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A many-fibre theory of airflow through a fibrous filter—II: Fluid inertia and fibre proximity
J. Aerosol Sci.. Vol. 17, No. 4, pp, 685~97, 1986. Printed in Great Britain.
0021-8502/86 $3.00+0.00 Pergamon Journals Ltd.
A M A N Y - F I B R E T H E O R Y O F AIRFLOW T H R O U G H A F I B R O U S FILTER--II: F L U I D INERTIA A N D FIBRE PROXIMITY R. C. BROWN Health and Safety Executive, Broad Lane, Sheffield $3 7HQ, U.K.
(Received 27 September 1985) Abstract--The problem of Stokes flow through a fibrou/; filter modelled as a two.dimensional array has previously been solved by direct application of the variational principle to a generalized stream function, expressed as a two-dimensional Fourier series in Cartesian coordinates. In this paper the effect of fluid inertia at low Reynolds number is described, by solution of the steady state Navier-Stokes equation in the form of a perturbation expansion, leading to a system of linear partial differential equations with solutions in the form of Fourier series. Pressure drops across model filters, calculated by this method, are compared with experimental results. Both theory and experiment illustrate that the lowest order correction to Darcy's Law is a cubic term in the flow velocity, and that this term is strongly influenced by the filter structure. The details of the flow pattern close to the fibre surface, in particular the presence or absence of standing eddies at finite Reynolds number, is shown to depend critically on the filter structure; eddies are shown to occur in Stokes flow when the fibre layers are sufficiently close together. NOMENCLATURE A Defined in text; equations (8) and (32) Coefficient in double Fourier series description of WM Defined in text; equations (32) and (33) b Mnk Coefficient in double Fourier series description of WMm CMnk Coefficient in double Fourier series description of WM¢F f,~ Defined in text; equation (5) F Defined in text; equations (37) and (45) F' Defined in text; equation (41) l Half spacing of array in y-direction L Defined in text; equation (45) P Pressure drop across a filter R Fibre radius Re Reynolds number, 2RpUo/~ t Filter thickness Uo Macroscopic fluid velocity U~, Uy Fluid velocity components xi,Yl Co-ordinates of a point where boundary conditions are applied Dimensionless group used as expansion parameter in perturbation series; defined by equation (12) Lagranglan multiplier Half spacing of array in x-direction Coefficient of dynamic viscosity 0 Rate of dissipation of energy Lagranglan multiplier #u Matrix element defined in text; equation (7) ~g Complete stream function Stream function for Stokes flow ¢ Complete dimensionless stream function ~o Dimensionless stream function for Stokes flows and zero order term in the perturbation expansion of ¢, CN Nth order term in the perturbation expansion of ~ at finite Reynolds number Particular integral part of ¢,, ¢ICF Complementary function part of ~'t a Mnk B
INTRODUCTION
It has been shown previously (Brown, 1984) that the stream function for Stokes flow through
a two-dimensional array can be calculated by applying the principle that the flow pattern will be such that the dissipation of energy due to viscous drag is a minimum, subject to the boundary conditions of the system. The trial stream function used was a double Fourier series, the coefficients of which were treated as variational parameters. The series were Crown copyright © 1986.
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R.C. BROWN
summed over terms with non-negative indices only, and different series were used to describe arrays in which adjacent layers had fibres with the same y-coordinates, the channel array, and those in which each fibre layer had fibres with y-coordinates at the mid-points of those in adjacent layers, the staggered array. In further developments of the model, it will prove convenient to express the two series in a common form which describes either of the two geometries, depending on conditions applied in the summation; and it will simplify later working if the sums are extended to include coefficients with negative indices. This generalized form of the stream function can be written • nny • o = U o y + U ol ~ ao ,,k
sin ~ -
krtx
cos 25i-
(I)
n,k
Equation (1) applies to macroscopic flow in the x-direction, and the presence of only cosine terms in x constrains the function to have the upstream--downstream symmetry that is characteristic of Stokes flow. If the sum is limited to even values of k, the stream function will have the symmetry appropriate to the channel array, as shown in Fig. 1. If n and k are either both even or both odd or, in other words, if (n + k) is even, the staggered symmetry results. In all of the working below, double sums written as in equation (1) are assumed to extend from minus to plus infinity and to have one of the conditions above applied. F L O W P A T T E R N AT ZERO R E Y N O L D S N U M B E R The solution of the Stokes flow problem has already been described in detail and so only a brief summary of it, to include the presence of negative indices in the Fourier series, will be given here. Substitution of equation (1) into the expression for the rate of dissipation of energy, equation (36), and integration results in an expression containing squared Fourier coefficients but no cross-terms. Products of coefficients with the same values of Inl and Ik I will occur but it can be assumed, without any loss of generality, that ao_nk = -- aonk;
aOn_k = a o . k.
(2)
The minimum value of the expression is found by differentiating it with respect to every subject to the constraint that the fluid velocity is zero at the surface of the fibres. When the method is used in practice, the stream function is constrained to vanish at a finite number, M, of points taken in pairs just outside and just inside the surface of the fibre at the origin. If the coordinates of one such point are (xi,yi) the corresponding boundary condition can be expressed as a o . ~,
nrty i
ao. k sin - - f -
y~+l~
kltxi
cos ~
= 0.
(3)
n.k
The conditions are included in the problem by means of Lagrangian undetermined multipliers {7i}, and the result is that l nrcy~ ao.k + - - ~ "ti sin ~ ink i
k~xi
cos ~
= 0 (4)
0
0
0
0
0
0
o
o
0
0
0
0
o
o
0
o
Fig. 1. Two-dimensionalrectangular array of fibres.
Filter airflow model: II
687
for all {a0nk}, where
(5) The {~} are given by the solutions of the following simultaneous equations:
)~,Tibttj=yj;
1 ~j<~M,
(6)
i
where all the matrix elements/~u are given by
12
nny~
#'J = .,kJ,~7- sin - - ~
nnyj
sin ~
knx~
cos - ~ e cos
knxj
2--~
(7)
Equations (4)-(7) can be formulated into a simple computer program using a standard library subroutine for the solution of the simultaneous equations (6). The fibre shape is specified by the choice of {x~, y~}, the filter packing fraction and degree of anisotropy by choice of the unit cell repeating lengths 8 and l, and its structural symmetry by the choice of the conditions applied to the (n,k) sum. The flow pattern for the smallest repeating element of a channel array is shown in Fig. 2. The pressure drop across such an array is related to the rate of dissipation of energy through it; in terms of the {a0,~ }, which are now assumed to take the values fixed by the variational calculations, the pressure drop across a thickness t is given by
P= U°rlizn4t8 ~, ao,kfnk.Z
(8)
n,lt
The accuracy that the model can achieve depends on the value of M and on the upper limits of Inl and Ikl in the series. The model is at its poorest close to the fibre surface, where the boundary conditions are applied and, as a rough guide, the approximation is good at a distance away from the fibre comparable with the tangential spacing of the points {xi,y~}. The n, k series should be truncated such that the period of the highest order component is comparable with this spacing. Those streamlines shown in Fig. 2 are well described if M = 6 (with the stream function constrained to vanish at three pairs of points in each quadrant), but if details of the flow pattern closer to the fibre surface were required, more points would be needed. In particular, a greater number of fixed points are required to give a good description of sharp edges on the fibre surface than are needed to describe a perfect cylinder. The upper limit of M that can be used is governed by the computation time allowed, though at large values of M the matrix [/~u] may appear to be singular because of limitations in the computer used. With the Xerox NUMSLIB library subroutine on a Sigma-6 computer, spurious singularity is observed at about M = 40.
FLOW T H R O U G H A TWO-DIMENSIONAL ARRAY AT FINITE REYNOLDS NUMBER The Reynolds number in airflow through most fibrous filters used in respirators is low. However, at high flow rates through relatively coarse filters of the sort used as size-selectors in respirable dust samplers (Brown, 1980) the Reynolds number may take a value of unity or more, and a description of airflow through such filters requires fluid inertia to be taken into account. The Navier-Stokes equation for airflow through a two-dimensional system, under steady state conditions, for an incompressible fluid with finite viscosity and finite inertia (or finite density, p) takes the following form, when expressed in terms of the stream function (Landau and Lifshitz, 1959): p~d~P 0V2~P 0~p 0V2~P V2V2~I' = ~ (-~y ~ dx -~y J" AS 17:4-C
(9)
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R.C. BROWN
If the stream function is expressed in a dimensionless form, ~b, such that W
(10)
ql - Uol'
equation (9) becomes
, rI
JOy
,}
~x
~x
~y
"
(1l)
The multiplier on the right-hand side of this non-linear equation is closely related to the Reynolds number as normally defined, and for brevity it will be written as follows: P Uol
q
= ~.
(12)
It is now assumed that, when = is small, $ can be expressed as a power series in =, which is the fundamental assumption of perturbation theory. = ~ ~NSN.
(13)
N
Substitution of equation (13) into equation (11) results in two power series in ~. Equating the multipliers of ~N gives rise to the following relationships between the {~'N}Zero order term
V2V21#o = 0.
(14)
As expected, the zero order term is simply the Stokes flow expression, which has already been calculated.
First order term
V2V2qq
0~'o0V2~o = 0y dx
0q'oc~V2~o 0x 0y
(15)
The first order term can be expressed entirely in terms of derivatives and products of the zero order term. N t h order term
V2V2~u =
(16) i = o t~y
0x
0x
c3y
The Nth order term can be expressed as binary products involving the derivatives of lower order terms only. In principle, therefore, the perturbation series can be worked out in simple sequence, using the results of the first N - 1 terms to calculate the Nth term. The right-hand side of each equation will be an explicit function o f x and y, resulting in a simple linear partial differential equation in ~ , the solution of which can be expressed as the sum of a particular integral, which satisfies the equation, without taking account of the boundary conditions, and a complementary function, which satisfies the equation obtained by setting the right-hand side of equation (16) equal to zero (the bibarmonic equation), but which is constrained so that the sum of the particular integral and itself satisfies the boundary conditions. It will be shown below that this can be put into practice in a s t r ~ t f o r w a r d way. Only the first order term will be calculated, but once this has been done it will be clear how the calculation may be extended to any order. The method of solution of an intractably complicated equation in the form of a series of linear equations, which have to be solved in sequence, was used, though in a different form, in the solution of boundary layer problems by Howarth (1938).
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689
Particular integral solution of first order term ~bo is given, in its dimensionless form, by substituting equation (1) into equation (10). Substitution of the resulting expression into the right-hand side of equation (15) gives
n3
k F f n ~ 2 + ( k )21 sin nny sin knx
)
[(m)2 (j)2]
n,k m,j
nj
n~y
x ~ d cos T
mk
mty
-2e---/ sin T
kTtx
cos ~
k~zx
sin ~
tony
sin T
j~x
sin 2e
mrcy
jnx
cos - - 1 - cos 2 T J '
(17)
Use of the well-known product rules for trigonometric functions, along with the symmetry of the a0. k series, enables the expression to be simplified considerably. V2V2~b' = T
ZaO.kq--;./~L\./7// + \
j ~/
sin ~
sin
~,k
+ n " ..kZ -.iZao,,kao-j
2--~-
[ ( m ) 2 (~e)21(nj-mk'~ 7 + J~--~l / sin (m - n)rcy s m. (j - k)nx l- T -
(18)
A change of variable in the sums over m and j may now be made, resulting in sine terms containing only one index. The doubly infinite series used to describe ~Oomake the formal working considerably easier, because if the series were summed over non-negative terms only, a change of variable would result in one index appearing in the summation limit of the other, and this would be a tiresome complication. The resulting form of the double series dictates that a particular integral of equation (18), ~'m, should be an odd function of x, unlike ~'0, which is even.
~.
dim = n=
~.,
--o0
k=
nrcy krtx bl.k sin ~ sin 2--e-
(19)
--o'3
Equation (19) may be substituted into the left-hand side of equation (18), and the {bin k } can be calculated by equating the coefficients of the sine terms on the two sides. k
n 2
+ n'k - nk' 2el )"
k 2
[- fn+n,'~2
i/k+k,'~21
)
J (20)
All of the {bt~k} in the particular integral can, therefore, be calculated as sums of binary products of the coefficients of 4o. If the appropriate conditions for limiting the values of (n', k') and (n + n', k + k') for the required array are imposed, it follows that the same conditions will apply to (n, k), although this could have been assumed from considerations of symmetry.
Complementary function All that is required of ~bm is that it should be one solution of the differential equation. A
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R.C. BROWN
complementary function is also needed. This function, q/ice, must satisfy the equation V2V21]IICF ~- O,
(21)
and, when taken with the particular integral, it must satisfy the boundary conditions. In order to do so, it must have the same symmetry as the particular integral, and the sums must be limited by the structure of the array to be described: nny krtx d/lc r ---- ~ cl. k sin --~- sin - ,./ 2e
(22)
The boundary conditions to be satisfied are that each fluid velocity component must vanish at the fibre surface. The procedure used in the solution of the Stokes flow problem can be used here because, although the concept of minimization of rate of dissipation of energy does not apply, the biharmonic equation is the Euler equation that is a necessary condition for the energy dissipation functional to be a minimum, as shown in the Appendix. Those values of {cx.k} that will give the minimum value of the functional, O, must be found when ~'~ce is substituted into equation (36). The boundary conditions are fixed at the points {xi, y~}, and they can be included, using Lagrangian undetermined multipliers, in exactly the same way as before. The condition at the point (x~, y~) is: nrcy i kTcx i (bl.k+Cl.k) sin - T - sin ~ = 0.
(23)
n,k
The symmetry of the function is such that the simplest form is obtained with the following assumptions: Cl _.k = --Cl.k; Cl._k = --Cl. k. (24) A similar condition is imposed on the {bl.k} by the form of equation (20). The minimization procedure, in this case, results in O {~f,~c~.k 0C~.k .k. 2
+
[ ~ (b,.k+C,.k) sin nny, knx,]~ ~/Yi. ..k ~ sin 2e j j = 0 .
(25)
The following simultaneous equations are obtained: M
~/aij=9~;
I~
(26)
i=1
where #ij
= ~ 1 sin nnYi nnyj kr~x i krcxj .,kf.k ----(-- sin ~ sin ~ sin 2e
(27)
9j depends on the particular integral nny i k~xi 9 j = ~ b l . k sin T sin 2----e-
(28)
n,k
The total stream function is the sum of the two parts. nny knx ~01 = ~ al.k sin - - 7 sin 2----e-'
(29)
al. k = ba.k +Ca. k for all n,k.
(30)
n,k
where Each higher order term can be found in a basically similar way. The particular integral coefficients will be given by equations like equation (20), but containing N binary series for the Nth order term. The complementary function is of the same order of complexity for every term.
Filter airflow model: II
691
RESULTS
There is no fixed limit for the order of term that can be calculated, but errors in calculation will be cumulative. Accuracy is ultimately limited by the computing time that can be devoted to the large sums in the particular integral. The results below apply to models in which M = 12, In ] ~ 30 and Ikl ~< 60. The exact shape of stream function for each order of term is interesting, and the first three of these are illustrated in Figs 2-4 for the channel array. Figure 2 is the zero order term (the Stokes flow pattern) for conditions in which e = 1 -- 10.0, R = 2.0. Only one quadrant is shown but this form, repeated, gives the complete picture of the stream function. The first order term is shown in Fig. 3. The form of rotational motion of the fluid that could give rise to standing eddies on the downstream side of the fibre is obvious here. The second order term, which is not easy to imagine, is shown in Fig. 4. This has the symmetry obvious from its form, and it has the effect of moving incipient vortices downstream. Unlike the first order term, it will have the effect of moving fluid streamlines closer to the fibre surface in the x = 0 plane. The third order term is of the same basic shape as the first order except that it is weaker, and the sign of rotation is reversed. The fourth order term has the same relationship with the second, and the series continues in this way. The first and second order terms for a staggered array are shown in Figs 5 and 6, which illustrate clearly the radical difference between the two geometries. The zero order term is not shown because, close to the fibre, it is indistinguishable from that for the channel array. Once the various terms have been calculated, it is straightforward to build up the complete stream function, for any value of a or Reynolds number, by using equation (13). The individual odd order components have upstream-downstream symmetry, and the even order components have antisymmetry, but the complete stream function has neither.
7 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i I
i i
OS "
Fig. 2. Stream lines corresponding to dimensionless stream function for Stokes flow; or zero order dimensionless stream function in perturbation expansion: e = l = 10, R = 2: Channel model. (The stream line at y = l has the value ~ = l.)
,)
iI
Fig. 3. Stream lines corresponding to first order dimensionless stream function in perturbation expansion: e = I = 10, R = 2: Channel model.
692
R . C . BROWN r
.
.
.
.
o.0
Fig. 4. Stream lines corresponding to second order dimensionless stream function in perturbation expansion: e = I = 10, R = 2: Channel model.
Fig. 5. Stream lines corresponding to first order dimensionless stream function in perturbation expansion: e = l = 10, R = 2: Staggered model.
o0 'oO
Fig. 6. Stream lines corresponding to second order dimensionless stream function in perturbation expansion: e = l = 10, R = 2: Staggered model.
The fundamental simplicity of the complete solution is made clear if it is written in the following way:
y
[
~'
0 = i + n~.k ,
+
even
N =
. mty
~NaNnk s i n
knx
--T- cos
N = 0
l
~, odd
~
~tNaN.k sin ~ - - sin 2e J"
(31)
1
Once the {aNnk} have been calculated, the stream function can be written down for any value of Re or ~ within the limits of convergence, without further calculation. The function has a simple analytical form, which should make it particularly easy to use in calculations of particle trajectories or studies of filtration. The stream lines for flow through a square two-dimensional channel array are shown in Fig. 7, with e = l = 10, R = 2 and a value o f a = 7, which corresponds to a Reynolds number, as usually defined, of 2.8. The Stokes flow solution for this array is the zero order term drawn in Fig. 2. The fundamental asymmetry in the flow pattern produced by the effect of fluid inertia is clearly shown in Fig. 7.
Filter airflowmodel: II
693
J
Fig. 7. Completestream functionfor square array: ,t = 7, Re = 2.8, e = l = 10, R = 2. PRESSURE DROP AT F I N I T E REYNOLDS NUMBER The pressure drop across a filter as a function of small Reynolds number can be obtained by substituting equation (31) into the energy dissipation equation (36). The orthogonality of the two series in equation (31) results in a pressure drop that is a series containing only odd powers of Re, a, or Uo: P = AUo+BU3o+O(U~).
(32)
When Uo is sufficiently small for higher powers to be neglected, equation (32) reverts to the Darcy Law/Stokes flow form, given in equation (8). At higher velocities, the first correction term is such that B = P21*~*t 8rI ~,f,,k(2ao,ka2~k + aani).
(33)
It is clear from the symmetry of the perturbation terms illustrated in Figs 2-6 that odd order terms will produce no net tangential stress at the fibre surface and that the general expression for pressure drop at finite Reynolds number will have the form of equation (32). In fact, such a result would follow from a more general treatment than that above, and a similar result follows from the calculations of Tamada and Fujikawa (1957) for a single perfectly periodic layer of fibres. An experimental test of the validity of equation (32) has been carried out on model filters consisting of stacks of thin metal plates, each photo-etched to form a regular grid. The plates were 200 #m thick, etched to give fibres that were rectangular in cross-section with a width of 250 #m. The distance I was 450/an, and the plates were spaced such that the distance e was 380 #m. Eighty such grids were stacked in both the channel and staggered symmetries, air was passed through the two model filters at velocities of up to 0.8 m s- t, and the pressure drop across each filter was monitored. The results are shown in Fig. 8, where theory and experiment are compared. The experimental results support the theoretical prediction that the lowest order correction to Darcy's law is a cubic term, and both theory and experiment show that the resistance of a filter at finite Reynolds number is highly dependent on its structure, even though the dependence at zero Reynolds number is weak. The discrepancy between experiment and theory, apparent in Fig. 8, is similar in nature to the discrepancy between the experimental measurements of pressure drop across a single layer of fibres carried out by Kirsch and Stechkina (1977) and the calculations of Tamada and Fujikawa (1957). The application of simple curve-fitting techniques to the experimental measurements of Jefferies (1974) on the pressure drop across coarse porous foams of three-dimensional structure shows that a linear plus cubic function for the velocity dependence of pressure drop gives a good fit, whereas a linear plus quadratic function does not. SEPARATION IN I N E R T I A L FLOW AND STOKES F L O W The variational model is at its weakest close to the fibre surface, where the boundary conditions are applied. Nevertheless, it is in this re#on that flow separation is most obvious,
694
R.C. BROWN 100
/
I
/
90
/ 80
/
7O
/ P
/
50
I
x
I x
/x
~4o
/ / x
30
®/*
20 10 j~epv
0.1
,
,
~
J
,
J
J
(12
0.3
0.4
0.5
0.6
0.7
0.8
A i r Velocit y ( m s-~ )
Fig. 8. Pressure drop as a function of filtration velocity for model filters: Channel model theoryC) ; channel model experiment--; staggered model theory---; staggered model experiment x.
and the model does illustrate the formation o f standing eddies on the downstream side o f the fibre. In the theoretical models shown, these appear at a b o u t u = 8.4 for the channel array, but the critical value o f ~ will depend on e and l. An illustration o f the flow patterns in the vicinity o f a single fibre is shown in Figs 9 and 10. Separation does not seem to occur in the staggered model, for these values, at least not when the perturbation series is truncated after the second order term. The flow pattern is strongly influenced by the geometry o f the system. F o r the channel array, separation will occur even in Stokes flow provided the fibre layers are close together. If is reduced, standing eddies appear on those faces o f the fibres that are closest to the next fibre layer, and as the spacing is further reduced, the eddies coalesce. I f the distance I is five fibre diameters, eddies start to form when e is a b o u t 1.04 fibre diameters, and the eddies
G2
0.1
Fig. 9. Flow pattern close to fibre surface:e = 1 = 10, R = 2, ,, = 9.4;Channel model.Q-point where boundary condition is applied. 0.2
O.1
Fig. 10. Flow pattern close to fibre surface: e = I = 10, R = 2, ct = 9.4; Staggered model,
Filter airflow model: II
695
coalesce when e is about 0.69 fibre diameters. Distinct and coalesced eddies are illustrated in Figs 11 and 12. The critical spacings are comparable in magnitude with those observed by Davis (1976) for two spheres whose common axis coincides with the macroscopic Stokes flow direction. The eddies observed in Stokes flow, unlike those at finite Re, have upstream-downstream symmetry. Figure 13 illustrates the convoluted pattern that occurs when flow is slightly inclined to thcaxis. Highly inclined flow eliminates the eddies altogether. The development of separation in Stokes flow can be simply explained. A general solution of the biharmonic equation, if boundary conditions are specified at x = 0 only, or if the filter consists of a single regular layer of flat fibres, is
etal.
~b = ~ a. sin
~-(
nnlxl'~/e x p / -
1 + ~
n~ t Ixl ) .
(34)
n
Each mode will decay in a distance in the x-direction comparable with its period in the ydirection. If another layer of fibres lies well within the decay length, the mode will be preserved, because it must be regenerated at the next fibre layer. This means that at certain layer spacings long-wavelength modes in the stream function will be preserved whilst shortwavelength modes decay. The finite thickness of the fibres will alter this simple picture, but an imbalance between long-wavelength and short-wavelength modes will still occur, and the result of this imbalance will be the symmetrical standing eddies illustrated in Figs 11 and 12. DISCUSSION
A variational approach to solution of Stokes flow through two-dimensional arrays of fibres is a simple way of obtaining detail about the flow pattern. The model has been extended to the case of steady state incompressible fluid flow at low but finite Reynolds number, by means of a perturbation expansion. The result is that the Nth order term satisfies a differential equation involving only terms of lower order. If the terms are calculated
Fig. 11. Distinct standing eddies in Stokes flow.
O.OO2
Fig. 12. Coalesced standing eddies in Stokes flow.
000
-0.06
Fig. 13. Convoluted flow pattern in Stokes flow.
696
R.C. BROWN
s e q u e n t i a l l y , t h e s e differential e q u a t i o n s b e c o m e t h e t y p e t h a t can be solved as the s u m o f a p a r t i c u l a r i n t e g r a l a n d a c o m p l e m e n t a r y f u n c t i o n , the f o r m e r o f w h i c h can be f o u n d by e q u a t i n g coefficients in F o u r i e r series, a n d the latter by explicit use o f the v a r i a t i o n a l principle. F o r the m e t h o d to be valid, b o t h t h e F o u r i e r series m a k i n g up the s t r e a m t u n c t i o n c o m p o n e n t s a n d t h e p e r t u r b a t i o n series itself m u s t c o n v e r g e . A f u n d a m e n t a l t r e a t m e n t o f this is b e y o n d the s c o p e o f this paper, b u t t h e s t r e a m l i n e s s h o w n in F i g s 2 - 6 are i m p e r c e p t i b l y a l t e r e d if the t r u n c a t i o n p o i n t s for the M , n a n d k s u m s a r e r e d u c e d f r o m 20, 30 a n d 60 to 6, 15 a n d 30; this suggests t h a t t h e series c o n v e r g e s q u i t e rapidly. T h e l i m i t i n g v a l u e o f ~t for which the p e r t u r b a t i o n series will c o n v e r g e m u s t d e p e n d o n e, l a n d the fibre d i m e n s i o n s . S t r a i g h t f o r w a r d e x a m i n a t i o n o f the v a r i o u s o r d e r s o f s t r e a m f u n c t i o n for the c h a n n e l a r r a y case in w h i c h e = l = 10, R = 2 suggests t h a t the series is c o n v e r g e n t at a R e y n o l d s n u m b e r o f a b o u t 4. H o w e v e r , the fact t h a t e r r o r s in s t r e a m f u n c t i o n s will be c u m u l a t i v e limits the c o n f i d e n c e we m a y h a v e in the a b s o l u t e v a l u e o f this result. Acknowledgement--The author wishes to thank Mr D. Wake of HSE, who assembled and tested the model filters.
REFERENCES Brown, R. C. (1980) J. Aerosol Sci. I!, 151-159. Brown, R. C. (1984) J. Aerosol ScL 15, 583-593. Burley, D. M. (1974) Studies in Optimisation. lntertext, Gt Yarmouth. Davis, A. M. J., O'Neill, M. E., Dorrepaal, J. and Ranger, K. B. (1976) J. Fluid Mech. 77, 625-644. Howarth, L (1938) Prec. R. Soc. A164, 547-569. Jefferies, R. A. (1974) Fibrous Materials for the Filtration of Gases. Publication No $14, Shirley Institute, Manchester. Kirsch, A. A. and Stechkina, J. B (1977) J. Aerosol Sci. 8, 301-308. Landau, L. D. and Lifshitz, E. M. (1959) Fluid Mechanics. Pergamon, London. Tamada, K. and Fujikawa, H. (1957) Q. JI Mech. appl. Math. 10, 425-432. APPENDIX PHYSICAL
AND MATHEMATICAL
FORMULATION PRINCIPLE
OF THE VARIATIONAL
In this and previous work (Brown, 1984) arguments have been based on the assumption that the solution of the biharmonic equation in the stream function, ~b,
V2V2~//~ 0,
(35)
is equivalent to finding the minimum of the following functional, which is the rate of dissipation of energy, O, in viscous flow, expressed in terms of the stream function 23,J\~x 2
t~y2 ]
\Sx~y/
Both of these approaches have a sound theoretical basis (Landau and Lifshitz, 1959) and so both are correct. However, in the present paper the solution of the biharmonic equation is required under circumstances in which the physical meaning of equation (36) is obscure, and so a mathematical justification is required if the method is to be used. Using the normal definition of stream function in cartesian coordinates and without any reference to energy arguments, or to its physical significance, the integrand in equation (36) can be shown to be proportional to the function below, expressed in terms of fluid velocity components Ux and Ux: F = 2 \ ~ - x / + 2 \ ~ - y / + \ ~ - y + ~'tx / .
(37)
With this change of variable, the implicit condition of a solenoidal flow field is lost, and so this condition must be introduced again into the argument 8Ux +SU~. V ..U . . . . . ~x ~y
0.
(38)
F is a function of x, y, U~, Uy and all their derivatives, and equation (38) is a non-integral constraint, and so the necessary conditions for an extremum of O are (Burley, 1974): t~F' ~ ?F' 8 ~F' = 0 09)
Filter airflow model: II
and
OF'
0
OF'
0
697
OF'
O,
(40)
where
r ' = Y+,l(x,y)~[OU,, ~ - x + OUy)oy .
(41)
Substitution of equation (41) into equations (39) and (40) results in
O--XX \4~-x + X,/+ ~y \ and
ay
O / OV,,
OUy'\
O/
Ox"
Ox /
Oy" Oy
Or.:,
--|2--+2--J+--{4--+2]
Oy
"~
/
= O.
(43)
2, the Lagrangian multiplier, can be eliminated by further partial differentiation and subtraction of equations (41) and (42), giving, after some easy manipulation, the biharmonic equation in ~, and establishing the required relationship. APPLICATION TO POTENTIAL FLOW Potential flow is described by Laplace's equation: v 2 ~ = o.
(44)
It is very easy to show that the solution of equation (44) is equivalent to finding the minimum of the functional
No constraints are necessary and the equivalent Lagrangian equation is simply
OF
0
OF
0
OF -
-
=
O.
(46)
Equation (46) leads directly to Laplace's equation. Energy arguments are not valid for potential flow, but equation (46) shows that the variational approach is sound. In an analogous problem in electrostatics, ~ can be identified with electrostatic potential, in which case equation (44) is equivalent to Gauss' Theorem in a charge-frec space, and L is proportional to the electrostatic energy, which can be shown on sound physical grounds to take the lowest possible value, consistent with the constraints that the system is subjected to. Potential flow is not a good description of the microscopic flow pattern through a fibre array under any circumstances. Nevertheless, airflow at effectively zero Reynolds number through filter media that are macroscopically homogeneous and isotropic in two dimensions does obey the potential flow equation, provided that attention is confined to the gross shape of the stream function, and details of flow at an individual fibre level are averaged out. In this case, L is related to the energy dissipation in the filter due to viscous drag and Darcy's law is implicit in the formulation. If the filter medium had periodically placed obstructions, the airflow pattern could be described by the Fourier series. The only differences between the solution of this problem and the Stokes flow problem would be that the boundary conditions would require only the normal component of the velocity to vanish at the surface of the obstructions, and that equation (5) would be replaced by n
2
k
2
0021-8502/86 $3.00+0.00 Pergamon Journals Ltd.
A M A N Y - F I B R E T H E O R Y O F AIRFLOW T H R O U G H A F I B R O U S FILTER--II: F L U I D INERTIA A N D FIBRE PROXIMITY R. C. BROWN Health and Safety Executive, Broad Lane, Sheffield $3 7HQ, U.K.
(Received 27 September 1985) Abstract--The problem of Stokes flow through a fibrou/; filter modelled as a two.dimensional array has previously been solved by direct application of the variational principle to a generalized stream function, expressed as a two-dimensional Fourier series in Cartesian coordinates. In this paper the effect of fluid inertia at low Reynolds number is described, by solution of the steady state Navier-Stokes equation in the form of a perturbation expansion, leading to a system of linear partial differential equations with solutions in the form of Fourier series. Pressure drops across model filters, calculated by this method, are compared with experimental results. Both theory and experiment illustrate that the lowest order correction to Darcy's Law is a cubic term in the flow velocity, and that this term is strongly influenced by the filter structure. The details of the flow pattern close to the fibre surface, in particular the presence or absence of standing eddies at finite Reynolds number, is shown to depend critically on the filter structure; eddies are shown to occur in Stokes flow when the fibre layers are sufficiently close together. NOMENCLATURE A Defined in text; equations (8) and (32) Coefficient in double Fourier series description of WM Defined in text; equations (32) and (33) b Mnk Coefficient in double Fourier series description of WMm CMnk Coefficient in double Fourier series description of WM¢F f,~ Defined in text; equation (5) F Defined in text; equations (37) and (45) F' Defined in text; equation (41) l Half spacing of array in y-direction L Defined in text; equation (45) P Pressure drop across a filter R Fibre radius Re Reynolds number, 2RpUo/~ t Filter thickness Uo Macroscopic fluid velocity U~, Uy Fluid velocity components xi,Yl Co-ordinates of a point where boundary conditions are applied Dimensionless group used as expansion parameter in perturbation series; defined by equation (12) Lagranglan multiplier Half spacing of array in x-direction Coefficient of dynamic viscosity 0 Rate of dissipation of energy Lagranglan multiplier #u Matrix element defined in text; equation (7) ~g Complete stream function Stream function for Stokes flow ¢ Complete dimensionless stream function ~o Dimensionless stream function for Stokes flows and zero order term in the perturbation expansion of ¢, CN Nth order term in the perturbation expansion of ~ at finite Reynolds number Particular integral part of ¢,, ¢ICF Complementary function part of ~'t a Mnk B
INTRODUCTION
It has been shown previously (Brown, 1984) that the stream function for Stokes flow through
a two-dimensional array can be calculated by applying the principle that the flow pattern will be such that the dissipation of energy due to viscous drag is a minimum, subject to the boundary conditions of the system. The trial stream function used was a double Fourier series, the coefficients of which were treated as variational parameters. The series were Crown copyright © 1986.
685
686
R.C. BROWN
summed over terms with non-negative indices only, and different series were used to describe arrays in which adjacent layers had fibres with the same y-coordinates, the channel array, and those in which each fibre layer had fibres with y-coordinates at the mid-points of those in adjacent layers, the staggered array. In further developments of the model, it will prove convenient to express the two series in a common form which describes either of the two geometries, depending on conditions applied in the summation; and it will simplify later working if the sums are extended to include coefficients with negative indices. This generalized form of the stream function can be written • nny • o = U o y + U ol ~ ao ,,k
sin ~ -
krtx
cos 25i-
(I)
n,k
Equation (1) applies to macroscopic flow in the x-direction, and the presence of only cosine terms in x constrains the function to have the upstream--downstream symmetry that is characteristic of Stokes flow. If the sum is limited to even values of k, the stream function will have the symmetry appropriate to the channel array, as shown in Fig. 1. If n and k are either both even or both odd or, in other words, if (n + k) is even, the staggered symmetry results. In all of the working below, double sums written as in equation (1) are assumed to extend from minus to plus infinity and to have one of the conditions above applied. F L O W P A T T E R N AT ZERO R E Y N O L D S N U M B E R The solution of the Stokes flow problem has already been described in detail and so only a brief summary of it, to include the presence of negative indices in the Fourier series, will be given here. Substitution of equation (1) into the expression for the rate of dissipation of energy, equation (36), and integration results in an expression containing squared Fourier coefficients but no cross-terms. Products of coefficients with the same values of Inl and Ik I will occur but it can be assumed, without any loss of generality, that ao_nk = -- aonk;
aOn_k = a o . k.
(2)
The minimum value of the expression is found by differentiating it with respect to every subject to the constraint that the fluid velocity is zero at the surface of the fibres. When the method is used in practice, the stream function is constrained to vanish at a finite number, M, of points taken in pairs just outside and just inside the surface of the fibre at the origin. If the coordinates of one such point are (xi,yi) the corresponding boundary condition can be expressed as a o . ~,
nrty i
ao. k sin - - f -
y~+l~
kltxi
cos ~
= 0.
(3)
n.k
The conditions are included in the problem by means of Lagrangian undetermined multipliers {7i}, and the result is that l nrcy~ ao.k + - - ~ "ti sin ~ ink i
k~xi
cos ~
= 0 (4)
0
0
0
0
0
0
o
o
0
0
0
0
o
o
0
o
Fig. 1. Two-dimensionalrectangular array of fibres.
Filter airflow model: II
687
for all {a0nk}, where
(5) The {~} are given by the solutions of the following simultaneous equations:
)~,Tibttj=yj;
1 ~j<~M,
(6)
i
where all the matrix elements/~u are given by
12
nny~
#'J = .,kJ,~7- sin - - ~
nnyj
sin ~
knx~
cos - ~ e cos
knxj
2--~
(7)
Equations (4)-(7) can be formulated into a simple computer program using a standard library subroutine for the solution of the simultaneous equations (6). The fibre shape is specified by the choice of {x~, y~}, the filter packing fraction and degree of anisotropy by choice of the unit cell repeating lengths 8 and l, and its structural symmetry by the choice of the conditions applied to the (n,k) sum. The flow pattern for the smallest repeating element of a channel array is shown in Fig. 2. The pressure drop across such an array is related to the rate of dissipation of energy through it; in terms of the {a0,~ }, which are now assumed to take the values fixed by the variational calculations, the pressure drop across a thickness t is given by
P= U°rlizn4t8 ~, ao,kfnk.Z
(8)
n,lt
The accuracy that the model can achieve depends on the value of M and on the upper limits of Inl and Ikl in the series. The model is at its poorest close to the fibre surface, where the boundary conditions are applied and, as a rough guide, the approximation is good at a distance away from the fibre comparable with the tangential spacing of the points {xi,y~}. The n, k series should be truncated such that the period of the highest order component is comparable with this spacing. Those streamlines shown in Fig. 2 are well described if M = 6 (with the stream function constrained to vanish at three pairs of points in each quadrant), but if details of the flow pattern closer to the fibre surface were required, more points would be needed. In particular, a greater number of fixed points are required to give a good description of sharp edges on the fibre surface than are needed to describe a perfect cylinder. The upper limit of M that can be used is governed by the computation time allowed, though at large values of M the matrix [/~u] may appear to be singular because of limitations in the computer used. With the Xerox NUMSLIB library subroutine on a Sigma-6 computer, spurious singularity is observed at about M = 40.
FLOW T H R O U G H A TWO-DIMENSIONAL ARRAY AT FINITE REYNOLDS NUMBER The Reynolds number in airflow through most fibrous filters used in respirators is low. However, at high flow rates through relatively coarse filters of the sort used as size-selectors in respirable dust samplers (Brown, 1980) the Reynolds number may take a value of unity or more, and a description of airflow through such filters requires fluid inertia to be taken into account. The Navier-Stokes equation for airflow through a two-dimensional system, under steady state conditions, for an incompressible fluid with finite viscosity and finite inertia (or finite density, p) takes the following form, when expressed in terms of the stream function (Landau and Lifshitz, 1959): p~d~P 0V2~P 0~p 0V2~P V2V2~I' = ~ (-~y ~ dx -~y J" AS 17:4-C
(9)
688
R.C. BROWN
If the stream function is expressed in a dimensionless form, ~b, such that W
(10)
ql - Uol'
equation (9) becomes
, rI
JOy
,}
~x
~x
~y
"
(1l)
The multiplier on the right-hand side of this non-linear equation is closely related to the Reynolds number as normally defined, and for brevity it will be written as follows: P Uol
q
= ~.
(12)
It is now assumed that, when = is small, $ can be expressed as a power series in =, which is the fundamental assumption of perturbation theory. = ~ ~NSN.
(13)
N
Substitution of equation (13) into equation (11) results in two power series in ~. Equating the multipliers of ~N gives rise to the following relationships between the {~'N}Zero order term
V2V21#o = 0.
(14)
As expected, the zero order term is simply the Stokes flow expression, which has already been calculated.
First order term
V2V2qq
0~'o0V2~o = 0y dx
0q'oc~V2~o 0x 0y
(15)
The first order term can be expressed entirely in terms of derivatives and products of the zero order term. N t h order term
V2V2~u =
(16) i = o t~y
0x
0x
c3y
The Nth order term can be expressed as binary products involving the derivatives of lower order terms only. In principle, therefore, the perturbation series can be worked out in simple sequence, using the results of the first N - 1 terms to calculate the Nth term. The right-hand side of each equation will be an explicit function o f x and y, resulting in a simple linear partial differential equation in ~ , the solution of which can be expressed as the sum of a particular integral, which satisfies the equation, without taking account of the boundary conditions, and a complementary function, which satisfies the equation obtained by setting the right-hand side of equation (16) equal to zero (the bibarmonic equation), but which is constrained so that the sum of the particular integral and itself satisfies the boundary conditions. It will be shown below that this can be put into practice in a s t r ~ t f o r w a r d way. Only the first order term will be calculated, but once this has been done it will be clear how the calculation may be extended to any order. The method of solution of an intractably complicated equation in the form of a series of linear equations, which have to be solved in sequence, was used, though in a different form, in the solution of boundary layer problems by Howarth (1938).
Filter airflowmodel:11
689
Particular integral solution of first order term ~bo is given, in its dimensionless form, by substituting equation (1) into equation (10). Substitution of the resulting expression into the right-hand side of equation (15) gives
n3
k F f n ~ 2 + ( k )21 sin nny sin knx
)
[(m)2 (j)2]
n,k m,j
nj
n~y
x ~ d cos T
mk
mty
-2e---/ sin T
kTtx
cos ~
k~zx
sin ~
tony
sin T
j~x
sin 2e
mrcy
jnx
cos - - 1 - cos 2 T J '
(17)
Use of the well-known product rules for trigonometric functions, along with the symmetry of the a0. k series, enables the expression to be simplified considerably. V2V2~b' = T
ZaO.kq--;./~L\./7// + \
j ~/
sin ~
sin
~,k
+ n " ..kZ -.iZao,,kao-j
2--~-
[ ( m ) 2 (~e)21(nj-mk'~ 7 + J~--~l / sin (m - n)rcy s m. (j - k)nx l- T -
(18)
A change of variable in the sums over m and j may now be made, resulting in sine terms containing only one index. The doubly infinite series used to describe ~Oomake the formal working considerably easier, because if the series were summed over non-negative terms only, a change of variable would result in one index appearing in the summation limit of the other, and this would be a tiresome complication. The resulting form of the double series dictates that a particular integral of equation (18), ~'m, should be an odd function of x, unlike ~'0, which is even.
~.
dim = n=
~.,
--o0
k=
nrcy krtx bl.k sin ~ sin 2--e-
(19)
--o'3
Equation (19) may be substituted into the left-hand side of equation (18), and the {bin k } can be calculated by equating the coefficients of the sine terms on the two sides. k
n 2
+ n'k - nk' 2el )"
k 2
[- fn+n,'~2
i/k+k,'~21
)
J (20)
All of the {bt~k} in the particular integral can, therefore, be calculated as sums of binary products of the coefficients of 4o. If the appropriate conditions for limiting the values of (n', k') and (n + n', k + k') for the required array are imposed, it follows that the same conditions will apply to (n, k), although this could have been assumed from considerations of symmetry.
Complementary function All that is required of ~bm is that it should be one solution of the differential equation. A
690
R.C. BROWN
complementary function is also needed. This function, q/ice, must satisfy the equation V2V21]IICF ~- O,
(21)
and, when taken with the particular integral, it must satisfy the boundary conditions. In order to do so, it must have the same symmetry as the particular integral, and the sums must be limited by the structure of the array to be described: nny krtx d/lc r ---- ~ cl. k sin --~- sin - ,./ 2e
(22)
The boundary conditions to be satisfied are that each fluid velocity component must vanish at the fibre surface. The procedure used in the solution of the Stokes flow problem can be used here because, although the concept of minimization of rate of dissipation of energy does not apply, the biharmonic equation is the Euler equation that is a necessary condition for the energy dissipation functional to be a minimum, as shown in the Appendix. Those values of {cx.k} that will give the minimum value of the functional, O, must be found when ~'~ce is substituted into equation (36). The boundary conditions are fixed at the points {xi, y~}, and they can be included, using Lagrangian undetermined multipliers, in exactly the same way as before. The condition at the point (x~, y~) is: nrcy i kTcx i (bl.k+Cl.k) sin - T - sin ~ = 0.
(23)
n,k
The symmetry of the function is such that the simplest form is obtained with the following assumptions: Cl _.k = --Cl.k; Cl._k = --Cl. k. (24) A similar condition is imposed on the {bl.k} by the form of equation (20). The minimization procedure, in this case, results in O {~f,~c~.k 0C~.k .k. 2
+
[ ~ (b,.k+C,.k) sin nny, knx,]~ ~/Yi. ..k ~ sin 2e j j = 0 .
(25)
The following simultaneous equations are obtained: M
~/aij=9~;
I~
(26)
i=1
where #ij
= ~ 1 sin nnYi nnyj kr~x i krcxj .,kf.k ----(-- sin ~ sin ~ sin 2e
(27)
9j depends on the particular integral nny i k~xi 9 j = ~ b l . k sin T sin 2----e-
(28)
n,k
The total stream function is the sum of the two parts. nny knx ~01 = ~ al.k sin - - 7 sin 2----e-'
(29)
al. k = ba.k +Ca. k for all n,k.
(30)
n,k
where Each higher order term can be found in a basically similar way. The particular integral coefficients will be given by equations like equation (20), but containing N binary series for the Nth order term. The complementary function is of the same order of complexity for every term.
Filter airflow model: II
691
RESULTS
There is no fixed limit for the order of term that can be calculated, but errors in calculation will be cumulative. Accuracy is ultimately limited by the computing time that can be devoted to the large sums in the particular integral. The results below apply to models in which M = 12, In ] ~ 30 and Ikl ~< 60. The exact shape of stream function for each order of term is interesting, and the first three of these are illustrated in Figs 2-4 for the channel array. Figure 2 is the zero order term (the Stokes flow pattern) for conditions in which e = 1 -- 10.0, R = 2.0. Only one quadrant is shown but this form, repeated, gives the complete picture of the stream function. The first order term is shown in Fig. 3. The form of rotational motion of the fluid that could give rise to standing eddies on the downstream side of the fibre is obvious here. The second order term, which is not easy to imagine, is shown in Fig. 4. This has the symmetry obvious from its form, and it has the effect of moving incipient vortices downstream. Unlike the first order term, it will have the effect of moving fluid streamlines closer to the fibre surface in the x = 0 plane. The third order term is of the same basic shape as the first order except that it is weaker, and the sign of rotation is reversed. The fourth order term has the same relationship with the second, and the series continues in this way. The first and second order terms for a staggered array are shown in Figs 5 and 6, which illustrate clearly the radical difference between the two geometries. The zero order term is not shown because, close to the fibre, it is indistinguishable from that for the channel array. Once the various terms have been calculated, it is straightforward to build up the complete stream function, for any value of a or Reynolds number, by using equation (13). The individual odd order components have upstream-downstream symmetry, and the even order components have antisymmetry, but the complete stream function has neither.
7 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i I
i i
OS "
Fig. 2. Stream lines corresponding to dimensionless stream function for Stokes flow; or zero order dimensionless stream function in perturbation expansion: e = l = 10, R = 2: Channel model. (The stream line at y = l has the value ~ = l.)
,)
iI
Fig. 3. Stream lines corresponding to first order dimensionless stream function in perturbation expansion: e = I = 10, R = 2: Channel model.
692
R . C . BROWN r
.
.
.
.
o.0
Fig. 4. Stream lines corresponding to second order dimensionless stream function in perturbation expansion: e = I = 10, R = 2: Channel model.
Fig. 5. Stream lines corresponding to first order dimensionless stream function in perturbation expansion: e = l = 10, R = 2: Staggered model.
o0 'oO
Fig. 6. Stream lines corresponding to second order dimensionless stream function in perturbation expansion: e = l = 10, R = 2: Staggered model.
The fundamental simplicity of the complete solution is made clear if it is written in the following way:
y
[
~'
0 = i + n~.k ,
+
even
N =
. mty
~NaNnk s i n
knx
--T- cos
N = 0
l
~, odd
~
~tNaN.k sin ~ - - sin 2e J"
(31)
1
Once the {aNnk} have been calculated, the stream function can be written down for any value of Re or ~ within the limits of convergence, without further calculation. The function has a simple analytical form, which should make it particularly easy to use in calculations of particle trajectories or studies of filtration. The stream lines for flow through a square two-dimensional channel array are shown in Fig. 7, with e = l = 10, R = 2 and a value o f a = 7, which corresponds to a Reynolds number, as usually defined, of 2.8. The Stokes flow solution for this array is the zero order term drawn in Fig. 2. The fundamental asymmetry in the flow pattern produced by the effect of fluid inertia is clearly shown in Fig. 7.
Filter airflowmodel: II
693
J
Fig. 7. Completestream functionfor square array: ,t = 7, Re = 2.8, e = l = 10, R = 2. PRESSURE DROP AT F I N I T E REYNOLDS NUMBER The pressure drop across a filter as a function of small Reynolds number can be obtained by substituting equation (31) into the energy dissipation equation (36). The orthogonality of the two series in equation (31) results in a pressure drop that is a series containing only odd powers of Re, a, or Uo: P = AUo+BU3o+O(U~).
(32)
When Uo is sufficiently small for higher powers to be neglected, equation (32) reverts to the Darcy Law/Stokes flow form, given in equation (8). At higher velocities, the first correction term is such that B = P21*~*t 8rI ~,f,,k(2ao,ka2~k + aani).
(33)
It is clear from the symmetry of the perturbation terms illustrated in Figs 2-6 that odd order terms will produce no net tangential stress at the fibre surface and that the general expression for pressure drop at finite Reynolds number will have the form of equation (32). In fact, such a result would follow from a more general treatment than that above, and a similar result follows from the calculations of Tamada and Fujikawa (1957) for a single perfectly periodic layer of fibres. An experimental test of the validity of equation (32) has been carried out on model filters consisting of stacks of thin metal plates, each photo-etched to form a regular grid. The plates were 200 #m thick, etched to give fibres that were rectangular in cross-section with a width of 250 #m. The distance I was 450/an, and the plates were spaced such that the distance e was 380 #m. Eighty such grids were stacked in both the channel and staggered symmetries, air was passed through the two model filters at velocities of up to 0.8 m s- t, and the pressure drop across each filter was monitored. The results are shown in Fig. 8, where theory and experiment are compared. The experimental results support the theoretical prediction that the lowest order correction to Darcy's law is a cubic term, and both theory and experiment show that the resistance of a filter at finite Reynolds number is highly dependent on its structure, even though the dependence at zero Reynolds number is weak. The discrepancy between experiment and theory, apparent in Fig. 8, is similar in nature to the discrepancy between the experimental measurements of pressure drop across a single layer of fibres carried out by Kirsch and Stechkina (1977) and the calculations of Tamada and Fujikawa (1957). The application of simple curve-fitting techniques to the experimental measurements of Jefferies (1974) on the pressure drop across coarse porous foams of three-dimensional structure shows that a linear plus cubic function for the velocity dependence of pressure drop gives a good fit, whereas a linear plus quadratic function does not. SEPARATION IN I N E R T I A L FLOW AND STOKES F L O W The variational model is at its weakest close to the fibre surface, where the boundary conditions are applied. Nevertheless, it is in this re#on that flow separation is most obvious,
694
R.C. BROWN 100
/
I
/
90
/ 80
/
7O
/ P
/
50
I
x
I x
/x
~4o
/ / x
30
®/*
20 10 j~epv
0.1
,
,
~
J
,
J
J
(12
0.3
0.4
0.5
0.6
0.7
0.8
A i r Velocit y ( m s-~ )
Fig. 8. Pressure drop as a function of filtration velocity for model filters: Channel model theoryC) ; channel model experiment--; staggered model theory---; staggered model experiment x.
and the model does illustrate the formation o f standing eddies on the downstream side o f the fibre. In the theoretical models shown, these appear at a b o u t u = 8.4 for the channel array, but the critical value o f ~ will depend on e and l. An illustration o f the flow patterns in the vicinity o f a single fibre is shown in Figs 9 and 10. Separation does not seem to occur in the staggered model, for these values, at least not when the perturbation series is truncated after the second order term. The flow pattern is strongly influenced by the geometry o f the system. F o r the channel array, separation will occur even in Stokes flow provided the fibre layers are close together. If is reduced, standing eddies appear on those faces o f the fibres that are closest to the next fibre layer, and as the spacing is further reduced, the eddies coalesce. I f the distance I is five fibre diameters, eddies start to form when e is a b o u t 1.04 fibre diameters, and the eddies
G2
0.1
Fig. 9. Flow pattern close to fibre surface:e = 1 = 10, R = 2, ,, = 9.4;Channel model.Q-point where boundary condition is applied. 0.2
O.1
Fig. 10. Flow pattern close to fibre surface: e = I = 10, R = 2, ct = 9.4; Staggered model,
Filter airflow model: II
695
coalesce when e is about 0.69 fibre diameters. Distinct and coalesced eddies are illustrated in Figs 11 and 12. The critical spacings are comparable in magnitude with those observed by Davis (1976) for two spheres whose common axis coincides with the macroscopic Stokes flow direction. The eddies observed in Stokes flow, unlike those at finite Re, have upstream-downstream symmetry. Figure 13 illustrates the convoluted pattern that occurs when flow is slightly inclined to thcaxis. Highly inclined flow eliminates the eddies altogether. The development of separation in Stokes flow can be simply explained. A general solution of the biharmonic equation, if boundary conditions are specified at x = 0 only, or if the filter consists of a single regular layer of flat fibres, is
etal.
~b = ~ a. sin
~-(
nnlxl'~/e x p / -
1 + ~
n~ t Ixl ) .
(34)
n
Each mode will decay in a distance in the x-direction comparable with its period in the ydirection. If another layer of fibres lies well within the decay length, the mode will be preserved, because it must be regenerated at the next fibre layer. This means that at certain layer spacings long-wavelength modes in the stream function will be preserved whilst shortwavelength modes decay. The finite thickness of the fibres will alter this simple picture, but an imbalance between long-wavelength and short-wavelength modes will still occur, and the result of this imbalance will be the symmetrical standing eddies illustrated in Figs 11 and 12. DISCUSSION
A variational approach to solution of Stokes flow through two-dimensional arrays of fibres is a simple way of obtaining detail about the flow pattern. The model has been extended to the case of steady state incompressible fluid flow at low but finite Reynolds number, by means of a perturbation expansion. The result is that the Nth order term satisfies a differential equation involving only terms of lower order. If the terms are calculated
Fig. 11. Distinct standing eddies in Stokes flow.
O.OO2
Fig. 12. Coalesced standing eddies in Stokes flow.
000
-0.06
Fig. 13. Convoluted flow pattern in Stokes flow.
696
R.C. BROWN
s e q u e n t i a l l y , t h e s e differential e q u a t i o n s b e c o m e t h e t y p e t h a t can be solved as the s u m o f a p a r t i c u l a r i n t e g r a l a n d a c o m p l e m e n t a r y f u n c t i o n , the f o r m e r o f w h i c h can be f o u n d by e q u a t i n g coefficients in F o u r i e r series, a n d the latter by explicit use o f the v a r i a t i o n a l principle. F o r the m e t h o d to be valid, b o t h t h e F o u r i e r series m a k i n g up the s t r e a m t u n c t i o n c o m p o n e n t s a n d t h e p e r t u r b a t i o n series itself m u s t c o n v e r g e . A f u n d a m e n t a l t r e a t m e n t o f this is b e y o n d the s c o p e o f this paper, b u t t h e s t r e a m l i n e s s h o w n in F i g s 2 - 6 are i m p e r c e p t i b l y a l t e r e d if the t r u n c a t i o n p o i n t s for the M , n a n d k s u m s a r e r e d u c e d f r o m 20, 30 a n d 60 to 6, 15 a n d 30; this suggests t h a t t h e series c o n v e r g e s q u i t e rapidly. T h e l i m i t i n g v a l u e o f ~t for which the p e r t u r b a t i o n series will c o n v e r g e m u s t d e p e n d o n e, l a n d the fibre d i m e n s i o n s . S t r a i g h t f o r w a r d e x a m i n a t i o n o f the v a r i o u s o r d e r s o f s t r e a m f u n c t i o n for the c h a n n e l a r r a y case in w h i c h e = l = 10, R = 2 suggests t h a t the series is c o n v e r g e n t at a R e y n o l d s n u m b e r o f a b o u t 4. H o w e v e r , the fact t h a t e r r o r s in s t r e a m f u n c t i o n s will be c u m u l a t i v e limits the c o n f i d e n c e we m a y h a v e in the a b s o l u t e v a l u e o f this result. Acknowledgement--The author wishes to thank Mr D. Wake of HSE, who assembled and tested the model filters.
REFERENCES Brown, R. C. (1980) J. Aerosol Sci. I!, 151-159. Brown, R. C. (1984) J. Aerosol ScL 15, 583-593. Burley, D. M. (1974) Studies in Optimisation. lntertext, Gt Yarmouth. Davis, A. M. J., O'Neill, M. E., Dorrepaal, J. and Ranger, K. B. (1976) J. Fluid Mech. 77, 625-644. Howarth, L (1938) Prec. R. Soc. A164, 547-569. Jefferies, R. A. (1974) Fibrous Materials for the Filtration of Gases. Publication No $14, Shirley Institute, Manchester. Kirsch, A. A. and Stechkina, J. B (1977) J. Aerosol Sci. 8, 301-308. Landau, L. D. and Lifshitz, E. M. (1959) Fluid Mechanics. Pergamon, London. Tamada, K. and Fujikawa, H. (1957) Q. JI Mech. appl. Math. 10, 425-432. APPENDIX PHYSICAL
AND MATHEMATICAL
FORMULATION PRINCIPLE
OF THE VARIATIONAL
In this and previous work (Brown, 1984) arguments have been based on the assumption that the solution of the biharmonic equation in the stream function, ~b,
V2V2~//~ 0,
(35)
is equivalent to finding the minimum of the following functional, which is the rate of dissipation of energy, O, in viscous flow, expressed in terms of the stream function 23,J\~x 2
t~y2 ]
\Sx~y/
Both of these approaches have a sound theoretical basis (Landau and Lifshitz, 1959) and so both are correct. However, in the present paper the solution of the biharmonic equation is required under circumstances in which the physical meaning of equation (36) is obscure, and so a mathematical justification is required if the method is to be used. Using the normal definition of stream function in cartesian coordinates and without any reference to energy arguments, or to its physical significance, the integrand in equation (36) can be shown to be proportional to the function below, expressed in terms of fluid velocity components Ux and Ux: F = 2 \ ~ - x / + 2 \ ~ - y / + \ ~ - y + ~'tx / .
(37)
With this change of variable, the implicit condition of a solenoidal flow field is lost, and so this condition must be introduced again into the argument 8Ux +SU~. V ..U . . . . . ~x ~y
0.
(38)
F is a function of x, y, U~, Uy and all their derivatives, and equation (38) is a non-integral constraint, and so the necessary conditions for an extremum of O are (Burley, 1974): t~F' ~ ?F' 8 ~F' = 0 09)
Filter airflow model: II
and
OF'
0
OF'
0
697
OF'
O,
(40)
where
r ' = Y+,l(x,y)~[OU,, ~ - x + OUy)oy .
(41)
Substitution of equation (41) into equations (39) and (40) results in
O--XX \4~-x + X,/+ ~y \ and
ay
O / OV,,
OUy'\
O/
Ox"
Ox /
Oy" Oy
Or.:,
--|2--+2--J+--{4--+2]
Oy
"~
/
= O.
(43)
2, the Lagrangian multiplier, can be eliminated by further partial differentiation and subtraction of equations (41) and (42), giving, after some easy manipulation, the biharmonic equation in ~, and establishing the required relationship. APPLICATION TO POTENTIAL FLOW Potential flow is described by Laplace's equation: v 2 ~ = o.
(44)
It is very easy to show that the solution of equation (44) is equivalent to finding the minimum of the functional
No constraints are necessary and the equivalent Lagrangian equation is simply
OF
0
OF
0
OF -
-
=
O.
(46)
Equation (46) leads directly to Laplace's equation. Energy arguments are not valid for potential flow, but equation (46) shows that the variational approach is sound. In an analogous problem in electrostatics, ~ can be identified with electrostatic potential, in which case equation (44) is equivalent to Gauss' Theorem in a charge-frec space, and L is proportional to the electrostatic energy, which can be shown on sound physical grounds to take the lowest possible value, consistent with the constraints that the system is subjected to. Potential flow is not a good description of the microscopic flow pattern through a fibre array under any circumstances. Nevertheless, airflow at effectively zero Reynolds number through filter media that are macroscopically homogeneous and isotropic in two dimensions does obey the potential flow equation, provided that attention is confined to the gross shape of the stream function, and details of flow at an individual fibre level are averaged out. In this case, L is related to the energy dissipation in the filter due to viscous drag and Darcy's law is implicit in the formulation. If the filter medium had periodically placed obstructions, the airflow pattern could be described by the Fourier series. The only differences between the solution of this problem and the Stokes flow problem would be that the boundary conditions would require only the normal component of the velocity to vanish at the surface of the obstructions, and that equation (5) would be replaced by n
2
k
2