Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979
199
REVIEW
Flow through Porous Media-the Ergun Equation Revisited I. F. Macdonald, M. S. El-Sayed, K. Mow, and F. A. L. Dullien” Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G 1
Two friction factor-Reynolds number correlations, Le., the nondimensional Forchheimer equation of Ahmed and Sunada and the modified Ergun equation, are tested statistically with a large number of experimental data. It is concluded that the physical basis of the Forchheimer equation appears to be accurate. The modified Ergun equation, while certainly not rigorous, can be expected to predict experimental results for unconsolidated medii with an accuracy of f50 % . The importance of correcting for deviations from spherical particle shape and of using a correct averaging procedure is noted. The form of the dependence on porosity of the viscous and inertial parameters is investigated empirically, and it is concluded that using c3,6, instead of c in the Ergun equation, would result in a better fit to the data points.
Introduction For flow through porous media, it is desirable to be able to predict the flow rate obtainable for a given energy input (usually measured as pressure drop) or to be able to predict the pressure drop necessary to achieve a specific flow rate. Pragmatically, the complexity of the flow pattern rules out a rigorous analytic solution of the problem and suggests that an empirical or quasiempirical correlation is the best one can hope for. Ideally, one would like to have a correlation with universal constants and in terms of easily measured properties of the porous medium and the flowing fluid. The various models published in the literature have been divided into the following categories in a recent review article by Dullien (1975): (1)phenomenological models; (2) models based on conduit flow: (i) Geometrical models, (ii) statistical models, and (iii) models utilizing the complete Navier-Stokes equation; and (3) models based on flow around submerged objects. These categories represent merely different approaches to the solution of one and the same problem and, therefore, it is natural that there is often a great deal of overlap between them. Perhaps the simplest correlation one can look to is the so-called Ahmed and Sunada (A-S) equation (1969) which may be considered both a phenomenological and a Navier-Stokes type model 1 H=1+NRe”
where
and
Here grad P is the pressure gradient across the bed; V , is the superficial velocity and is given by V , = Q / A where Q is volumetric flow rate and A is surface area of the porous medium perpendicular to the flow direction. p and are the density and viscosity, respectively, of the fluid, 0019-7874/79/1018-0199$01.00/0
and a and p are model parameters to be established empirically. In practice, the values of cy and /3 are determined by rearranging eq l to the form --grad
P
P
vo
-a+p(3) PVO P and performing a least-squares fitting procedure. The Ahmed-Sunada equation is simply a rearrangement of the Forchheimer equation (1901);equivalent expressions have been looked at by Ergun (1952) and by Green and Duwez (1951). The A-S equation is not merely a truncated power series for AP as a function of V,, as suggested by the actions of Johnson and Taliaferro (1938) and Firoozabadi and Katz (1976), who have either added a cubic term or substituted exponent “n” for “2” in the second term of eq 4 below. Rather it is based on the general physical model (Dullien, 1975) which argues that the flow resistance is the sum of a viscous resistance giving the linear term in the relation -grad P = C,Vo + CzVo2
(4)
and an inertial resistance giving the quadratic term. Therefore, the test of the ability of the A-S equation to fit the data adequately is a test of this model. Green and Duwez (1951) and Geertsma (1974) have found that model adequate. There are a number of drawbacks to the A-S equation which in our view outweigh the advantage of its simplicity. The most serious shortcoming is the lack of parameters characterizing the porous medium. Therefore, the parameters a and p must be functions of the medium rather than universal constants, and accordingly, must be empirically established for each separate medium. Indeed, the parameters are not even dimensionless, having dimension L-* and L-l, respectively. A second drawback is that the dimensionless variables H and NRe”contain the to-be-determined parameters a and p as well as experimentally measurable quantities. Thus individual ( H ,NR~’? points cannot be plotted without a preliminary analysis in which (AP, V,,) data for the particular medium are fit to eq 3. Finally, data must be available for both the viscous-dominated and the inertial-dominated regimes of flow in order that a and both be accurately established, 0 1979 American Chemical Society
200
Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979
which is necessary since both are needed to determine H and N R e ” . In practice, as we shall see, the results can be relatively insensitive to the accuracy of cy and P. Probably the best known equation which includes a model of the porous media is the Ergun equation (1952). Here, we use what we call the Modified Ergun (M-E) equation which is simply the Ergun equation with the specific constants 150 and 1.75 replaced by A and B , respectively. The equation is €3 A ( l - -E) Fk‘= B 4(5) 1-€ NRe‘ ~
or
where (7) and
where -E is the porosity of the porous medium and De, is an appropriate characteristic length for the medium. One choice for D, is the equivalent mean sphere diameter D,, defined as
where up/spis the volume-to-surface ratio for the particulate system. An equivalent choice is
where D, is the diameter of a different hypothetical sphere having the average volume of the actual particles, and &, the sphericity, is the ratio of surface area of the hypothetical sphere to the average surface area of the actual particles. From an inspection of the A-S and M-E relations, we see that a and @ are related to A and B by (1 - -E)2 a=A(11) t3Deq2 and
The M-E equation includes the same general model of fluid behavior as the A-S equation, but, in addition, is based on a model for the porous medium which characterizes the medium in terms of the porosity, e (expressed explicitly in the form of appropriate porosity functions) and an average characteristic length, De, (six times the volume-to-surface ratio of the system). In this paper, we use a variety of literature data of varying degrees of completeness and levels of accuracy to assess these models. As discussed in the following sections, the data indicate that the A-S equation is adequate for the description of the data. The verdict on the M-E equation is not quite conclusive, but it can be plausibly argued that for unconsolidated media the equation is
Figure 1. Test of the Ahmd-Sunada equation with the data of Gupte (1970). (In each of Figures 1-8, one-third to one-half of the data pointa completely overlap those shown and were omitted for clarity.)
satisfactory for most purposes. Test of Models The following data from the literature are used in the evaluation of these equations. Data Set No. 1. The data of Gupte (1970) (Rumpf and Gupte, 1971) for media consisting of narrow size distribution, spherical glass beads packed in a “uniformly random” manner. Eight sets of experimental data with porosities ranging from 0.366 to 0.64 have been used. D,, f t for all data sets. is constant at 2.22 X Data Set No. 2. The data of Kyan et al. (1970) for flow through beds of cylindrical fibers. Porosities range from 0.682 to 0,919. Fiber diameters are uniform for a given bed f t to 9.2 X ft. and range from 2.6 X Data Set No. 3. The data of Dudgeon (1966) for a wide variety of coarse granular media consisting of spherical marble mixtures, sand and gravel mixtures, and ground Blue Metal mixtures. With the exception of the marble data, the media consist of mixtures of several sizes of irregularly shaped particles. Porosities range from 0.367 to 0.515 and De, varies by a factor of about 200. Data Set No. 4. The data of Fancher and Lewis (1933) for flow through a variety of consolidated media. Porosities range from 0.123 to 0.378 and De, varies by a factor of about 10. Data Set No. 5. The unpublished data of Pahl (1975) for a wide variety of cylindrical packings. Particle sizes are uniform for a given bed and range from 0.0062 to 0.012 ft. Length-to-diameter ratios range from 0.37 to 7.42. Porosities range from 0.32 to 0.59. Data Set No. 6. The data of Doering (1955), Matthies (1956), and Luther et al. (1971) for a variety of materials. Porosities range from 0.33 to 0.80. A. Ahmed-Sunada Equation Plots of H vs. NRe”for the first four sets of data are shown in Figures 1-4. The values of cy and and their 95% confidence intervals are given in Tables I-IV. As can be seen in the Figures, the A-S equation appears to fit the data very well. Analysis of the results points up the need for better and more extensive data. From the tables, we see that only for the Gupte data are the confidence intervals on both CY and @ quite narrow. For almost all data sets, the confidence interval on one or the other parameter is reasonably narrow. Investigation shows that the wide confidence interval on the other parameter is not due to failure of the model; rather, it is due to lack of data for the flow region where the term in eq 4 containing that
Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979
201
Model Parameters for the Data of Gupte
Table I. -
a ,f t - 2
x
95% confidence interval,
material 0.124 0.168 0.266 0.405 0.675 1.053 1.564 2.684
1
2 3 4 5 6 7 8
x
0, ft-I
95% confidence interval,
*%
10-3
r%
0.60 0.25 0.36 0.65 0.34 0.56 0.43 0.58
0.674 0.819 1.365 1.846 3.179 3.975 6.120 8.678
0.73 0.34 0.46 0.94 0.47 0.97 1.09 3.15
De , f t x E
P 03
Aa
Bb
0.640 0.612 0.566 0.501 0.480 0.436 0.409 0.366
2.221 2.221 2.221 2.221 2.221 2.221 2.221 2.221
124 126 126 131 136 135 151 162
1.09 1.07 1.27 1.03 1.50 1.30 1.57 1.49
no. of data points 15 15 15 15 15 15 14 13
__
117
'A= 133. bB=1.29. Model Parameters for the Data of Kyan et al.
Table 11. -
material
10-8
+%
4, f t - ' x 10-3
5 nylon 4 nylon 11dacron 3 dacron 1 glass 2 glass 1 0 glass
39.5 24.7 28.4 12.1 10.4 27.7 19.1
2.7 1.3 1.5 1.1 2.2 1.9 1.1
103.0 94.4 64.7 40.9 6.68 -0.104 -5.67
a, f t - l
-
no. no. no. no. no. no. no.
95% confidence inter-
'A=194.
x
95% confidence interval,
De , f t X
k%
E
Po3
26.0 6.1 15 4.2 34 9500.0 37.0
0.682 0.765 0.820 0.829 0.919 0.868 0.895
0.138 0.0984 0.0640 0.0886 0.0394 0.0394 0.0394
no. of data points
Bb
A' 236 194 198 184 192 161 192
14.1 17.7 12.7 12.1 2.52 -0.020 -1.524
12 18 11 18 16 16 14 105
B= 8.22
10'h
1 SYMBOL
. 0
0
10.'
D-Z
id1 Nb;. =
PVO P T
'kp
IO0
Figure 2. Test of the b e d - S u n a d a equation with the data of Kyan et al. (1970).
Nie =
PVOB PQ
parameter is dominant or even significant. This is borne out by the good fit observed in the figures even for the data sets where one parameter is not well established. We conclude that the physical model underlying the A-S equation is adequate. B. Modified Ergun Equation There is considerably greater need for additional data to verify the tentative conclusions we reach here. First, one can expect greater uncertainty in the values of A and E than in the values of a and P, as the latter reflect uncertainty in the measurement of aP and V , only, whereas the former reflect uncertainty in the measurement of not only AP and Vobut also De, and e . Second, irregularly shaped particles are usually sized by screen analysis, and the relation between De, and mesh size is not known. Third, when a distribution of particle sizes is used, information on the distribution is often lacking. Nonetheless, a comparison of the existing data and the M-E equation is quite informative.
Figure 3. Test of the Ahmed-Sunada equation with the data of Dudgeon (1966). SYMBOL 2MATERIAL 4-
7-
A
I
Figure 4. Test of the Ahmed-Sunada equation with the data of Fancher and Lewis (1933).
202
Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979
Table 111
A. Model Parameters for the Marble Data of Dudgeon
cy,
material 16 19 20 17 22 2
95% confidence interval,
ft-l x 10-8
0.00942 0.00393 0.00636 0.00375 0.00401 0.0041 6
4 , ft-'
x
95% confidence interval,
De , f t X
*%
10-3
+%
E
'io3
Aa
Bb
8.7 11.1 12.3 15.8 17.9 5.0
0.296 0.194 0.310 0.195 0.129 0.257
1.10
0.369 0.415 0.372 0.369 0.385 0.379
52.4 52.4 52.4 81.6 95.0 70.0
326 226 228 315 546 287
1.24 1.24 1.33 1.27 1.14 1.58
1.73 1.56 1.32 1.55 1.83
no. of data points 20 13 11 16 18 16 94
B. Model Parameters for the Sand and Gravel Data of Dudgeon
mater- a , f t - ? x ial 10-8
-
4 6 13 12 10 18
5.03 0.490 0.0209 0.00694 0.00359 0.000950
95% confidence interval, 4, ft-' x r% 10-3 1.71 1.38 9.0 9.3 11.0 17.0
95% confidence interval,
17.5 6.23 0.710 0.456 0.144 0.0414
D,,' f t X
r%
E
10'
46.0 3.31 1.80 0.86 0.93 1.00
0.387 0.418 0.367 0.372 0.369 0.406
1.52 6.11 51.5 81.6 173 360
$s
A
B
no. of data points
1.0 0.6 0.6 0.6 0.6 0.6
179 142 246 217 488 841
2.62 2.87 1.71 1.83 1.19 1.01
23 20 18 13 17 17
assumed
BI& 2.52 4.78 2.86 3.05 1.99 1.68
179 394 684 604 1360 2340
108
C. Model Parameters for the Blue Metal Data of Dudgeon 8 9 15 21 1 14 11
0.121 0.0570 0.0110 0.00470 0.00735 0.00470 0.00191
4.18 3.76 4.05 7.5 2.48 11.0 25
3.02 1.80 0.754 0.290 0.545 0.364 0.413
2.65 1.36 0.65 1.09 0.34 0.66 0.64
0.477 0.458 0.428 0.515 0.455 0.438 0.483
254 565 442 459 560 781 939
7.27 17.4 41.0 41.0 49.0 79.0 108
4.55 5.54 4.24 3.34 4.61 4.30 9.72
0.6 0.6 0.6 0.6 0.6 0.6 0.6
91.3 203 159 165 202 281 338
2.73 3.33 2.54 2.01 2.77 2.58 5.83
16 17 15 13 35 11 13 120
1
\'\
MATERIAL SYMBOL -
'h
-
18011-d+B
Fl
1
I-.
Nk,
LOWER BRANCH l E 0 1 5 0
UPPER BRANCH IEO I5b: 25 % ENVELOPE 2 50 % ENVELOPE
2
I
I
1 lo-2
~
1 10-1
I IO0
-N=k- e 1-6
1
101
4
p "0 Deq p(1-c)
Figure 5. Test of the modified Ergun equation with the data of Gupte (1970).
B.l Gupte Data. The Gupte data are the most comprehensive and precise data looked at. This is the only set of data resulting in well-established values of both a and fl for each medium used. It is also least challenging since the bed is composed of uniform-sized, smooth, spherical particles. Values of A and B calculated from the
a and 0values previously determined, plus the 6 and D,
values, are given in Table I and the Fk1c3/(l - e)] vs. N&'/(l- c) values are plotted in Figure 5. The data seem to indicate that A and B are weakly dependent on t, rather than constant, suggesting that the t dependence of the model may not be quite right. Indeed, better fit to these
Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979
203
..
_ MATERNAL _ _SYMBOL _ * 5 NILCN NILSN
.4
I , ,C L i C 9 3 N
., *,
1*CFC*
6..lj;
2 GLASS
910 $ L A S S
0 0
a
A 7
L O W E R B R A N C H I E O ,501
101-
U P P E R BRP.NCH I E P 1 5 b l
-----
t 2 5 9 . ENVELOPE t SO 9. E N V E L O P E
I 10-2
10-1
100
10
I02
NRe = pVoDeq -
I-
