for the ranges 0 < F0 c F,,, 0 < F0 < Fo, (where F,,, and F% are the generalized transition times for heat and mass transfer potentials) and the other valid for the ranges F,, > FO,, Fo > F&. In the first phase when heat transfer is assumed to lag behind the mass transfer, we get a relation
for boundary conditions of the first kind, Lu >
1 1 + dCoKi_ IKi,,
for boundary conditions of the second kind, and Lu2+(4KoKi, - Bi,(Bi, -6))Lu -4BCc0 (Bi, - 6)(I- e)KoKi, for boundary conditions of the third kind.
UES Vd.
14.No. 11-A
K. N. RAI and R.N. PANDEY
916
The above inequalities are reversed if the mass transfer lags behind the heat transfer in the tirst phase. The inequalities seem to modify the criterion of Luikov[3]. In the first part of the present work, we discuss the case of heat and mass transfer in the body with boundary conditions of the second kind, completely neglecting Posnov criterion term. Secondly we discuss the problem with the boundary conditions of the third kind. Towards the end we have discussed a problem where the diffusivity of heat is a linear function of heat transfer potential and the diffusivity of mass is a linear function of mass transfer potential. This non-linear problem has been studied with the boundary conditions of the first hind and the solutions have been obtained for both the cases; viz. when heat transfer precedes mass transfer and vice versa. The results thus obtained have been graphically depicted. The numerical results have been obtained on a TDC 12 computer.
2.APPLICATIONOFTHEMETHODTOSOLUTIONOF A PROBLEM WITH 3OUNDARY CONDITIONSOFTHESECONDKIND
Let us consider a porous body in the form of an infinite cylinder of radius R. Initially, the transfer potentials for heat and mass are assumed to be constant. Further, the fluxes of these potentials are assumed to be uniform over the surface of the cylinder. Neglecting the thermal diffusion, we have determined the temperature and mass transfer potential distributions. The differenti~ equations govemi~ the process of heat and mass transfer may be expressed as follows:
, 1
+ t.P(CmlCqMxe).,
W), = a,W,), and
O
W),
= Um(X@,)_x
f>O.
(2.1)
Since the initial distributions of transfer potentials are assumed to be constant, we have T(x, 0) = TO e(x,o)= 80 1 ’
(2.2)
OsxsR.
The conditions over the surface of the body can be set in the form: -A,T.x(R,t)+q, A,fI,(R,t)+q,
=0 t>O =0 I ’
(2.3)
and the co~itions due to symmetry can be written in the form: T,(O,C)=O B,(O,C)=O
I
(2.4)
t’“*
’
In order to simplify the solution of the problem, it is desirable to present eqns (2.1H2.4) in dimensio~ess form. On in~~uci~ the ~ensionless variables and simiiarity criteria defined in the nomenclature list. we can write them as follows: wed., = wbh - dwx~2).lb ih (xe,, 1.X we2x, =
edx,0) = e,(x, e,.d,
e2A1,
I
,
o<
0) = 0, 0 s
x
x
<
I;
I.
Fd = Ki, F,,>O Fd= Kim ’
3
1
. , Fo,o
(2.5) (2.6) (2.7)
and e,.,(o,
m = e2do, m = 0.
F. > 0.
(2.8)
Boundary layer
approachto heataadmasstrmsfer
9-n
SOLUTION
first phase
We define a quantity a(&), called the dimensionlesspenetrationdistance. Its properties are such that for X>8(F0), the body for all practical purposes, is at equilibrium and there is no transfer of potentials beyond this point. Let us assume that for any value of generalized time F0 the distributions of potentials for heat and mass have their influence inside the body up to distances S,(FJ and &(FO) respectively, measured from its surface. Thus, the conditions at X= l-S, and X= I-& are &(I -S,,Fo)=O
@,x(1 - 61, Fo) = 0
I
(2.9)
and
1
&( 1 -- 81, &.x(1 82,F0) Fo) == 00 .
(2.10)
Let us assume that dimensionless potentials are approximated by second degree polynomials in the space variable x &=Ao+A,(l-X)+Az(l-X)* e* = Bo + B,( 1- X) + B*(l
- X)’
1 ’
(2.11)
where A, A,, A2, B,,, B, and B2 may be functions of generalized time F. and the similarity criteria. By malting use of eqns (2.7), (2.9H2.1 l), the expressions for potential distributions in the first phase come out to be: Ki, 6,(X, Fd = ~(6, Ki,,, 0,(X, Fd = 2~2 (62
- 1+ X)’
(2.12) -
1+
JO’
Integrating (2.5), between the limits X = 1 and X = 1 - S2 and using eqns (2.7)*, (2.10) and (2.12)*, we get a lirst order differential equation for S2 as
d&
82(3S2- 8) do, = - 24 Lo.
(2.13)
Integrating (2.13) under the initial condition S*(O)= 0 we get S:-46:+24LuoFo=O.
(2.14)
While integrating the heat transfer eqn (2.5), from X = 1 to X = I- &, we have to consider two cases, namely, 6, > S2 and 6, < &. Case 1 (6, > 63. In this case the heat balance integral under the condition (2.9) and the condition e,,(i - 6,. 0) = 0 can be written as
I
1-s,
(xeI)PodX
= -
1
e,xx(x_1+ lLUKO~*.XlX-I*
(2.15)
Substituting the value of 8, from (2.12), and using conditions (2.7)1, a Grst order differential equation for S, is obtained in the form &(3SI - 8) $=
0
- 24(1- eLuK&i, /Ki,,),
(2.16)
978
K. N. RAI and R. N. PANDEY
which reduces to the following form after integration 8: - 48: + 24(1 - eLuKoKi, /Ki,)FO = 0.
(2.17)
Case 2 (6, < 63. In this case the heat balance integral under the conditions (2.7), and (2.9), can be written as [‘-“(X&).,dX=
-Kiq+tLrtKoKi,,,[l-(l-&)(1-2)],
(2.18)
where the value of &xIx_I-~, from the mass transfer potential (2.12), has been used. Substituting the value of & from (2.12),, a fist order differential equation for S1 is obtained in the form -l+eLuK+-“(l-(l-S+?))].
(2.19)
Using Picard’s method of successive approximation the value of 6, up to the first approximation is 2 S,=6
Ki,,, 1-eLuKo$Tm
(2.20)
where
(eKoKi,,,/Kiq)‘+&
l/z >3.
(2.21)
From (2.14), (2.17) and (2.21), we have Theorem 1. If mass transfer lags behind the heat transfer it envisages the relation Lu c
1
(2.22)
1 + eKoKi, IKi,
and the inequality sign is reversed if the heat transfer lags behind the mass transfer. Second phase When the energy or mass penetration depth reaches the axis of the cylinder, the idea of penetration distance fails and therefore we have to take into account the boundary conditions at X = 0, and the profiles have to be redetermined to include the effect of this boundary. Case 1 (6, > &). Let us assume that the temperature and mass penetration depths reach the axis at generalized times F,, and Fg respectively. Thus, the initial distributions of the potentials for the second phase become
1
0,(X, F,,) = 3 K&X*
e,(x, Fg) = i Ki,,,X* ’
(2.23)
To determine the distribution of potentials in this phase, we again assume polynomials of second degree in X That is &=Ab+A:X+A:X* eZ=B:,+B:X+B:X2
1 ’
(2.24)
Boundarylayer approach to beat and mass transfer
979
where A:, A :, A ;, B;, B ‘I and I?; may be functions of generalized time F0 and similarity criteria. On using eqns (1.7), (1.8) and (1.24), we get 0, = AA+iKia’ t12= B:+iKi,,,X2
1
(2.25)
.
Further, on integrating eqns (2.5) from X = 0 to X = 1 and using eqns (2.7), (2.8) and (2.25), we get = 2Ki, (1 - CLuKoKi,,,/Ki, )
$ 0
3
0 =
2LuKi,,,
1
(2.26)
Integrating eqns (2.26) under the initial conditions given in (2.23), we obtain A : = 2Ki, (1 - ELuKoKi, IKi, )(Fo - F,,) BA = ZLuKi,,, (F. - Fg)
(2.27)
On substituting the values of Ah and Bh from (2.27) into eqns (2.29, we obtain the transfer potential for heat in the form 2F0-$l-2X7-2~LuKo(Kim/Ki,)Fo
(2.28)
3
and transfer potential for mass in the form 0,(X, Fo) = Kim 2LuFo - il - 2X2)].
(2.29)
Case 2 (S, < 6,). The distribution of temperature in this case is similar to (2.28) except that Fo, has to be replaced by Fo, whose approximate value can be obtained from (2.20) by putting S, = 1 at F. = Fo, and then the potential distribution of temperature in dimensionless form is
1
Kim (F. - F,,,) + i K&X2. 13,(x,Fo) = 2Ki, 1 - lLUKOpi,
(2.30)
The mass transfer potential distribution will be the same as in the S, > S2case because the mass transfer equation is independent of the heat transfer equation. COMPARISON
WITH EXACT
SOLUTION
The exact solutions to this problem are given by Luikov and Mikhailov[3] in the form: &(X, Fo) = Ki, 2Fo -$l
- 2X2) - m$,U’~$~~p~~~‘F~)]
cLuKoKi,,, + Lu _ 1 [ 2(1 - Lu)Fo - $, #
[exp (-cL ‘Jd - exp t-c1 %A)]
I
(2.31)
and e,(x, Fo) = Ki,,, 2LuFo - $1 - 2X2) - “2, $f&$
II
exp (-P,,~LuFoI].
(2.32)
K. N. RAI and R. N. PANDEY
980
In the quasi-steady state, the distribution of potentials of heat and mass are exactly the same as (2.28) and (2.29) which are the approximate solutions found by the boundary layer method. Equation (2.30) when compared with (2.28) shows that F% has taken the place of {8[1 - ELUKO (Ki,,, /Ki,)]}-‘. As F,, is a function of transfer coefficients this difference between the two equations will depend upon them. 3. APPLICATION
OF THE METHOD CONDITIONS
TO A PROBLEM
OF THE THIRD
WITH BOUNDARY
KIND
Here we discuss the previous problem when the surface of the cylinder is subjected to boundary conditions of third kind for heat transfer and boundary conditions of second kind for mass transfer. The boundary conditions at the surface will now be of the form -A,T,(R,t)+a[T,-T(R,t)]-(l-E)@J, =o A,e.X(R, f) + A,&T.X(R, t) + qm = 0I ’
t >o,
(3.1)
whereas the other conditions remain the same as in the previous problem. Making use of dimensionless variables defined in the nomenclature list, eqn (3.1) can be written as e,~,(l,F,,)-Bi,[1-8,(1,Fo)l+(1-e)LuKoKi~ e,,(l,Fo)-P.el.x(l,Fo)-Ki,
=O F,,>O =O I ’
(3.2)
Solution The Posnov criterion represents the internal process of heat and mass transfer. It affects only the mass transfer potential fields and for small values of generalized time F0 it has a negligible
influence. Thus in the tlrst phase it can be neglected, whereas in the second phase it will be kept in the transfer equations and the boundary conditions. When P,, = 0, (3.2), becomes
e,,,( 1, Fo) = Kim.
(3.3)
First phase
Applying the same procedure as in the previous problem, the transfer potentials for heat and mass in the first phase come out to be
e,(x, Fd =
(3.4)
and Ki,,, 6,(X, FO)= 2s~ (62 - 1+ x)‘*
(3.5)
The non-dimensional mass transfer depth is again given by the eqn (2.14). For the determination of 6,. we have to consider two cases; Case 1 (6, > &). In this case S, is given by the differential equation 24cLuKoKi,,, Bi, - (1 - c)LuKoKi,
s + 2(LuKoKim -SC) ’ eLuKoKi,,,Bi, I*
(3.6)
Integrating the above differential equation under the initial condition 61(O)= 0, we get an implicit relation 26: - a&,
+
a, log (1 + add + a2 1% ( I+ 5’ B1dl ’ )=alFo,
(3.7)
Boundarylayerapproachto heatand mass transfer
981
where
2 2 + (LuKoKi,,, - B&)ao a’=7 C eLuKoKi,Bi, I’ 2(2Bi, + l)eLuKoKi,,, a2= ((1 - r)LuKoKi,,, - Bi,)Bi, ’ 12eLuKoKi,,,Bi, a3 = Bi, - (1 - l)LuKoKi, ’ ~LuKoKi~B~ a’= Z(LuKoKi,,,- Bi,)’ Case 2 (S, < 8,). Here the difference equation for 61 is given by (8, - 4)B’ da: 2(5~- 3, - 2(2 + Bilged a = - 24 + ~~_~_~~~~~‘[l-(l-~,)(l-~)]. I
(3.8)
The value of St can be obtained by substituti~ the value of SZfrom eqn (2.14). We have also derived the values of S, in case 1 and in case 2 using an approximate method. The Picard’s method of successive approximations for case 1 leads to the following value in the fkst approx~tion ““(Ba
24(LuKoKi,,, - Si,)F,, -6)(Biq -(I-c)LuKoKi,)’
(3.9)
For Case 2, we get
lLuKoKiJJ, -_ 24 ” - (Bi, - 6) EBi, - (1 - r)LuKoKi,,, - ’ F”
(3.10)
to a first approximation using the same method as above, where 2eKoKi,,, +
(2eKoKi,,,)2-y(&
- (1 - r )LuKoKi,Y]
(Bi, - 6)(Biq - (I- e)LuKoKi,) The value of S2 upto the tirst approximation is &: = 6LuFc,.
(3.11)
(3.12)
From (3.9), (3.11) and (3.12), we have Theorem 2. For small values of generalized time, if mass transfer lags behind the heat transfer it envisages the relation 1 Lu2 + (Bi, _ 6xl _ 6)KoKi, [(4KoKi,,, - Bi,(Bi, - 6))Lu - 4Bi,] > 0
(3.13)
and the inequality sign is reversed if the heat transfer lags behind the mass transfer. Second phase Case 1 (6, > S2). The initial distribution of potentials in the second phase are
MY,
Fo,) =
Si, - (1 - e)LuKoKi,,, x2 2+Bi,
(3.14)
K. N. RAI and R.N. PANDEY
!a2
and (3.15) where
Here we include the effect of thermal diffusion by retaining 6, in both the transfer equations and boundary conditions. Hence we consider (3.18) and
w_xlx.
(xtq, = a, (xe, ), + am&
(3.19)
On using the dimensionless variables and similarity criteria defined in the nomenclature list, the above differential equations reduce to the following form
(xe,).,=(I + ddw,wel.x).x
- hx4xe2.x).x
(3.20)
and
(xe2),lb =LU [(xe,,1, -
P. wel.x).xi.
(3.21)
In this case also we assume polynomials of second degree in X for non-dimensional temperature and moisture transfer potentials. Proceeding exactly as before, we get temperature and mass transfer potential in the nondimensional form as
KI,- ArLuKoKi, e,(x,Fo)=~-(l-~)L~Ko~~~ 2
+ Si, - (1 - t)LuKoKi, 2+Bi,
-i
(1 - X2 + 2/B&)
lLuKoKi,
1
(X2 - I- 2/B&) exp
(3.22)
and ez(x, F,,) = Ki,,, ZLuF,, - $1 - 2X’) + i lLuK0p.x~ Si, -(1-c)LuKoKi,
1
- 2 rLuKoKi,
1 )
(3.23) Cuse 2 (6, < 63. Expressions similar to (3.22) and (3.23) can also be arrived at for the transfer potentials of heat and mass in the second phase. The only change that can occurs is that FO, has to be replaced by F%. The approximate value of generalized transition time FO, can be obtained from (3.10) by putting 6, = 1 at F,= F%.
983
Boundary layer approach to heat and mass transfer 4. A NON-LINEAR
PROBLEM
In this section we discuss the transfer of heat and mass in an infinite cylindrical porous body of finite radius with the boundary conditions of first kind, assuming the thermal diffusivity and mass diiusivity to be linear functions of temperature and mass transfer potentials respectively. The differential equations governing the process of simultaneous heat and mass transfer in an infinite cylindrical porous body with thermophysical characteristic dependent on temperature and moisture transfer potentials are (XT), = (XUJJ,
+
+xe), 4
,
t>o.
O
(4.1)
(xfl),, = (XU,BJJ + &h.L~.xL 1
The boundary conditions are: T(R,f)= e(R,r)=
TY
I
e: ’ l’O*
(4.2)
The initial conditions and the conditions due to symmetry are given by (2.2) and (2.4) respectively. We assume the following dependence of thermophysical properties: u, = u,~(I+ od u, = ~,o(l+ a2e2)1 *
(4.3)
On introducing the dimensionless variables and similarity criteria defined in nomenclature list, we can write (4.1) and (4.2) as (xe,)., = (x(1 + alel)el.x)x - dwxe2).po (xe,),, = hh((x(i
+ r2e2)e2x)x -Ml
, O
+ ~2e2)xed.X)
I
Fo>O,
and e,(l,Fo)=e2(1,FO)=
1, FdO
(4.5)
respectively. SOLUTION
first phase Adopting a procedure similar to that used in the earlier problem, the distributions of potentials for heat and mass in the first phase are e,(X, Fd = (‘I -i’
3’
e,(X,F,,) = ( s2 -i+ “)’ ’
(4.6)
where S, and a2 are the thicknesses of heat and mass layers respectively. Case 1 (8, >S2). In this case the thicknesses of heat and mass layers are given by the following differential equations (& qhF=da:
-2q1+
a,)[1 + eKoLU*(P” - &lS2)1,
0 da: = _ 24~~~[1+ az + P, (( 1 - a2)(1- S2/W(S2lSd - Cl+ u2)s2hl* (s2 _ 2) dF 0
(4.7) (4.8)
K. N. RAI and R. N. PANDEY
984
Using
picard’s method of successive approximation the value of 81 and 82 upto the first approximation are s: = 12(1+ a,)[ 1 - EKOLU*t uoo- P” )lFll
(4.9)
s:= 12Luo Lffr~-~Cr*+l/Um)lFo [ where Uoo=lim $ Fo-40 2 = &EKO + P&/(1 + 02) +
((-do
+ (P”cr2/(1+ u2))* + W”(dh
+ (l/l + a2)) +
l/Lu*))ln].
(4.10) Case 2 (sI < s2). In this case we assume the mass diffusivity to be constant whereas the rjiflusivity of heat follows the same linear relationship as expressed by (4.3)1.The eqn (4.3)~in this case is written as a, = amo.
(4.11)
Here the thicknesses of heat and mass layers are given by the following differential equations (6, - 2) dF = - 24[1 + uj+ eKoLuo(((10
d6;
(62-2)@=
0
-24Luo[l
S2M
- CR/S,) -
1)031/S,)
+ Pn))l,
-P,S,/S,l.
(4.12) (4.13)
Using Picard’s method of successive approximation, the values of 6, and S2 upto a first approximation are s: =,12[lf a1 - eLuoKo(V&- P”)lPO 9 s: = 12Luo[l- P”/voolFo I
(4.14)
where
voo=lim (S&32) F&
1 zJ.+ P’+~l~~~)(l+u,+eKoLu&))‘~]. = 2( 1+ lKO) [ (
(4.15)
From (4.10) and (4.15) we have Theorem 3. if mass transfer is assumed to lag behind the heat transfer the following relation should be satisfied (4.16) The inequality sign is reversed if the heat transfer lags behind the mass transfer. Second phase Case 1 (S, > cS~).The initial distribution of potentials in the second phase are fL(x, Fo,) = @2(X, FoJ = x2,
(4.17)
where PO, and Fg are the generalized transition times for heat and mass transfer potentials respectively.
Boundary layerapproachto heatandmasstransfer
985
In this case, we again assume the profiles of the same form as given in (2.24) and, adopting the same procedure as before, we get two simultaneous differential equations for the determination of A; and B;
dA; dF= 0
-b,A$+b,B:
$=
b,(l’,,A;- B;),
o
.
b, = 8Lu0(1+ a& bz = b&o, b,=@(l+a,)+bJ’.). Solving (4.18) and (4.19) under the initial condition (4.17), we get the values of A i and B; which on substitution in the profiles determine the distributions as given below: 6,(X, F0) = 1 + (X2- 1)A;
(4.20)
0,(X, F0) = 1 + (X2- l)B;, where A; = C, exp(-m,F0)+ C,= --$(m,-
Gexp(-m#‘0),
b,) exp (-m&J
C2 =-$(m, - b,)exp(-m,Fg)+
+ b2 exp
(-mJW1,
blexp(-mtFd1,
m, = $b, + b,+ (-l)‘((bl + ba)2-4Lu*)‘T, d, = (ml-
b,)exp(-ml&-
m2Fo,)-(ml-
i = 1, 29
b,)exP(-mzBo,-mlB0Js
B;= C,exp(-m,Fo)+C.exp(-mtB0), C, = --$m,C,,=-$(m,-
b,)exp(-m#0,)+ b,)exp(-m$‘0,)+
bJ’0 exp(-m&J], IQ”, exp(-ml%)]
2
and d, = (m, -bJ exp (-mlBo, - m2Fs) - (m2 - bd exp (-m2Fo, - mtFd. Case 2 (8, < S2). Expressions similar to (4.20) can also be arraived at for the transfer potentials of heat and mass in the second phase. The only change that occurs, is that a2 has to be made equal to zero and F,, and F% have to be replaced by FO, and For respectively. The approximate value of generalized transition time Fo, can be obtained from (4.14), by putting 8, = 1 at F. = Fo, and similarly that of F% from (4.14)~by & = 1 at F. = Fo.. ANALYSISOF THE SOLUTION Case 1 (6, > 63. The mean values of the trasnsfer potentials in an infinite cylindrical porous body will be obtained from the relation (0,(X, Fo)) = 2 1’ X&(X, Fo) dx,
i = 1.2.
(4.21)
K. N. RAI and R. N. PANDEY Table 1. Effect of variability of Lu* on Fo,/F4, (e,)/(&) and (a(e,))/aF,/a(e,)/aF P. = 0.1 at different generalized time
o.aea
0.1
0.416
0.8
0.666
0.6
Table 2.
0.5
1.6ll
O.T8T
1.0
l.ao6
0.666
1.5
1.064
0.666
a.0
1.066
0.646
0.5
1.a65
O.T66
1.0
l.oT5
0.6T6
1.5
1.666
0.664
6.0
1.6OT
0.*66
0.6
1.011
0.6TO
1.0
1.006
O.S@6
1.5
1.000
0.666
a.0
l.ooo
0.*66
for c = 0.5, Ko = 1.5,
Effect of variability of 6, Ko and P, on I%,,&, (8,)/(&) and (a(0,)/dF~a(&)/aF,,) for Lu* = 0.4 at different generalized time
0.1, 1.1, 0.1
0.a.
1.6,
0.1
0.6, 1.6, 0.1
0.6, 0.6, 0.1
0.5. 1.5, 0.0
0.415
0.464
0.665
0.440
0.640
0.6
1.616
O.Tl6
1.0
1.040
0.664
1.6
1.006
0.6T6
a.0
1,009
0.666
0.5
l.lT6
O.TT6
1.0
1.064
0,614
1.5
l.OOT
0.6T6
a.0
l.ooL
0.666
0.6
1.15T
0.661
1.0
1.666
0.666
1.6
1.006
0.664
6.0
1.001
0.666
0.6
1.606
0.?66
1.0
1.666
0.606
1.6
1.006
0.6TT
a.0
1.006
0.666
0.5
1.110
0.666
1.0
l.oao
0.646
1.6
1.004
0.666
a.0
l.ooO
0.66T
Boundary layer approach to heat and mass transfer
987
5 I.0
1.5
I 2.0
‘=o -
Fig. 1.Effect of variability of Lu*on (e,(x, F&andf8(6&
Fo))/dFo)forc = O.f,Ko = 1.5andPn = 0.1.
On using the expressions for 6(X, FO)and 8,(X, I?,) from the eqns (4.2Ohand (4.20)2respectively, we obtain (6,(X,Fo)) = 1 - 0.5A; (0,(X, Fo)) = I- 0.5B;
3
(4.22)
in the liiht of (4.21). The generalized time rate of transfer potentials for heat and mass can be obtained by differentiati~ eqns (4.22), and (4.22)z with respect to the generalized time Fe. Thus, we find (4.23) and
a(e2$ofi” =f[m,C,
exp (-mJro) + m2G exp (-m2F,)l.
Table 1 shows the influence of modified Luikov number Lu* on the ratios Fo,/&,
(4.24) 03J/XO2)
0.5
I
I
r’
,f
f --c
“,L_
/
pnsx0.1
-0.5
I.0
Pn-0.0
C =0.5
Fo -
Ko-I
.5
I.5
Pn.0.I
Pn-O.,
KO’1.S
e -0.3
C -0.5
K0.1.5
Pn=O.l
Kool.5
Ko’0.S
Pn=O.l
Pn 50.1
Fii. 2. Effect of variability of c, Ko and Pn on (0,(x, F,,)) and @(13,(x, Fo))/dFo) for Lu = 0.4.
0.
/’
’
K0’1.5
E’0.3
C=O.l
K0’1.5
E’O.5
I
/
/
/
I
1.0
I 1.5
Fo Fig. 3. Effect of variability of Lu* on (6’,(x, FJ) and (8(&(x, F&/S,) K,= l.SandP. =O.l.
5
/
/
,
for c = 0.5,
2.0
,
1.0
Fo -
1”s
2-O
990
K. N. RAl and R. N. PANDEY
and [a(e,)/aF,]/[a(e,)/aF,] for different values of generalized time F0 in the range 0.5 < Fu < 2.0 for case 1 (8, > &). It is observed that as Lu* increases, the ratio of generalized transition times increases and the two transition times become comparable at Lu * = 0.6, whereas for a fixed value of generalized time F0 the ratio of the average transfer potentials decreases and the two average transfer potentials become comparable at Lu * = 0.6. Also the ratio of the generalized time rates of average transfer potentials for heat and mass increases with increase of Lu* and again the two generalized time rates of the average transfer potentials become comparable for Lu * = 0.6. The influence of q Ko and P. over these ratios are shown in Table 2. In the case of 6 and Ko, a similar behaviour is observed but the influence of these criteria is less significant than that of Lu*. Further, it is seen that the effect of P,, is opposite to that of Lu*, c and Ko. Figures 1-5 show the influence of Lu*, E, Ko and P. on the average transfer potentials and the generalized time rate of average transfer potentials. The most important influence on the transfer potentials for the heat and mass is that of the modified Luikov number Lu *. For Lu * > 0.6, the mass transfer potential lags behind the heat transfer potential and for Lu * > 0.6, the mass transfer potential precedes the heat transfer potential. NOMENCLATURE thermal diffusivity coefficient diffusion coefficient of moisture in the material specific heat capacity of the material specific isothermal mass capacity of the material beat flux 4m mass flux R radius of the cylinder time heat transfer potential temperature of the surrounding atmospohere coordinate perpendicular to the central axis of the cylinder mass transfer potential equilibrium value of mass transfer potential specific heat of evaporation Soret coefficient thermal conductivity moisture conductivity heat transfer coefficient Dimensionlessvariables X=x/R
dimensionless length coordinate
F~=[a;“;;“R:
e,(x,m = v-
75~~ e,(x, Fd = (e, - e)/Ae
dimensionless temperature dimensionless mass transfer potential
Similarity crireria Luikov number La = a, /a, Lu’ = L&(1 t u*)/U+ (I,) modified Luikov number phase change criterion Kossovich number Ko = p(W~XA0lAT; P. = &(AT/AB) Posnov number Kirpichev criterion for heat transfer Ki, = (@)/(&AT) Kirpichev criterion for mass transfer Ki, = (q,r)/(A,W Biot criterion for heat transfer Si. = (aR)/(A,)
7-f- To boundary conditions of the first kind AT=
To
[ T, - To
boundary conditions of the second kind boundary conditions of the third kind
boundary conditions boundary conditions boundary conditions partial derivative of
of of of ( )
the first kind the second the third kind with respect to X. REFERENCES
[I] I. J. KUMAR and H. N. NARANG, Inl. 1. Hear Mass Tmnsfcr 10, 1095 (1967). [2] T. R. GOODMAN, Adaances in Hear Transfer, Vol. 1. Academic Press, New York (1964). [3] A. V. LUIKOV and Yu. A. MIKHAILOV, meory of Energy and Mass Tmnsfer. Pergamon Press, Oxford (1%~). (Received I6 Lkcem ber 1974)
h.
PriUed iu Goal
fkih
BOUNDARY LAYER APPROACH TO HEAT AND MASS TRANSFER IN POROUS BODIES OF CYLINDRICAL GEOMETRY K. N. RAI and R. N. PANDEY Applied Mathematics Section, Institute of Technology, Banaras Hindu University,
Varanasi-221005.
India
Ahstraet-Integral balance method is applied to determine the transfer potentisds of heat and nmsa inporous bodies of simple geometry. The solutions thus obtained are compared with the exact solutions of the linear problems. Further, the method is utilized in developing solutions for a noo-linear probkm also. Criterial influence over the transfer potentials is shown graphically. The advantage of the present method is to provide quick approximate results to non-linear problems involving temperature and moisture dependent physical properties of the medium. 1. INTRODUCTION
A PROCESS engineer is often faced with the problems involving simultaneous heat and mass transfer in porous media. For instance, the dehydration of food-stuff, drying of wood by infrared rays, drying of diatomic slabs, mineral wool and autoclave concrete, which are of immense industrial importance, involve simultaneous transfer of heat and mass during their processing. Usually the shape of the porous matrix in which the transfer of heat and mass takes place is quite complicated and cannot be easily described by means of well defined boundaries; it is therefore proper to consider them in simple geometrical configurations like cylinder, sphere and plate for the analytaical solution. In general, the differential equations governing the coupled phenomena of heat and mass transfer in porous media are complicated in nature. Kumar and Narang[ l] have developed a technique which is an extension of Goodman’s[2] technique to obtain the solutions of certain problems in coupled phenomena of heat and mass transfer in an infinite porous plate of finite thickness. Here we have used a similar technique to obtain approximate solutions of heat and mass transfer potentials in an infinite cylindrical porous body with Merent types of boundary conditions and constant initial distribution of potentials. In this technique, it is assumed that a thermal boundary layer and a mass boundary layer exist, whose thicknesses grow with time. The thicknesses of these layers are specified by the surfaces where there is no heat and mass transfer and the temperature and concentration have their uniform initial vaiues. Therefore, as long as the thicknesses of these layers are less than the radius of the cylinder, it behaves as an infinite medium, because the boundary conditions at the axis of the cylinder do not matter. At the transition times-when the thicknesses of two layers are separately equal to the radius of the cylinder-the boundary conditions prescribed on the surface of the cylinder come into play. Therefore, for the solution, these problems are split up into two phases-one valid for the ranges 0 < F0 c F,,, 0 < F0 < Fo, (where F,,, and F% are the generalized transition times for heat and mass transfer potentials) and the other valid for the ranges F,, > FO,, Fo > F&. In the first phase when heat transfer is assumed to lag behind the mass transfer, we get a relation
for boundary conditions of the first kind, Lu >
1 1 + dCoKi_ IKi,,
for boundary conditions of the second kind, and Lu2+(4KoKi, - Bi,(Bi, -6))Lu -4BCc0 (Bi, - 6)(I- e)KoKi, for boundary conditions of the third kind.
UES Vd.
14.No. 11-A
K. N. RAI and R.N. PANDEY
916
The above inequalities are reversed if the mass transfer lags behind the heat transfer in the tirst phase. The inequalities seem to modify the criterion of Luikov[3]. In the first part of the present work, we discuss the case of heat and mass transfer in the body with boundary conditions of the second kind, completely neglecting Posnov criterion term. Secondly we discuss the problem with the boundary conditions of the third kind. Towards the end we have discussed a problem where the diffusivity of heat is a linear function of heat transfer potential and the diffusivity of mass is a linear function of mass transfer potential. This non-linear problem has been studied with the boundary conditions of the first hind and the solutions have been obtained for both the cases; viz. when heat transfer precedes mass transfer and vice versa. The results thus obtained have been graphically depicted. The numerical results have been obtained on a TDC 12 computer.
2.APPLICATIONOFTHEMETHODTOSOLUTIONOF A PROBLEM WITH 3OUNDARY CONDITIONSOFTHESECONDKIND
Let us consider a porous body in the form of an infinite cylinder of radius R. Initially, the transfer potentials for heat and mass are assumed to be constant. Further, the fluxes of these potentials are assumed to be uniform over the surface of the cylinder. Neglecting the thermal diffusion, we have determined the temperature and mass transfer potential distributions. The differenti~ equations govemi~ the process of heat and mass transfer may be expressed as follows:
, 1
+ t.P(CmlCqMxe).,
W), = a,W,), and
O
W),
= Um(X@,)_x
f>O.
(2.1)
Since the initial distributions of transfer potentials are assumed to be constant, we have T(x, 0) = TO e(x,o)= 80 1 ’
(2.2)
OsxsR.
The conditions over the surface of the body can be set in the form: -A,T.x(R,t)+q, A,fI,(R,t)+q,
=0 t>O =0 I ’
(2.3)
and the co~itions due to symmetry can be written in the form: T,(O,C)=O B,(O,C)=O
I
(2.4)
t’“*
’
In order to simplify the solution of the problem, it is desirable to present eqns (2.1H2.4) in dimensio~ess form. On in~~uci~ the ~ensionless variables and simiiarity criteria defined in the nomenclature list. we can write them as follows: wed., = wbh - dwx~2).lb ih (xe,, 1.X we2x, =
edx,0) = e,(x, e,.d,
e2A1,
I
,
o<
0) = 0, 0 s
x
x
<
I;
I.
Fd = Ki, F,,>O Fd= Kim ’
3
1
. , Fo,o
(2.5) (2.6) (2.7)
and e,.,(o,
m = e2do, m = 0.
F. > 0.
(2.8)
Boundary layer
approachto heataadmasstrmsfer
9-n
SOLUTION
first phase
We define a quantity a(&), called the dimensionlesspenetrationdistance. Its properties are such that for X>8(F0), the body for all practical purposes, is at equilibrium and there is no transfer of potentials beyond this point. Let us assume that for any value of generalized time F0 the distributions of potentials for heat and mass have their influence inside the body up to distances S,(FJ and &(FO) respectively, measured from its surface. Thus, the conditions at X= l-S, and X= I-& are &(I -S,,Fo)=O
@,x(1 - 61, Fo) = 0
I
(2.9)
and
1
&( 1 -- 81, &.x(1 82,F0) Fo) == 00 .
(2.10)
Let us assume that dimensionless potentials are approximated by second degree polynomials in the space variable x &=Ao+A,(l-X)+Az(l-X)* e* = Bo + B,( 1- X) + B*(l
- X)’
1 ’
(2.11)
where A, A,, A2, B,,, B, and B2 may be functions of generalized time F. and the similarity criteria. By malting use of eqns (2.7), (2.9H2.1 l), the expressions for potential distributions in the first phase come out to be: Ki, 6,(X, Fd = ~(6, Ki,,, 0,(X, Fd = 2~2 (62
- 1+ X)’
(2.12) -
1+
JO’
Integrating (2.5), between the limits X = 1 and X = 1 - S2 and using eqns (2.7)*, (2.10) and (2.12)*, we get a lirst order differential equation for S2 as
d&
82(3S2- 8) do, = - 24 Lo.
(2.13)
Integrating (2.13) under the initial condition S*(O)= 0 we get S:-46:+24LuoFo=O.
(2.14)
While integrating the heat transfer eqn (2.5), from X = 1 to X = I- &, we have to consider two cases, namely, 6, > S2 and 6, < &. Case 1 (6, > 63. In this case the heat balance integral under the condition (2.9) and the condition e,,(i - 6,. 0) = 0 can be written as
I
1-s,
(xeI)PodX
= -
1
e,xx(x_1+ lLUKO~*.XlX-I*
(2.15)
Substituting the value of 8, from (2.12), and using conditions (2.7)1, a Grst order differential equation for S, is obtained in the form &(3SI - 8) $=
0
- 24(1- eLuK&i, /Ki,,),
(2.16)
978
K. N. RAI and R. N. PANDEY
which reduces to the following form after integration 8: - 48: + 24(1 - eLuKoKi, /Ki,)FO = 0.
(2.17)
Case 2 (6, < 63. In this case the heat balance integral under the conditions (2.7), and (2.9), can be written as [‘-“(X&).,dX=
-Kiq+tLrtKoKi,,,[l-(l-&)(1-2)],
(2.18)
where the value of &xIx_I-~, from the mass transfer potential (2.12), has been used. Substituting the value of & from (2.12),, a fist order differential equation for S1 is obtained in the form -l+eLuK+-“(l-(l-S+?))].
(2.19)
Using Picard’s method of successive approximation the value of 6, up to the first approximation is 2 S,=6
Ki,,, 1-eLuKo$Tm
(2.20)
where
(eKoKi,,,/Kiq)‘+&
l/z >3.
(2.21)
From (2.14), (2.17) and (2.21), we have Theorem 1. If mass transfer lags behind the heat transfer it envisages the relation Lu c
1
(2.22)
1 + eKoKi, IKi,
and the inequality sign is reversed if the heat transfer lags behind the mass transfer. Second phase When the energy or mass penetration depth reaches the axis of the cylinder, the idea of penetration distance fails and therefore we have to take into account the boundary conditions at X = 0, and the profiles have to be redetermined to include the effect of this boundary. Case 1 (6, > &). Let us assume that the temperature and mass penetration depths reach the axis at generalized times F,, and Fg respectively. Thus, the initial distributions of the potentials for the second phase become
1
0,(X, F,,) = 3 K&X*
e,(x, Fg) = i Ki,,,X* ’
(2.23)
To determine the distribution of potentials in this phase, we again assume polynomials of second degree in X That is &=Ab+A:X+A:X* eZ=B:,+B:X+B:X2
1 ’
(2.24)
Boundarylayer approach to beat and mass transfer
979
where A:, A :, A ;, B;, B ‘I and I?; may be functions of generalized time F0 and similarity criteria. On using eqns (1.7), (1.8) and (1.24), we get 0, = AA+iKia’ t12= B:+iKi,,,X2
1
(2.25)
.
Further, on integrating eqns (2.5) from X = 0 to X = 1 and using eqns (2.7), (2.8) and (2.25), we get = 2Ki, (1 - CLuKoKi,,,/Ki, )
$ 0
3
0 =
2LuKi,,,
1
(2.26)
Integrating eqns (2.26) under the initial conditions given in (2.23), we obtain A : = 2Ki, (1 - ELuKoKi, IKi, )(Fo - F,,) BA = ZLuKi,,, (F. - Fg)
(2.27)
On substituting the values of Ah and Bh from (2.27) into eqns (2.29, we obtain the transfer potential for heat in the form 2F0-$l-2X7-2~LuKo(Kim/Ki,)Fo
(2.28)
3
and transfer potential for mass in the form 0,(X, Fo) = Kim 2LuFo - il - 2X2)].
(2.29)
Case 2 (S, < 6,). The distribution of temperature in this case is similar to (2.28) except that Fo, has to be replaced by Fo, whose approximate value can be obtained from (2.20) by putting S, = 1 at F. = Fo, and then the potential distribution of temperature in dimensionless form is
1
Kim (F. - F,,,) + i K&X2. 13,(x,Fo) = 2Ki, 1 - lLUKOpi,
(2.30)
The mass transfer potential distribution will be the same as in the S, > S2case because the mass transfer equation is independent of the heat transfer equation. COMPARISON
WITH EXACT
SOLUTION
The exact solutions to this problem are given by Luikov and Mikhailov[3] in the form: &(X, Fo) = Ki, 2Fo -$l
- 2X2) - m$,U’~$~~p~~~‘F~)]
cLuKoKi,,, + Lu _ 1 [ 2(1 - Lu)Fo - $, #
[exp (-cL ‘Jd - exp t-c1 %A)]
I
(2.31)
and e,(x, Fo) = Ki,,, 2LuFo - $1 - 2X2) - “2, $f&$
II
exp (-P,,~LuFoI].
(2.32)
K. N. RAI and R. N. PANDEY
980
In the quasi-steady state, the distribution of potentials of heat and mass are exactly the same as (2.28) and (2.29) which are the approximate solutions found by the boundary layer method. Equation (2.30) when compared with (2.28) shows that F% has taken the place of {8[1 - ELUKO (Ki,,, /Ki,)]}-‘. As F,, is a function of transfer coefficients this difference between the two equations will depend upon them. 3. APPLICATION
OF THE METHOD CONDITIONS
TO A PROBLEM
OF THE THIRD
WITH BOUNDARY
KIND
Here we discuss the previous problem when the surface of the cylinder is subjected to boundary conditions of third kind for heat transfer and boundary conditions of second kind for mass transfer. The boundary conditions at the surface will now be of the form -A,T,(R,t)+a[T,-T(R,t)]-(l-E)@J, =o A,e.X(R, f) + A,&T.X(R, t) + qm = 0I ’
t >o,
(3.1)
whereas the other conditions remain the same as in the previous problem. Making use of dimensionless variables defined in the nomenclature list, eqn (3.1) can be written as e,~,(l,F,,)-Bi,[1-8,(1,Fo)l+(1-e)LuKoKi~ e,,(l,Fo)-P.el.x(l,Fo)-Ki,
=O F,,>O =O I ’
(3.2)
Solution The Posnov criterion represents the internal process of heat and mass transfer. It affects only the mass transfer potential fields and for small values of generalized time F0 it has a negligible
influence. Thus in the tlrst phase it can be neglected, whereas in the second phase it will be kept in the transfer equations and the boundary conditions. When P,, = 0, (3.2), becomes
e,,,( 1, Fo) = Kim.
(3.3)
First phase
Applying the same procedure as in the previous problem, the transfer potentials for heat and mass in the first phase come out to be
e,(x, Fd =
(3.4)
and Ki,,, 6,(X, FO)= 2s~ (62 - 1+ x)‘*
(3.5)
The non-dimensional mass transfer depth is again given by the eqn (2.14). For the determination of 6,. we have to consider two cases; Case 1 (6, > &). In this case S, is given by the differential equation 24cLuKoKi,,, Bi, - (1 - c)LuKoKi,
s + 2(LuKoKim -SC) ’ eLuKoKi,,,Bi, I*
(3.6)
Integrating the above differential equation under the initial condition 61(O)= 0, we get an implicit relation 26: - a&,
+
a, log (1 + add + a2 1% ( I+ 5’ B1dl ’ )=alFo,
(3.7)
Boundarylayerapproachto heatand mass transfer
981
where
2 2 + (LuKoKi,,, - B&)ao a’=7 C eLuKoKi,Bi, I’ 2(2Bi, + l)eLuKoKi,,, a2= ((1 - r)LuKoKi,,, - Bi,)Bi, ’ 12eLuKoKi,,,Bi, a3 = Bi, - (1 - l)LuKoKi, ’ ~LuKoKi~B~ a’= Z(LuKoKi,,,- Bi,)’ Case 2 (S, < 8,). Here the difference equation for 61 is given by (8, - 4)B’ da: 2(5~- 3, - 2(2 + Bilged a = - 24 + ~~_~_~~~~~‘[l-(l-~,)(l-~)]. I
(3.8)
The value of St can be obtained by substituti~ the value of SZfrom eqn (2.14). We have also derived the values of S, in case 1 and in case 2 using an approximate method. The Picard’s method of successive approximations for case 1 leads to the following value in the fkst approx~tion ““(Ba
24(LuKoKi,,, - Si,)F,, -6)(Biq -(I-c)LuKoKi,)’
(3.9)
For Case 2, we get
lLuKoKiJJ, -_ 24 ” - (Bi, - 6) EBi, - (1 - r)LuKoKi,,, - ’ F”
(3.10)
to a first approximation using the same method as above, where 2eKoKi,,, +
(2eKoKi,,,)2-y(&
- (1 - r )LuKoKi,Y]
(Bi, - 6)(Biq - (I- e)LuKoKi,) The value of S2 upto the tirst approximation is &: = 6LuFc,.
(3.11)
(3.12)
From (3.9), (3.11) and (3.12), we have Theorem 2. For small values of generalized time, if mass transfer lags behind the heat transfer it envisages the relation 1 Lu2 + (Bi, _ 6xl _ 6)KoKi, [(4KoKi,,, - Bi,(Bi, - 6))Lu - 4Bi,] > 0
(3.13)
and the inequality sign is reversed if the heat transfer lags behind the mass transfer. Second phase Case 1 (6, > S2). The initial distribution of potentials in the second phase are
MY,
Fo,) =
Si, - (1 - e)LuKoKi,,, x2 2+Bi,
(3.14)
K. N. RAI and R.N. PANDEY
!a2
and (3.15) where
Here we include the effect of thermal diffusion by retaining 6, in both the transfer equations and boundary conditions. Hence we consider (3.18) and
w_xlx.
(xtq, = a, (xe, ), + am&
(3.19)
On using the dimensionless variables and similarity criteria defined in the nomenclature list, the above differential equations reduce to the following form
(xe,).,=(I + ddw,wel.x).x
- hx4xe2.x).x
(3.20)
and
(xe2),lb =LU [(xe,,1, -
P. wel.x).xi.
(3.21)
In this case also we assume polynomials of second degree in X for non-dimensional temperature and moisture transfer potentials. Proceeding exactly as before, we get temperature and mass transfer potential in the nondimensional form as
KI,- ArLuKoKi, e,(x,Fo)=~-(l-~)L~Ko~~~ 2
+ Si, - (1 - t)LuKoKi, 2+Bi,
-i
(1 - X2 + 2/B&)
lLuKoKi,
1
(X2 - I- 2/B&) exp
(3.22)
and ez(x, F,,) = Ki,,, ZLuF,, - $1 - 2X’) + i lLuK0p.x~ Si, -(1-c)LuKoKi,
1
- 2 rLuKoKi,
1 )
(3.23) Cuse 2 (6, < 63. Expressions similar to (3.22) and (3.23) can also be arrived at for the transfer potentials of heat and mass in the second phase. The only change that can occurs is that FO, has to be replaced by F%. The approximate value of generalized transition time FO, can be obtained from (3.10) by putting 6, = 1 at F,= F%.
983
Boundary layer approach to heat and mass transfer 4. A NON-LINEAR
PROBLEM
In this section we discuss the transfer of heat and mass in an infinite cylindrical porous body of finite radius with the boundary conditions of first kind, assuming the thermal diffusivity and mass diiusivity to be linear functions of temperature and mass transfer potentials respectively. The differential equations governing the process of simultaneous heat and mass transfer in an infinite cylindrical porous body with thermophysical characteristic dependent on temperature and moisture transfer potentials are (XT), = (XUJJ,
+
+xe), 4
,
t>o.
O
(4.1)
(xfl),, = (XU,BJJ + &h.L~.xL 1
The boundary conditions are: T(R,f)= e(R,r)=
TY
I
e: ’ l’O*
(4.2)
The initial conditions and the conditions due to symmetry are given by (2.2) and (2.4) respectively. We assume the following dependence of thermophysical properties: u, = u,~(I+ od u, = ~,o(l+ a2e2)1 *
(4.3)
On introducing the dimensionless variables and similarity criteria defined in nomenclature list, we can write (4.1) and (4.2) as (xe,)., = (x(1 + alel)el.x)x - dwxe2).po (xe,),, = hh((x(i
+ r2e2)e2x)x -Ml
, O
+ ~2e2)xed.X)
I
Fo>O,
and e,(l,Fo)=e2(1,FO)=
1, FdO
(4.5)
respectively. SOLUTION
first phase Adopting a procedure similar to that used in the earlier problem, the distributions of potentials for heat and mass in the first phase are e,(X, Fd = (‘I -i’
3’
e,(X,F,,) = ( s2 -i+ “)’ ’
(4.6)
where S, and a2 are the thicknesses of heat and mass layers respectively. Case 1 (8, >S2). In this case the thicknesses of heat and mass layers are given by the following differential equations (& qhF=da:
-2q1+
a,)[1 + eKoLU*(P” - &lS2)1,
0 da: = _ 24~~~[1+ az + P, (( 1 - a2)(1- S2/W(S2lSd - Cl+ u2)s2hl* (s2 _ 2) dF 0
(4.7) (4.8)
K. N. RAI and R. N. PANDEY
984
Using
picard’s method of successive approximation the value of 81 and 82 upto the first approximation are s: = 12(1+ a,)[ 1 - EKOLU*t uoo- P” )lFll
(4.9)
s:= 12Luo Lffr~-~Cr*+l/Um)lFo [ where Uoo=lim $ Fo-40 2 = &EKO + P&/(1 + 02) +
((-do
+ (P”cr2/(1+ u2))* + W”(dh
+ (l/l + a2)) +
l/Lu*))ln].
(4.10) Case 2 (sI < s2). In this case we assume the mass diffusivity to be constant whereas the rjiflusivity of heat follows the same linear relationship as expressed by (4.3)1.The eqn (4.3)~in this case is written as a, = amo.
(4.11)
Here the thicknesses of heat and mass layers are given by the following differential equations (6, - 2) dF = - 24[1 + uj+ eKoLuo(((10
d6;
(62-2)@=
0
-24Luo[l
S2M
- CR/S,) -
1)031/S,)
+ Pn))l,
-P,S,/S,l.
(4.12) (4.13)
Using Picard’s method of successive approximation, the values of 6, and S2 upto a first approximation are s: =,12[lf a1 - eLuoKo(V&- P”)lPO 9 s: = 12Luo[l- P”/voolFo I
(4.14)
where
voo=lim (S&32) F&
1 zJ.+ P’+~l~~~)(l+u,+eKoLu&))‘~]. = 2( 1+ lKO) [ (
(4.15)
From (4.10) and (4.15) we have Theorem 3. if mass transfer is assumed to lag behind the heat transfer the following relation should be satisfied (4.16) The inequality sign is reversed if the heat transfer lags behind the mass transfer. Second phase Case 1 (S, > cS~).The initial distribution of potentials in the second phase are fL(x, Fo,) = @2(X, FoJ = x2,
(4.17)
where PO, and Fg are the generalized transition times for heat and mass transfer potentials respectively.
Boundary layerapproachto heatandmasstransfer
985
In this case, we again assume the profiles of the same form as given in (2.24) and, adopting the same procedure as before, we get two simultaneous differential equations for the determination of A; and B;
dA; dF= 0
-b,A$+b,B:
$=
b,(l’,,A;- B;),
o
.
b, = 8Lu0(1+ a& bz = b&o, b,=@(l+a,)+bJ’.). Solving (4.18) and (4.19) under the initial condition (4.17), we get the values of A i and B; which on substitution in the profiles determine the distributions as given below: 6,(X, F0) = 1 + (X2- 1)A;
(4.20)
0,(X, F0) = 1 + (X2- l)B;, where A; = C, exp(-m,F0)+ C,= --$(m,-
Gexp(-m#‘0),
b,) exp (-m&J
C2 =-$(m, - b,)exp(-m,Fg)+
+ b2 exp
(-mJW1,
blexp(-mtFd1,
m, = $b, + b,+ (-l)‘((bl + ba)2-4Lu*)‘T, d, = (ml-
b,)exp(-ml&-
m2Fo,)-(ml-
i = 1, 29
b,)exP(-mzBo,-mlB0Js
B;= C,exp(-m,Fo)+C.exp(-mtB0), C, = --$m,C,,=-$(m,-
b,)exp(-m#0,)+ b,)exp(-m$‘0,)+
bJ’0 exp(-m&J], IQ”, exp(-ml%)]
2
and d, = (m, -bJ exp (-mlBo, - m2Fs) - (m2 - bd exp (-m2Fo, - mtFd. Case 2 (8, < S2). Expressions similar to (4.20) can also be arraived at for the transfer potentials of heat and mass in the second phase. The only change that occurs, is that a2 has to be made equal to zero and F,, and F% have to be replaced by FO, and For respectively. The approximate value of generalized transition time Fo, can be obtained from (4.14), by putting 8, = 1 at F. = Fo, and similarly that of F% from (4.14)~by & = 1 at F. = Fo.. ANALYSISOF THE SOLUTION Case 1 (6, > 63. The mean values of the trasnsfer potentials in an infinite cylindrical porous body will be obtained from the relation (0,(X, Fo)) = 2 1’ X&(X, Fo) dx,
i = 1.2.
(4.21)
K. N. RAI and R. N. PANDEY Table 1. Effect of variability of Lu* on Fo,/F4, (e,)/(&) and (a(e,))/aF,/a(e,)/aF P. = 0.1 at different generalized time
o.aea
0.1
0.416
0.8
0.666
0.6
Table 2.
0.5
1.6ll
O.T8T
1.0
l.ao6
0.666
1.5
1.064
0.666
a.0
1.066
0.646
0.5
1.a65
O.T66
1.0
l.oT5
0.6T6
1.5
1.666
0.664
6.0
1.6OT
0.*66
0.6
1.011
0.6TO
1.0
1.006
O.S@6
1.5
1.000
0.666
a.0
l.ooo
0.*66
for c = 0.5, Ko = 1.5,
Effect of variability of 6, Ko and P, on I%,,&, (8,)/(&) and (a(0,)/dF~a(&)/aF,,) for Lu* = 0.4 at different generalized time
0.1, 1.1, 0.1
0.a.
1.6,
0.1
0.6, 1.6, 0.1
0.6, 0.6, 0.1
0.5. 1.5, 0.0
0.415
0.464
0.665
0.440
0.640
0.6
1.616
O.Tl6
1.0
1.040
0.664
1.6
1.006
0.6T6
a.0
1,009
0.666
0.5
l.lT6
O.TT6
1.0
1.064
0,614
1.5
l.OOT
0.6T6
a.0
l.ooL
0.666
0.6
1.15T
0.661
1.0
1.666
0.666
1.6
1.006
0.664
6.0
1.001
0.666
0.6
1.606
0.?66
1.0
1.666
0.606
1.6
1.006
0.6TT
a.0
1.006
0.666
0.5
1.110
0.666
1.0
l.oao
0.646
1.6
1.004
0.666
a.0
l.ooO
0.66T
Boundary layer approach to heat and mass transfer
987
5 I.0
1.5
I 2.0
‘=o -
Fig. 1.Effect of variability of Lu*on (e,(x, F&andf8(6&
Fo))/dFo)forc = O.f,Ko = 1.5andPn = 0.1.
On using the expressions for 6(X, FO)and 8,(X, I?,) from the eqns (4.2Ohand (4.20)2respectively, we obtain (6,(X,Fo)) = 1 - 0.5A; (0,(X, Fo)) = I- 0.5B;
3
(4.22)
in the liiht of (4.21). The generalized time rate of transfer potentials for heat and mass can be obtained by differentiati~ eqns (4.22), and (4.22)z with respect to the generalized time Fe. Thus, we find (4.23) and
a(e2$ofi” =f[m,C,
exp (-mJro) + m2G exp (-m2F,)l.
Table 1 shows the influence of modified Luikov number Lu* on the ratios Fo,/&,
(4.24) 03J/XO2)
0.5
I
I
r’
,f
f --c
“,L_
/
pnsx0.1
-0.5
I.0
Pn-0.0
C =0.5
Fo -
Ko-I
.5
I.5
Pn.0.I
Pn-O.,
KO’1.S
e -0.3
C -0.5
K0.1.5
Pn=O.l
Kool.5
Ko’0.S
Pn=O.l
Pn 50.1
Fii. 2. Effect of variability of c, Ko and Pn on (0,(x, F,,)) and @(13,(x, Fo))/dFo) for Lu = 0.4.
0.
/’
’
K0’1.5
E’0.3
C=O.l
K0’1.5
E’O.5
I
/
/
/
I
1.0
I 1.5
Fo Fig. 3. Effect of variability of Lu* on (6’,(x, FJ) and (8(&(x, F&/S,) K,= l.SandP. =O.l.
5
/
/
,
for c = 0.5,
2.0
,
1.0
Fo -
1”s
2-O
990
K. N. RAl and R. N. PANDEY
and [a(e,)/aF,]/[a(e,)/aF,] for different values of generalized time F0 in the range 0.5 < Fu < 2.0 for case 1 (8, > &). It is observed that as Lu* increases, the ratio of generalized transition times increases and the two transition times become comparable at Lu * = 0.6, whereas for a fixed value of generalized time F0 the ratio of the average transfer potentials decreases and the two average transfer potentials become comparable at Lu * = 0.6. Also the ratio of the generalized time rates of average transfer potentials for heat and mass increases with increase of Lu* and again the two generalized time rates of the average transfer potentials become comparable for Lu * = 0.6. The influence of q Ko and P. over these ratios are shown in Table 2. In the case of 6 and Ko, a similar behaviour is observed but the influence of these criteria is less significant than that of Lu*. Further, it is seen that the effect of P,, is opposite to that of Lu*, c and Ko. Figures 1-5 show the influence of Lu*, E, Ko and P. on the average transfer potentials and the generalized time rate of average transfer potentials. The most important influence on the transfer potentials for the heat and mass is that of the modified Luikov number Lu *. For Lu * > 0.6, the mass transfer potential lags behind the heat transfer potential and for Lu * > 0.6, the mass transfer potential precedes the heat transfer potential. NOMENCLATURE thermal diffusivity coefficient diffusion coefficient of moisture in the material specific heat capacity of the material specific isothermal mass capacity of the material beat flux 4m mass flux R radius of the cylinder time heat transfer potential temperature of the surrounding atmospohere coordinate perpendicular to the central axis of the cylinder mass transfer potential equilibrium value of mass transfer potential specific heat of evaporation Soret coefficient thermal conductivity moisture conductivity heat transfer coefficient Dimensionlessvariables X=x/R
dimensionless length coordinate
F~=[a;“;;“R:
e,(x,m = v-
75~~ e,(x, Fd = (e, - e)/Ae
dimensionless temperature dimensionless mass transfer potential
Similarity crireria Luikov number La = a, /a, Lu’ = L&(1 t u*)/U+ (I,) modified Luikov number phase change criterion Kossovich number Ko = p(W~XA0lAT; P. = &(AT/AB) Posnov number Kirpichev criterion for heat transfer Ki, = (@)/(&AT) Kirpichev criterion for mass transfer Ki, = (q,r)/(A,W Biot criterion for heat transfer Si. = (aR)/(A,)
7-f- To boundary conditions of the first kind AT=
To
[ T, - To
boundary conditions of the second kind boundary conditions of the third kind
boundary conditions boundary conditions boundary conditions partial derivative of
of of of ( )
the first kind the second the third kind with respect to X. REFERENCES
[I] I. J. KUMAR and H. N. NARANG, Inl. 1. Hear Mass Tmnsfcr 10, 1095 (1967). [2] T. R. GOODMAN, Adaances in Hear Transfer, Vol. 1. Academic Press, New York (1964). [3] A. V. LUIKOV and Yu. A. MIKHAILOV, meory of Energy and Mass Tmnsfer. Pergamon Press, Oxford (1%~). (Received I6 Lkcem ber 1974)