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A note on the pressure drop across a fibrous filter modelled as a symmetrical array of cylinders
J. Aerosol Sci., Vol. 19, No. 3, pp. 385-386, 1988
0021-8502/88 $3.00 + 0.00 Pergamon Press plc
Printed in Great Britain.
T E C H N I C A L NOTE A NOTE ON THE PRESSURE DROP ACROSS A FIBROUS FILTER MODELLED AS A SYMMETRICAL ARRAY OF CYLINDERS
D.B. Department *Schools
Ingham,
P.J. Heggs*
and M.L. Hildyard
of Applied Mathematical Studies, University of Leeds of Chemical Engineering, University of Bradford
(Received 4Februa~ 1988)
The problem of the removal of suspended particles from a fluid is of great importance in many industries, such as the dust removal from the atmosphere of a coal mine and the removal of small metallic grains from the lubricating oil of an internal combustion engine. In most cases a fibrous filter is used consisting of many threadlike fibres which act to obstruct the flow through the filter, causing the solid particles to become attached to the fibres and hence removed from the fluid. In a recent paper, Brown (1984) examined the small Reynolds number creeping flow through such a filter which was modelled as an assemblage or array of cylinders with each cylinder representing a fibre of the filter. Each cylinder in the assemblage (which was infinite in extent) was assumed to have a radius r with axes that were both mutually parallel a n d p e r p e n d i c u l a r to the bulk flow of fluid through the array. The flow through both square and staggered arrays of cylinders was treated using the variational principle described by Helmholtz (1856) and results for the stream function and pressure drop across the assemblage were presented in terms of a double Fourier series. The pressure drop Ap across a length d of the assemblage was given by the equation,
Ap =
~U d o r2
f(c)
(I)
where f(c) is a non-dimensional function of the packing fraction, c, U is the supero ficial velocity and p the viscosity of the fluid. Results were presented for f(c) for the flow through square and staggered arrays of cylinders and were found to be in reasonable agreement with previously published results. Hildyard et al (1985) examined the same flow problems as Brown (1984) solving the governing fluid flow equations for the stream function ~ and vorticity ~ using the boundary element method (BEM). Using finite difference techniques on ~ , Ingham et al (1987) extended the work of Hildyard et al (1985) to determine the pressure drop Ap across the arrays. Results for Ap were presented as functions of the minimum passage between the surfaces of the cylinders of the appropriate arrays. The main aim of this note is to demonstrate how accurately Ap can be calculated using the BEM. Using the techniques described by Hildyard et al (1985) and Ingham et al (1987), values for f(c) are calculated for various values of c and are shown and compared with the results presented by Brown (1984) in Table I.
Table c O. 002 0. 005 0.01 0.02 0.05 0.I 0.2 0.5 0.6
I:
The function
f(c)
according
Square Array Brown Ingham et al 0.00344 0.0106 0.0256 0.0653 0.250 0.800 3.35 91.9 -
0.00344 0.0104 0.0254 0.0645 0.247 0.788 3.28 84.8 337.0 385
to the model.
Staggered array Brown Ingham et al 0.00352 0.0109 0.0265 0.0682 0.266 0.873 3.76 101.3
0.00352 0.0107 0.0262 0.0672 0.262 0.858 3.67 93.3 356.0
386
"l'eclmiclll Note
It is observed that there is excellent agreement between the results obtained by the BEM and the technique described by Brown (1984) for small values of c but there is a growing discrepancy as c increases. This is to be expected because of the restrictively small number of points (namely 20) used to represent the cylindrical fibres in the collocation technique used by Brown (1984). In the present study it was found that for c =0.1, 0.2 and 0.5, 140 boundary elements around the surface of the cylinder were sufficient to ensure accuracy to three significant figures. Ingham et al (1987) pointed out that for the case where the staggered array is simply the square array rotated through 45 ° , then by the linearity of the governing equations the corresponding values of f(c) must be equal for each value of c. The BEM was used to obtain the results for such a staggered array and were found to be identical to those presented in table I using the BEM for the,square array for each value of c. Brown (1984) also presented results for the pressure drop in terms of f(c) for the flow parallel to a symmetrical square array of cylinders, again solving the governing equations using the variational principle. In the present study, the same results are calculated using the BEM and are compared with those obtained by Brown (1984) in Table 2. Table 2. The function
f(c)
for the parallel flow through a square array of cylindrical fibres
c
Brown
0.002 0.005 0.01 0.02 0.05 0.1 0.2 0.5
0.00171 0.00528 0.0128 0.0326 0.123 0.381 1.39 7.17
Ingham et al 0.00171 0.00530 0.0129 0.0329 0.127 0.421 2.00 11.5
As with the results in Table I, the results in Table 2 show a good agreement for small values of c with a growing discrepancy for larger values of c. This is to be expected for the same reasons as described for Table I. It was again found that for c = 0.1, 0.2 and 0.5, 140 boundary elements around the surface of the cylinder were sufficient to ensure convergence to three significant figures. In conclusion we have illustrated how the BEM enables an accurate treatment of the viscous flow through a cylinder array to be made in a computationally fast and economical manner. The excellent agreement between the results obtained for the flow through the square array and the staggered array obtained by rotating the square array through 45 = demonstrate how accurately such problems can be treated using the BEM. The extension to cylinders with arbitrary cross sections is i~nediate using this formulation, an extension which has proved difficult and computationally expensive using previously published techniques. It is hoped that this note will lead to a greater understanding of the nature of the flow through fibrous filters. Extensions are planned to examine the entrance effects of the flow into such filters and to consider finite Reynolds number flow effects. REFERENCES Brown, R.C. A many fibre model of airflow through a fibrous filter, J. Aerosol Sci. Vol. 15, 1984, pp. 583-593. Hildyard, M.L,, Ingham, D.B., Heggs, P.J. and Kelmanson, M.A. Integral equation solution of viscous flow through a fibrous filter, Proc. of the 7th International conference on Boundary Elements, Italy, 1985; 2:81-94. Ingham, D.B., Heggs, P.J., and Hildyard, M.L. The evaluation of the pressure drop across a filter using the Boundary Element method, accepted for publication in the Journal of Mathematical Engineering in Industry.
0021-8502/88 $3.00 + 0.00 Pergamon Press plc
Printed in Great Britain.
T E C H N I C A L NOTE A NOTE ON THE PRESSURE DROP ACROSS A FIBROUS FILTER MODELLED AS A SYMMETRICAL ARRAY OF CYLINDERS
D.B. Department *Schools
Ingham,
P.J. Heggs*
and M.L. Hildyard
of Applied Mathematical Studies, University of Leeds of Chemical Engineering, University of Bradford
(Received 4Februa~ 1988)
The problem of the removal of suspended particles from a fluid is of great importance in many industries, such as the dust removal from the atmosphere of a coal mine and the removal of small metallic grains from the lubricating oil of an internal combustion engine. In most cases a fibrous filter is used consisting of many threadlike fibres which act to obstruct the flow through the filter, causing the solid particles to become attached to the fibres and hence removed from the fluid. In a recent paper, Brown (1984) examined the small Reynolds number creeping flow through such a filter which was modelled as an assemblage or array of cylinders with each cylinder representing a fibre of the filter. Each cylinder in the assemblage (which was infinite in extent) was assumed to have a radius r with axes that were both mutually parallel a n d p e r p e n d i c u l a r to the bulk flow of fluid through the array. The flow through both square and staggered arrays of cylinders was treated using the variational principle described by Helmholtz (1856) and results for the stream function and pressure drop across the assemblage were presented in terms of a double Fourier series. The pressure drop Ap across a length d of the assemblage was given by the equation,
Ap =
~U d o r2
f(c)
(I)
where f(c) is a non-dimensional function of the packing fraction, c, U is the supero ficial velocity and p the viscosity of the fluid. Results were presented for f(c) for the flow through square and staggered arrays of cylinders and were found to be in reasonable agreement with previously published results. Hildyard et al (1985) examined the same flow problems as Brown (1984) solving the governing fluid flow equations for the stream function ~ and vorticity ~ using the boundary element method (BEM). Using finite difference techniques on ~ , Ingham et al (1987) extended the work of Hildyard et al (1985) to determine the pressure drop Ap across the arrays. Results for Ap were presented as functions of the minimum passage between the surfaces of the cylinders of the appropriate arrays. The main aim of this note is to demonstrate how accurately Ap can be calculated using the BEM. Using the techniques described by Hildyard et al (1985) and Ingham et al (1987), values for f(c) are calculated for various values of c and are shown and compared with the results presented by Brown (1984) in Table I.
Table c O. 002 0. 005 0.01 0.02 0.05 0.I 0.2 0.5 0.6
I:
The function
f(c)
according
Square Array Brown Ingham et al 0.00344 0.0106 0.0256 0.0653 0.250 0.800 3.35 91.9 -
0.00344 0.0104 0.0254 0.0645 0.247 0.788 3.28 84.8 337.0 385
to the model.
Staggered array Brown Ingham et al 0.00352 0.0109 0.0265 0.0682 0.266 0.873 3.76 101.3
0.00352 0.0107 0.0262 0.0672 0.262 0.858 3.67 93.3 356.0
386
"l'eclmiclll Note
It is observed that there is excellent agreement between the results obtained by the BEM and the technique described by Brown (1984) for small values of c but there is a growing discrepancy as c increases. This is to be expected because of the restrictively small number of points (namely 20) used to represent the cylindrical fibres in the collocation technique used by Brown (1984). In the present study it was found that for c =0.1, 0.2 and 0.5, 140 boundary elements around the surface of the cylinder were sufficient to ensure accuracy to three significant figures. Ingham et al (1987) pointed out that for the case where the staggered array is simply the square array rotated through 45 ° , then by the linearity of the governing equations the corresponding values of f(c) must be equal for each value of c. The BEM was used to obtain the results for such a staggered array and were found to be identical to those presented in table I using the BEM for the,square array for each value of c. Brown (1984) also presented results for the pressure drop in terms of f(c) for the flow parallel to a symmetrical square array of cylinders, again solving the governing equations using the variational principle. In the present study, the same results are calculated using the BEM and are compared with those obtained by Brown (1984) in Table 2. Table 2. The function
f(c)
for the parallel flow through a square array of cylindrical fibres
c
Brown
0.002 0.005 0.01 0.02 0.05 0.1 0.2 0.5
0.00171 0.00528 0.0128 0.0326 0.123 0.381 1.39 7.17
Ingham et al 0.00171 0.00530 0.0129 0.0329 0.127 0.421 2.00 11.5
As with the results in Table I, the results in Table 2 show a good agreement for small values of c with a growing discrepancy for larger values of c. This is to be expected for the same reasons as described for Table I. It was again found that for c = 0.1, 0.2 and 0.5, 140 boundary elements around the surface of the cylinder were sufficient to ensure convergence to three significant figures. In conclusion we have illustrated how the BEM enables an accurate treatment of the viscous flow through a cylinder array to be made in a computationally fast and economical manner. The excellent agreement between the results obtained for the flow through the square array and the staggered array obtained by rotating the square array through 45 = demonstrate how accurately such problems can be treated using the BEM. The extension to cylinders with arbitrary cross sections is i~nediate using this formulation, an extension which has proved difficult and computationally expensive using previously published techniques. It is hoped that this note will lead to a greater understanding of the nature of the flow through fibrous filters. Extensions are planned to examine the entrance effects of the flow into such filters and to consider finite Reynolds number flow effects. REFERENCES Brown, R.C. A many fibre model of airflow through a fibrous filter, J. Aerosol Sci. Vol. 15, 1984, pp. 583-593. Hildyard, M.L,, Ingham, D.B., Heggs, P.J. and Kelmanson, M.A. Integral equation solution of viscous flow through a fibrous filter, Proc. of the 7th International conference on Boundary Elements, Italy, 1985; 2:81-94. Ingham, D.B., Heggs, P.J., and Hildyard, M.L. The evaluation of the pressure drop across a filter using the Boundary Element method, accepted for publication in the Journal of Mathematical Engineering in Industry.