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temperature gradient induced natural convection
It=t..I. Heut A,las.~ l'runslt'r. Printed in Great Britain
Vol. 3b. No. 4. pp 1097 I112. 1993
0017-931(I/9356.00+0.00 ,i, 1993 Pergamon Press Lid
Analysis of evaporation in the presence of composition/temperature gradient induced natural convection S H O U - S H I N G H S I E H t and N A N - H U E I K U O Department of Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan 80424, R.O.C. (Receit,ed 16 Jull' 1991 and in final form 7 January 1992) Abstract--A numerical analysis is performed to investigate the effects of the presence of combined composition and temperature gradient induccd natural convection for evaporation from an open-topped tube which is partially filled with a liquid. Numerical solutions were carried out for water as the evaporation vapor and air as the gas. The present numerical results obtained at Gr r = l04 and GrL = 105 with Gr~ = 10", 105 and 106, respectively, for a completely dry case (i.e. ~b = 0) and a relatively humid case (i.e. q~ = 90%). Results were presented in terms of streamlines and isotherms as well as isohaline distributions. Numerical data show that there is no definite distribution among these three distributions as GrM is increased. However, the change seems to follow a definite trend and it is reasonable.
1_ I N T R O D U C T I O N
THIS PAPER concerns the natural convection that is induced in a binary g a s - v a p o r mixture due to evaporation at a liquid surface which b o u n d s the mixture from below. Such a situation is c o m p h c a t e d due to the s i m u l t a n e o u s presence o f differences in temperature and variations in concentration_ The physical system to be considered here is an o p e n - t o p p e d circular tube that is partially filled with a volatile liquid. The partial pressure of the v a p o r in the a m b i e n t is m a i n t a i n e d at a lower value than that at the liquid surface, so that e v a p o r a t i o n occurs. T h e t e m p e r a t u r e condition was chosen to simulate an oil storage tank exposed to the a t m o s p h e r e a m b i e n t for the petroleum industry. A survey of the literature revealed very few prior studies of n a t u r a l convection either for simultaneous presence o f differences in t e m p e r a t u r e and variations in c o n c e n t r a t i o n or for sole mass transfer problems. S p a r r o w et al. [1] carried out a numerical study for e v a p o r a t i o n from an o p e n - t o p p e d vertical tube which is partially filled with a liquid. Later, N u n e z and Sparrow [2] presented an analytical model for isothermal and n o n i s o t h e r m a l e v a p o r a t i o n o f a liquid from a partially filled cylindrical tube. S u b s e q u e n t work reported by S p a r r o w and N u n e z [3] was performed to provide a critical test o f analytical/numerical models_ In spite o f this, only c o m p o s i t i o n - i n d u c e d natural convection was investigated_ A g g a r w a l et al_ [4] studied numerically b u o y a n c y - d r i v e n flows o f gases a b o v e liquids in a c o m m o n enclosure with n o n u n i f o r m heat-
t-Author to whom correspondence should be addressed.
ing from above. However, the paper only considered the differences in temperature. In view of the foregoing discussion and an extensive review o f the literature, it indicates that, indeed, no effort has been made to study e v a p o r a t i o n in the simultaneous presence of c o m p o s i t i o n and temperatureinduced n a t u r e convection. It is believed that a substantial body or result is needed in order to have a complete description of heat transfer, mass transfer and flow fields in this i m p o r t a n t area o f practical applications. F o r instance, the e v a p o r a t i o n loss of an oil storage tank when it is exposed to the atmosphere. Therefore, the main objective of this work is to investigate the effect o f coupled thermal and mass diffusion on the velocity, temperature and conc e n t r a t i o n fields in the natural convection flow of a gas-liquid v a p o r mixture in a finite o p e n - t o p p e d circular tube.
2. M A T H E M A T I C A L F O R M U L A T I O N
The system being studied is an o p e n - t o p p e d vertical circular tube o f finite length partially filled with liquid water (see Fig. 1 for details). The liquid surface and circumferential surfaces of the tube are m a i n t a i n e d at a uniform t e m p e r a t u r e T~ higher than that of the b o t t o m wall (To). This particular flow has obvious applications in the petroleum industry for the design a n d p e r f o r m a n c e assessment of oil storage tanks. The moist air in the a m b i e n t is d r a w n into the tube by the simultaneous action o f c o m b i n e d b u o y a n c y forces due to t e m p e r a t u r e and c o n c e n t r a t i o n gradients. Since the molecular weight of water v a p o r is smaller than that of air, the b u o y a n c y force due to mass transfer acts upwardly. As a result, in the present study the flow
1097
1098
S.-S. HSIEH and N.-H. K u o
NOMENCLATURE
Ar ArL Ar(;
aspect ratio, H / R aspect ratio for liquid phase, L/R aspect ratio for gas phase, h/R ( = Ar-- Aru)
B
(C~.,- C~ =)(] -o~,~.)/C~
specific heat [J kg- ~ K-~] mass diffusivity [m-' s-~] buoyancy per unit mass [m s -'] acceleration due to gravity [m s -'] Grashof number for liquid phase, gfl(T~. - To)R ~/v 2 GrT Grashof number for heat transfer, g( T~_ -- To)R a/v2T~ Gr M Grashof number for mass transfer, g ( M 2 / M . - I)(I - m . ~ ) R 3 / v ~h distance between liquid surface and tube opening ( = H - L) [m] hf~ latent heat of vaporization [J k g - ~] H tube height [m] k thermal conductivity [W m - ~ K ~] L height of liquid pool [m] M molecular weight [kg kmol- ~] rn evaporation rate [kg s-~] ~h" evaporation rate per unit area [kg m - -' s- i] p rescaled pressure (p" + p~gx) [ k g m - i s i] p' pressure [kg m - i s - ~] P dimensionless pressure, pR ~-/pv~Pr Prandtl number, v/~ r radial coordinate [m] R tube radius [m] universal gas constant [kg m= kmol - t s -2 K - I ] Cp D Fh g GrL
induced by the buoyancy force of mass transfer is retarded/counteracted due to T~ > To. The governing equations for the present case are the incompressible steady Navier-Stokes equations along with conservation of component and energy for two-dimensional flow_ The Boussinesq approximations (the concentration of water vapor in the mixture being very low and the temperature nonuniformity in the system being small) are employed to include the buoyancy effects due to temperature and concentration gradients_ As stated earlier, the key feature of the analysis is the buoyancy term which appears in the x-momentum equation. At any point (x,r), the local body force Fu per unit mass can be written a s ax
evaporation Reynolds number, 4rh/2pnR Schmidt number, v/D temperature [K] axial velocity [m s ~] dimensionless axial velocity, uR/v radial velocity [m s- ~] dimensionless radial velocity, vR/v dimensionless mass fraction,
x X X"
axial coordinate [m] dimensionless axial coordinate, x / R dimensionless axial coordinate from interface, X - ArL.
(o~,-
co, ~ ) / ( 1 - ~ , ~. )
Greek symbols thermal diffusivity [m 2 s ~] thermal expansion coefficient [K-~] dimensionless radial coordinate, r/R 0 dimensionless temperature, (T~ - T)/(T,. - To) kinematic viscosity [m: s '] density [kg m 3] P relative humidity (.O mass fraction. Subscripts 0 at initial condition or in the ground 1 component I, vapor 2 component 2, air in the ambient G gas phase I in the interface L liquid phase.
Ox
Pg
Ifp = ( p ' + p ~ g x ) , then equation (1) becomes
pg 1 -
(2)
in which p and p~ denote the densities of the mixture at point (x, r) and in the ambient at the open top of the tube, respectively. 2.1. Governin 9 equations 2.1.1. Liquid phase. Using the Boussinesq approximation, p~ can be written as p~ ~ p [ l - f l ( T ~ - - T ) ] .
(3)
Therefore, the buoyancy force in the liquid phase can be represented as : pg l - -
pr
pFb
Re Sc T u U v V W
--
~
p g f l ( r ~ : - T). (4)
(I)
After introducing the following dimensionless variables,
Effects of composition and temperature induced natural convection for evaporation
Symmetric
Equations (5)-(8) constitute a coupled nonlinear system for U, V, P and 0. The two prescribed parameters appearing in equations (6) and (8) are the Grashof number (GrL) and the Prandtl number (PrL). 2.1_2. Gas phase. The mixture can be treated as a perfect gas, so that
OXlS
t
O=O,W=O r
I
Gas p~se
Interfece 7
C>ro
pM P = ~"
O= O,w = w
I
8=0
I099
P~-
tr~M~ RT
(9)
Since the variation of the pressure is very much smaller, this implies that the following approximation stands
/.-/
Liquid p~se
0o) Therefore,
f
U,X
To,O=l
, v
and
FIG. 1. Physical system of the present study.
x X=~,
r ~=~,
T~ - T 0-T~-To'
uR U=~-,
t,R V = - - ,v
P-
pR 2 pV 2 "
v 9fl(T~ - To)R 3 P r L = - and GrL-y2
Subscripts c and t denote the buoyancy due to concentration and temperature gradient alone, respectively It was assumed that the LHS ofequation (10) could be decoupled :
= [(,-
the liquid phase is governed by the following equations Continuity
au
1 a
+ ~ ~(~v) = o
(51
The mixture's molecular weight can be expressed in terms of the molecular weights M. and M_, of the components I and 2 (the vapor and the gas, respectively) and of their mass fractions ~o~ and ~o,_: M~ ~1M2+(1 - w , ) M , l-- M - = 1 - w l ~ M 2 + ( l _ ~ , ~ ) M i
X-Momentum 0 a ~(uu)+ ~1 ~(~vv)
aP 02U 1 0 [ aU'~ ~+~X~_+~L~)-GrcO
-
+(,-
(6) The substitution of equation (11) into equation (2) with the above assumption yields
l-Momentum 0 1 0 ~X ( v v ) + ~ O~ (~w)
-
aP
~2V
I O(
OV~
a~+bY~-~+?N\ N / - ~
V
(7)
Energy equation
y(uo
2l_ + ~-~ L~'~-)J. (8)
Again, after introducing the following dimensionless variables,
x X=R,
r ~=?,
U=
uR v '
vR V=--,v
P-
pR 2 pv 2'
I100 O-
S_-S. HSmH and N.-H. Kuo --T~ - T
W__ O)I --OJlx
T~ --To"
Pr G ~
1-~,~ Grr --
V -
?0 .~:0, OU a~-=0, v=o, ~=0
Y
Sc = ~ ,
g(T., - To)R v "-T.,
on the peripheral side = 1.
9(~/i-
V=0,
0=0.
(20)
2_22 Gasphase. The boundary conditions are
and
1,2
~3U ~3~=0,
X=Ar,
c""-c""- (i-
c.
the governing equations for the gas phase can be written as :
~=0,
OU
8(=0,
V=O,
0=0,
?0
V=0,
?~=0,
W=0
~W
-~-~= 0
(21)
(22)
on the peripheral side
Continuity OU
1
ax+~ ~(~v)=0
(13)
X-Momentum
a ~ ( V V ) + ~ 100~ ( ~ v v ) = - a xc3P O~-U
1 O ( OU~_GrTO+GrM W
+U~-+Ta~t ac)
(14)
~-Momentum c~
U=0,
I ) ( l - - m , , )R 3
Gr M =
B-
(19)
1 c~
ax(UV)+ ~ ~(~w)
~=1,
v=o,
u=o,
0=0,
/~W ?~=0.
(2.3)
Based on the extensive discussion reported in Sparrow et al. [1], a realistic boundary condition was employed as formulated in equation (21). 2.2.3. Interlace. The surface of the liquid (X = Arl) was considered as being impermeable to the gas. In accordance with the impermeability condition, Ihe naturally occurring diffusive flow of gas to the interface (diffusion from higher to lower concentration) must be balanced by a convective flow of gas away from the interface_ This balance yields (at X = Art_ and any r) t~CO -~
~P
?")2V
I ~ /" OV~
V
p,_u = pD 0 X '
in which D is the mass diffusion coefficient_ Since a9, = 1 -~o~ and p = p,+p~_, equation (24), in dimcnsionless form, becomes
Energy equation
,
B(a0aw a0T( (]6) Concentration equation
uo., -
W~ = /Fc~-'w
I 0
--seLax'--" + ~ ( ¢ ~ 0 ]
(17)
Equations (13)-(17) constitute another coupled nonlinear system for five variables U, V, P. 0 and W_ The four prescribed parameters are the Grashof numbers (GrT, GrM), the Prandtl number (PrG) and the Schmidt number (Sc)_ 2.2. Boundary conditions 2.2.1. Liquid phase. The boundary conditions are U=0,
V=0,
0=l
(18)
z(0w)
l Z w, Sc \ a x JG.,
(25)
where W l = ( o h - - m l , ) / ( l - - c o l + ). The interracial mass fraction o)~ is obtained by assuming the gasliquid interface to be in thermodynamic equilibrium. that is,
~(uw)+ ~~(c.vw)
X=0,
('_'.4)
Mtpl M2(p--pO+ MIPE
(26)
where p~ stands for the corresponding saturation pressure at 0~. Equation (25) serves as the boundary condition for interface evaporation velocity. The boundary conditions for Vat the interface can be derived from the shear force balance between the liquid and gas phases at the interface [5] ; in dimensionless form, we have
\.o} \ox/, For temperature boundary conditions, at X = ArL,
Effects of composition and temperature induced natural convection for evaporation from the energy balance at the interface, the interracial dimensionless temperature can be obtained as --kc, ~
+pc;,hrg (T~_ - To) -
kL
~X
"
(28)
In the present study, energy transport at the interface in the presence of mass transfer depends on two related factors. They are the gas temperature gradient at the interface and the rate of mass transfer. 3. N U M E R I C A L P R O C E D U R E
The governing differential equations (5)-(8), (13)(17) and their boundary conditions were solved as an adaptation of an elliptic finite difference scheme ( S I M P L E C ) set forth in ref. [6]. Two computational domains are employed, the liquid and the gas phase are treated separately and the conditions (equations (27) and (28)) at the interface are used to match the velocity and temperature fields from both phases. The numerical methodology for treating the gas-liquid interface was exactly as in Prata and Sparrow [5]. Through Table I, the grid dependence examination was made. Finally, a 32 × 32 hybrid nonuniform/uniform grid was chosen_ The layout of the grid in the liquid phase and the gas phase was arranged as follows : (a) liquid phase X-direction, AXIng. = 0.05ARE (nonuniform) AX,,m = 0.0083ArL (nonuniform)
1101
the computational region. The under relaxation values were 0.5 (for U), 0.6 (for V), 0.7 (for 0 and IV), and 0.8 (for P) respectively. The parameterization of the solutions will be presented and discussed_ As stated earlier, the numerical solution was performed for water as the evaporating vapor and air as the gas. For this system, the Schmidt number Sc is 0.6 [7]. The reference condition is chosen at T~ = 37:C, To = 23"C, with an associated GrL = 105 and Grv = 104, GrM = 10 d, 105 and 106, respectively, and a system pressure of I arm, while o9, = 0.04 and w,:,. = 0.0 (completely dry ambient). For the second case, to, = 0.04 and co,:~ = 0.036 (ambient relative humidity of 90%) with the same corresponding values for GrM. The physical properties of the working medium (water/air) were chosen at T = 7-,+ 1 / 3 ( T ~ - T ~ ) and o9 = o9,~,+ 1/3(o9~-o9,~) [8] instead of the conventional average weighted values. The present numerical calculations were performed on a 486 IBM compatible PC.
4. RESULTS A N D D I S C U S S I O N
The results to be presented and discussed will begin with the rate ofevaporation. Subsequently, to provide a basic understanding as well as to provide further insights, streamline, isothermal and isohaline maps, velocity and temperature profiles, and mass fraction profiles will be presented and discussed. 4.1. Rate o f evaporation The rate of evaporation at a differential area dA on the liquid surface (x = L) can be written as
C-direction, A~ = 0.05 (uniform) (b) gas phase
0o9,
X-direction, AXm,~ = 0.0489ArG (nonuniform) AXmi. = 0-0083ArG (nonuniform) ~-direction, A~ = 0.05 (uniform). A variable G is said to be convergent if its residual is smaller than a pre-assigned value. The residual is defined as
where both the convective and diffusive contributions have been included. By making use of equation (24) with O t o 2 / O X = - & o , / O X and noting that p = p, +p,_, the expression in equation (30) becomes pudA
G"-G"/3 = max G"
I
i,j
(29)
(31)
where dA = 2nr dr. Therefore, the rate ofevaporation can be expressed as :
where n refers to the nth iteration and i, j stands for the node position. In the above expression, the residual/3 represents the maximum values throughout
rh =
f:
pu2rcr dr.
Table I. Grid dependence examination for evaporation rate Reynolds number, absolute maximum stream function for liquid and gas flow, interface average temperature and bulk concentration
Re I~Llm~ I~olm~, 01 Wb
(30)
17x7
22x 12
27x 17
0.006121 0.454384 0.069731 0.001614 0-001833
0_006148 0.649866 0.052742 0.003758 0.001843
0.006154 0.709724 0.053888 0.003653 0_001844
32x22 0_006159 0.737772 0.058255 0.003585 0.001845
37x27 0_006156 0.749904 0.055292 0.003638 0.001845
42X32 0.006160 0.737324 0.058444 0.003565 0.001846
(32)
1102
S.-S. HSIEH and N.-H. Kuo
The Reynolds n u m b e r can be o b t a i n e d in the following by incorporating n7 and the tube radius R for the tube flow: Re -
Re = 4
Curves are included representing the present results for c o m b i n e d temperature and c o m p o s i t i o n gradients and c o m p o s i t i o n gradient alone, as well as those of the Stefan solution. E x a m i n a t i o n of the results of G r L = 10 5 a n d G r T = 10 4 for three different GrM = 10 4, l0 t , and 10 ~' for the completely dry and humid cases, shown in Figs. 3(a) and (b), indicates that the G r a s h o f n u m b e r for mass transfer has a great influence on the e v a p o r a t i o n rate_ It is also seen, except for GrM = l 0 6 a n d co,~ = 0.0, that the evaporation rate decreases m o n o t o n i c a l l y as the distance between the liquid surface and the tube opening increases. This b e h a v i o r reflects the increased resistance to mass transfer associated with the greater transport length. On further e x a m i n a t i o n of Fig. 3(a), at GrM = 104, it is seen that b u o y a n c y e v a p o r a t i o n rates, either due to a sole c o m p o s i t i o n gradient or double diffusion, are nearly the same as those for the Stefan problem at the completely dry ambient. The result at GrM = l0 t coincides with that given in S p a r r o w et al. [1], in which it was found that e v a p o r a t i o n rates are 1-5 times those o f the Stefan solutions. This is partly caused by the effect o f the thermal and solutal buoyancies, which are c o m p a r a b l e (GrM = Grv = 104), and partly by the c o u n t e r a c t i n g effect of thermal buoyancy. As GrM is increased to 10 -~, the discrepancy a m o n g these three solutions remains up to a distance between the liquid surface and the tube opening of 4.5_ A m o n g the discrepancy, double diffusion gives the highest e v a p o r a t i o n rate to three times the Stefan value even though the thermal gradient reduces buoyancy. C o n c e n t r a t i o n gradient induced b u o y a n c y gives the values in between Stefan and double diffusion. At GrM = 10 ~', this trend is quite different and the superiority is switched. However, c o m p o s i t i o n gradient driven convection gives higher values than those
(33)
2r~Rll
I0'
U~ d~.
(34)
E q u a t i o n (34) was evaluated using the numerical integral scheme As m e n t i o n e d before, several grid sizes were tested and a c o m p a r i s o n o f the results a m o n g these c o m p u t a t i o n s is given in Table 1 for the e v a p o r a t i o n rate Reynolds number, the absolute m a x i m u m stream funcLion for liquid flow and gas flow, the interface average t e m p e r a t u r e and the bulk v a p o r concentration. It is noted that the differences in the e v a p o r a t i o n Reynolds numbers, absolute m a x i m u m stream functions, interface temperatures and bulk mass fractions are less than 1%, by using 3 2 x 2 2 and 4 2 x 3 2 grids. Accordingly, the comp u t a t i o n for the 32 x 22 grid is sufficient to u n d e r s t a n d the heat and mass characteristics in the flow. In addition to the grid dependence examination, the present numerical scheme was used to redevelop the case of Sparrow et al. [1] for b o t h the completely dry a m b i e n t (i.e. w,~ = 0 . 0 ) and very humid (i.e. ~o,~ = 0.036) a m b i e n t conditions. The c o m p a r i s o n shown in Fig. 2 reveals good agreement between b o t h results. The numerical results for the e v a p o r a t i o n rate are presented in Fig. 3, for Ar L = 2 , G r L = 10 ', GrT = 104, and co, = 0.04 for two a m b i e n t conditions (one for the dry and one for the humid case) with GrM = 104 increased to GrM = I06. In this figure, the e v a p o r a t i o n Reynolds n u m b e r is plotted as a function of the g a s - v a p o r space, which is in terms of Ar~;.
%==0 036 (~=90%)
w,® = 00ldry)
0 2,~
00iO
~
- -
0 20
0 16
I
Present study r r o w e t o~ [ I ]
o 008
OOO6
(xl ~ 0 i2 -0004
ii
~ P4
ii ~
~ oo8 0¸002
0 04 ~,== 0 036 {~ = 90%1
O
2
4
6
8
nO
Ar G = h/t?
FiG 2. Rerun Sparrow et al. [I] for GrM = 104, o), = 0.04 (saturation), and for o.h~ = 0.0 (4; = 0%), tol,~ = 0.036 (~b = 90%).
Effects of composition and temperature induced natural convection for evaporation O 08
1103
00C~ X
S t e fc~n soLutiOf~
....
Bouy~ncy cono~ntra
0 06
due to tlon gradient
OCX3~
\
000zl
0 04
Gr M : i0 4
002
6r~, 0
:
-4
IO*
-'4
i ~ . [ . . , I ' i i I i i i I i i i
0
•
oh6-
~.
Ol2
J
'
J I
' '
'
J
'
~
'
t ' i i I
~
J
'
I
.
L
i
I
i
r
0 008
20
0
O00E
O0(>q
o
-?
08
0 002
oo4
Gr u : 10 5
~M=I
o
i i ,
I
'
'
'
I
J
i i
I
,
i
i I , i
L
2 o
J
,
I
'
'
,
J
i
i
i
o 020
I
6
I
2
0o
0
B
0 OO8
0
n
00,61 12
~=I0
GFM:
10 6
,,,~,',~,;,~,r,~,~ 2
4
6
. . . . . . . . . . . . . . . . . . . . 8
I0
0
2
q
6
8
Ar e : h/R
(a) ~b : O*/.
(u.,, : 0 0 )
(b) qb: 90*Io (w,®: 0 0~,6)
FIG. 3. Evaporation rates for Ar, = 2, Gre = 10~, Grv = 104, o.h = 0.04 (saturation) and: (a) (o,, = 0.0 (~b = 0%); (b) co,, 0.036 (qb 90%). =
of double diffusion. The enhancement stems from the dominance of the buoyancy-driven convection motions either by composition gradient alone or double diffusion, yielding evaporation rates up to 1_5-4.0 times the Stefan value. The peaks for composition gradient convection/double diffusion all happend at Arc, = 6. The reason for this strange phenomenon still seems unclear at this stage. However, it is certain that the physical interaction due to mass and temperature gradients makes the problelm more complex than that discussed in Sparrow et al. [1]_ The evaporation results in Fig_ 3(b) convey a different message compared to Fig. 3(a). Here, the buoyancy evaporation rates for both thermal and solutal convection and solutal convection alone are, except for GrM = 10 ~, the same as those for the Stefan problem. Following Sparrow et al. [1], the natural convection is unaffected in this case due to the small difference between the vapor mass fractions at the liquid surface and in the ambient (i.e. oJ.-eo.~, = 0•004), as is also reflected by the values of GrM = 104 and 105 . However, the present study of double diffusion gives slightly lower values than those of the Stefan and sole composition gradient solutions as Arc, increases. This is quite obvious because the present temperature gradients cause a counteracting effect in evaporation rates which results in an evaporation rate decrease• Moreover, at GrM = 106, a crossover point occurs at Aro ~ 3, where evaporation
=
rates due to double diffusion are higher than those of the composition gradient alone when ArG >~ 3. Furtherrnore, it is found that, keeping the gas-vapor region fixed, the evaporation rate seems to he unchanged as the liquid space is varied (not shown here). 4_2_ Streamline distribution, isothermal map and isohaline distribution The streamline distribution information to be presented includes the results of the concentration gradient alone and double diffusion convection at two extreme ambient conditions• They are shown in Figs. 4 and 5, respectively. Moreover, to observe the gas-vapor space influences on the flow pattern, Fig. 6 is now added. Figure 4 shows three different Grashof numbers for mass transfer, GrM = 104, 105 and 106 for the dry and humid cases• Only the gas-vapor region is shown. For the completely dry ambient shown in Fig. 4(a), at GrM = 10 ", the streamline distribution indicates that the pattern of fluid flow is similar to a uniform upward flow. The evaporation vapor is represented by the streamlines which originated at the gas-liquid interface• As GrM is increased to 105, the presence of a U-shaped flow loop occurs. This indicates that the flow is downward in the outer annular and upward in the core, which is similar to that observed by Sparrow et al. [1]. Note that the downflowing fluid very nearly penetrates to the surface of
1104
S.-S. HSlEH and N.-H. Kuo GG,, = IOs
GA, = I0 ~
GG,, = I0 ~
Value
~b VaLue 9
r
I 4E-2
7: I I E - 2
6
@ VaLue 9:1 BE+0
9 -44E+0
7 i 3E+O
7 - I 7E-I-O a
5: - I 7E+ I
5:77E-3
5: 9 : ~ - -
:5 4 5 E - : 5
:5: 5 2 E - I i- i 4E-I
5 - 2 :SE+ I
o: 2 8E - 2
O: 5 8 E - 2
I: 8 5 E - 4
tl
I
I -29E+I
X' V"
_
I
--~'0
g
(a) ~5= 0 %
(wj:= O)
Gfr~ = I 05
GrM = I 0 4
G',"r,~ = I 0 '~
I
i VaLue i
9 I 5E-3
VoLue " 9: 1 8 E + O
~5 VaLue
i'
9: I 5 E - 5 6
7 liE-5! .5 7 7 E - 4 !
:5:4:5E-4 [
11117
? I IE-5 5 : 7 8E-4
7 : l 4E+0 5:9 7E-I
5 : 4 5E-4
5: 5 4 E - I
Ar G
I: I I E - I
i. B 7E-5
l a 7E-S i
c: 2 7 E - 3 /
4i" "=-0
-="- O
g
r
(b)q~=90%
¢
(%®=0036)
FIG. 4. Streamline distribution, concentration gradicn[ only, for A r e = 2, tu~ = 0.04 (saturation) and : (a)
Oil, = 0 . 0 (Q/) = 0 % ) :
the liquid, whereupon it turns and sweeps across the surface as it moves radially inward. As can be seen in the figure, the circulating flow, upon approaching the tube axis, wells up from the liquid surface and streams upward to the tube opening. At GrM = 106, a small cell exists at the bottom of the gas-vapor region and, consequently, the U-shaped flow loop appears to be broken near the bottom of the gas phase. On the contrary, the flow is upward in the outer annulus and downward in the core. The strength of the streamline becomes stronger a s G r M is increased. This can be seen for the absolute value of the streamline. On the other hand, for the humid case, shown in Fig. 4(b), the flow patterns are similar to those of the completely dry case, except for an occurrence delay for the U-shaped loop in the GrM effect. This is perhaps because the present temperature gradient counteracts the mass gradient and such an effect is significant in the humid case. Figure 5 shows the streamline maps for both gas-vapor and liquid regions for GrM = 104, 105, and 106, respectively, at the two ambient cases. Unlike composition gradient driven convection (see, for instance, Fig. 4 at GrM = 10~), the flow from the U-
(b)
O) 1," =
0.036 (~b = 90%).
shaped flow loop (see, for instance, Fig. 5 at GrM = 105) is swept in the outer annulus and downward in the core_ After several trial runs, it is found that this phenomenon disappears when Grc < l02 and GrT < 10. The streamline map at GrM = l04 and 105 in g a s - v a p o r region in Fig. 5(b) shows a dual cell recirculation. One thing worthy of mention here is that when the strength of streamline becomes weaker, in accordance with the shear force balance between liquid and gas flow at the interface, a recirculation cell is formed in the gas region adjacent to the interface (at GrM = l04 in Fig. 5(a), and GrM = l04 and 105 in Fig. 5(b)). The above statements provide adequate reasons to explain why, for the humid case, the evaporation rates due to concentration gradient alone are a little higher than those of double diffusive convection, which may not always be the case as would be expected (see, for instance, Fig. 3(a) for Gru = l05 and Fig. 3(b) for GrM = 106). F o r the humid ambient case, in the liquid region, the streamline distribution displays dual recirculation cells for GrM = l04, 105 and l06. However, for the completely dry case, such behavior only occurs
Effects of composition and temperature induced natural convection for ew*poration
Gr = IOs
GrM = 10'
~Vatue]
~b Volue 9: 45E-I 9 -4 2E+O 7: 1.6C~FO 5;- I 6E~ I
7:-I .I E-~O 5: I.BE+O 3:-2.5E+0 h-5.1E+O O: 4.5E-2
il' ( I
5: - 2.2E+
illltll// ))/llllll A~ il\\~llll
I
qbValue
9:-2.8E-I 7:-6.5E-I 5: - 1.4E-I-O
9-
Art.
2
5E+O
7: 13E÷O 5:-4.3E-I
5:-1.4E+0
3: - I 9E÷O
I :-1.8E+O ,
I : - 3 4 E'I'O
4
xTo C, ~
rG
h - 2 8E+I
¢ V a t ~ "~'-
~b Vatue---~lj ~
5.-7 4E-I
6 r M :106
1
'I
1105
°
/"L ,
X ¸
~
(o)~b= 0% 1,,.o~==01 GrM:
104
GrM = IO 6
G r M = IO 5
,0 ',,bLue
Value 9: 4. 9E - 2 ' 7:3.6E-2 I 5: 2. 5E - 2 I 3:99E-3
9:4.8E-2 7: 5.5E- 2 5: 2.2E- 2 5:9.0E-3 1:-3.9E- 3
,atG
qS Votue
,:-zoE-, i I t ( v m ~ I
9:-I IE-I 7:-2.6E-I 5:-4.1E-I 3:-5.5E-I h-7.0E-I X
ArL
Art_
o
C
(b) qb:90*/. 1to.=:0.036)
FIG. 5. Streamline distribution for A r t = 2, A r G = 2, Gr L = 10~, Grv = 104, ~t4 = 0.04 (saturation) and: (a) co., = 0.0 (~b = 0%): (b) oJ,,. = 0.036 (q5 = 90%).
at GrM = 104. Since the liquid phase space does not affect the e v a p o r a t i o n rate at all, as stated earlier, the present discrepancy will not be discussed. F u r t h e r study m a y include this aspect. Figure 6 presents the effect of Arc (shortest/longest tube) o n the flow pattern for the completely dry case. T h e flow pattern in the liquid phase seems to be i n d e p e n d e n t o f the variation of A r G . However, the flow p a t t e r n s in the g a s - v a p o r region do change slightly. This change confirms the previous finding that the tube opening in-
crease reflects the increased resistance to mass transfer. Figure 7 shows the isotherm c o n t o u r maps corresponding to the flow discussed earlier in this section. Generally speaking, a low temperature distribution was found at the g a s - v a p o r and liquid interface except at the b o t t o m o f the tube. F o r the liquid phase in Fig. 7(b), the thermal field is stratified fairly linearly in the vertical direction in most o f the flow domain, especially in the liquid phase for a relatively humid case. In addition, for the humid case, it seems there is
1106
S.-S. HSIEH and N.-H. Kuo ArG =I0
•,4r G = 2
Value
3:-I 8E-3 1:-52E-2
+Vol. ~: 7 : 4~.,~E-~ 6E-2 ~5:28E-2 :'~-~
II
\
~ II / II / II
~
~
1
[
!~,
0 ~, Volae
.
Value
9.-27E- } 7-47E-I i
~:-" ~-' ~//k\k\%UJ/llll ':-2~-' i/\KkUJ/Ill
5-4 E-I ' ~ \ \ \ X / I l l
~ - ~ ° ~E"I-O -' 5-1.3
~:-~.~-,
X,
X
o{;
! \ \71 )kill ^
~
of, (a)
F I G . 6. S t r e a m l i n e
-~
(b)
distribution f o r Ar, = 2. Gq = I 0 ' . Gr T = I 0 ~. Gr,~ = 10 ~. m, = 0 . 0 4 ~ul~ = 0 . 0 (q~ = 0 % ) a n d : (a) A r c = 2 ; ( b ) A r e = 10.
no interaction with the gas-vapor phase. However, it is important to note that radial temperature gradients exist. These radial temperature gradients, although small in magnitude, tend to be more pronounced in the regions closer to the tube wall. On the other hand, for the completely dry case, the temperature in the liquid phase does somehow interact with the gasvapor temperature distribution. This phenomenon becomes more distinct as GrM is increased to 106. Further inspection of Fig_ 7(a) reveals that the isothermal map pattern initially (at Gr M = 104) behaves as stratified and later becomes a plume rise occupying almost the entire portion of the right half of the tube.
(saturation).
This plume rise becomes more distinct and, consequently, results in the plume getting smaller as GrM is increased to 10~. This can be seen by comparing the different plots in Fig. 7(a). This change somehow becomes moderate/delayed when compared to the corresponding plots for a relatively humid condition. The radial temperature gradients with plume rise phenomenon are essential to support the vertical velocity shear as evidenced by the corresponding streamline map, which results in convective heat transport evaporation due to the dominance of the evaporation in the main body of the flow field. Consequently, a narrow zone or annulus of intense vertical tem-
Effects or composition and temperature induced natural convection for evaporation ~r~ : I0 ~ 0
: I0 ~
Gr~
Gr
0 Value '
OVatue
9:3 7E-2
9 I TE-I 7 I .:3E-I
9:27E-I 7:2 IE-I
5 : 8 BE-2 :3.4.9E-- 2 I 98E-:3 O: OOE+O
514E-I 3 79E-2
I
\
I 2 2E-:3 0 00E+O L~'~'~5~X
I~'o
5E-i
76
5E-I
54
5E-I
I 1 5E-2
0V a t u ~ ~
8 ,0
I OE+O
9B
iO
I
I OE+O
9:8 5E-I
9:8 76
0
lit a
0:00E+O
\]
0 'v~tue ~0
: I0 6
0
0 VaLue
~ , ~-~ !
1107
5 4 5E-I
fL
:3 2 5 E - I
:32 5E-I
I 52E-2
I 5 5E-2 C 00E+O X
0 00E+O X
C (w,=:O)
to~q6:O%
G7-M :iO 4
0 VaLue
7 4:3E-:3 i ~
Oi/b"
~
, :3:3~-o , ~ - 2 . ~ \
o:oo
0 VaLue
0
~:3 E~-6E-:3 : 3 ~ .;-._:3 .. o
GT~ = ,0 G
GT~ : ~0 ~
\ \ Ii
\1
o
9:~ ~E-:3~ ' ~ ~ :3E-:3 i ~ ~ ~.~-:3i-~ :3:, ~E-:3i~
~
, :3:3E-~ ~ \
o: o o ,o
~-
/ 1 olA,o \ II \ \ '
7 I 3E-21
~
5:9oE-:3i :3 5.OE-3 i ,
/ \ |
~Ac~
/
\ I
(~:00E+O ~
J~_\
~|1
=l
0 VaLue I
O voLue I
i
I
iO I OE+O i 9:8 . S E - I I, 7 : 6 5E- I I 5:3: 42 . 55EE-- ' I Ji, 1:5.9E- 2 ~ 0 00E+O X c
~0 I OE+O
o AlL
9:8 5E-I 7 6 5E-I
l
5 : 4 5E-I :3: 2 . 5 E - I
~: 5 8E-2 00E+O X
0 VaLue [
i tO
i i
, '4rL
I
i OE+O i
9 8 5E:-I ! 7 6.5E- I I
~rL
5 4.5E-I I :3 2 . 5 E - l i I / 5 5E-2 0 0 0 E + O ~---:3_-5_~_~
i, -
=7~
x F - = ~
..
i 1b~:90%
(,,a = 0 0 3 6 1
FIG. 7. Isothermal-line distribution for ArL = 2, A r c = 2, Gr L = 105, Grv = 104, wl = 0.04 (saturation) a n d : (a) ~o~ = 0.0 (~b = 0 % ) ; (b) o ~ = 0.036 (4) = 90%).
p e r a t u r e as well as radial g r a d i e n t s exist near the tube wall. T h e isohaline d i s t r i b u t i o n s h o w n in Fig. 8 generally indicates that the c o n c e n t r a t i o n s h o w s a n e a r linear stratification in the vertical d i r e c t i o n at GrM = 104. As GrM is increased f u r t h e r (GrM= 105, q~ = 9 0 % a n d GrM = 106, ~b = 9 0 % ) , a radial c o n c e n t r a t i o n gradi-
ent exists_ At
GrM =
10 ~, the effect o f solutal b u o y a n c y out-
weighs the thermal effect, resulting in plume-like concentration distributions occupying a comparatively larger portion of the gas phase. Due to a high relative humidity for the case shown in Fig. 8(b) the concentration fields are still substantially linearly straUfled in the vertical direction, even though GrM is increased to 105_ Until GrM = 1 0 6 , the layered structure vanishes, and the solutally driven flow comes into play in much of the gas phase.
1108
S.-S. HSlEH and N.-H. Kuo GrM r I0 ~
GrM
: iOs
GrM : I0 G
~ o ~
I
w i value L
~ l VaLue
i_-3~
I F
9 3 34E-2 i7 51 7 7 E - 2 i
9: 3.32E- 2 [
,4,"s 7
5
~ 7
98oE- i 97E- 3 0 000E+O
9
~c
i
~
232E-2 i 161E-2 i
3
3 891E-.3,
0
I. I 7 8 E - 3 ! C: OOOE+O I
i
i
(a) qb:O*/.
: I0"
G,,"M : I0 s
T
Vo l Lie i
(w, =0)
ua. Votue
G.,-M= I0 '~
o
LI
f ~ v°wei
T
o / ~
~---3 l
9 : 3 91E-21
9::3 91E-2
7 3.84E-2 [
A~
'
~
5
~
5 : 3 76E-2
5 3 76E-2 i I
__7
3.3 69E-2
3 3 6~E-2 '
2I O: 3GOE-2 I
7. 3 8 4 E - 2
1 362E
__
-9--
I
i: 3 6 2 E - 2 ,
O: 3 6 0 E - 2
-°
'
9--
[ 7.:iiii:?V,'ll?
C
-o
C
'
(1~) qb : 90':'/o (,~, = : 0 056)
FIG. 8. lsohaline distribution for ArL = 2, Arm = 2, G r L 10', G r r 104, w. = 0.04 (saturation) and: (a) ~,~., = 0.0 ((k = 0%): (b) (o,, = 0.036 (q5 = 90% ). =
4.3. Velocity profile, tenlperature distribution and mass ./i'acrion profile The evolution of the velocity fields and temperature distribution as well as mass Fraction along the tube is respectively set forth in Figs. 9, 10 and II for the shortest of the investigated tube lengths, namely Arc; = 2. Each figure displays a succession of velocity/ temperature profiles and mass fractions parameterized by the dimensionless axial coordinate X'. On the ordinate, the velocity (temperature, concentration) is embedded in the dimensionless group uD/v ((T~ -- T ) / ( T ~ - To), ~ol) which can be regarded as a local Reynolds number (local temperature, local mass fraction), The most noteworthy feature of Fig. 9 is the two-lobed nature of the velocity profile. This shape is the same as that reported in Sparrow et al. [1]_ In the central core of the tube, which extends from the eenterline (( = 0) to ~ ~ 0.6, the U velocity is negative, indicating that flow is drawn into the liquid surface. This downward-directioned flow is drawn into the open top of the tube from the ambient. On the other hand, in the annulus which spans between the outer rim of the core and the tube wall, the flow is directed upward toward the open end of the tube
=
as evidenced by the positive values of U. Generally speaking, it is interesting to note that the crossover from negative U to positive U occurs at virtually the same radial position at all axial elevations and, in addition, the crossover point is hardly affected by the gas-vapor space ratio A%. Another significant feature of Fig. 9 is the magnitude of the velocities that are induced by natural convection. For both ambient cases, as GrM is increased, the velocity magnitudes (absolute values) increase. To provide a prospective for the velocity magnitudes displayed in these figures, it may be noted that the evaporation Reynolds number corresponding to the completely dry case is approximately 0.07 at GrM = 104 and Ar G = 2. The evaporation Reynolds number can be written as ffD/v, where ti is the mean velocity of the net downflow. In Fig. 9(a), the maximum downward velocity occurs at the center with the axial eievation at X ' = 0.28 for GrM = 104 and is characterized by uD/v ~ 2.5, which is about 35 times larger than that (uD/v ~ 0_07) for the corresponding result of Fig_ 4(a). From the preceding discussion, it is clear that certain portions of the flow which streams upward through the annulus are evaporating vapor, while a
Effects of composition and temperature induced natural convection for evaporation
Grr, :
1109
IO 4
08---_
05._~-03.-~-4 -2
0 -4
0.28
0.68
1.26
200
05
i i i ] i i i ] i i i ] i i i ] i i I
25
I',,t
t illiit
JiJliil
I G G6:IO
~
~,,. =10 {3 8
,
- j c
05
%
O 3
-25
O "3
-
75
_o
i • Io.%Io.%I,. 12%1
"Io.%, Io. I,
-O 5
r i J I I I I I I i I I I I I I I I I
I oot
.... J J I I I I I I I I L I I III
I I1
2GC 25
3 ~
Gr,i : I0'~
4
Gr,., : IO 6
Syrn~¢~.
ililililililllllllt 02
04
06
75
08
0
~: (a)(:#:O°/o FIG.
9.
Velocity profile
for
(L,.,
I
2
~
4
5
I t t t [ i i i i i i [ i i i i i i i i O C2 C4 06 08
fir
=0)
',b)
A r L = ~,9 A r G = =,3 GrL =
105,
~:90°/0
GrT =
(~J®:0056)
[04, ")l
=
0.04
(saturation)
and:
(a) (o~, = 0.0 (~b = 0%); (b) (,)~ = 0.036 (~b = 90%).
certain portion of the a n n u l a r flow consists o f fluid that was d r a w n into the tube t h r o u g h the core. This indicates the presence of a U - s h a p e d flow loop, with the downflow leg in the core and the upflow leg in the annulus. On further inspection of Fig. 9, one can find that, for the completely dry case and GrM = 104, these studies indicate an evolution of the velocity profiles, which is virtually complete in the g a s - v a p o r regions_ This o b s e r v a t i o n is witnessed by noting that there are gradual a n d steady changes between the profiles from X ' = 0 to 0.28. This indicates t h a t the m a x i m u m velocity (absolute value) occurs at X ' = 0.28. However, for the r e m a i n i n g figures of Fig. 9, it is f o u n d that the m a x i m u m velocity occurs at the centerline of the tube and axial location at X ' = 0.68, except for Fig. 9(b) at Gr M = 105 ( X ' = 0.28) which is not the top of the tube, T h e reason for this is not clear at this stage.
Figure 10 represents the c o r r e s p o n d i n g temperature distributions of Fig. 9. Generally speaking, the evolution of the t e m p e r a t u r e profile is virtually complete in the upper half o f the g a s - v a p o r region. As GrM is increased to 10 ~, s h o w n in Fig. 10(a), it is n o t e w o r t h y that the temperature gradients near the vertical walls become steeper_ Profiles of the v a p o r mass fraction co ~are presented in Fig. 1 I. This figure is the c o u n t e r p a r t of the velocity and t e m p e r a t u r e plots (Figs. 9 and 10). In the figure, curves are plotted at a succession of axial elevations. Based on the present b o u n d a r y condition, one can find that the o)~ profile at the tube opening (X' = 2) is horizontal and c,J ~~ = 0 or 0.036 all across the opening, which is also confirmed by Sparrow et al. [1]. At any axial elevation, the v a p o r mass fraction increases from the tube axis to the wall. This trend is in accordance with the flow patterns illustrated in Fig. 5. The
I110
S.-S. HSIEH and N.-H. Kuo
O(:108
0.05
"lo , Io%,I, %,I=%oq O 04-
~
0 03"
--I
O O I O71 1.31 ~.O0
= 10°
0 02 I
OOi
O
° ~ ° - ~ ' N I s " ~ l ' I 2 I 3. I * I 5 1 SI ~ -" I o IO.3, IO.7, I,.3.,Izoo]
0.0~
'
I' 1o
Io%,I, 12:ol
OOO6 ~ I
I ii
°°?
\, 0.002
~
4
O - ill/ 51- i ] l J l l l J r ] l 0.4
O020~
G,-M : I0 s
jlj
GF. : l O S ~
0.3 O 012 O 2--
I -Iolo3.,Io7,1 ,[2ool Symlx~. I
2
3
4
5
O I --
000B
0{ ~l ill I ill ill I ~llJJl 0 2
0~
06
08
Sym~c4. I
X 0
2
3
4
5
Iolo3.,IoT, l, ,] ool )
02
4 04
~ 06
08
I
= rlR
(o) (;b:0% ((.,a,®:0)
(b)qb:0°/o (~,®:0.036)
FIG. 10. Temperature profile for A r L = 2, Arc; = 2, Gr L = l0 s, GrT = 10~, ~o~ = 0.04 (saturation) and : (a) o~, = 0.0 (q~ = 0%); (b) o)., = 0.036 (4) = 90%).
extent of the radial variations is accentuated for the completely dry case, as can be verified by Fig. I 1. 5. C O N C L U S I O N
The nature of natural convection in the presence of composition/temperature gradients has been numerically studied for an open-topped tube which is partially filled with a liquid to broaden our basic understanding of this type of convective heat and mass transfer. The most significant contributions of the present study are the following. (I) The present theoretical analysis of combined heat and mass transfer for an open-topped tube which is partially filled with water has made it possible to examine for the first time the effect of a number of
parameters_ For instance, Gr M, GrT, Gr L and A r G for two extreme ambient conditions (4) = 0 and 90%). (2) The streamline distribution indicates that a Ushaped flow loop most likely occurs at GrM /> I05- The upflow leg of the loop is formed in the core of the tube and the do .wnflow leg is formed in the annular region adjacant to the tube wall. At GrM = 104, this loop does not exist, and two toroidal recirculation zones occur instead. (3) Isotherms corresponding to the flow field in remark (2) show that the thermal field is stratified fairly linearly in the vertical direction in most of the flow domain, especially in the liquid phase. Radial temperature gradients are found to exist and, although small in magnitude, they tend to be more pronounced in the regions closer to the tube wall.
Effects ofcomposition and temperature induced natural convection for evaporation £'¸04
O
040
i
1
2
,
q
G r M : 104
1111
2
0 o39 _
3
3
0 038---
O. 02. 4
I
q
Gre =I0J
0 037d
I=';
1" I0 , I0%,I,',, I=%ol
5
0
i
i i
i j
i
t i
i
i ~ i
i
i
i
i
i
i
"IoZ3, Io , 0056 t
I
I
I i I I
I
I
I i
I
I I
I I
I
I
I
I
0040
004
-i ,-4
o Io.23,I0. , I, 3,1.001
o
0
I 2
039
!
0 02
0"038I
Gr., :
I0 ~
4
0 C3T 6 f M : I0 ~
o
5 0
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
0 036 !
I
004
'
I
I I I
o
I
o.
I I / * I I I I I I I I I
O0~lO
C 039 ~
Is
1;I03,1o,, '4
1.31
-
f
2
0.02'
0 C38
OCI
I J I J I I
02
i
I
04
i
i J I
06
I
I
z ~ I
08
I
I
I
51
0 036
!0
,
x"
I I I I
0
02
o I i
I
J I
0'I
I I I I
06
i
08
I
I
I
O
= rlR
(a) 4):0*./° (%.=001
(b)
~:90°/o
w,=:00361
FIG. I 1. Profile of the vapor mass fraction for ArL = 2, Arc; = 2, G r L 105. Grr = 104, "h = 0.04 (saturation) and: (a) ,J., = 0.0 (4) = 0%); (b) ~,~)1, = 0.036 (95 = 90%). =
Moreover, there is an interaction with the g a s - v a p o r phase for the liquid phase at GrM = 10 5 and a plume rise occurs in the g a s - v a p o r phase. This is essential to support the vertical velocity shear as evidenced by the c o r r e s p o n d i n g streamline map, which results in convective heat transport evaporation. (4) The isohaline distribution indicates that the concentration shows a near linear stratification in the vertical direction at Gr M = 104. As GrM is increased to 10 5, a radial concentration gradient exists. The temperature field is qualitatively akin to the solutal d o m a i n convection and the overall velocity in the bulk o f the tube is fairly upward near the tube wall. (5) The aspect ratio Ara seems to have no significant influence on the velocity field as well as the temperature and concentration distribution at a definite G r a s h o f n u m b e r (Gr T, Gr L, GrM), which is essential to support the vertical velocity shear. This
explanation shows that convective heat transport is d o m i n a n t for evaporation. (6) The evaporation rate decreases as the gas vapor region aspect ratio increases for Grr~, ~< 10 5, as confirmed by Sparrow et al. [I]. Furthermore, the temperature gradient seems to have certain different influences on the evaporation rates at different ambient conditions. (7) The effect o f the variations o f G r a s h o f n u m b e r for mass transfer Grr, is not clearly noted. F r o m Figs. 4 and 5, it is not easy to obtain a definite trend to describe the influence as GrM is varied at a fixed Grt and GrT-
Ack,owledgements--The authors are grateful to the referee who provided detailed and constructive comments. This work was supported in part by a research grant from China Petroleum Refinery Company (Taiwan, R.O.C.).
1112
S.-S. HSIEH and N.-H. K u o REFERENCES
1. E. M. Sparrow, G. A. Nunez and A. T. Prata. Analysis of ewtporation in the presence of composition-induced natural convection, htt. J. Heat Mass Tran.~/~'r 28, 1451 1460 (1985). 2. G. A. Nunez and E M. Sparrow, Model and solutions for isothermal and nonisothermal evaporation from a partially filled tube. htt. J. Heat Mass Tran.~/~,r 33, 461 477 (1988). 3. E. M. Sparrow and G. A. Nunez, Experiments on isothermal and nonisothevmal evaporation from partially filled, open-topped vertical tubes, Int. J. tteat Mass Transl e t 3 3 , 1345 1355 (1988). 4. S K AggarwaI, J. lyengarand W. A. Sirignano. Enclosed
5.
6.
7.
8.
gas and liquid with nonuniform heating from above, h~t. J. Heat Mass Tran_~'[~,r29, 1593-1604 (1986). A. T. Prata and E. M. Sparrow, Evaporation mass and heat transfer from a lid-driven cavity, Nztmer. H e , t Tr~msl e t 13, 27 48 (1988). J. P. Van Doormaal and G. D. Raithby, Enhancements of the SIMPLE method for predicting incompressible fluid flows, Namer. Heat Tran.~Jer 7, 147 163 (1984). R. C. Reid, J. M. Prausnitz and T. K. Sherwood, The, Properties r~['Gases and Liquid~'. McGraw-H ill, New York (1966). L. C. Chow and J. N. Chang, Evaporation of water into a laminar steam of air and superheated steam, hit. J. Heat Mass Tran.~'lk,r 26, 147 163 (1983).
Vol. 3b. No. 4. pp 1097 I112. 1993
0017-931(I/9356.00+0.00 ,i, 1993 Pergamon Press Lid
Analysis of evaporation in the presence of composition/temperature gradient induced natural convection S H O U - S H I N G H S I E H t and N A N - H U E I K U O Department of Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan 80424, R.O.C. (Receit,ed 16 Jull' 1991 and in final form 7 January 1992) Abstract--A numerical analysis is performed to investigate the effects of the presence of combined composition and temperature gradient induccd natural convection for evaporation from an open-topped tube which is partially filled with a liquid. Numerical solutions were carried out for water as the evaporation vapor and air as the gas. The present numerical results obtained at Gr r = l04 and GrL = 105 with Gr~ = 10", 105 and 106, respectively, for a completely dry case (i.e. ~b = 0) and a relatively humid case (i.e. q~ = 90%). Results were presented in terms of streamlines and isotherms as well as isohaline distributions. Numerical data show that there is no definite distribution among these three distributions as GrM is increased. However, the change seems to follow a definite trend and it is reasonable.
1_ I N T R O D U C T I O N
THIS PAPER concerns the natural convection that is induced in a binary g a s - v a p o r mixture due to evaporation at a liquid surface which b o u n d s the mixture from below. Such a situation is c o m p h c a t e d due to the s i m u l t a n e o u s presence o f differences in temperature and variations in concentration_ The physical system to be considered here is an o p e n - t o p p e d circular tube that is partially filled with a volatile liquid. The partial pressure of the v a p o r in the a m b i e n t is m a i n t a i n e d at a lower value than that at the liquid surface, so that e v a p o r a t i o n occurs. T h e t e m p e r a t u r e condition was chosen to simulate an oil storage tank exposed to the a t m o s p h e r e a m b i e n t for the petroleum industry. A survey of the literature revealed very few prior studies of n a t u r a l convection either for simultaneous presence o f differences in t e m p e r a t u r e and variations in c o n c e n t r a t i o n or for sole mass transfer problems. S p a r r o w et al. [1] carried out a numerical study for e v a p o r a t i o n from an o p e n - t o p p e d vertical tube which is partially filled with a liquid. Later, N u n e z and Sparrow [2] presented an analytical model for isothermal and n o n i s o t h e r m a l e v a p o r a t i o n o f a liquid from a partially filled cylindrical tube. S u b s e q u e n t work reported by S p a r r o w and N u n e z [3] was performed to provide a critical test o f analytical/numerical models_ In spite o f this, only c o m p o s i t i o n - i n d u c e d natural convection was investigated_ A g g a r w a l et al_ [4] studied numerically b u o y a n c y - d r i v e n flows o f gases a b o v e liquids in a c o m m o n enclosure with n o n u n i f o r m heat-
t-Author to whom correspondence should be addressed.
ing from above. However, the paper only considered the differences in temperature. In view of the foregoing discussion and an extensive review o f the literature, it indicates that, indeed, no effort has been made to study e v a p o r a t i o n in the simultaneous presence of c o m p o s i t i o n and temperatureinduced n a t u r e convection. It is believed that a substantial body or result is needed in order to have a complete description of heat transfer, mass transfer and flow fields in this i m p o r t a n t area o f practical applications. F o r instance, the e v a p o r a t i o n loss of an oil storage tank when it is exposed to the atmosphere. Therefore, the main objective of this work is to investigate the effect o f coupled thermal and mass diffusion on the velocity, temperature and conc e n t r a t i o n fields in the natural convection flow of a gas-liquid v a p o r mixture in a finite o p e n - t o p p e d circular tube.
2. M A T H E M A T I C A L F O R M U L A T I O N
The system being studied is an o p e n - t o p p e d vertical circular tube o f finite length partially filled with liquid water (see Fig. 1 for details). The liquid surface and circumferential surfaces of the tube are m a i n t a i n e d at a uniform t e m p e r a t u r e T~ higher than that of the b o t t o m wall (To). This particular flow has obvious applications in the petroleum industry for the design a n d p e r f o r m a n c e assessment of oil storage tanks. The moist air in the a m b i e n t is d r a w n into the tube by the simultaneous action o f c o m b i n e d b u o y a n c y forces due to t e m p e r a t u r e and c o n c e n t r a t i o n gradients. Since the molecular weight of water v a p o r is smaller than that of air, the b u o y a n c y force due to mass transfer acts upwardly. As a result, in the present study the flow
1097
1098
S.-S. HSIEH and N.-H. K u o
NOMENCLATURE
Ar ArL Ar(;
aspect ratio, H / R aspect ratio for liquid phase, L/R aspect ratio for gas phase, h/R ( = Ar-- Aru)
B
(C~.,- C~ =)(] -o~,~.)/C~
specific heat [J kg- ~ K-~] mass diffusivity [m-' s-~] buoyancy per unit mass [m s -'] acceleration due to gravity [m s -'] Grashof number for liquid phase, gfl(T~. - To)R ~/v 2 GrT Grashof number for heat transfer, g( T~_ -- To)R a/v2T~ Gr M Grashof number for mass transfer, g ( M 2 / M . - I)(I - m . ~ ) R 3 / v ~h distance between liquid surface and tube opening ( = H - L) [m] hf~ latent heat of vaporization [J k g - ~] H tube height [m] k thermal conductivity [W m - ~ K ~] L height of liquid pool [m] M molecular weight [kg kmol- ~] rn evaporation rate [kg s-~] ~h" evaporation rate per unit area [kg m - -' s- i] p rescaled pressure (p" + p~gx) [ k g m - i s i] p' pressure [kg m - i s - ~] P dimensionless pressure, pR ~-/pv~Pr Prandtl number, v/~ r radial coordinate [m] R tube radius [m] universal gas constant [kg m= kmol - t s -2 K - I ] Cp D Fh g GrL
induced by the buoyancy force of mass transfer is retarded/counteracted due to T~ > To. The governing equations for the present case are the incompressible steady Navier-Stokes equations along with conservation of component and energy for two-dimensional flow_ The Boussinesq approximations (the concentration of water vapor in the mixture being very low and the temperature nonuniformity in the system being small) are employed to include the buoyancy effects due to temperature and concentration gradients_ As stated earlier, the key feature of the analysis is the buoyancy term which appears in the x-momentum equation. At any point (x,r), the local body force Fu per unit mass can be written a s ax
evaporation Reynolds number, 4rh/2pnR Schmidt number, v/D temperature [K] axial velocity [m s ~] dimensionless axial velocity, uR/v radial velocity [m s- ~] dimensionless radial velocity, vR/v dimensionless mass fraction,
x X X"
axial coordinate [m] dimensionless axial coordinate, x / R dimensionless axial coordinate from interface, X - ArL.
(o~,-
co, ~ ) / ( 1 - ~ , ~. )
Greek symbols thermal diffusivity [m 2 s ~] thermal expansion coefficient [K-~] dimensionless radial coordinate, r/R 0 dimensionless temperature, (T~ - T)/(T,. - To) kinematic viscosity [m: s '] density [kg m 3] P relative humidity (.O mass fraction. Subscripts 0 at initial condition or in the ground 1 component I, vapor 2 component 2, air in the ambient G gas phase I in the interface L liquid phase.
Ox
Pg
Ifp = ( p ' + p ~ g x ) , then equation (1) becomes
pg 1 -
(2)
in which p and p~ denote the densities of the mixture at point (x, r) and in the ambient at the open top of the tube, respectively. 2.1. Governin 9 equations 2.1.1. Liquid phase. Using the Boussinesq approximation, p~ can be written as p~ ~ p [ l - f l ( T ~ - - T ) ] .
(3)
Therefore, the buoyancy force in the liquid phase can be represented as : pg l - -
pr
pFb
Re Sc T u U v V W
--
~
p g f l ( r ~ : - T). (4)
(I)
After introducing the following dimensionless variables,
Effects of composition and temperature induced natural convection for evaporation
Symmetric
Equations (5)-(8) constitute a coupled nonlinear system for U, V, P and 0. The two prescribed parameters appearing in equations (6) and (8) are the Grashof number (GrL) and the Prandtl number (PrL). 2.1_2. Gas phase. The mixture can be treated as a perfect gas, so that
OXlS
t
O=O,W=O r
I
Gas p~se
Interfece 7
C>ro
pM P = ~"
O= O,w = w
I
8=0
I099
P~-
tr~M~ RT
(9)
Since the variation of the pressure is very much smaller, this implies that the following approximation stands
/.-/
Liquid p~se
0o) Therefore,
f
U,X
To,O=l
, v
and
FIG. 1. Physical system of the present study.
x X=~,
r ~=~,
T~ - T 0-T~-To'
uR U=~-,
t,R V = - - ,v
P-
pR 2 pV 2 "
v 9fl(T~ - To)R 3 P r L = - and GrL-y2
Subscripts c and t denote the buoyancy due to concentration and temperature gradient alone, respectively It was assumed that the LHS ofequation (10) could be decoupled :
= [(,-
the liquid phase is governed by the following equations Continuity
au
1 a
+ ~ ~(~v) = o
(51
The mixture's molecular weight can be expressed in terms of the molecular weights M. and M_, of the components I and 2 (the vapor and the gas, respectively) and of their mass fractions ~o~ and ~o,_: M~ ~1M2+(1 - w , ) M , l-- M - = 1 - w l ~ M 2 + ( l _ ~ , ~ ) M i
X-Momentum 0 a ~(uu)+ ~1 ~(~vv)
aP 02U 1 0 [ aU'~ ~+~X~_+~L~)-GrcO
-
+(,-
(6) The substitution of equation (11) into equation (2) with the above assumption yields
l-Momentum 0 1 0 ~X ( v v ) + ~ O~ (~w)
-
aP
~2V
I O(
OV~
a~+bY~-~+?N\ N / - ~
V
(7)
Energy equation
y(uo
2l_ + ~-~ L~'~-)J. (8)
Again, after introducing the following dimensionless variables,
x X=R,
r ~=?,
U=
uR v '
vR V=--,v
P-
pR 2 pv 2'
I100 O-
S_-S. HSmH and N.-H. Kuo --T~ - T
W__ O)I --OJlx
T~ --To"
Pr G ~
1-~,~ Grr --
V -
?0 .~:0, OU a~-=0, v=o, ~=0
Y
Sc = ~ ,
g(T., - To)R v "-T.,
on the peripheral side = 1.
9(~/i-
V=0,
0=0.
(20)
2_22 Gasphase. The boundary conditions are
and
1,2
~3U ~3~=0,
X=Ar,
c""-c""- (i-
c.
the governing equations for the gas phase can be written as :
~=0,
OU
8(=0,
V=O,
0=0,
?0
V=0,
?~=0,
W=0
~W
-~-~= 0
(21)
(22)
on the peripheral side
Continuity OU
1
ax+~ ~(~v)=0
(13)
X-Momentum
a ~ ( V V ) + ~ 100~ ( ~ v v ) = - a xc3P O~-U
1 O ( OU~_GrTO+GrM W
+U~-+Ta~t ac)
(14)
~-Momentum c~
U=0,
I ) ( l - - m , , )R 3
Gr M =
B-
(19)
1 c~
ax(UV)+ ~ ~(~w)
~=1,
v=o,
u=o,
0=0,
/~W ?~=0.
(2.3)
Based on the extensive discussion reported in Sparrow et al. [1], a realistic boundary condition was employed as formulated in equation (21). 2.2.3. Interlace. The surface of the liquid (X = Arl) was considered as being impermeable to the gas. In accordance with the impermeability condition, Ihe naturally occurring diffusive flow of gas to the interface (diffusion from higher to lower concentration) must be balanced by a convective flow of gas away from the interface_ This balance yields (at X = Art_ and any r) t~CO -~
~P
?")2V
I ~ /" OV~
V
p,_u = pD 0 X '
in which D is the mass diffusion coefficient_ Since a9, = 1 -~o~ and p = p,+p~_, equation (24), in dimcnsionless form, becomes
Energy equation
,
B(a0aw a0T( (]6) Concentration equation
uo., -
W~ = /Fc~-'w
I 0
--seLax'--" + ~ ( ¢ ~ 0 ]
(17)
Equations (13)-(17) constitute another coupled nonlinear system for five variables U, V, P. 0 and W_ The four prescribed parameters are the Grashof numbers (GrT, GrM), the Prandtl number (PrG) and the Schmidt number (Sc)_ 2.2. Boundary conditions 2.2.1. Liquid phase. The boundary conditions are U=0,
V=0,
0=l
(18)
z(0w)
l Z w, Sc \ a x JG.,
(25)
where W l = ( o h - - m l , ) / ( l - - c o l + ). The interracial mass fraction o)~ is obtained by assuming the gasliquid interface to be in thermodynamic equilibrium. that is,
~(uw)+ ~~(c.vw)
X=0,
('_'.4)
Mtpl M2(p--pO+ MIPE
(26)
where p~ stands for the corresponding saturation pressure at 0~. Equation (25) serves as the boundary condition for interface evaporation velocity. The boundary conditions for Vat the interface can be derived from the shear force balance between the liquid and gas phases at the interface [5] ; in dimensionless form, we have
\.o} \ox/, For temperature boundary conditions, at X = ArL,
Effects of composition and temperature induced natural convection for evaporation from the energy balance at the interface, the interracial dimensionless temperature can be obtained as --kc, ~
+pc;,hrg (T~_ - To) -
kL
~X
"
(28)
In the present study, energy transport at the interface in the presence of mass transfer depends on two related factors. They are the gas temperature gradient at the interface and the rate of mass transfer. 3. N U M E R I C A L P R O C E D U R E
The governing differential equations (5)-(8), (13)(17) and their boundary conditions were solved as an adaptation of an elliptic finite difference scheme ( S I M P L E C ) set forth in ref. [6]. Two computational domains are employed, the liquid and the gas phase are treated separately and the conditions (equations (27) and (28)) at the interface are used to match the velocity and temperature fields from both phases. The numerical methodology for treating the gas-liquid interface was exactly as in Prata and Sparrow [5]. Through Table I, the grid dependence examination was made. Finally, a 32 × 32 hybrid nonuniform/uniform grid was chosen_ The layout of the grid in the liquid phase and the gas phase was arranged as follows : (a) liquid phase X-direction, AXIng. = 0.05ARE (nonuniform) AX,,m = 0.0083ArL (nonuniform)
1101
the computational region. The under relaxation values were 0.5 (for U), 0.6 (for V), 0.7 (for 0 and IV), and 0.8 (for P) respectively. The parameterization of the solutions will be presented and discussed_ As stated earlier, the numerical solution was performed for water as the evaporating vapor and air as the gas. For this system, the Schmidt number Sc is 0.6 [7]. The reference condition is chosen at T~ = 37:C, To = 23"C, with an associated GrL = 105 and Grv = 104, GrM = 10 d, 105 and 106, respectively, and a system pressure of I arm, while o9, = 0.04 and w,:,. = 0.0 (completely dry ambient). For the second case, to, = 0.04 and co,:~ = 0.036 (ambient relative humidity of 90%) with the same corresponding values for GrM. The physical properties of the working medium (water/air) were chosen at T = 7-,+ 1 / 3 ( T ~ - T ~ ) and o9 = o9,~,+ 1/3(o9~-o9,~) [8] instead of the conventional average weighted values. The present numerical calculations were performed on a 486 IBM compatible PC.
4. RESULTS A N D D I S C U S S I O N
The results to be presented and discussed will begin with the rate ofevaporation. Subsequently, to provide a basic understanding as well as to provide further insights, streamline, isothermal and isohaline maps, velocity and temperature profiles, and mass fraction profiles will be presented and discussed. 4.1. Rate o f evaporation The rate of evaporation at a differential area dA on the liquid surface (x = L) can be written as
C-direction, A~ = 0.05 (uniform) (b) gas phase
0o9,
X-direction, AXm,~ = 0.0489ArG (nonuniform) AXmi. = 0-0083ArG (nonuniform) ~-direction, A~ = 0.05 (uniform). A variable G is said to be convergent if its residual is smaller than a pre-assigned value. The residual is defined as
where both the convective and diffusive contributions have been included. By making use of equation (24) with O t o 2 / O X = - & o , / O X and noting that p = p, +p,_, the expression in equation (30) becomes pudA
G"-G"/3 = max G"
I
i,j
(29)
(31)
where dA = 2nr dr. Therefore, the rate ofevaporation can be expressed as :
where n refers to the nth iteration and i, j stands for the node position. In the above expression, the residual/3 represents the maximum values throughout
rh =
f:
pu2rcr dr.
Table I. Grid dependence examination for evaporation rate Reynolds number, absolute maximum stream function for liquid and gas flow, interface average temperature and bulk concentration
Re I~Llm~ I~olm~, 01 Wb
(30)
17x7
22x 12
27x 17
0.006121 0.454384 0.069731 0.001614 0-001833
0_006148 0.649866 0.052742 0.003758 0.001843
0.006154 0.709724 0.053888 0.003653 0_001844
32x22 0_006159 0.737772 0.058255 0.003585 0.001845
37x27 0_006156 0.749904 0.055292 0.003638 0.001845
42X32 0.006160 0.737324 0.058444 0.003565 0.001846
(32)
1102
S.-S. HSIEH and N.-H. Kuo
The Reynolds n u m b e r can be o b t a i n e d in the following by incorporating n7 and the tube radius R for the tube flow: Re -
Re = 4
Curves are included representing the present results for c o m b i n e d temperature and c o m p o s i t i o n gradients and c o m p o s i t i o n gradient alone, as well as those of the Stefan solution. E x a m i n a t i o n of the results of G r L = 10 5 a n d G r T = 10 4 for three different GrM = 10 4, l0 t , and 10 ~' for the completely dry and humid cases, shown in Figs. 3(a) and (b), indicates that the G r a s h o f n u m b e r for mass transfer has a great influence on the e v a p o r a t i o n rate_ It is also seen, except for GrM = l 0 6 a n d co,~ = 0.0, that the evaporation rate decreases m o n o t o n i c a l l y as the distance between the liquid surface and the tube opening increases. This b e h a v i o r reflects the increased resistance to mass transfer associated with the greater transport length. On further e x a m i n a t i o n of Fig. 3(a), at GrM = 104, it is seen that b u o y a n c y e v a p o r a t i o n rates, either due to a sole c o m p o s i t i o n gradient or double diffusion, are nearly the same as those for the Stefan problem at the completely dry ambient. The result at GrM = l0 t coincides with that given in S p a r r o w et al. [1], in which it was found that e v a p o r a t i o n rates are 1-5 times those o f the Stefan solutions. This is partly caused by the effect o f the thermal and solutal buoyancies, which are c o m p a r a b l e (GrM = Grv = 104), and partly by the c o u n t e r a c t i n g effect of thermal buoyancy. As GrM is increased to 10 -~, the discrepancy a m o n g these three solutions remains up to a distance between the liquid surface and the tube opening of 4.5_ A m o n g the discrepancy, double diffusion gives the highest e v a p o r a t i o n rate to three times the Stefan value even though the thermal gradient reduces buoyancy. C o n c e n t r a t i o n gradient induced b u o y a n c y gives the values in between Stefan and double diffusion. At GrM = 10 ~', this trend is quite different and the superiority is switched. However, c o m p o s i t i o n gradient driven convection gives higher values than those
(33)
2r~Rll
I0'
U~ d~.
(34)
E q u a t i o n (34) was evaluated using the numerical integral scheme As m e n t i o n e d before, several grid sizes were tested and a c o m p a r i s o n o f the results a m o n g these c o m p u t a t i o n s is given in Table 1 for the e v a p o r a t i o n rate Reynolds number, the absolute m a x i m u m stream funcLion for liquid flow and gas flow, the interface average t e m p e r a t u r e and the bulk v a p o r concentration. It is noted that the differences in the e v a p o r a t i o n Reynolds numbers, absolute m a x i m u m stream functions, interface temperatures and bulk mass fractions are less than 1%, by using 3 2 x 2 2 and 4 2 x 3 2 grids. Accordingly, the comp u t a t i o n for the 32 x 22 grid is sufficient to u n d e r s t a n d the heat and mass characteristics in the flow. In addition to the grid dependence examination, the present numerical scheme was used to redevelop the case of Sparrow et al. [1] for b o t h the completely dry a m b i e n t (i.e. w,~ = 0 . 0 ) and very humid (i.e. ~o,~ = 0.036) a m b i e n t conditions. The c o m p a r i s o n shown in Fig. 2 reveals good agreement between b o t h results. The numerical results for the e v a p o r a t i o n rate are presented in Fig. 3, for Ar L = 2 , G r L = 10 ', GrT = 104, and co, = 0.04 for two a m b i e n t conditions (one for the dry and one for the humid case) with GrM = 104 increased to GrM = I06. In this figure, the e v a p o r a t i o n Reynolds n u m b e r is plotted as a function of the g a s - v a p o r space, which is in terms of Ar~;.
%==0 036 (~=90%)
w,® = 00ldry)
0 2,~
00iO
~
- -
0 20
0 16
I
Present study r r o w e t o~ [ I ]
o 008
OOO6
(xl ~ 0 i2 -0004
ii
~ P4
ii ~
~ oo8 0¸002
0 04 ~,== 0 036 {~ = 90%1
O
2
4
6
8
nO
Ar G = h/t?
FiG 2. Rerun Sparrow et al. [I] for GrM = 104, o), = 0.04 (saturation), and for o.h~ = 0.0 (4; = 0%), tol,~ = 0.036 (~b = 90%).
Effects of composition and temperature induced natural convection for evaporation O 08
1103
00C~ X
S t e fc~n soLutiOf~
....
Bouy~ncy cono~ntra
0 06
due to tlon gradient
OCX3~
\
000zl
0 04
Gr M : i0 4
002
6r~, 0
:
-4
IO*
-'4
i ~ . [ . . , I ' i i I i i i I i i i
0
•
oh6-
~.
Ol2
J
'
J I
' '
'
J
'
~
'
t ' i i I
~
J
'
I
.
L
i
I
i
r
0 008
20
0
O00E
O0(>q
o
-?
08
0 002
oo4
Gr u : 10 5
~M=I
o
i i ,
I
'
'
'
I
J
i i
I
,
i
i I , i
L
2 o
J
,
I
'
'
,
J
i
i
i
o 020
I
6
I
2
0o
0
B
0 OO8
0
n
00,61 12
~=I0
GFM:
10 6
,,,~,',~,;,~,r,~,~ 2
4
6
. . . . . . . . . . . . . . . . . . . . 8
I0
0
2
q
6
8
Ar e : h/R
(a) ~b : O*/.
(u.,, : 0 0 )
(b) qb: 90*Io (w,®: 0 0~,6)
FIG. 3. Evaporation rates for Ar, = 2, Gre = 10~, Grv = 104, o.h = 0.04 (saturation) and: (a) (o,, = 0.0 (~b = 0%); (b) co,, 0.036 (qb 90%). =
of double diffusion. The enhancement stems from the dominance of the buoyancy-driven convection motions either by composition gradient alone or double diffusion, yielding evaporation rates up to 1_5-4.0 times the Stefan value. The peaks for composition gradient convection/double diffusion all happend at Arc, = 6. The reason for this strange phenomenon still seems unclear at this stage. However, it is certain that the physical interaction due to mass and temperature gradients makes the problelm more complex than that discussed in Sparrow et al. [1]_ The evaporation results in Fig_ 3(b) convey a different message compared to Fig. 3(a). Here, the buoyancy evaporation rates for both thermal and solutal convection and solutal convection alone are, except for GrM = 10 ~, the same as those for the Stefan problem. Following Sparrow et al. [1], the natural convection is unaffected in this case due to the small difference between the vapor mass fractions at the liquid surface and in the ambient (i.e. oJ.-eo.~, = 0•004), as is also reflected by the values of GrM = 104 and 105 . However, the present study of double diffusion gives slightly lower values than those of the Stefan and sole composition gradient solutions as Arc, increases. This is quite obvious because the present temperature gradients cause a counteracting effect in evaporation rates which results in an evaporation rate decrease• Moreover, at GrM = 106, a crossover point occurs at Aro ~ 3, where evaporation
=
rates due to double diffusion are higher than those of the composition gradient alone when ArG >~ 3. Furtherrnore, it is found that, keeping the gas-vapor region fixed, the evaporation rate seems to he unchanged as the liquid space is varied (not shown here). 4_2_ Streamline distribution, isothermal map and isohaline distribution The streamline distribution information to be presented includes the results of the concentration gradient alone and double diffusion convection at two extreme ambient conditions• They are shown in Figs. 4 and 5, respectively. Moreover, to observe the gas-vapor space influences on the flow pattern, Fig. 6 is now added. Figure 4 shows three different Grashof numbers for mass transfer, GrM = 104, 105 and 106 for the dry and humid cases• Only the gas-vapor region is shown. For the completely dry ambient shown in Fig. 4(a), at GrM = 10 ", the streamline distribution indicates that the pattern of fluid flow is similar to a uniform upward flow. The evaporation vapor is represented by the streamlines which originated at the gas-liquid interface• As GrM is increased to 105, the presence of a U-shaped flow loop occurs. This indicates that the flow is downward in the outer annular and upward in the core, which is similar to that observed by Sparrow et al. [1]. Note that the downflowing fluid very nearly penetrates to the surface of
1104
S.-S. HSlEH and N.-H. Kuo GG,, = IOs
GA, = I0 ~
GG,, = I0 ~
Value
~b VaLue 9
r
I 4E-2
7: I I E - 2
6
@ VaLue 9:1 BE+0
9 -44E+0
7 i 3E+O
7 - I 7E-I-O a
5: - I 7E+ I
5:77E-3
5: 9 : ~ - -
:5 4 5 E - : 5
:5: 5 2 E - I i- i 4E-I
5 - 2 :SE+ I
o: 2 8E - 2
O: 5 8 E - 2
I: 8 5 E - 4
tl
I
I -29E+I
X' V"
_
I
--~'0
g
(a) ~5= 0 %
(wj:= O)
Gfr~ = I 05
GrM = I 0 4
G',"r,~ = I 0 '~
I
i VaLue i
9 I 5E-3
VoLue " 9: 1 8 E + O
~5 VaLue
i'
9: I 5 E - 5 6
7 liE-5! .5 7 7 E - 4 !
:5:4:5E-4 [
11117
? I IE-5 5 : 7 8E-4
7 : l 4E+0 5:9 7E-I
5 : 4 5E-4
5: 5 4 E - I
Ar G
I: I I E - I
i. B 7E-5
l a 7E-S i
c: 2 7 E - 3 /
4i" "=-0
-="- O
g
r
(b)q~=90%
¢
(%®=0036)
FIG. 4. Streamline distribution, concentration gradicn[ only, for A r e = 2, tu~ = 0.04 (saturation) and : (a)
Oil, = 0 . 0 (Q/) = 0 % ) :
the liquid, whereupon it turns and sweeps across the surface as it moves radially inward. As can be seen in the figure, the circulating flow, upon approaching the tube axis, wells up from the liquid surface and streams upward to the tube opening. At GrM = 106, a small cell exists at the bottom of the gas-vapor region and, consequently, the U-shaped flow loop appears to be broken near the bottom of the gas phase. On the contrary, the flow is upward in the outer annulus and downward in the core. The strength of the streamline becomes stronger a s G r M is increased. This can be seen for the absolute value of the streamline. On the other hand, for the humid case, shown in Fig. 4(b), the flow patterns are similar to those of the completely dry case, except for an occurrence delay for the U-shaped loop in the GrM effect. This is perhaps because the present temperature gradient counteracts the mass gradient and such an effect is significant in the humid case. Figure 5 shows the streamline maps for both gas-vapor and liquid regions for GrM = 104, 105, and 106, respectively, at the two ambient cases. Unlike composition gradient driven convection (see, for instance, Fig. 4 at GrM = 10~), the flow from the U-
(b)
O) 1," =
0.036 (~b = 90%).
shaped flow loop (see, for instance, Fig. 5 at GrM = 105) is swept in the outer annulus and downward in the core_ After several trial runs, it is found that this phenomenon disappears when Grc < l02 and GrT < 10. The streamline map at GrM = l04 and 105 in g a s - v a p o r region in Fig. 5(b) shows a dual cell recirculation. One thing worthy of mention here is that when the strength of streamline becomes weaker, in accordance with the shear force balance between liquid and gas flow at the interface, a recirculation cell is formed in the gas region adjacent to the interface (at GrM = l04 in Fig. 5(a), and GrM = l04 and 105 in Fig. 5(b)). The above statements provide adequate reasons to explain why, for the humid case, the evaporation rates due to concentration gradient alone are a little higher than those of double diffusive convection, which may not always be the case as would be expected (see, for instance, Fig. 3(a) for Gru = l05 and Fig. 3(b) for GrM = 106). F o r the humid ambient case, in the liquid region, the streamline distribution displays dual recirculation cells for GrM = l04, 105 and l06. However, for the completely dry case, such behavior only occurs
Effects of composition and temperature induced natural convection for ew*poration
Gr = IOs
GrM = 10'
~Vatue]
~b Volue 9: 45E-I 9 -4 2E+O 7: 1.6C~FO 5;- I 6E~ I
7:-I .I E-~O 5: I.BE+O 3:-2.5E+0 h-5.1E+O O: 4.5E-2
il' ( I
5: - 2.2E+
illltll// ))/llllll A~ il\\~llll
I
qbValue
9:-2.8E-I 7:-6.5E-I 5: - 1.4E-I-O
9-
Art.
2
5E+O
7: 13E÷O 5:-4.3E-I
5:-1.4E+0
3: - I 9E÷O
I :-1.8E+O ,
I : - 3 4 E'I'O
4
xTo C, ~
rG
h - 2 8E+I
¢ V a t ~ "~'-
~b Vatue---~lj ~
5.-7 4E-I
6 r M :106
1
'I
1105
°
/"L ,
X ¸
~
(o)~b= 0% 1,,.o~==01 GrM:
104
GrM = IO 6
G r M = IO 5
,0 ',,bLue
Value 9: 4. 9E - 2 ' 7:3.6E-2 I 5: 2. 5E - 2 I 3:99E-3
9:4.8E-2 7: 5.5E- 2 5: 2.2E- 2 5:9.0E-3 1:-3.9E- 3
,atG
qS Votue
,:-zoE-, i I t ( v m ~ I
9:-I IE-I 7:-2.6E-I 5:-4.1E-I 3:-5.5E-I h-7.0E-I X
ArL
Art_
o
C
(b) qb:90*/. 1to.=:0.036)
FIG. 5. Streamline distribution for A r t = 2, A r G = 2, Gr L = 10~, Grv = 104, ~t4 = 0.04 (saturation) and: (a) co., = 0.0 (~b = 0%): (b) oJ,,. = 0.036 (q5 = 90%).
at GrM = 104. Since the liquid phase space does not affect the e v a p o r a t i o n rate at all, as stated earlier, the present discrepancy will not be discussed. F u r t h e r study m a y include this aspect. Figure 6 presents the effect of Arc (shortest/longest tube) o n the flow pattern for the completely dry case. T h e flow pattern in the liquid phase seems to be i n d e p e n d e n t o f the variation of A r G . However, the flow p a t t e r n s in the g a s - v a p o r region do change slightly. This change confirms the previous finding that the tube opening in-
crease reflects the increased resistance to mass transfer. Figure 7 shows the isotherm c o n t o u r maps corresponding to the flow discussed earlier in this section. Generally speaking, a low temperature distribution was found at the g a s - v a p o r and liquid interface except at the b o t t o m o f the tube. F o r the liquid phase in Fig. 7(b), the thermal field is stratified fairly linearly in the vertical direction in most o f the flow domain, especially in the liquid phase for a relatively humid case. In addition, for the humid case, it seems there is
1106
S.-S. HSIEH and N.-H. Kuo ArG =I0
•,4r G = 2
Value
3:-I 8E-3 1:-52E-2
+Vol. ~: 7 : 4~.,~E-~ 6E-2 ~5:28E-2 :'~-~
II
\
~ II / II / II
~
~
1
[
!~,
0 ~, Volae
.
Value
9.-27E- } 7-47E-I i
~:-" ~-' ~//k\k\%UJ/llll ':-2~-' i/\KkUJ/Ill
5-4 E-I ' ~ \ \ \ X / I l l
~ - ~ ° ~E"I-O -' 5-1.3
~:-~.~-,
X,
X
o{;
! \ \71 )kill ^
~
of, (a)
F I G . 6. S t r e a m l i n e
-~
(b)
distribution f o r Ar, = 2. Gq = I 0 ' . Gr T = I 0 ~. Gr,~ = 10 ~. m, = 0 . 0 4 ~ul~ = 0 . 0 (q~ = 0 % ) a n d : (a) A r c = 2 ; ( b ) A r e = 10.
no interaction with the gas-vapor phase. However, it is important to note that radial temperature gradients exist. These radial temperature gradients, although small in magnitude, tend to be more pronounced in the regions closer to the tube wall. On the other hand, for the completely dry case, the temperature in the liquid phase does somehow interact with the gasvapor temperature distribution. This phenomenon becomes more distinct as GrM is increased to 106. Further inspection of Fig_ 7(a) reveals that the isothermal map pattern initially (at Gr M = 104) behaves as stratified and later becomes a plume rise occupying almost the entire portion of the right half of the tube.
(saturation).
This plume rise becomes more distinct and, consequently, results in the plume getting smaller as GrM is increased to 10~. This can be seen by comparing the different plots in Fig. 7(a). This change somehow becomes moderate/delayed when compared to the corresponding plots for a relatively humid condition. The radial temperature gradients with plume rise phenomenon are essential to support the vertical velocity shear as evidenced by the corresponding streamline map, which results in convective heat transport evaporation due to the dominance of the evaporation in the main body of the flow field. Consequently, a narrow zone or annulus of intense vertical tem-
Effects or composition and temperature induced natural convection for evaporation ~r~ : I0 ~ 0
: I0 ~
Gr~
Gr
0 Value '
OVatue
9:3 7E-2
9 I TE-I 7 I .:3E-I
9:27E-I 7:2 IE-I
5 : 8 BE-2 :3.4.9E-- 2 I 98E-:3 O: OOE+O
514E-I 3 79E-2
I
\
I 2 2E-:3 0 00E+O L~'~'~5~X
I~'o
5E-i
76
5E-I
54
5E-I
I 1 5E-2
0V a t u ~ ~
8 ,0
I OE+O
9B
iO
I
I OE+O
9:8 5E-I
9:8 76
0
lit a
0:00E+O
\]
0 'v~tue ~0
: I0 6
0
0 VaLue
~ , ~-~ !
1107
5 4 5E-I
fL
:3 2 5 E - I
:32 5E-I
I 52E-2
I 5 5E-2 C 00E+O X
0 00E+O X
C (w,=:O)
to~q6:O%
G7-M :iO 4
0 VaLue
7 4:3E-:3 i ~
Oi/b"
~
, :3:3~-o , ~ - 2 . ~ \
o:oo
0 VaLue
0
~:3 E~-6E-:3 : 3 ~ .;-._:3 .. o
GT~ = ,0 G
GT~ : ~0 ~
\ \ Ii
\1
o
9:~ ~E-:3~ ' ~ ~ :3E-:3 i ~ ~ ~.~-:3i-~ :3:, ~E-:3i~
~
, :3:3E-~ ~ \
o: o o ,o
~-
/ 1 olA,o \ II \ \ '
7 I 3E-21
~
5:9oE-:3i :3 5.OE-3 i ,
/ \ |
~Ac~
/
\ I
(~:00E+O ~
J~_\
~|1
=l
0 VaLue I
O voLue I
i
I
iO I OE+O i 9:8 . S E - I I, 7 : 6 5E- I I 5:3: 42 . 55EE-- ' I Ji, 1:5.9E- 2 ~ 0 00E+O X c
~0 I OE+O
o AlL
9:8 5E-I 7 6 5E-I
l
5 : 4 5E-I :3: 2 . 5 E - I
~: 5 8E-2 00E+O X
0 VaLue [
i tO
i i
, '4rL
I
i OE+O i
9 8 5E:-I ! 7 6.5E- I I
~rL
5 4.5E-I I :3 2 . 5 E - l i I / 5 5E-2 0 0 0 E + O ~---:3_-5_~_~
i, -
=7~
x F - = ~
..
i 1b~:90%
(,,a = 0 0 3 6 1
FIG. 7. Isothermal-line distribution for ArL = 2, A r c = 2, Gr L = 105, Grv = 104, wl = 0.04 (saturation) a n d : (a) ~o~ = 0.0 (~b = 0 % ) ; (b) o ~ = 0.036 (4) = 90%).
p e r a t u r e as well as radial g r a d i e n t s exist near the tube wall. T h e isohaline d i s t r i b u t i o n s h o w n in Fig. 8 generally indicates that the c o n c e n t r a t i o n s h o w s a n e a r linear stratification in the vertical d i r e c t i o n at GrM = 104. As GrM is increased f u r t h e r (GrM= 105, q~ = 9 0 % a n d GrM = 106, ~b = 9 0 % ) , a radial c o n c e n t r a t i o n gradi-
ent exists_ At
GrM =
10 ~, the effect o f solutal b u o y a n c y out-
weighs the thermal effect, resulting in plume-like concentration distributions occupying a comparatively larger portion of the gas phase. Due to a high relative humidity for the case shown in Fig. 8(b) the concentration fields are still substantially linearly straUfled in the vertical direction, even though GrM is increased to 105_ Until GrM = 1 0 6 , the layered structure vanishes, and the solutally driven flow comes into play in much of the gas phase.
1108
S.-S. HSlEH and N.-H. Kuo GrM r I0 ~
GrM
: iOs
GrM : I0 G
~ o ~
I
w i value L
~ l VaLue
i_-3~
I F
9 3 34E-2 i7 51 7 7 E - 2 i
9: 3.32E- 2 [
,4,"s 7
5
~ 7
98oE- i 97E- 3 0 000E+O
9
~c
i
~
232E-2 i 161E-2 i
3
3 891E-.3,
0
I. I 7 8 E - 3 ! C: OOOE+O I
i
i
(a) qb:O*/.
: I0"
G,,"M : I0 s
T
Vo l Lie i
(w, =0)
ua. Votue
G.,-M= I0 '~
o
LI
f ~ v°wei
T
o / ~
~---3 l
9 : 3 91E-21
9::3 91E-2
7 3.84E-2 [
A~
'
~
5
~
5 : 3 76E-2
5 3 76E-2 i I
__7
3.3 69E-2
3 3 6~E-2 '
2I O: 3GOE-2 I
7. 3 8 4 E - 2
1 362E
__
-9--
I
i: 3 6 2 E - 2 ,
O: 3 6 0 E - 2
-°
'
9--
[ 7.:iiii:?V,'ll?
C
-o
C
'
(1~) qb : 90':'/o (,~, = : 0 056)
FIG. 8. lsohaline distribution for ArL = 2, Arm = 2, G r L 10', G r r 104, w. = 0.04 (saturation) and: (a) ~,~., = 0.0 ((k = 0%): (b) (o,, = 0.036 (q5 = 90% ). =
4.3. Velocity profile, tenlperature distribution and mass ./i'acrion profile The evolution of the velocity fields and temperature distribution as well as mass Fraction along the tube is respectively set forth in Figs. 9, 10 and II for the shortest of the investigated tube lengths, namely Arc; = 2. Each figure displays a succession of velocity/ temperature profiles and mass fractions parameterized by the dimensionless axial coordinate X'. On the ordinate, the velocity (temperature, concentration) is embedded in the dimensionless group uD/v ((T~ -- T ) / ( T ~ - To), ~ol) which can be regarded as a local Reynolds number (local temperature, local mass fraction), The most noteworthy feature of Fig. 9 is the two-lobed nature of the velocity profile. This shape is the same as that reported in Sparrow et al. [1]_ In the central core of the tube, which extends from the eenterline (( = 0) to ~ ~ 0.6, the U velocity is negative, indicating that flow is drawn into the liquid surface. This downward-directioned flow is drawn into the open top of the tube from the ambient. On the other hand, in the annulus which spans between the outer rim of the core and the tube wall, the flow is directed upward toward the open end of the tube
=
as evidenced by the positive values of U. Generally speaking, it is interesting to note that the crossover from negative U to positive U occurs at virtually the same radial position at all axial elevations and, in addition, the crossover point is hardly affected by the gas-vapor space ratio A%. Another significant feature of Fig. 9 is the magnitude of the velocities that are induced by natural convection. For both ambient cases, as GrM is increased, the velocity magnitudes (absolute values) increase. To provide a prospective for the velocity magnitudes displayed in these figures, it may be noted that the evaporation Reynolds number corresponding to the completely dry case is approximately 0.07 at GrM = 104 and Ar G = 2. The evaporation Reynolds number can be written as ffD/v, where ti is the mean velocity of the net downflow. In Fig. 9(a), the maximum downward velocity occurs at the center with the axial eievation at X ' = 0.28 for GrM = 104 and is characterized by uD/v ~ 2.5, which is about 35 times larger than that (uD/v ~ 0_07) for the corresponding result of Fig_ 4(a). From the preceding discussion, it is clear that certain portions of the flow which streams upward through the annulus are evaporating vapor, while a
Effects of composition and temperature induced natural convection for evaporation
Grr, :
1109
IO 4
08---_
05._~-03.-~-4 -2
0 -4
0.28
0.68
1.26
200
05
i i i ] i i i ] i i i ] i i i ] i i I
25
I',,t
t illiit
JiJliil
I G G6:IO
~
~,,. =10 {3 8
,
- j c
05
%
O 3
-25
O "3
-
75
_o
i • Io.%Io.%I,. 12%1
"Io.%, Io. I,
-O 5
r i J I I I I I I i I I I I I I I I I
I oot
.... J J I I I I I I I I L I I III
I I1
2GC 25
3 ~
Gr,i : I0'~
4
Gr,., : IO 6
Syrn~¢~.
ililililililllllllt 02
04
06
75
08
0
~: (a)(:#:O°/o FIG.
9.
Velocity profile
for
(L,.,
I
2
~
4
5
I t t t [ i i i i i i [ i i i i i i i i O C2 C4 06 08
fir
=0)
',b)
A r L = ~,9 A r G = =,3 GrL =
105,
~:90°/0
GrT =
(~J®:0056)
[04, ")l
=
0.04
(saturation)
and:
(a) (o~, = 0.0 (~b = 0%); (b) (,)~ = 0.036 (~b = 90%).
certain portion of the a n n u l a r flow consists o f fluid that was d r a w n into the tube t h r o u g h the core. This indicates the presence of a U - s h a p e d flow loop, with the downflow leg in the core and the upflow leg in the annulus. On further inspection of Fig. 9, one can find that, for the completely dry case and GrM = 104, these studies indicate an evolution of the velocity profiles, which is virtually complete in the g a s - v a p o r regions_ This o b s e r v a t i o n is witnessed by noting that there are gradual a n d steady changes between the profiles from X ' = 0 to 0.28. This indicates t h a t the m a x i m u m velocity (absolute value) occurs at X ' = 0.28. However, for the r e m a i n i n g figures of Fig. 9, it is f o u n d that the m a x i m u m velocity occurs at the centerline of the tube and axial location at X ' = 0.68, except for Fig. 9(b) at Gr M = 105 ( X ' = 0.28) which is not the top of the tube, T h e reason for this is not clear at this stage.
Figure 10 represents the c o r r e s p o n d i n g temperature distributions of Fig. 9. Generally speaking, the evolution of the t e m p e r a t u r e profile is virtually complete in the upper half o f the g a s - v a p o r region. As GrM is increased to 10 ~, s h o w n in Fig. 10(a), it is n o t e w o r t h y that the temperature gradients near the vertical walls become steeper_ Profiles of the v a p o r mass fraction co ~are presented in Fig. 1 I. This figure is the c o u n t e r p a r t of the velocity and t e m p e r a t u r e plots (Figs. 9 and 10). In the figure, curves are plotted at a succession of axial elevations. Based on the present b o u n d a r y condition, one can find that the o)~ profile at the tube opening (X' = 2) is horizontal and c,J ~~ = 0 or 0.036 all across the opening, which is also confirmed by Sparrow et al. [1]. At any axial elevation, the v a p o r mass fraction increases from the tube axis to the wall. This trend is in accordance with the flow patterns illustrated in Fig. 5. The
I110
S.-S. HSIEH and N.-H. Kuo
O(:108
0.05
"lo , Io%,I, %,I=%oq O 04-
~
0 03"
--I
O O I O71 1.31 ~.O0
= 10°
0 02 I
OOi
O
° ~ ° - ~ ' N I s " ~ l ' I 2 I 3. I * I 5 1 SI ~ -" I o IO.3, IO.7, I,.3.,Izoo]
0.0~
'
I' 1o
Io%,I, 12:ol
OOO6 ~ I
I ii
°°?
\, 0.002
~
4
O - ill/ 51- i ] l J l l l J r ] l 0.4
O020~
G,-M : I0 s
jlj
GF. : l O S ~
0.3 O 012 O 2--
I -Iolo3.,Io7,1 ,[2ool Symlx~. I
2
3
4
5
O I --
000B
0{ ~l ill I ill ill I ~llJJl 0 2
0~
06
08
Sym~c4. I
X 0
2
3
4
5
Iolo3.,IoT, l, ,] ool )
02
4 04
~ 06
08
I
= rlR
(o) (;b:0% ((.,a,®:0)
(b)qb:0°/o (~,®:0.036)
FIG. 10. Temperature profile for A r L = 2, Arc; = 2, Gr L = l0 s, GrT = 10~, ~o~ = 0.04 (saturation) and : (a) o~, = 0.0 (q~ = 0%); (b) o)., = 0.036 (4) = 90%).
extent of the radial variations is accentuated for the completely dry case, as can be verified by Fig. I 1. 5. C O N C L U S I O N
The nature of natural convection in the presence of composition/temperature gradients has been numerically studied for an open-topped tube which is partially filled with a liquid to broaden our basic understanding of this type of convective heat and mass transfer. The most significant contributions of the present study are the following. (I) The present theoretical analysis of combined heat and mass transfer for an open-topped tube which is partially filled with water has made it possible to examine for the first time the effect of a number of
parameters_ For instance, Gr M, GrT, Gr L and A r G for two extreme ambient conditions (4) = 0 and 90%). (2) The streamline distribution indicates that a Ushaped flow loop most likely occurs at GrM /> I05- The upflow leg of the loop is formed in the core of the tube and the do .wnflow leg is formed in the annular region adjacant to the tube wall. At GrM = 104, this loop does not exist, and two toroidal recirculation zones occur instead. (3) Isotherms corresponding to the flow field in remark (2) show that the thermal field is stratified fairly linearly in the vertical direction in most of the flow domain, especially in the liquid phase. Radial temperature gradients are found to exist and, although small in magnitude, they tend to be more pronounced in the regions closer to the tube wall.
Effects ofcomposition and temperature induced natural convection for evaporation £'¸04
O
040
i
1
2
,
q
G r M : 104
1111
2
0 o39 _
3
3
0 038---
O. 02. 4
I
q
Gre =I0J
0 037d
I=';
1" I0 , I0%,I,',, I=%ol
5
0
i
i i
i j
i
t i
i
i ~ i
i
i
i
i
i
i
"IoZ3, Io , 0056 t
I
I
I i I I
I
I
I i
I
I I
I I
I
I
I
I
0040
004
-i ,-4
o Io.23,I0. , I, 3,1.001
o
0
I 2
039
!
0 02
0"038I
Gr., :
I0 ~
4
0 C3T 6 f M : I0 ~
o
5 0
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
0 036 !
I
004
'
I
I I I
o
I
o.
I I / * I I I I I I I I I
O0~lO
C 039 ~
Is
1;I03,1o,, '4
1.31
-
f
2
0.02'
0 C38
OCI
I J I J I I
02
i
I
04
i
i J I
06
I
I
z ~ I
08
I
I
I
51
0 036
!0
,
x"
I I I I
0
02
o I i
I
J I
0'I
I I I I
06
i
08
I
I
I
O
= rlR
(a) 4):0*./° (%.=001
(b)
~:90°/o
w,=:00361
FIG. I 1. Profile of the vapor mass fraction for ArL = 2, Arc; = 2, G r L 105. Grr = 104, "h = 0.04 (saturation) and: (a) ,J., = 0.0 (4) = 0%); (b) ~,~)1, = 0.036 (95 = 90%). =
Moreover, there is an interaction with the g a s - v a p o r phase for the liquid phase at GrM = 10 5 and a plume rise occurs in the g a s - v a p o r phase. This is essential to support the vertical velocity shear as evidenced by the c o r r e s p o n d i n g streamline map, which results in convective heat transport evaporation. (4) The isohaline distribution indicates that the concentration shows a near linear stratification in the vertical direction at Gr M = 104. As GrM is increased to 10 5, a radial concentration gradient exists. The temperature field is qualitatively akin to the solutal d o m a i n convection and the overall velocity in the bulk o f the tube is fairly upward near the tube wall. (5) The aspect ratio Ara seems to have no significant influence on the velocity field as well as the temperature and concentration distribution at a definite G r a s h o f n u m b e r (Gr T, Gr L, GrM), which is essential to support the vertical velocity shear. This
explanation shows that convective heat transport is d o m i n a n t for evaporation. (6) The evaporation rate decreases as the gas vapor region aspect ratio increases for Grr~, ~< 10 5, as confirmed by Sparrow et al. [I]. Furthermore, the temperature gradient seems to have certain different influences on the evaporation rates at different ambient conditions. (7) The effect o f the variations o f G r a s h o f n u m b e r for mass transfer Grr, is not clearly noted. F r o m Figs. 4 and 5, it is not easy to obtain a definite trend to describe the influence as GrM is varied at a fixed Grt and GrT-
Ack,owledgements--The authors are grateful to the referee who provided detailed and constructive comments. This work was supported in part by a research grant from China Petroleum Refinery Company (Taiwan, R.O.C.).
1112
S.-S. HSIEH and N.-H. K u o REFERENCES
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