Revista Brasileira de Produtos Agroindustriais, Campina Grande, v.4, n.1, p.41-50, 2002 41 MASS DIFFUSION INSIDE PROLATE SPHERICAL SOLIDS:
AN ANALYTICAL SOLUTION
Vital Araújo Barbosa de Oliveira1, Antonio Gilson Barbosa de Lima2
ABSTRACT
The analytical solution of the transient mass diffusion equation in prolate spherical coordinates by considering constant transport coefficient and convective boundary conditions is presented. The solution is obtained by the variables separation method. The formal solution is applied to predict the average moisture content and moisture content distribution of a prolate spherical solid (ellipsoid of revolution) during the drying process. Analytical results are compared with numerical results that are reported in the literature and good agreement was obtained.
Keywords: drying, formal solution, mass, diffusion, elliptical geometry
DIFUSÃO DE MASSA NO INTERIOR DE SÓLIDOS ESFEROIDAIS PROLATOS: UMA SOLUÇÃO ANALÍTICA
RESUMO
A solução analítica da equação de difusão de massa em coordenadas esferoidais prolata considerando coeficiente de difusão constante e condição de contorno convectiva é apresentada. A solução é obtida usando o método da separação de variáveis. A metodologia é aplicada para predizer o teor de umidade médio e a distribuição do teor de umidade, de um sólido esferoidal prolato (elipsóide de revolução), durante o processo de secagem. Resultados analíticos são comparados com resultados numéricos reportados na literatura e uma boa concordância foi obtida.
Palavras-chave: secagem, solução exata, massa, difusão, geometria elíptica
INTRODUCTION
The formal solution of the diffusion equation has been obtained from various boundary conditions with constant or variable diffusion coefficient, in homogeneous or heterogeneous and isotropic or anysotropic bodies, and in steady or unsteady cases. The partial differential equation for non steady-state mass diffusion has been solved to mass transfer in bodies with single geometry, like
plates, cylinders and spheres (Luikov, 1968; Skelland, 1974 and Crank, 1992). Norminton’s and Blackwell’s (1964), Haji-Sheikh’s & Sparrow (1966), Alassar’s (1999) and Limass et al. (1999) works can be cited, for example, to complex geometry and constant boundary conditions.
Norminton & Blackwell (1964) present an analytical solution to predict the heat flow in the half-space around of an isothermal thin circular disk.
_________________________
1. Mestre em Engenharia Mecânica, Departamento de Engenharia Mecânica, CCT, Universidade Federal da Paraíba (UFPB), CEP 58109-970, Cx. Postal 10069, Campina Grande-PB, Brasil. Fone (083) 310-1317
2. Professor Doutor do Departamento de Engenharia Mecânica, CCT, Universidade Federal da Paraíba (UFPB), CEP 58109-970, Cx. Postal 10069, Campina Grande-PB, Brasil. Fone (083) 310-1317, e-mail: [email protected]
42
Haji-Sheikh & Sparrow (1966) gave an analytical solution to the heat transfer equation in a prolate spheroid body with constant temperature at the surface, using an elliptical coordinate system in two-dimensional cases, but the results of the temperature in the center and focal point are given alone.
Lima et al. (1999), presented an analytical solution to predict the mass transfer inside a prolate spheroid. They considered constant properties and equilibrium boundary conditions at the solid surface. As application, results of the moisture content distribution inside of solid as well as of the average moisture content for an aspect ratio is presented.
The objective of this work is to develop an analytical solution to describe the moisture transport in a continuous medium, by utilizing the prolate spheroid coordinate system in two-dimensional cases, considering convective boundary conditions at the surface of the solid.
MATHEMATICAL MODEL
This mass diffusion equation in the short form is given by:
M
D
t
M
(1)
where D is the diffusion coefficient, M is the moisture content and t is the time.
Depending on the geometrical form of the body, a coordinate system, adequate to describe the domain in study, must be selected. In the specific case of ellipsoid of revolution, the adequate one is the prolate spherical system. The relations between the cartesian (x, y, z) and the prolate spherical ( , , ) co-ordinate systems are given by Haji-Sheikh & Sparrow (1966):
x= L sinh sin cos
y= L sinh sin sin
(2a-c) z= L cosh cos
where L is the focal length equal to (L2 2
-L1 2
)1/2. An ellipsoid of revolution scheme is shown in Figure 1.
z
y
x
L
2L
1L
Figure 1- Characteristics of a prolate spherical
solid
Defining =cosh , =cos and = cos , the metrics coefficient and the Laplacian to the new coordinate system can be obtained using the mathematical relations which are given by Abramowitz & Stegun (1972). Utilizing the metrics coefficients, the variables , and and the differentiation’s rules, the mass diffusion equation can be written:
M t L D M 1 1 2 2 2 2 2 21 2 1 2 L D M (3) M D L 2 2 2 2 2 1 1 1 1
For a situation with symmetry around the z axis, it is: M t L D M 1 1 2 2 2 2 1 1 2 2 2 2 L D M (4)
According to Figure 2, the = o (constant), o>1 is an elongated ellipsoid of revolution with bigger axis of length L and smaller length axis L( 2-1)1/2. The surfaces constants are a prolate spheroids confocal family and they have their
43 common center at the origin. The degenerate
surface =1 is the curve that links the center (z=0) to the focal point (z=L). The surface = o (constant), o<1, is an asymptotic cone whose two sheets hyperboloid of revolution generating by line passes through the origin and it is inclined at the angle =cos-1 to the z-axis. The degenerate surface =1 is part of the axis z>L.
The initial and boundary conditions of equation (4) are: M( , ,0)=Mo = constant (5a) e f m f 2 2 2 M t , , M h M 1 L D (5b)
where f=L2/L at the surface of the solid, hm is the mass transfer coefficient and Me is the equilibrium moisture content.
Using the separation of variables solution method we can write M( , ,t)= ( , ) (t). The solution of the equation (4) is then, (Haji-Sheikh and Sparrow, 1966):
M= ( , )exp(-c2Dt/L2) (6) where c is constant.
Assuming that the diffusion coefficient is constant and applying the Equation (6) to Equation (4), we have: c L L 2 2 2 2 2 2 1 1 1 1 2 2 2 2 L (7)
which can be written in a short form as follows:
2 2 2 0 c L
(8)
Assuming that ( , )= ( ). ( ), putting it into Equation (8), and separating the variables, two ordinary differential equation are obtained:
d d d d b c 1 2 2 2 0 (9) d d d d b c 1 2 2 2 0 (10)
In Equations (9) and (10), b is the separation constant or eigenvalues. These two equations are exactly in the same form, as a function of , varies between 0 and the singular point +1, while as a function of , varies between the singular point +1 and L2/L. The solution of the angular function ( ) is expressed in terms of a Legendre function series of the first kind (Pn), while the radial function ( ) is obtained from a spherical Bessel functions series of the first kind of order n (jn). The solution of the Equations (9) and (10) are given by:
constant constant y z 1 0 L 1 1 Rotation axis to angle
44
m n m n n n m n c d d j c n m ( , ) , . . , 0 1 0 1 2 m n m n n c d c P ( , ) , ( ). ( ) 0 (11a-b) r
d
r 2,m rb d
n r m, rd
r 2,m0
(12) where: rr
r
c
r
r
2
1
2
5 2
3
2 ; rr r
c
r
r
r r
2
1
1
2
1 2
3
1
2 ; (13a-c) rr r
c
r
r
1
2
3 2
1
2 with r=0, 2, 4... .The bn values are given by the transcendental equations as follows:
U (bn) = U1(bn) + U2(bn) = 0 (14) with: U b b b b n n n n n n n n n 1 2 2 4 ... (15a) U b b b n n n n n n n 2 2 2 4 4 ... (15b) and n n n c n n n n 2 2 4 2 1 2 1 2 1 2 3 ; 2 n n n c n n n 1 2 1 1 2 1 2 3 0 2 ;
The technique utilized in the Equation (14) to determine the bn coefficients is called continued fraction technique (Stratton et al., 1941; Stratton et al., 1956). This technique has been used to determine the eigenvalues to c 8.0. When c 10.0, the eigenvalues are obtained through an asymptotic expansion. The asymptotic development of bn is
given by the successive approximations method, as follows: 2 6 3 4 4 2 2 2 2 3 7 2 5 2 3 1 2 2 3 2 2 1 2 c n n n c n n n n n c n bn 3 10 2 3 4 5 2 453 1321 1278 962 165 66 c n n n n n 4 12 2 3 4 5 6 2 4425 13349 18478 10510 5885 75 252 c n n n n n n 6 5 20 3 5 7 2 1 2 2241599 1 2 1043961 1 2 61529 1 2 527 c O c n n n n (16) where O (c-6) represents the error order.
A convergent series for dn,m can be obtained to a discrete set of values of the eigenvalues bn. There are two sets of finite solutions, one for even values of n, the other for odd values. The lowest value of bn corresponds to n=0, the next to n=2, (Morse & Feshbach, 1953), so, the set that corresponds to even value of n was used in this work. The values of the coefficients dn,m are different, depending on the normalization adopted scheme. The utilized criterion by the authors is presented below: 1 2 2 2 1 2 2 2 2 0 2 r r r m r n n r r r d n n n ! ! ! ! ! ! , (17) for r=0,2,... and n=0,2,... .
The Equation (17) together with the Equations (16) or (12), allows the complete determination of the coefficients dn,m. The index n is into all the cases 0. We have Pn( )=0 For n<0 indicating that the series really begin at n=0.
The condition that restricts the values of the bn in the differential equations is reflected in Equation (12) as a requirement that the ratio of the coefficients dn,m/dn-2,m 0, when n (Stratton et al.,1956). Observing that the coefficients c, b and d, they must be obtained and satisfy the Equation (5b) at the surface of the prolate spherical ( =L2/L). This condition is given by:
L
/
L
1
1
Bi
1
2 f m 2 2 f m (18)With the determined dn,m coefficients, the formal solution of the problem is given by:
45 2 , 0 1 , , m k mk e A M t M
,
,
2 2 mk m mk m L t D cc
c
e
mk (19)The coefficients Amk are obtained from the orthogonal conditions. Substituting the initial condition Equation (5a) in Eq. (19), it is obtained:
Mo Me Amk m cmk m cmk k m , , , 1 0 2 (20)
Multiplying both sides of Equation (20) by p(cpk, ) p(cpk, )( 2- 2) and integrating in a quarter of the ellipsoid volume, it is obtained:
p cpk p cpk Mo M d de L L , , 2 2 1 0 1 2
1 0 1 2 2 2 , 0 1 2 , , L L pk p pk p m k c c
d d c c Amk m mk, m mk, (21)
where the integration and the sum operations were exchanged.
Considering that the integration in Equation (20) can be made term by term, and the orthogonality of the functions, the unique term in the right side that supply an integral that´s different from zero, is the term with m=p. For m=p, the result is: A c c M M d d c c d d mk m mk m mk o e m mk m mk L L L L , , , , 2 2 1 0 1 2 2 2 1 0 1 2 2 (22) where the denominator is the norm of ( m m)(
2 -2
).
Defining the following dimensionless parameters: e o e *
M
M
M
M
M
; 2 L Dt t ;D
L
h
Bi
m (23) It can be writen the Equation (18) as follows2 , 0 1 2 m k t c mk e mk Me Mo A M , , m mk mk m c c (24) The average moisture content of the solid can be calculated as follows:
1 0 L L 1 2 2 1 0 L L 1 2 2 2 , 0 m k1 mk m mk m t c e o mk 2 2 2 mk d d d d , c , c e M M A M (25) where the denominator is the total solid volume in the prolate spherical coordinate system.
RESULTS AND DISCUSSIONS
As application, the methodology was used to predict the drying kinetics and moisture content distribution of a prolate spheroid with aspect ratio L2/L1=2.0 and Bi=1.0. Table 1 presents the c values, roots of the radial spherical function, for =L2/L, the eigenvalues b of the expansion coefficients dn,m and coefficients Am.k and, finally, the obtained values for the orthogonality criterion to radial and angular functions.
Three computational code were implemented, utilizing the software Mathematica . To obtain the values of the c, b, dn,m, and Am,k coefficients and the orthogonality conditions of the function of the final presented solution in the Equation (19). It can be observed that the acquirement of this analytical solution requires a very hard work and an excessive number of computational work hours, besides its comparison with the numerical solution that’s given by Lima (1999) and Lima & Nebra (2000).
Some obtained results with the computational code for given conditions were exhaustively compared with the supplied results in works of Flammer (1957), Haji-Sheikh & Sparrow (1966), Stratton et al. (1941), and Abramowitz & Stegun (1972). The given values in the Table 1 can be used to reproduce the results that are shown in this work and to help investigators to validate computational codes in future works.
Results of this work were compared with numerical results for an ellipsoid (L2/L1=1.1), with Bi infinite given by Haji-Sheikh & Sparrow (1966) to validate the mathematical model Figure 3 shows the comparison between the concentration ratio at the center and focal point of a prolate spheroid as a function of Fo that’s defined as Fo=Dt/L1
2
. Almost complete concordance exists between the results, like can be observed.
Figure 4 shows the comparison between the average moisture content as a function of Fourier number during the drying process for a prolate spheroid with aspect ratio L2/L1=2.0 and Bi=1.0, which was obtained in this work and numerical results that were reported by Lima (1999) and Lima & Nebra (2000).
46
Figure 5 illustrates the moisture content distribution inside the prolate spheroid through the use of different tons for three Fourier numbers. The moisture content changes with the changing angular and radial coordinates. The comparison of the graphs indicates that the increasing of the values of causes the moisture content decrease for any at any Fo. The moisture content profile decrease in any point for increasing values of Fo ( , ), what indicates indicating that the moisture flux occurs from center to the surface. The strong moisture content dependence with the radial and angular coordinates it can be also observed. The concentration dependence with the angular coordinate is slightly larger than its dependence with the radial coordinate. In this case, the dimensionless moisture content is decreasing with
the increase of , for all values of Fo. It is verified that the moisture content gradients are high, except the ones for the regions near the center of the body. It is verified that the surfaces of and constant, are not spherical, but they present approximately an elliptical behaviour.
It is verified that the concentration ratio decreases faster in the extremity of the z-axis (z=L2). This effect decays to the end of y-axis (z=L1). This behaviour occurs in all types of ellipsoids, and it increases proportionally to the aspect ratio. In order, the behaviour of the moisture content with the angular coordinate is different from a sphere (L2/L1=1.0), where symmetry exists relative to this coordinate. This difference will increase with the increase L2/L1.
0.01 0.10 1.00 Fo=Dt/L1^2 0.00 0.20 0.40 0.60 0.80 1.00 (M -M e) /( M o -M e) This work Center Focal point
Haji-Sheikh and Sparrow (1966) Center
Focal point
L2/L1=1.1
Figure 3 - Comparison between the moisture content ratio in the center and focal point of a spheroid with
L2/L1=1.1, that’s given by the authors and Haji-Sheikh & Sparrow (1966)
0.00 1.00 2.00 3.00 4.00 5.00 Fo=Dt/L12 0.00 0.20 0.40 0.60 0.80 1.00 _ _ M* Bi=1.0 L2/L1=1.0 (Analytical, Luikov, 1968) L2/L1=2.0 (Analytical, This work) L2/L1=2.0 (Numerical, Lima, 1999)
Figure 4 - Dimensionless average moisture content as a function of Fourier number of a spheroid with
47
Table 1 - Values of the coefficients and orthogonality criterion of the spherical functions for L2/L1=2.0 and Bi=1.0
m k c b A Mo Me mk m m p p L L d d 2 2 1 2 0 1 / , m p 1 1,953419 1,08354 1,48481 -0,0004899991 2 7,47987 6,69892 -0,866435 0,00330227 3 10,2890584 9,55454 0,342731 0,00290506 0 4 12,9668292 12,2296 15,468 -0,00156183 5 15,6714269 14,9323 -26,6011 0,000930949 6 18,3848365 17,6443 29,1506 -0,000531135 7 21,10095 20,3593 -24,519 0,00026297 8 23,8186 23,7939 -9,15009 -0,0000915305 1 5,81391 39,3376 0,0109559 -0,000164286 2 13,0637331 61,462 0,127735 0,00769859 3 15,7245503 74,7909 17,2802 0,000591925 2 4 18,396268 88,165 -23,7898 -0,000347971 5 21,1103237 101,711 21,4571 0,000202749 6 23,816222 115,283 -14,8797 -0,00010286 7 26,532186 128,869 8,43806 0,0000385393 1 10,26605 80,089 1,57958 0,000406412 2 15,69248 129,670 0,383505 0,000147741 3 18,4858417 154,969 11,5816 0,00021217 4 4 21,10981 178,686 -16,8422 -0,000118301 5 23,808902 203,053 12,9801 0,0000591328 6 26,5200414 227,511 -7,21616 -0,0000195042 7 29,2384852 252,022 11,842 0,0000337982 1 12,58607 137,593 1,50672 0,000247829 2 18,11000 211,265 -0,082394 -0,0000192767 3 21,24923 252,551 3,86001 0,0000823914 6 4 23,83100 286,383 -9,7495 -0,0000432522 5 26,517288 321,515 7,42446 0,0000230514 6 29,20961 356,679 17,3211 0,000148078 7 34,90268 430,936 -7,62993 -0,000224578 1 15,01682 210,010 1,91373 0,000187496 2 20,22141 301,763 -0,772457 -0,0000422199 3 23,9797148 366,791 0,387092 0,0000154751 8 4 26,54714 410,966 -1,67017 4,47683,10E-7 5 29,26756 457,641 20,0474 0,000105134 6 35,66107 567,020 3,96735 0,0000539277 7 39,52014 632,911 -12,1622 -0,0000827413 1 17,49912 297,357 2,41766 0,00020642 2 22,22500 401,371 -1,13313 -0,0000385566 3 26,6785033 497,183 -0,290447 -0,0000462464 10 4 29,18105 550,598 -5,85693 -0,000307788 5 35,27147 679,930 -4,87921 -0,0000827867 6 39,39431 767,163 10,6966 0,0000700765 7 43,07483 844,907 -11,6956 -0,0000372207 1 20,01203 399,693 4,28054 0,000190972 2 24,44630 516,960 -1,10305 -0,0000271166 3 29,32767 642,774 6,30253 0,000150271 12 4 35,32351 795,359 2,89170 0,000066225 5 39,53762 901,959 -4,53534 -0,0000553565 6 43,28633 996,524 3,32747 0,0000215678 1 22,54517 517,067 5,47110 0,000251751 2 26,79318 648,431 -1,03888 -0,0000197972 14 3 30,71061 766,522 -5,93359 -0,000116087 4 35,40961 906,288 -1,72847 -0,0000250977 5 39,29675 1021,03 1,22652 -5,16498.10-6 6 42,81976 1124,60 3,41665 0,0000216238
Figure 6 illustrates the moisture content distribution inside the solid with aspect ratios L2/L1=1.5 and L2/L1=2.0 in Fo=0.122 and Bi infinity. The highest moisture gradients are found as expected in comparison with the case to Bi=1.0.
Figures 7 and 8 show the moisture content as a function of the radial and angular coordinates for various Fourier’s numbers during the drying process.
48
0.00 0.20 0.40 0.60 0.80 1.00 y 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 z 0.00 0.20 0.40 0.60 0.80 1.00 y 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 z 0.00 0.20 0.40 0.60 0.80 1.00 y 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 z
(a) Fo=0.122 (b) Fo=0.366 (c) Fo=0.732
Figure 5 – Moisture content distribution inside the prolate spheroid with aspect ratio L2/L1=2.0 and Bi=1.0 during the drying process. Fo=Dt/L1
2 0.00 0.50 1.00 y 0.00 0.50 1.00 1.50 z (a) 0.00 0.50 1.00 y 0.00 0.50 1.00 1.50 2.00 z (b)
Figure 6 – Moisture content distribution inside the prolate spheroid with aspect ratios (a) L2/L1=1.5 and (b) L2/L1=2.0 to Fo=0.122 and Bi infinity.
49 1.00 1.04 1.08 1.12 1.16 0.00 0.20 0.40 0.60 0.80 1.00 M * L2/L1=2.0 Bi=1.0 Fo=0.0366 Fo=0.0732 Fo=0.1464 Fo=0.2440 Fo=0.3660 Fo=0.5490 Fo=0.7930 Fo=1.2200 Fo=1.5860
Figure 7 – Dimensionless moisture content as a function of in =0,0 ( 0 y L1) and various Fourier’s numbers, L2/L1=2,0 and Bi=1,0.
0.00 0.20 0.40 0.60 0.80 1.00 0.00 0.20 0.40 0.60 0.80 1.00 M * L2/L1=2.0 Bi=1.0 Fo=0.0366 Fo=0.0732 Fo=0.1464 Fo=0.2440 Fo=0.3660 Fo=0.5490 Fo=0.7930 Fo=1.2200 Fo=1.5860
Figure 8 - Dimensionless moisture content as a function of in =1,0 ( 0 z L) and various Fourier’s numbers, L2/L1=2,0 and Bi=1,0.
The results for ellipsoid, calculated with the coefficients shown in Table 1, present a small error for any Fourier’s number. This difference can be attributed to the instability of the Bessel functions for small Fourier’s numbers, and to the successive approximation of the presented method here. This last problem can be solved using a higher number of terms in the determination of the coefficients. Some others results are noted: the moisture content strongly depends on the Fo and the equilibrium moisture content is achieved at Fo 5.0.
CONCLUSIONS
The present theory was applied to an ellipsoid with L2/L1=2.0 treated here as an application of the general method. It indicates that
the method can be solved directly by use of the Equation (24) and (25), with determined the eigenvalues and spherical coefficients. The method does not require any particular geometry form in study changing from sphere to cylinder besides ellipsoids of revolution. The analytical solution presented here can be used to obtain results that describe the transient phenomena, in particular, moisture content distribution and drying kinetics, in bodies with spherical, cylindrical and elliptical geometries, considering the diffusion coefficient constant and the mass diffusion as sole mechanism of moisture migration. As obtained the solution is referred to the case with convective boundary condition at the surface, it can be used to validate numerical solutions, which can be extended to cases with less restrictive conditions.
The mean value of moisture content is particularly useful when the model is used to
50
determine the diffusion coefficient from drying kinetic experimental data. The used dimensionless coordinates, moisture ratio and Fourier’s number, were adequate to get general results, to be applied to any case of heat or mass transfer. The moisture content is strongly influenced by the Fourier’s number in any position in the interior of the spheroid. The equilibrium moisture content is approached, at any point of the body, to Fo=Dt/L12 5.0 (L2/L1=2.0 and Bi=1.0) according to the results reported in the literature. The dimensionless moisture content decreases faster in the extremity of the z axis (z=L2) what and decays to the end of y axis (z=L1) indicates that the regions near the z=L2 dry first.
ACKNOWLEDGEMENTS
The authors would like to express their
thanks to CAPES (Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior, Brazil) and CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), for its financial support nº 476457/2001-7 to this work.
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