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Diffusion to flow down an incline with surface resistance
Pergamon Press.
Chemical Engineering Science, 197 1, Vol. 26, pp. 799-808.
Printed in Great Britain.
Diffusion to flow down an incline with surface resistance ABRAHAM TAMIR Department of Chemical Engineering, University of the Negev, Beer-Sheva, Israel and YEHUDA
TAITEL
Department of Engineering Sciences, Tel Aviv University, Tel Aviv, Israel (Receiued 13 July 1970) Abstract-A solution of the ditfusion equation with a constant surface resistance, so far being neglected, is presented for a flow down an incline. An exact solution is obtained in terms of the eigenfunctions for which ten eigenvalues are tabulated. In addition, an approximate solution based on the integral method is obtained and found to be in excellent agreement with the exact solution (7 per cent maximum discrepancy). Also, previously published simplified solutions to this problem are checked and their error is evaluated. INTRODUCTION
of mass transfer to a liquid film have so far been reported for the case of thermodynamic equilibrium at the interface. This problem was considered as early as 1940 by Vyazovov [l] and by Emmert and Pigford[2]. In 1969. Olbrich[3] provided a detailed analysis and solved for the ten sets of eigenvalues and the eigenfunction coefficients. Very recently, Rotem [4] solved the diffusion equation taking into account the axial diffusion term. From his analysis one may observe that for Peclet numbers larger than 100 (which satisfies most practical cases) the neglect of the axial diffusion is indeed justified. In this article, we provide a solution for the case of heat or mass transfer for a flow down an incline (or falling film) which takes into account constant interfacial resistance. Our result could be used for engineering calculations which include the following cases: (a) Condensation and evaporation with interfacial resistance. (b) Absorption and desorption with interfacial resistance. (c) Direct contact heat transfer for which case the transfer of heat from the gas to the interface SOLUTIONS
of the flow can be approximated by a constant heat transfer coefficient. (d) Direct contact condensation of pure vapor on a dissimilar liquid film, again, provided that this complicated process, which may include nucleate condensation [51, is represented in terms of a constant transfer coefficient. It is interesting to observe that the need for such a solution has been raised in the past for experimental analysis of inter-facial resistance studies. Emmett et al. and Raimondi et al. [2,6] examined the so-called accomodation coefficient (Y which is related to the interfacial resistance through the Knudsen equation by
In their analysis they accounted for the interfacial resistance by the use of the simple series resistances formula 1 _A+1 k
kL
k’
(2)
In this equation kL was taken from the constant interfacial concentration analysis (no surface
799
A. TAMIR
and Y. TAITEL
resistance). This procedure may be considered, a priori, only as an approximation. Another approximate method was used by Chiang and Toor[71. They simply used the result of an exact solution for the case of semi-intinite falling film with uniform velocity. The solution in this case is restricted to the region near the leading edge (short contact time). Though, in general it seems that interfacial resistance to adsorption is quite low [7,8] many authors indicate that contamination or the addition of surface active agents may increase the interfacial resistance to mass transfer considerably[2,6,9,10] and to the extent that the analysis presented here will be useful. PHYSICAL
MODEL
AND
BASIC
where:
and
c, is the solute concentration at the interface that would exist in equilibrium with the actual partial pressure of the gas. In the case of heat transfer the Schmidt number is replaced by the Prandtl number and cm is simply the bulk temperature of the vapor. The boundary conditions are:
EQUATIONS
8=1
We consider here a fully developed laminar stream- over an inclined plane. The velocity distribution for this flow is well documented and yields[ll]: U = u,7)(2--7))
=#Ii?-/(2--r))
at
z=O
(7)
at
7j=O
(8)
g=O
(3)
:=-Be
at
r)= 1.
where r) = y/6 and 6 = ( 3pI’/p2g sin y ) l13. The dimensionless conservation diffusion equation associated with the system described in Fig. 1, is:
The last boundary condition expresses the lack of thermodynamic equilibrium at the interface. B is a dimensionless interfacial transfer coefficient defined as
B=!@
q(2-4=$
B,kis
or
K
D'
(10)
SOLUTION The
problem was solved by two methods: an exact solution and an integral method. Exact solution We assume a separation of the form, Nz,r,)
= i
of variables solution
A,H,(7))e-An’z
n=1
This form yields a “Sturm-Liouville cribed by: 5$+
Fig. 1. Physical model and coordinate system.
800
q(2---17)h,VY, = 0
(11) set” des-
(12)
Diffusion to flow down an incline with surface resistance
The A,,‘sare chosen to satisfy Eq. (14) and are the roots of the equation
with the boundary conditions dH dq
A=O
at
7)=0
(13) B( l-&2)
-Q&2+
and dH dq
A+BH,=O
at
7)=1.
(14)
A power series solution yields: H,(q)
(15)
= % GlV
where a, is: a,=
= 0. (17)
g a,(n+B) n=6
1, a,=a,=O,
a3= -$A,2, n=4,5...m.
a, = An2“>i~_~j+,
(16)
Equation (17) was solved using numerical means for AI to A5. For higher orders of A,,it was found more efficient to obtain the eigenvalues through the numerical integration of Eq. (12) using the Runge-Kutta method. The results for the first ten eigenvalues for B ranging from 10s to 10e3 are reported in Table 1. Table 2 contains the associate eigenfunctions at r) = 1 and Table 3 tabulates the values of A, obtained through the initial condition, Eq. (7) and the Sturm-Liouville orthogonality condition. The local average dimensionless concentration
Table 1. Eigenvalues A, 1 106 104 103 102 10 5 2 1 0.5 o-1 10-z 10-s
2.2631 2.2628 2.2601 2.2330 1.9952 1.7917 1.4105 l-0971 0.8181 0.3828 0.1223 O-03872
4
3
2 6.2977 6.2969 6.28% 6.1278 5.6373 5.2559 4.7897 4.5610 4.4297 4.3166 4.2902 4.2875
IO.3077 10.3064 10.2946 10.1781 9.3472 8.952 1 8.5946 8.4535 8.3796 8.3 190 8.3053 8.3039
6
5
14.3128 14.3110 14.2946 14.1336 13.1390 12.7852 12.5124 12.4129 12.3621 12.3209 12.3116 12.3107
22.3181 22.3152 22.2897 22.0416 20.8857 20.6199 20.4413 20.3795 20.3484 20.3234 20.3177 20.3172
18.3159 18.3136 18.2926 18.0878 169918 16.6852 16.4684 16.3921 16.3534 16.3223 16.3 153 16.3146
7 26.3 197 26.3 163 26.2826 259956 24.8064 24.5745 24.4229 24.3712 24.345 1 24.3242 24.3 195 24.3189
9
8 30.3209 30.3171 30.2824 29.9502 28.7466 28.5416 284098 28.3652 28.3428 28.3248 28.3208 28.3204
10
34.3219 34.3 175 34.2782 339055 326993 32.5 157 32.3999 32.3608 32.3411 32.3254 32.3218 32.3215
38.3227 38.3178 38.2740 37.8617 366614 36.4957 36.3922 36.3573 36.3398 36.3258 36.3227 36.3224
Table 2. Constants A, 1
10 5 2 1 0.5 0.1 10-1 10-S
2
1.33819 1.33818 1.33813 1.3373 1.3108 1.2700 I.1795 I.1109 1.0621 I.0136 0.9999 09999
t-0.,1718
-0.54552 -0.54551 -0.54541 -0.5435 -0.4758 -0.3904 -0.2413 -0.1443 -0.07939 -0*,1718t -0.11753 -0.,1554
3 0.35890 0.35889 0.35881 0.3564 0.2664 0.1857 o-09161 O-04885 O-02518 0.*5137 0.35274 0.45455
4 -0.27208 -0.27207 -0.27199 - 0.2689 -0.1691 -0.1053 -0G4728 -0.02439 -0.01238 -0.%2489 -0.S2392 -0*,2115
5
6
7
8
0.22111 0.22110 0.22102 0.2172 0.1156 0.06718 0.02886 0.01469 0.007399 0+83 0.,1457 0.,1184
-0.18730 -0.18729 -0.16299 -0.1826 -0.08350 -O+M47 -0.01951 -0m9867 -0m4964 -0*,1001 -O-Q49 1 -0.,1515
0.16311 0.16310 0.16299 0.1576 O&Z89 0.03408 0.01409 om7117 0.003568 0.37157 0.47289 0*,9277
-0.14487 -0.14486 -0.14476 -0.1386 -0.40908 -0.02614 -0~01070 -0.005385 -0*002700 -0~~5363 -Oe45461 -0+455
= -0~001718.
801 CES Vol. 26 No. 6-D
9 0.13058 0.13057 0.13047 0.1235 0.03932 0.02063 om8413 0.004229 0.002 114 0.,4232 0.,3542 0.52864
10 -0.11907 -0.11905 -0.11895 -0.1112 -0.03223 -0:01675 -0m68O3 -0+03412 -0GO1706 -0.,3382 -0.43507 -O-S723
A. TAMIR Table
Bn
1
106 104 103 102 10 5 2 1 0.5 0.1 10-Z 10”
0.0 0*,2015t oQO13 0*01992 0.1779 0.3118 0.5476 0.7158 0.8769 0.%36 09999 0.9999
2
and Y. TAITEL
3. Eigenfunctions
3
4
5
-:.:4713
:.;7122
-0+709 -0.04655 -0.3736 -0.5561 -0.7162 -0.7628 -0.7791 -0.7866 -0.7875 -0.7876
0.,7116 0.07012 0.4788 0.6158 0.6910 0.7063 0.7109 0.7129 0.7131 0.7132
0.0 -0.39379 -0.,9369 -0*09189 -0.5283 -0.6208 -06603 -06674 -06694 -0.6703 -0.6704 -0-6704
0.0 0*,1153 0.01152 0.1123 0.5489 0.6119 0.6354 0.6393 0w.U 064093 064098 064099
H,
(1)
6 0.0 -0.*1361 -0.01152 -0.1315 -0.5557 -0.5999 -0.6152 -0.6177 -0.6184 -0.61873 -0.61876 -0.61877
7 0.0 O-,1562 0.01560 0.1496 0.5557 0.5879 0.5986 06003 0.6008 060101 0601036 0601038
8
9
0.0 -0*,1759 -0.01756 -0.1668 -0.5525 -0.5768 -0.5846 -0.5858 -0.5862 -0.58634 -0.586361 -0.586363
10
0.0 0*,1951 0.01948 0.1829 0.5478 0.5667 0.5726 0.5735 0.5737 0.57387 O-573887 0.573888
0.0 -0.,2139 -0.02136 -0.1981 -0.5241 -0.5574 -0.5620 -0.5628 -0.5629 -0.563059 -0.563069 -0.563070
to*,2015 = 0~0002015.
(or temperature) is:
For the two limiting cases Eq. (2 1) yields Sh,,, = B
(22)
and LVIZ*m = $A?. The local dimensionless concentration gradient is calculated to yield
+ae
=-B
arl 1)=1
5 A,H,(
1) e-Am’z (19)
The results of Eq. (23) are given in Table 4 for a widerangeofB.
n=1
Table 4. Asymdotic
and the local overall mass (or heat) transfer expressed by the Sherwood (or Nusselt) numbers is:
Cm
fil Ad% ( 1)eP*‘Z
2
e-*““’
E
Z
Qi
= AImI ha2 n=1
(20)
The average Sherwood (or Nusselt) number applicable to mass or heat transfer calculations with the logarithmic mean temperature difference formula is:
105 104 103 102 10 5 2 1 0.5 0.2 0.1 0.01 0~001 0
THE
(21)
(23)
values of Sh and B
Exact solution Sh=S?;
Integral solution Sh=i%
3.41441 3.41435 3.41353 340526 3.32413 2.65382 2*14006 1.32642 0.80241 044615 0~19091 0.09769 0@0997 owO99 0
3.63636 3.63623 3.63504 3.62319 3.50877 2+6667 2.10526 1.29032 0.7843 1 O-43956 0.18957 0.09732 0@0997 OGJO99 0
INTEGRAL
METHOD
For the solution of this problem by the integral method, it is convenient to place the x axis along the interface (see Fig. 1) and to express the
802
Diffusion to flow down an incline with surface resistance
dimensionless temperature Thus Eq. (4) takesthe form:
8’ as
The integral form of the conservation d -L*[e’(1-,+)]d17+Bl~$(l--I*2) % I
8’ = 1 - 19. where z1 is the value of z corresponding to the transition from Eq. (29) to (31). This transition occurs when r*(=2#/BB) reaches unity. Using Eqs. (26) (28) and (29) we obtain (24) equation is:
I -- I
ao' =F
_Et
de'
a7 Se (25)
Near the leading edge (small z) we assume a polinomial @=t”(z)[l+2($+($]
z1=
8’ = e;+F(z)[r)+o*S$].
(27)
=B(l--8;).
[+B2-Q][B-2-3[B-‘-
l] +aS[/3-3l] -zB4 = 0
11
(33)
Using the afore-mentioned equations the overall local and average values of Sh can be calculated. The results are: For t* < 1 (z c zl) P Sh = 1 ; “.4(U$’
(34)
(1-P)”
BP
(28)
1+0.4(1--P)4 (1-P)” (B/S3 P #] lnB- [&B2-#f~P-2 +g[p-“-
l] -3[P-“-
Table 5. z1 as a function of B(r* = 1)
l]
B
(29)
co
105 104 1F 101 10 5 2
where a=;.
(32)
II -&w-
2B F1 = 2+B
The solution of Eq. (25) subject to the temperature profiles (26) and (27) is straightforward. For the region where ?* < 1 (the thermal boundary layer has not yet reached the wall) we obtain: [$B2--41 lnB-
+&4w3-
where B1 (equal to 2/(2 + B) ) is the value of B at z = zl. z1 is a function of B and its values are listed in Table 5. Likewise F, is the value of F at z1 and it is equal to:
In addition, f3: must satisfy the inter-facial resistance condition, namely
ae’
11
(26)
once t* = 17we obtain
x=F(z)
[&-&]lM+j&+%-2-
(30)
This equation yields B(z) implicitly and it is solved easily by an iterative procedure. For the region further down stream we get, L,-Z F = F, e 11/W2/3E (31)
A.5 0.2 0.1 0.01 0
803
Zl
0.07083 or 17/240 omO83 OWO86 OW115 0.07379 0.09136 0.10267 0*11964 0.13087 0.13897 0.14513 0.14748 0~15000 m
111.
(35)
A. TAMIR
Fort*
and Y. TAITEL
= 1 (z > zl)
1 1 11 B+40
Sh=
-
Sh = Sh[l-zJz]
-,ln2
(36)
[&*$-I.
(37)
At z = 0, Sh = Sh = B. For large z we also obtain, as expected, that Sh = Sh and Eq. (37) is identical to (36) for z + m. Comparison between the results obtained by the integral method and the exact solution is given in Table 4. RESULTS
AND
DISCUSSION
Tables 1, 2 and 3 cover the numerical results of the exact solution for ten values of the eigenvalues in, the coefficient A, and the eigenfunction H,(l) respectively and for a wide range of the interfacial resistance (B = 10e3 to 10s). This information will enable the reader to calculate values such as F, 8, Sh and a (or Nu and Nu) without repeating our calculations. In the case of large B our results are in excellent agreement with those obtained by Rotem[4] for Peclet numbers larger than 100 (in which case axial diffusion can be neglected).
A comparison between the results of the exact and integral method solutions indicates an excellent agreement. In Table 4 the asymptotic values of Sh or ,!?&are reported, for B in the range of zero to infinity. The discrepancy never exceeds 7 per cent. A comparison for a wider range is displayed in Figs. 2, 3 where Sh and Sh are plotted vs. z. Also here the deviation between the exact and approximate solution is usually in the range of 4 per cent. The ease in which results are obtained through the integral method makes it a very attractive and powerful tool for engineering calculation. For the region near the leading edge the only difficulty is an iterative solution of a transcendental equation. For z > z1 however, the results for Sh as well as % are given explicitly and very simply (Eqs. (36), (37)). Figures 4 and 5 display the results in the form of the ratio Sh/(Shs=,)e,act and s / (.%B=m)exact as a function of z. In addition, we include here two approximate solutions and the experimental data obtained by Emmert and Pigford[2]. The simplest approximate approach is obtained through the use of a “series connection” rule given by Eq. (2) where kL is taken from the solution of no surface resistance (constant surface concentration). Since the interfacial concentration is not constant, the results
Exact solution
-
----Integral
solution B
0.001
oco5
0.a
0x35
0.5
01
I
5
Z
Fig. 2. Local overall Sherwood number for exact and integral solutions.
804
IO
Diffusion to flow down an incline with surface resistance
-
Exact sdution --- Integral solution
Fig. 3. Average overall Sherwood number for exact and integral solutions.
--‘----
Exact solution Small 2 solution (Ep(38)) Series connection(Eq(2))
B
1%) 30
IO ____________________--__________
1 a
5
_------_-----_____-__-___-
2
&+==
I 05 I
0001
0005
I
0 01
I
I11111
I
01
0.05
I
I
I111111
05
I
2
2
Fig. 4. Comparison between Sherwood numbers: Exact, small z and series connection solutions.
are not accurate. Fortunately, we can observe that the agreement between the exact and approximate methods is surprisingly good for quite a wide and approximate methods is surprisingly good for quite a wide range of interfacial resistance (B) and position (z), especially for
the Sh. As an example, we add here the data points obtained by Emmert and Pigford[2]. In their analysis they used this method to evaluate their experimental interfacial resistances and calculated the associate accomodation coefficients. Our analysis indicates that their data are in
A. TAMIR and Y. TAITEL
-
Exact
----Series
solution connection
Absorption
(Es.12))
and desorption
Doto of Ftef.[21:.-4 I ;
:! Y
s
I&
0.5
0 OCQI
O-005
0.01
005
01
0.5
I
2
Fig. 5. Comparison between average Sherwood numbers: Exact and series connection solutions.
the range of B and z to yield about 4 per cent error in Sh and about 12 per cent in Sh. Figure 4 contains also results of an approximate method .used by Chiang and Toor[7] who adopted a solution for short contact times. Such a model of semi infinite film with uniform velocity yields the following equation [ 123:
The accuracy and range of applicability of two approximate methods used before is checked. The “series summation” approach was found reasonably accurate for all ranges of B and z tested. Equation (38), applicable near the leading edge, yields reasonable results for small z, depending on the value of B.
B exp ( B2.z) erfc (Bdz)
Shzz; = Ji [erf (r)/2dz)
The Sh is obtained from Eq. (2 1) using the above iI Figure 4 indicates quantitatively the range of applicability of Eq. (38) for the range tested. SUMMARY
AND
(38)
+ exp (Bq + B2z) erfc (q/24~ + B~z)]
dn
NOTATION
A,, B
CONCLUSIONS
The problem of heat or mass transfer for a flow down an incline, with interfacial resistance is considered. Solutions are obtained via an exact solution and an approximate integral method. The agreement between the exact and approximate solutions is excellent. go6
constants dimensionless interfacial transfer coefficient c local solute concentration in the liquid c, solute concentration in the entering liquid c, actual solute concentration at the interface cm solute concentration at the interface that would exist in equilibrium with the actual partial pressure of the gas and related to it by Henry’s law
Diffusion to flow down an incline with surface resistance
D F FI g h ht H H, k k ki kL K M NU Pe R Re SC Sh I t* T u
diffusion coefficient of solute in the liquid dimensionless concentration gradient proportional to the local mass or heat value of F at t* = 1 acceleration of gravity local overall heat transfer coefficient interfacial heat transfer coefficient Henry’s law constant eigenfunction local overall mass transfer coefficient average overall mass transfer coefficient interfacial mass transfer coefficient local mass transfer coefficient of the liquid thermal conductivity molecular weight of solute Nusselt number, hiYK Peclet number universal gas constant Reynolds number; 46iiplp = 4P/p Schmidt number Sherwood number thickness in the “leading edge region” dimensionless t absolute temperature local velocity
n U, x y z
mean velocity u at the liquid surface coordinate in direction of flow coordinate perpendicular to x dimensionless coordinate in direction flow z1 value of z at t* = 1
Greek symbols
defined by Eq. (30) inclination angle; see Fig. 1 thickness of film dimensionless coordinate; y/6 eigenvalues viscosity density mass rate of flow per unit width dimensionless concentration or temperature Functions erf
erfc exp
error function l-et-f exponent
REFERENCES [1] [2] [3] [4]
[S] [6] [7] [8] [9] [lo] [1 I] [12]
of
VYAZOVOVV. V.,J. Tech.Phys. U.S.S.R. 194010 1519. EMMERT R. E. and PIGFORD R. L., Chem. Engng Prog. 1954 50 87. OLBRICH W. E. and WILD J. D., Chem. Engng Sci. 1969 24 25. ROTEM Z. and NEILSON J. E., Can. J. them. Engng 1969 47 341. SKYES J. A. and MARCHELLO J. M., Ind. Engng Chem. Process Design Dev. 1970 9 63. RAIMONDI P. andTOOR H. L.,A.I.Ch.E.JI 19595 86. CHIANG S. H. andTOOR H. L.,A.I.Ch.E.JI 1959 5 165. SCRIVEN L. E. and PIGFORD R. L.,A.I.Ch.E.Jf 1959 5 397. SCRIVEN L. E. and PIGFORD R. L.,A.I.Ch.E.J11958 4 438. BURNETTJ. R. andHIMMELBLAU D. M.,A.Z.Ch.E.JI 1970 16 185. BIRD R. B., STEWART W. E. and LIGHTFOOT E. N., Transporr Phenomena. Wiley, New York 1960. CARSLAW H. S. and JAEGER J. C., Conduction of Heat in Solids, p. 72. Oxford University Press, London 1969.
Resume-Une solution de l’equation de diffusion avec une resistance de surface constante qui a et6 jusqu’ici negligee, est present&e pour un Ccoulement le long d’un plan incline. Une solution exacte est obtenue en terme de fonctions propres pour lesquelles dix valeurs propres sont calculees. En outre, une solution approximative basee sur la mtthode integrale est obtenue et on trouve qu’elle s’accorde parfaitement B la solution exacte (desaccord maxi. 7 pour cent). Des solutions simplifites de ce problbme ayant fait l’objet de publications anttrieures sent v6rifiees et leur erreur est tvaluee. Zusammenfassung- Es wird eine Lijsung der Diffusionsgleichung mit einem, bisher vemachlisigten, konstanten Obertllachenwiderstand fiir eine Stromung entlang eines Gefdles dargelegt. Es wird eine exakte Losung, ausgedriickt durch die Eigenfunktionen, erhalten und eine Tabelle zeigt zehn Eigenwerte dafiir. Dariiber hinaus wird eine Niiherungsbung, die auf der Integralmethode
A. TAMIR
and Y. TAITEL
beruht, erhalten, und es wird ausgezeichnete ubereinstimmung (Abweichung maximal 7 prozent) mit der exakten Liisung festgestellt. Femer werden ftiher vetiffentlichte, vereinfachte LGsungen dieses Problems Gberpriift und ihre Fehler abgeschltzt.
808
Chemical Engineering Science, 197 1, Vol. 26, pp. 799-808.
Printed in Great Britain.
Diffusion to flow down an incline with surface resistance ABRAHAM TAMIR Department of Chemical Engineering, University of the Negev, Beer-Sheva, Israel and YEHUDA
TAITEL
Department of Engineering Sciences, Tel Aviv University, Tel Aviv, Israel (Receiued 13 July 1970) Abstract-A solution of the ditfusion equation with a constant surface resistance, so far being neglected, is presented for a flow down an incline. An exact solution is obtained in terms of the eigenfunctions for which ten eigenvalues are tabulated. In addition, an approximate solution based on the integral method is obtained and found to be in excellent agreement with the exact solution (7 per cent maximum discrepancy). Also, previously published simplified solutions to this problem are checked and their error is evaluated. INTRODUCTION
of mass transfer to a liquid film have so far been reported for the case of thermodynamic equilibrium at the interface. This problem was considered as early as 1940 by Vyazovov [l] and by Emmert and Pigford[2]. In 1969. Olbrich[3] provided a detailed analysis and solved for the ten sets of eigenvalues and the eigenfunction coefficients. Very recently, Rotem [4] solved the diffusion equation taking into account the axial diffusion term. From his analysis one may observe that for Peclet numbers larger than 100 (which satisfies most practical cases) the neglect of the axial diffusion is indeed justified. In this article, we provide a solution for the case of heat or mass transfer for a flow down an incline (or falling film) which takes into account constant interfacial resistance. Our result could be used for engineering calculations which include the following cases: (a) Condensation and evaporation with interfacial resistance. (b) Absorption and desorption with interfacial resistance. (c) Direct contact heat transfer for which case the transfer of heat from the gas to the interface SOLUTIONS
of the flow can be approximated by a constant heat transfer coefficient. (d) Direct contact condensation of pure vapor on a dissimilar liquid film, again, provided that this complicated process, which may include nucleate condensation [51, is represented in terms of a constant transfer coefficient. It is interesting to observe that the need for such a solution has been raised in the past for experimental analysis of inter-facial resistance studies. Emmett et al. and Raimondi et al. [2,6] examined the so-called accomodation coefficient (Y which is related to the interfacial resistance through the Knudsen equation by
In their analysis they accounted for the interfacial resistance by the use of the simple series resistances formula 1 _A+1 k
kL
k’
(2)
In this equation kL was taken from the constant interfacial concentration analysis (no surface
799
A. TAMIR
and Y. TAITEL
resistance). This procedure may be considered, a priori, only as an approximation. Another approximate method was used by Chiang and Toor[71. They simply used the result of an exact solution for the case of semi-intinite falling film with uniform velocity. The solution in this case is restricted to the region near the leading edge (short contact time). Though, in general it seems that interfacial resistance to adsorption is quite low [7,8] many authors indicate that contamination or the addition of surface active agents may increase the interfacial resistance to mass transfer considerably[2,6,9,10] and to the extent that the analysis presented here will be useful. PHYSICAL
MODEL
AND
BASIC
where:
and
c, is the solute concentration at the interface that would exist in equilibrium with the actual partial pressure of the gas. In the case of heat transfer the Schmidt number is replaced by the Prandtl number and cm is simply the bulk temperature of the vapor. The boundary conditions are:
EQUATIONS
8=1
We consider here a fully developed laminar stream- over an inclined plane. The velocity distribution for this flow is well documented and yields[ll]: U = u,7)(2--7))
=#Ii?-/(2--r))
at
z=O
(7)
at
7j=O
(8)
g=O
(3)
:=-Be
at
r)= 1.
where r) = y/6 and 6 = ( 3pI’/p2g sin y ) l13. The dimensionless conservation diffusion equation associated with the system described in Fig. 1, is:
The last boundary condition expresses the lack of thermodynamic equilibrium at the interface. B is a dimensionless interfacial transfer coefficient defined as
B=!@
q(2-4=$
B,kis
or
K
D'
(10)
SOLUTION The
problem was solved by two methods: an exact solution and an integral method. Exact solution We assume a separation of the form, Nz,r,)
= i
of variables solution
A,H,(7))e-An’z
n=1
This form yields a “Sturm-Liouville cribed by: 5$+
Fig. 1. Physical model and coordinate system.
800
q(2---17)h,VY, = 0
(11) set” des-
(12)
Diffusion to flow down an incline with surface resistance
The A,,‘sare chosen to satisfy Eq. (14) and are the roots of the equation
with the boundary conditions dH dq
A=O
at
7)=0
(13) B( l-&2)
-Q&2+
and dH dq
A+BH,=O
at
7)=1.
(14)
A power series solution yields: H,(q)
(15)
= % GlV
where a, is: a,=
= 0. (17)
g a,(n+B) n=6
1, a,=a,=O,
a3= -$A,2, n=4,5...m.
a, = An2“>i~_~j+,
(16)
Equation (17) was solved using numerical means for AI to A5. For higher orders of A,,it was found more efficient to obtain the eigenvalues through the numerical integration of Eq. (12) using the Runge-Kutta method. The results for the first ten eigenvalues for B ranging from 10s to 10e3 are reported in Table 1. Table 2 contains the associate eigenfunctions at r) = 1 and Table 3 tabulates the values of A, obtained through the initial condition, Eq. (7) and the Sturm-Liouville orthogonality condition. The local average dimensionless concentration
Table 1. Eigenvalues A, 1 106 104 103 102 10 5 2 1 0.5 o-1 10-z 10-s
2.2631 2.2628 2.2601 2.2330 1.9952 1.7917 1.4105 l-0971 0.8181 0.3828 0.1223 O-03872
4
3
2 6.2977 6.2969 6.28% 6.1278 5.6373 5.2559 4.7897 4.5610 4.4297 4.3166 4.2902 4.2875
IO.3077 10.3064 10.2946 10.1781 9.3472 8.952 1 8.5946 8.4535 8.3796 8.3 190 8.3053 8.3039
6
5
14.3128 14.3110 14.2946 14.1336 13.1390 12.7852 12.5124 12.4129 12.3621 12.3209 12.3116 12.3107
22.3181 22.3152 22.2897 22.0416 20.8857 20.6199 20.4413 20.3795 20.3484 20.3234 20.3177 20.3172
18.3159 18.3136 18.2926 18.0878 169918 16.6852 16.4684 16.3921 16.3534 16.3223 16.3 153 16.3146
7 26.3 197 26.3 163 26.2826 259956 24.8064 24.5745 24.4229 24.3712 24.345 1 24.3242 24.3 195 24.3189
9
8 30.3209 30.3171 30.2824 29.9502 28.7466 28.5416 284098 28.3652 28.3428 28.3248 28.3208 28.3204
10
34.3219 34.3 175 34.2782 339055 326993 32.5 157 32.3999 32.3608 32.3411 32.3254 32.3218 32.3215
38.3227 38.3178 38.2740 37.8617 366614 36.4957 36.3922 36.3573 36.3398 36.3258 36.3227 36.3224
Table 2. Constants A, 1
10 5 2 1 0.5 0.1 10-1 10-S
2
1.33819 1.33818 1.33813 1.3373 1.3108 1.2700 I.1795 I.1109 1.0621 I.0136 0.9999 09999
t-0.,1718
-0.54552 -0.54551 -0.54541 -0.5435 -0.4758 -0.3904 -0.2413 -0.1443 -0.07939 -0*,1718t -0.11753 -0.,1554
3 0.35890 0.35889 0.35881 0.3564 0.2664 0.1857 o-09161 O-04885 O-02518 0.*5137 0.35274 0.45455
4 -0.27208 -0.27207 -0.27199 - 0.2689 -0.1691 -0.1053 -0G4728 -0.02439 -0.01238 -0.%2489 -0.S2392 -0*,2115
5
6
7
8
0.22111 0.22110 0.22102 0.2172 0.1156 0.06718 0.02886 0.01469 0.007399 0+83 0.,1457 0.,1184
-0.18730 -0.18729 -0.16299 -0.1826 -0.08350 -O+M47 -0.01951 -0m9867 -0m4964 -0*,1001 -O-Q49 1 -0.,1515
0.16311 0.16310 0.16299 0.1576 O&Z89 0.03408 0.01409 om7117 0.003568 0.37157 0.47289 0*,9277
-0.14487 -0.14486 -0.14476 -0.1386 -0.40908 -0.02614 -0~01070 -0.005385 -0*002700 -0~~5363 -Oe45461 -0+455
= -0~001718.
801 CES Vol. 26 No. 6-D
9 0.13058 0.13057 0.13047 0.1235 0.03932 0.02063 om8413 0.004229 0.002 114 0.,4232 0.,3542 0.52864
10 -0.11907 -0.11905 -0.11895 -0.1112 -0.03223 -0:01675 -0m68O3 -0+03412 -0GO1706 -0.,3382 -0.43507 -O-S723
A. TAMIR Table
Bn
1
106 104 103 102 10 5 2 1 0.5 0.1 10-Z 10”
0.0 0*,2015t oQO13 0*01992 0.1779 0.3118 0.5476 0.7158 0.8769 0.%36 09999 0.9999
2
and Y. TAITEL
3. Eigenfunctions
3
4
5
-:.:4713
:.;7122
-0+709 -0.04655 -0.3736 -0.5561 -0.7162 -0.7628 -0.7791 -0.7866 -0.7875 -0.7876
0.,7116 0.07012 0.4788 0.6158 0.6910 0.7063 0.7109 0.7129 0.7131 0.7132
0.0 -0.39379 -0.,9369 -0*09189 -0.5283 -0.6208 -06603 -06674 -06694 -0.6703 -0.6704 -0-6704
0.0 0*,1153 0.01152 0.1123 0.5489 0.6119 0.6354 0.6393 0w.U 064093 064098 064099
H,
(1)
6 0.0 -0.*1361 -0.01152 -0.1315 -0.5557 -0.5999 -0.6152 -0.6177 -0.6184 -0.61873 -0.61876 -0.61877
7 0.0 O-,1562 0.01560 0.1496 0.5557 0.5879 0.5986 06003 0.6008 060101 0601036 0601038
8
9
0.0 -0*,1759 -0.01756 -0.1668 -0.5525 -0.5768 -0.5846 -0.5858 -0.5862 -0.58634 -0.586361 -0.586363
10
0.0 0*,1951 0.01948 0.1829 0.5478 0.5667 0.5726 0.5735 0.5737 0.57387 O-573887 0.573888
0.0 -0.,2139 -0.02136 -0.1981 -0.5241 -0.5574 -0.5620 -0.5628 -0.5629 -0.563059 -0.563069 -0.563070
to*,2015 = 0~0002015.
(or temperature) is:
For the two limiting cases Eq. (2 1) yields Sh,,, = B
(22)
and LVIZ*m = $A?. The local dimensionless concentration gradient is calculated to yield
+ae
=-B
arl 1)=1
5 A,H,(
1) e-Am’z (19)
The results of Eq. (23) are given in Table 4 for a widerangeofB.
n=1
Table 4. Asymdotic
and the local overall mass (or heat) transfer expressed by the Sherwood (or Nusselt) numbers is:
Cm
fil Ad% ( 1)eP*‘Z
2
e-*““’
E
Z
Qi
= AImI ha2 n=1
(20)
The average Sherwood (or Nusselt) number applicable to mass or heat transfer calculations with the logarithmic mean temperature difference formula is:
105 104 103 102 10 5 2 1 0.5 0.2 0.1 0.01 0~001 0
THE
(21)
(23)
values of Sh and B
Exact solution Sh=S?;
Integral solution Sh=i%
3.41441 3.41435 3.41353 340526 3.32413 2.65382 2*14006 1.32642 0.80241 044615 0~19091 0.09769 0@0997 owO99 0
3.63636 3.63623 3.63504 3.62319 3.50877 2+6667 2.10526 1.29032 0.7843 1 O-43956 0.18957 0.09732 0@0997 OGJO99 0
INTEGRAL
METHOD
For the solution of this problem by the integral method, it is convenient to place the x axis along the interface (see Fig. 1) and to express the
802
Diffusion to flow down an incline with surface resistance
dimensionless temperature Thus Eq. (4) takesthe form:
8’ as
The integral form of the conservation d -L*[e’(1-,+)]d17+Bl~$(l--I*2) % I
8’ = 1 - 19. where z1 is the value of z corresponding to the transition from Eq. (29) to (31). This transition occurs when r*(=2#/BB) reaches unity. Using Eqs. (26) (28) and (29) we obtain (24) equation is:
I -- I
ao' =F
_Et
de'
a7 Se (25)
Near the leading edge (small z) we assume a polinomial @=t”(z)[l+2($+($]
z1=
8’ = e;+F(z)[r)+o*S$].
(27)
=B(l--8;).
[+B2-Q][B-2-3[B-‘-
l] +aS[/3-3l] -zB4 = 0
11
(33)
Using the afore-mentioned equations the overall local and average values of Sh can be calculated. The results are: For t* < 1 (z c zl) P Sh = 1 ; “.4(U$’
(34)
(1-P)”
BP
(28)
1+0.4(1--P)4 (1-P)” (B/S3 P #] lnB- [&B2-#f~P-2 +g[p-“-
l] -3[P-“-
Table 5. z1 as a function of B(r* = 1)
l]
B
(29)
co
105 104 1F 101 10 5 2
where a=;.
(32)
II -&w-
2B F1 = 2+B
The solution of Eq. (25) subject to the temperature profiles (26) and (27) is straightforward. For the region where ?* < 1 (the thermal boundary layer has not yet reached the wall) we obtain: [$B2--41 lnB-
+&4w3-
where B1 (equal to 2/(2 + B) ) is the value of B at z = zl. z1 is a function of B and its values are listed in Table 5. Likewise F, is the value of F at z1 and it is equal to:
In addition, f3: must satisfy the inter-facial resistance condition, namely
ae’
11
(26)
once t* = 17we obtain
x=F(z)
[&-&]lM+j&+%-2-
(30)
This equation yields B(z) implicitly and it is solved easily by an iterative procedure. For the region further down stream we get, L,-Z F = F, e 11/W2/3E (31)
A.5 0.2 0.1 0.01 0
803
Zl
0.07083 or 17/240 omO83 OWO86 OW115 0.07379 0.09136 0.10267 0*11964 0.13087 0.13897 0.14513 0.14748 0~15000 m
111.
(35)
A. TAMIR
Fort*
and Y. TAITEL
= 1 (z > zl)
1 1 11 B+40
Sh=
-
Sh = Sh[l-zJz]
-,ln2
(36)
[&*$-I.
(37)
At z = 0, Sh = Sh = B. For large z we also obtain, as expected, that Sh = Sh and Eq. (37) is identical to (36) for z + m. Comparison between the results obtained by the integral method and the exact solution is given in Table 4. RESULTS
AND
DISCUSSION
Tables 1, 2 and 3 cover the numerical results of the exact solution for ten values of the eigenvalues in, the coefficient A, and the eigenfunction H,(l) respectively and for a wide range of the interfacial resistance (B = 10e3 to 10s). This information will enable the reader to calculate values such as F, 8, Sh and a (or Nu and Nu) without repeating our calculations. In the case of large B our results are in excellent agreement with those obtained by Rotem[4] for Peclet numbers larger than 100 (in which case axial diffusion can be neglected).
A comparison between the results of the exact and integral method solutions indicates an excellent agreement. In Table 4 the asymptotic values of Sh or ,!?&are reported, for B in the range of zero to infinity. The discrepancy never exceeds 7 per cent. A comparison for a wider range is displayed in Figs. 2, 3 where Sh and Sh are plotted vs. z. Also here the deviation between the exact and approximate solution is usually in the range of 4 per cent. The ease in which results are obtained through the integral method makes it a very attractive and powerful tool for engineering calculation. For the region near the leading edge the only difficulty is an iterative solution of a transcendental equation. For z > z1 however, the results for Sh as well as % are given explicitly and very simply (Eqs. (36), (37)). Figures 4 and 5 display the results in the form of the ratio Sh/(Shs=,)e,act and s / (.%B=m)exact as a function of z. In addition, we include here two approximate solutions and the experimental data obtained by Emmert and Pigford[2]. The simplest approximate approach is obtained through the use of a “series connection” rule given by Eq. (2) where kL is taken from the solution of no surface resistance (constant surface concentration). Since the interfacial concentration is not constant, the results
Exact solution
-
----Integral
solution B
0.001
oco5
0.a
0x35
0.5
01
I
5
Z
Fig. 2. Local overall Sherwood number for exact and integral solutions.
804
IO
Diffusion to flow down an incline with surface resistance
-
Exact sdution --- Integral solution
Fig. 3. Average overall Sherwood number for exact and integral solutions.
--‘----
Exact solution Small 2 solution (Ep(38)) Series connection(Eq(2))
B
1%) 30
IO ____________________--__________
1 a
5
_------_-----_____-__-___-
2
&+==
I 05 I
0001
0005
I
0 01
I
I11111
I
01
0.05
I
I
I111111
05
I
2
2
Fig. 4. Comparison between Sherwood numbers: Exact, small z and series connection solutions.
are not accurate. Fortunately, we can observe that the agreement between the exact and approximate methods is surprisingly good for quite a wide and approximate methods is surprisingly good for quite a wide range of interfacial resistance (B) and position (z), especially for
the Sh. As an example, we add here the data points obtained by Emmert and Pigford[2]. In their analysis they used this method to evaluate their experimental interfacial resistances and calculated the associate accomodation coefficients. Our analysis indicates that their data are in
A. TAMIR and Y. TAITEL
-
Exact
----Series
solution connection
Absorption
(Es.12))
and desorption
Doto of Ftef.[21:.-4 I ;
:! Y
s
I&
0.5
0 OCQI
O-005
0.01
005
01
0.5
I
2
Fig. 5. Comparison between average Sherwood numbers: Exact and series connection solutions.
the range of B and z to yield about 4 per cent error in Sh and about 12 per cent in Sh. Figure 4 contains also results of an approximate method .used by Chiang and Toor[7] who adopted a solution for short contact times. Such a model of semi infinite film with uniform velocity yields the following equation [ 123:
The accuracy and range of applicability of two approximate methods used before is checked. The “series summation” approach was found reasonably accurate for all ranges of B and z tested. Equation (38), applicable near the leading edge, yields reasonable results for small z, depending on the value of B.
B exp ( B2.z) erfc (Bdz)
Shzz; = Ji [erf (r)/2dz)
The Sh is obtained from Eq. (2 1) using the above iI Figure 4 indicates quantitatively the range of applicability of Eq. (38) for the range tested. SUMMARY
AND
(38)
+ exp (Bq + B2z) erfc (q/24~ + B~z)]
dn
NOTATION
A,, B
CONCLUSIONS
The problem of heat or mass transfer for a flow down an incline, with interfacial resistance is considered. Solutions are obtained via an exact solution and an approximate integral method. The agreement between the exact and approximate solutions is excellent. go6
constants dimensionless interfacial transfer coefficient c local solute concentration in the liquid c, solute concentration in the entering liquid c, actual solute concentration at the interface cm solute concentration at the interface that would exist in equilibrium with the actual partial pressure of the gas and related to it by Henry’s law
Diffusion to flow down an incline with surface resistance
D F FI g h ht H H, k k ki kL K M NU Pe R Re SC Sh I t* T u
diffusion coefficient of solute in the liquid dimensionless concentration gradient proportional to the local mass or heat value of F at t* = 1 acceleration of gravity local overall heat transfer coefficient interfacial heat transfer coefficient Henry’s law constant eigenfunction local overall mass transfer coefficient average overall mass transfer coefficient interfacial mass transfer coefficient local mass transfer coefficient of the liquid thermal conductivity molecular weight of solute Nusselt number, hiYK Peclet number universal gas constant Reynolds number; 46iiplp = 4P/p Schmidt number Sherwood number thickness in the “leading edge region” dimensionless t absolute temperature local velocity
n U, x y z
mean velocity u at the liquid surface coordinate in direction of flow coordinate perpendicular to x dimensionless coordinate in direction flow z1 value of z at t* = 1
Greek symbols
defined by Eq. (30) inclination angle; see Fig. 1 thickness of film dimensionless coordinate; y/6 eigenvalues viscosity density mass rate of flow per unit width dimensionless concentration or temperature Functions erf
erfc exp
error function l-et-f exponent
REFERENCES [1] [2] [3] [4]
[S] [6] [7] [8] [9] [lo] [1 I] [12]
of
VYAZOVOVV. V.,J. Tech.Phys. U.S.S.R. 194010 1519. EMMERT R. E. and PIGFORD R. L., Chem. Engng Prog. 1954 50 87. OLBRICH W. E. and WILD J. D., Chem. Engng Sci. 1969 24 25. ROTEM Z. and NEILSON J. E., Can. J. them. Engng 1969 47 341. SKYES J. A. and MARCHELLO J. M., Ind. Engng Chem. Process Design Dev. 1970 9 63. RAIMONDI P. andTOOR H. L.,A.I.Ch.E.JI 19595 86. CHIANG S. H. andTOOR H. L.,A.I.Ch.E.JI 1959 5 165. SCRIVEN L. E. and PIGFORD R. L.,A.I.Ch.E.Jf 1959 5 397. SCRIVEN L. E. and PIGFORD R. L.,A.I.Ch.E.J11958 4 438. BURNETTJ. R. andHIMMELBLAU D. M.,A.Z.Ch.E.JI 1970 16 185. BIRD R. B., STEWART W. E. and LIGHTFOOT E. N., Transporr Phenomena. Wiley, New York 1960. CARSLAW H. S. and JAEGER J. C., Conduction of Heat in Solids, p. 72. Oxford University Press, London 1969.
Resume-Une solution de l’equation de diffusion avec une resistance de surface constante qui a et6 jusqu’ici negligee, est present&e pour un Ccoulement le long d’un plan incline. Une solution exacte est obtenue en terme de fonctions propres pour lesquelles dix valeurs propres sont calculees. En outre, une solution approximative basee sur la mtthode integrale est obtenue et on trouve qu’elle s’accorde parfaitement B la solution exacte (desaccord maxi. 7 pour cent). Des solutions simplifites de ce problbme ayant fait l’objet de publications anttrieures sent v6rifiees et leur erreur est tvaluee. Zusammenfassung- Es wird eine Lijsung der Diffusionsgleichung mit einem, bisher vemachlisigten, konstanten Obertllachenwiderstand fiir eine Stromung entlang eines Gefdles dargelegt. Es wird eine exakte Losung, ausgedriickt durch die Eigenfunktionen, erhalten und eine Tabelle zeigt zehn Eigenwerte dafiir. Dariiber hinaus wird eine Niiherungsbung, die auf der Integralmethode
A. TAMIR
and Y. TAITEL
beruht, erhalten, und es wird ausgezeichnete ubereinstimmung (Abweichung maximal 7 prozent) mit der exakten Liisung festgestellt. Femer werden ftiher vetiffentlichte, vereinfachte LGsungen dieses Problems Gberpriift und ihre Fehler abgeschltzt.
808