General combined model for the hydrodynamic behaviour of fixed and fluidised granular beds

Water Research 111 (2017) 163e176

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General combined model for the hydrodynamic behaviour of fixed and fluidised granular beds Garry Hoyland Bluewater Bio Ltd, Winchester House, 259-269 Old Marylebone Road, London, NW1 5RA, UK

a r t i c l e i n f o

a b s t r a c t

Article history: Received 19 July 2016 Received in revised form 9 December 2016 Accepted 2 January 2017 Available online 3 January 2017

This work describes the derivation of a general mathematical model applicable to both fixed and fluidised granular beds, operating within the full hydrodynamic spectrum from viscous to inertial flows. The fundamental insight for the derivation of the model is that practical fluidised beds and fixed beds have similar hydrodynamic properties. The validity of the general model is demonstrated for fluid fractions up to 0.90. A crucial development in the general model is the replacement of hydraulic diameter, which has served as the size descriptor of flow paths in most fixed-bed models derived since the advent of the classic Blake-Kozeny equation. The new, replacement expression is based on the physical structure of the cross section of random porous beds. In addition, the general model contains a tortuosity factor, derived from the results of previous works involving computational fluid dynamics, to correct flow path length and fluid velocity. The model is constructed using regression analysis of experimental data from six previous major works and tested against previous models. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Blake Kozeny Fixed Fluidised Wall Model

1. Introduction The work described here stems from a need to improve further the understanding of the hydrodynamic principles relating to water flow through granular filter beds operating in fixed mode and regenerative, fluidised mode. This has led to the development of a General Combined Model, GCM, applicable to both fixed and fluidised modes operating over the full hydrodynamic spectrum. The GCM relates to fixed and fluidised packings comprising smooth, solid and spherical or near-spherical granules that have a similar packing geometry. Without further development, the GCM is not applicable to packings comprising other particle shapes such as cylindrical pipes and prisms. 1.1. Contents Section 1.2 describes historical models derived for fixed and fluidised beds, relevant to this work. The GCM has been developed from conventional hydrodynamic principles, which are described in Chapter 2. The prime development of this work, described in Chapter 3, relates to the derivation of theoretical relationships that

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.watres.2017.01.008 0043-1354/© 2017 Elsevier Ltd. All rights reserved.

characterise (i) the magnitude of the interstitial velocity, U, of the fluid passing through convoluted flow paths in random porous networks of fixed and fluidised beds and (ii) the physical size of the cross-sectional area of the flow paths. The same relationships apply to both types of beds, over the practical range of fluid fractions, E. As shown in Chapter 6, the GCM is applicable to E values
0.90. Chapter 4 tests the validity of derived sensitivities of pressure loss to E in fixed beds and fluid velocity to E in fluidised beds, using previously published experimental data. Chapter 5 constructs the GCM and uses previously published data from several major sources to determine the values of coefficients. Chapter 6 compares GCM predictions of the relationships between pressure loss and Reynolds number in fixed beds and between E and Reynolds number in fluidised beds with experimental data published in one of the major sources. 1.2. Historical models Historical models for fixed and fluidised beds have largely been developed in parallel, leading currently to separate models with different structures for each bed type. An exception is the modelling of Foscolo et al. (1983) who developed models for fixed and fluidised beds using a common methodology. The following subsection discusses a selection of models relevant to this work.

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G. Hoyland / Water Research 111 (2017) 163e176

K

Nomenclature

Kl Dimensional symbols Ax (m2) area of flow stream's cross-section Ab (m2) total cross-sectional area of bed Aw (m2) the total wetted wall area of a conduit C a context-dependent dimensioned or dimensionless constant Dc (m) diameter of pipe. Dx (m) hydraulic diameter equal to Ax/Lw Dm (m) diameter of sphere, the volume of which Vm Dv (m) diameter of bed L (m) path length of flow stream L0 (m) longitudinal length of fixed bed Lb (m) longitudinal length of fixed or expanded bed, equal to L0 for fixed bed Lw (m) wetted perimeter of a flow stream's cross-section Lx (m) characteristic dimension of the cross-section of a conduit Lu (m) side length of cubic packing unit Sm (m2/m3) the total surface area of granules divided by total volume of granules Sb (m2/m3) total specific surface area of a bed, including area of vessel wall Sv (m2/m3) specific surface area of vessel U (m/s) average velocity along the axis of a conduit, or average interstitial velocity along the axis of a convoluted conduit in a bed Ub (m/s) superficial (empty-bed) velocity through bed Uf (m/s) free settling velocity of granules Vm (m3) total volume of granules in a bed divided by the number of granules Vu (m3) volume of a cubic packing unit Vv (m3) total volume of the conduit or bed Dimensionless symbols Ar Archimedes number CI95 95-percentile, multiplicative, confidence interval De density number equal to (rm  r)/r and Ar/Ga E0 fluid fraction of fixed bed E overall fluid fraction of bed, equal to E0 for fixed bed f( ) context-dependent function Ga Galilei number Ga# a modifiied form of Galilei number

1.2.1. Fixed beds A major advance in the modelling of fixed beds containing granules and other media was made by Blake (1922) who applied conventional hydrodynamics principles to porous beds. Further described in Chapter 3, his model is:

Q ¼ K Rex s

(1.1)

where

Q ¼ Dp E3

.  Lb r Ub2 Sm ð1eEÞ

(1.2)

and

Res ¼ K Ub r=ðm Sm ð1eEÞ Þ:

Kt Q Rd Re Ref Rem Res Red T

a general hydraulic constant, with a value dependent on context the hydraulic constant for viscous flow, with a value dependent on context the hydraulic constant for inertial flow, with a value dependent on context coefficient in Felice and Kehlenbeck equation diameter ratio equal to Dm/Dv Reynolds number equal to Lx U r/m Reynolds number equal to Dm Uf r/m Reynolds number equal to Dm Ub r/m Reynolds number equal to Ub r/(m Sm(1  E)) Reynolds number equal to Dm Ub r/(m (1  E)) tortuosity of flow path through a granular bed

Greek symbols Q friction factor derived by Blake for fixed beds Qd analogous to Q, with Sm replaced with Dm F friction factor in GCM for fixed and fluidised beds Dp (Pa) loss in dynamic pressure over path length Dh (m) loss in head of fluid over path length, equal to Dp/(g r) m (Pa s) viscosity of fluid j sphericity equal to ratio of surface area of sphere to that of a granule where the sphere and granule have the same volume r (kg/m3)density of fluid rm (kg/m3) material density of granules t (s) retention time of fluid in bed u the wall-effect factor derived by Carman ua the wall-effect factor in the fixed component of the GCM ub the wall-effect factor in the fluidised component of the GCM u1, u2 and u3 are context-sensitive wall-effect factors Exponents m exponent in GCM, dependent on mode of bed n the expansion exponent in the Richardson and Zaki fluidisation model, when Rd ¼ 0 N the expansion exponent in the Richardson and Zaki fluidisation model, when Rd > 0 x context-dependent exponent y context-dependent exponent in Equations (4.5) and (4.6)

The dimensionless variables, Q, K and Res are friction factor, empirical hydrodynamic constant and a Reynolds number respectively. Res is occasionally referred to as the Blake number. When the x exponent ¼ 0.2, Equation (1.1) gives a reasonable fit to experimental data over a limited range of Rem values of from 0.1 to 4. Making an analogy with flow through pipes, the x value is directly related to Reynolds number, decreasing from a theoretical value of 1 for viscous flow to a value of 0 for inertial flow. Kozeny (1927) independently derived a form of Equation (1.1) specifically for viscous flow. His equation may also be obtained by substituting 1 for the exponent, x, in Blake's equation giving:

Dp = Lb ¼ Kl m Ub S2m ð1eEÞ2

.

E3 :

(1.4)

(1.3) Kozeny (1927) and Donat (1929) showed experimentally that

G. Hoyland / Water Research 111 (2017) 163e176

this equation could predict the trend in pressure gradient with E, over an E range of from 0.4 to 0.6, but did not derive a robust value for the hydrodynamic constant, Kl. Equation (1.4) is generally called the Blake-Kozeny equation. Carman (1937) derived a more general form of Equation (1.1), which increased applicability to a wider range of Res values covering viscous and turbulent regimes. Evaluating the constants from experimental data, obtained from published works, Carman's equation is: 0:1 Q ¼ 5Re1 : s þ 0:4Res

(1.5)

Equation (1.5), generally called the Carman-Kozeny equation, reduces to the Blake-Kozeny equation, when Res values are comparatively small and the value of Kl in Equation (1.4) is made equal to 5. The experimental data used by Carman covered a range of Rem values of from 0.2 to 4 000, and the transition from viscous to turbulent flow occurred in the Rem range of from 3 to 8. Carman (1937) realised that vessel walls increase pressure loss by increasing shearing area between solid material and fluid. He proposed the following wall-effect factor, u, but did not include the factor in Equation (1.5).

u ¼ 1 þ 4Rd =ð6ð1eEÞ Þ

(1.6)

where Rd is the ratio of granule diameter to vessel diameter. He assumed pressure loss is proportional to factor, u. Inspection of Equation (1.6) shows that, u, is the ratio of the total surface area of vessel and spherical granules to area of granules. At the root of Equation (1.5) is a modified form of an equation derived by de Prony (circa 1800) to model pressure gradients, Dp/L, in water flowing through circular pipes. The Prony equation (also referred to as the Forchheimer equation) is:

Dp Dc =L ¼ C1 U þ C2 U 2

(1.7)

where C1 and C2 are dimensioned empirical constants characteristic of any particular system. The first and second terms on the righthand side of this equation relate to viscous regime and inertial regime respectively. Many historical models for fixed beds, derived by numerous workers including Carman, have been based on the hydrodynamic principles inherent to Equations (1.1)e(1.4) and on the original or modified Prony equation. In general, these models may be expressed in the following generalised form: x Qd ¼ C1 Re1 d þ C2 Red

165

where Red ¼ Dm Ub r/(m(1  E)), Qd ¼ Dp E3 Dm/(Lb r U2b(1  E) and C1 and C2 are dimensionless constants. After the time of Blake and Carman, it has been conventional to expressed friction factor and Reynolds number in terms of Dm instead of Sm, where Sm ¼ 6/Dm for spherical granules. Table 1 lists just a few published models, most of which were specifically derived for spherical or near-spherical granules. As a minimum, experimental data used in the derivation of each model includes some for spherical granules. A comparison of the C1 values in Table 1 shows significant variation. As explained by MacDonald et al. (1979), some of this variation can be attributed to the range of Reynolds number values in the experimental data. The data in Table 1 suggest that the C1 values are roughly inversely related to the minimum value of Rem in the experimental data. As suggested by Erdim et al. (2015), the true value of Kl for spherical particles is probably in the range of 150e180. Further work is needed to clarify this aspect of the models. Some of the models contain wall-effect factors of different types. For example, Mehta and Hawley (1969) used the Carman factor, u, but applied u2 to the viscous term and u to the inertial term. Conversely, Erdim et al. (2015) found from a detailed statistical analysis of their experimental data, obtained using beds of spherical particles, that wall effect is not significant. Chapter 5 investigates wall effect further. The model derived by Foscolo et al. (1983) suggests that the relationship between Qd and Red does not correctly predict the sensitivity of pressure loss to E. Their model is based on relating drag force acting on spherical granules in fixed beds to pressure loss determined from the Ergun model. Also, their model addresses the impact of tortuosity on the lengths of flow paths through beds. When E ¼ 0.4, Foscolo's model defaults to Ergun‘s model. Chapter 3 derives the sensitivity of pressure loss to E for insertion into the GCM. Rose (1945) and MacDonald et al. (1979) considered that models based on Equation (1.8) do not necessarily give good fits to experimental data covering both viscous and turbulent regimes, particularly those based on the unmodified Prony equation (such as the Ergun model). Rose (1945) improved the statistical fit of experimental data by introducing an extra Reynolds number term on the right-hand side of Equation (1.8) and making x ¼ 0. This modelling approach has been subsequently used by several workers, including Montillet et al. (2007), who derived their model specifically for spherical particles and, in addition, introduced a wall-effect factor based on Rd. Montillet's model, based on experimental data covering Rem values of from 10 to 2 600, is as follows:

(1.8)

Table 1 Values of constants in historical models vary significantly. Source

C1

C2

x

Rem range

Carman (1937) Ergun (1952) Hanley and Heggs, 1968 Mehta and Hawley (1969) Hicks (1970) Tallmadge (1970) KTA (1981) Foscolo et al. (1983) Foumeny et al. (1993) Eisfeld and Schnitzlein, 2001 Yu et al. (2002) Allen et al. (2013) Harrison et al. (2013)

180 150 368 150 u2 0 150 160 17.3/(E1.8(1  E)) 130 154 u2 203 262 120 u21 u1 ¼ 1 þ p Rd/(6(1  E)) 160

2.87 1.75 1.24 1.75 u 6.8 4.2 3.1 0.336/E1.8 (0.335 þ 2.28 Rd)1 u(1.15 R2d þ 0.87)2 1.95 4.58 4.63 u2 u2 ¼ 1  p2 Rd(1  0.5 Rd)/24 2.81

0.10 0 0 0 0.2 0.17 0.10 0 3 0 0 0.12 0.17

0.2e4000 0.7e2500 200e13 000 0.1e6 180e36 000 0.06e60 000 0.06e60 000 Not applicable 3e5000 0.01e18000 750e 2500 50e8000 0.32e7700

0.096

1.2 to 2100

Erdim et al. (2015)

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G. Hoyland / Water Research 111 (2017) 163e176



1000ðRed ð1eEÞ Þ1 þ 60ðRed ð1eEÞ Þ0:5 þ 12 Qd ¼ KR0:20 d

 (1.9)

where K equals 0.061 when E < 0.4 and equals 0.050 when E  0.4. Montillet et al. suggested the model is valid for 0.02 < Rd < 0.28; albeit the minimum value of Rd in the experiment data is 0.063. A feature of Montillet's model is that pressure loss is directly related to vessel diameter, instead of inversely related as in most other models. When Rd ¼ 0.05 and E ¼ 0.38, Equation (1.9) reduces to: 0:5 Qd ¼ 179 Re1 þ 1:33: d þ 8:5 Red

(1.10)

The Ergun model is probably the most widely used in scientific and engineering practice. In spite of its popularity, it is generally understood the model over-predicts pressure loss in random beds of smooth spheres when Rem > 500. Its popularity may be partly due to its ease of use, which stems from its quadratic structure, making Red and Qd analytically related in both directions. 1.2.2. Fluidised beds With regard to fluidised beds, Wilhem and Kwark (1948) showed experimentally that pressure loss through any particular fluidised bed is approximately constant, determined by Archimedes principle, as follows:

Dp=Lb ¼ ð1eEÞðrm erÞ

(1.11)

With regard to expansion of fluidised beds, Lewis et al. (1949) derived a graphical relationship for spherical granules, covering a Rem range of from 0.2 to 20. Their relationship is as follows:

Ar E4:65

.  K Re2m ¼ fðRem Þ

(1.12)

Their graph shows the value of the left-hand side of Equation (1.12) declining with Rem value. Ar is Archimedes number, defined in Equation (5.8). In the viscous region occurring at Rem values less than about 1, f(Rem) is equal to Re1 m so that Equation (1.12) reduces to:

Ar E4:65 ¼ K Rem :

(1.13)

Lewis and co-workers showed that their graphical relationship (Equation (1.12)) could apply to beds fluidised with air, as well as water, but only when ‘slugging’ was not observed in the bed. Richardson and Zaki (1954) performed a wide range of hindered-settling and fluidisation experiments using granules comprising smooth spheres and other shapes and different liquids. They developed the following equation for fluidisation of smooth spherical granules:

. Ub Uf ¼ Rem =Ref ¼ u1 EN :

(1.14)

where

N ¼ u2 n

(1.15)

and

  n ¼ f Ref :

(1.16)

Parameters, u1 (
1) and u2 (1) are wall-effect factors expressed as functions of Rd. In addition, factor, u2, is an inverse function of Ref. When Rd ¼ 0, both factors are equal to unity, and Equation (1.14) reduces to:

. Ub Uf ¼ En :

(1.17)

Analysing their experimental data, Richardson and Zaki showed the value of n for viscous flow was 4.65, in agreement with the value obtained by Lewis et al. (1949). In addition, they found that n decreased to a minimum value of 2.39, when Ref  500. Khan and Richardson (1989) developed further the Richardson and Zaki model, by deriving revised expressions for estimating values of the three parameters n, u1 and u2. In place of Equations (1.15) and (1.16), their model comprises Equation (1.18), as follows:

. ð4:8eNÞ=ðNe2:4Þ ¼ 0:043 Ar0:57 u3

(1.18)

where

u3 ¼ 1

 . : 1e1:24 R0:27 d

(1.19)

They also revised the expression for u1, giving:

u1 ¼ 1e1:15 R0:6 d

(1.20)

Further, Khan and Richardson (1989) derived an expression, based on Ar number, for estimating the value of Ref (or Uf), which is an input parameter in Equation (1.14). The above models developed by Richardson and co-workers are probably the most widely used in scientific and engineering practice. Whereas the Richardson and Zaki model, with or without the Khan and Richardson development, suggests fluidisation (E, Ub) curves are sensitive to Rd, results of experiments performed by Wilhem and Kwark (1948) and by Loeffler and Ruth (1959) suggest that fluidisation curves are insensitive to Rd. Similarly, Felice and Kehlenbeck (2000) performed hindered settling experiments to investigate the effect of Rd on the Richardson and Zaki parameters, n, u1 u2. Hindered-settling and fluidisation are analogous and give identical or very similar (E, Ub) curves, as the results of dual experiments performed by Lewis et al. (1949) and Loeffler and Ruth (1959) demonstrate. Felice and Kehlenbeck performed their settling experiments using spherical granules and vessels of different diameters, giving a range of Rd values of from 0.03 to 0.21. They also used granules and fluids with different physical properties to broaden the experimental range, and Uf values were measured in a large diameter vessel to eliminate any wall effect. Values of Ref ranged from 2 to 50, indicating turbulent settling. The results obtained by Felice and Kehlenbeck suggest that fluidisation and hindered settling curves may be modelled by the following equation:

. Ub Uf ¼ Q En

(1.21)

where the exponent, n, may be estimated from Equation (1.16). They also determined that Q is a constant with values in the range of from 0.7 to 0.95 but could not find any definitive relationship between Q and Rd or any other parameter. Similar hindered settling results were obtained by Felice and Parodi (1996) and Chong et al. (1979) for viscous flow. Loeffler and Ruth (1959) used a semi-empirical method to derive the following graphical relationship for fluidisation curves in both the viscous and turbulent regimes:

Ar E3

.   18 Rem 5:7ð1eEÞ þ E2 ¼ f ðRem Þ

(1.22)

Loeffler and Ruth's graph relates the left-hand side of Equation (1.22) to Rem, over the Rem range of from 0.005 to 1000. For viscous

G. Hoyland / Water Research 111 (2017) 163e176

flow, the graph shows that f(Rem) ¼ 1, which on substitution into Equation (1.22) gives:

. .  5:7ð1eEÞ þ E2 : Ub Uf ¼ E 3

(1.23)

The graph also shows that the transition from viscous to turbulent flow occurs at Rem values of around 1, consistent with the results of Lewis et al. (1949). Foscolo et al. (1983) extended their model for fixed beds to fluidised beds. They trimmed their model using the expansion exponents, n, obtained experimentally by Richardson and Zaki (1954). Their fluidised bed model comprises three separate equations, covering the viscous regime (Ref < 0.2), turbulent regime, and inertial regime (Re > 500), as follows. For the viscous regime:

. .  3:33ð1eEÞ þ E3 : Ub Uf ¼ E 4

(1.24)

Equation (1.24) is similar in form to Equation (1.23). For the turbulent regime:

.     0:5 . Ub Uf ¼ 4 K Ref 1 þ K Ref E4:8 þ 1 e1 2 K Ref (1.25) where K ¼ 0.0194. For the inertial regime:

. . 0:5 3:55ð1eEÞ þ E3 Ub Uf ¼ E 2 :

(1.26)

All three equations predict Ub ¼ Uf when E ¼ 1. 2. Basic hydrodynamic principles The structures of fixed and fluidised granular beds, with E values less than an upper limit, may be compared to a multitude of convoluted conduits, with pores forming frequent cross-connections. The conduits and pores form a continuous, random, porous network. In fluidised beds, with E values greater than the upper limit, form drag becomes more prominent, changing the hydrodynamic properties. Chapter 6 shows that the upper limit of E, controlling the hydrodynamics, is approximately 0.90. The basic hydrodynamic equations presented in this chapter relate to steady flow of incompressible fluids through conduits, which may be convoluted conduits in fixed and fluidised bed or straight pipes that have no specific cross-sectional shape. The equations presented in this chapter are used in Chapters 3 to 5 for construction of the GCM.

Pressure-loss relationships for viscous flow and inertial flow through conduits may be derived from the following force balances:

Dp=L ¼ Kt r U 2

.

Dx :

continuously with position along the conduits. The average velocity, U, is constant with longitudinal position along the bed. The variable, Dx, is called the hydraulic diameter, which is defined herein as follows:

Dx ¼ Ax =Lw ¼ Vv =Aw :

(2.3)

The variables Kl and Kt are dimensionless hydrodynamic constants, which are generally estimated experimentally for conduits with irregular cross-sectional shapes. The value of Kt in Equation (2.2) increases with wall roughness. Allen et al. (2013) have experimentally investigated the effect on pressure loss through fixed beds of 16 mm diameter spheres with smooth and roughened surfaces. In separate experiments, they stuck sand grains with average diameters of 0.21 mm and 0.71 mm to the surfaces of the smooth spheres. They found that surface roughness increased pressure loss by some 45% at Rem z 4000 decreasing to 20% at Rem z 60, reflecting the pattern typically observed for flow through pipes. However, the pressure loss for the larger grains was only slightly greater than that for the smaller grains. The value of Lx in Equation (2.1) characterises the size of the cross-section of any particular conduit and, in principle, may be any definitive characteristic. It follows that the value of Kl depends on the definition applied to Lx. For irregular cross sections, general hydrodynamic practice is to make Lx equal (or proportional) to Dx, although this is not necessarily the best assumption for all crosssectional shapes. The data in Table 2 shows the relationship between Kl and Lx for several regular conduits and three definitions of Lx. These definitions are A1/2 x , Dx and Lw, which relate to each other via Equation (2.3). The data were derived from the theoretical results obtained by Tamayol and Bahrami (2009), who solved the relevant differential Poisson equations using analytical integration. Comparing the Kl values in Table 2 shows that A1/2 and Dx give the most x consistent values for the different regular cross-sections. As explained later in Chapter 3, A1/2 x is the preferred substitution for Lx in this work, instead of the conventional preference for Dx.

2.2. Composite variables When applied to porous beds, four of the variables in Equations (2.1) to (2.3) have special properties in that they are functions of other variables. These four variables, Dx, Lx, L and U, called composite variables in this work, are the building blocks of the GCM.

2.3. Dimensionless groups Equations (2.1) and (2.2) may be expressed in the following dimensionless forms:

2.1. Viscous and inertial equations

Dp=L ¼ Kl m U=ðLx Dx Þ

167

Table 2 A1/2 x and Dx give reasonably consistent values of Kl. Shape

(2.1) (2.2)

In the context of flow through a straight pipe, U is the axial velocity averaged over the pipe's cross-section. In the context of flow through convoluted conduits in a porous bed, U is the interstitial velocity in the axial direction of the conduits averaged over the conduits' cross sections. The axial directions of individual conduits and the values of interstitial velocities fluctuate

Equilateral triangle Square Regular hexagon Circlea Ellipse (axis ratio ¼ 2) Rectangle (side ratio ¼ 2) Average (lg mean) Standard deviation (multiplicative)

Value of Kl Lx ¼ A1/2 x

Lx ¼ Dx

Lx ¼ Lw

7.60 7.09 7.01 4 p1/2 ¼ 7.09 8.13 8.25 7.51 1.076

1.67 1.77 1.88 2 2.10 1.94 1.89 1.088

34.6 28.3 26.1 8 p ¼ 25.1 19.4 35.0 27.5 1.248

Note: a The Kl values for a circle are exact, leading to the Hagen-Poiseuille equation.

168

G. Hoyland / Water Research 111 (2017) 163e176 Table 3 Historical definitions of composite variables.

F ¼ Kl Re

(2.4)

F ¼ Kt Re2

(2.5)

Common

Porous bed

(2.6)

L U Dx Lx

Lb Ub/E E/Sm(1  E) Dx

Composite variable

where

F ¼ Ga# Dh=L

Source

e Dupuis and Dupuis (1863) Blake (1922), Kozeny (1927)

and

Re ¼ Lx U r=m:

(2.7)

F is the friction factor preferred in this work. In contrast to the Blake friction factor, Q, F contains only one dependent variable, namely Dh/L. Separation of dependent variables into different dimensionless groups has benefits for the regression analysis described in Chapter 5. Equation (2.6) contains the dimensionless group, Ga#, which is a modified Galilei number, defined as follows: .

Ga# ¼ L2x Dx r2 g m2 :

independently assumed Lx was equal to Dx, for a porous bed, according to normal engineering practice. Substituting the porous-bed versions of the composite variables from Table 3 into Equation (2.1) yields the Blake-Kozeny equation for viscous flow, and substituting into Equation (2.2) yields the Blake equation (with x ¼ 0) for inertial flow, which is as follows:

.

Dp =Lb ¼ Kt r Ub2 Sm ð1eEÞ E3

(3.1)

(2.8) 3.2. Definitions in GCM

2.4. Core hydrodynamic equation As indicated in Section 1.2, there are numerous models for fixed and fluidised beds. The core hydrodynamic equation in the GCM for both fixed and fluidised beds is as follows:

Fm

m  ¼ ðKl ReÞm þ Kt Re2

(2.9)

where m < 1.0. Equation (2.9) is preferred for the following reasons.  Equation (2.9) satisfies the theoretical hydrodynamic relationships between pressure loss and velocity in the viscous and inertial regimes.  The curvature of the relationship between F and Re may be adjusted by changing the value of the exponent, m, which varies with bed type. When m ¼ 1, Equation (2.9) defaults to the Prony equation.  The definitions of F and Re are formed using the compound variables, as indicated in Chapter 3.  Equation (2.9) contains only three degrees of freedom and has a quadratic structure.  Most importantly, experimental data relating to fixed and fluidised beds fit well to Equation (2.9), as indicated in Chapter 5. 3. Definitions of composite variables This chapter derives the definitions of the composite variables used in the GCM and compares these variables with those used historically in fixed bed models. 3.1. Historical definitions Table 3 lists the historical definitions of the composite variables for fixed beds. Determined by bed geometry, the expression for Dx is incontrovertible. Blake derived the expression by considering a bed's fluid fraction as a single conduit, whereas Kozeny obtained the same expression by assuming the fluid passes through a multitude of essentially separate conduits. Both Blake and Kozeny

Carman (1937) realised that convoluted conduits in fixed beds increase path length of the fluid. The actual length of the flow path is expressed in terms of a tortuosity parameter, T, as follows:

L ¼ Lb T:

(3.2)

Tortuosity is clearly an inverse function of E, reducing to a value of 1 when E ¼ 1. Carman also realised that the length of a convoluted path determines the interstitial velocity, U, of the fluid following the path. Retention time, t, in any particular bed is the volume of fluid in the bed divided by the superficial (or empty bed) volumetric flow rate, as follows:

t ¼ Lb E=Ub

(3.3)

The interstitial fluid velocity, U, averaged over the cross section is therefore given by:

U ¼ L=t ¼ T Ub =E

(3.4)

Kozeny (1927) also understood that convoluted channels increased path length; but he and Carman believed that the effect of the convolutions on pressure loss was included by default in their fixed-bed models. 3.2.1. Expression for Lx As explained in Chapter 2, A1/2 x , an exclusive property of a conduit's cross-section, is the preferred substitution for Lx. The expression for A1/2 x is derived from a consideration of the physical properties of randomly-structured porous beds. White and Walton (1937) explained that any regularlystructured porous bed of single-sized spheres comprises adjacent packing units in the shape of rectangular prisms. Each packing unit contains a single sphere and the associated free space, albeit the sphere may be segmented into several pieces depending on the packing pattern. In contrast, adjacent packing units in random porous beds have different shapes and sizes, distributed normally around a mean shape and a mean size. Since most beds contain multitudes of granules, it is reasonable to hypothesis that the mean packing unit along any conduit (or flow path) is a cube. Like packing units in regularly-structured beds, the cube contains a single

G. Hoyland / Water Research 111 (2017) 163e176

granule and associated free space. Also, it is hypothesised that the randomness gives rise to mean packing units, which are cubic, even when granules vary in shape and size. It is further assumed that the number of conduits is equal or proportional to the number of packing units. Containing one granule and associated free-space, the volume, Vu, of the mean packing unit is given by:

Vu ¼ Vm =ð1eEÞ

(3.5)

where Vm is the average volume of a granule, equal to the total volume of all granules in a bed divided by the number of granules. Since the shape of the mean packing unit is cubic, the average side length, Lu, of the unit is:

Lu ¼ ðVm =ð1eEÞ Þ1=3 :

(3.6)

The cross-sectional area of a bed, Ab, is given by:

Ab ¼ L2u N

(3.7)

where N is number of packing units in any particular cross section. Assuming the number of packing units and conduits are equal, the cross-sectional area, Ax, of each conduit is:

Ax N ¼ E Ab :

(3.8)

It follows from Equations (3.6) to (3.8) that: 2=3

Ax ¼ Vm

. E ð1eEÞ2=3 :

(3.9)

Equal to A1/2 x , Lx is then given by:

. 1=3 Lx ¼ Vm E1=2 ð1eEÞ1=3 :

(3.10)

Substituting the diameter, Dm, of a sphere of volume, Vm, into Equation (3.10) gives the expression for Lx, as follows:

. Lx ¼ ðp=6Þ1=3 Dm E1=2 ð1eEÞ1=3 :

(3.11)

3.2.2. Expression for tortuosity The expression for hydrodynamic (or hydraulic) tortuosity in the GCM is derived from computational numerical solutions of the

169

Navier-Stokes equations previously described by other workers. Duda et al. (2011) determined tortuosity for viscous flow in simulated porous beds using the Lattice Boltzmann Method of solution. The beds were made two dimensional to obtain accurate solutions within practical computational time limits. Comprising 1000 increments in each direction, plus additional entrance and exit lengths, the beds were filled with randomly placed 10  10 squares, representing granules, giving E values from 0.375 to 0.99 in different runs. The squares could overlap to simulate realistic packing. Matyka and Koza (2012) performed a similar numerical analysis using stepped circles rather than squares to represent granules. Saomoto and Katagiri (2015) also determined hydrodynamic tortuosity, using the COMSOLR numerical solution package. In contrast, they assumed non-overlapping granules, both squares and circles. Duda et al. (2011) and Matyka and Koza (2012) considered options for calculating hydrodynamic tortuosity from the integrated results. They concluded that the most accurate definition of T was an integral form of Equation (3.4), integrated over the fluid fraction of the bed to determine the average value of T. In contrast, Saomoto and Katagiri (2015) calculated tortuosity from an integral form of Equation (3.2), in which L is the average length of streamlines. Fig. 1 shows numerically-derived tortuosity values plotted against fluid fraction, for the three groups of workers. Saomoto and Katagiri (2015) obtained lower values of tortuosity compared with the other two groups of workers, suggesting that their lower values were probably caused by their assumption of non-overlapping granules. It is considered that T values obtained by Duda and co-workers and Matyka and Koza are the more realistic. The solid line in Fig. 1 is a fit to these particular T values, for E  0.4. The equation describing this line is as follows:

. T ¼ 1 E7=12

(3.12)

where the exponent is expressed as a fraction for compatibility with other exponents in the GCM. The form of Equation (3.12), complies with the Archie (1942) law, which relates to diffusive rather than hydrodynamic tortuosity; nevertheless Equation (3.12) gives a good fit to the accumulated numerical data. Foscolo et al. (1983) also used this form of relationship to model hydrodynamic tortuosity in fixed and fluidised beds. For the GCM, Equation (3.12) applies to fixed and fluidised beds operating under all hydrodynamic regimes. 3.3. Summary of definitions The GCM defines the composite variables in terms of Dm rather than Sm. The relationship between these two granule parameters in the GCM is:

Sm ¼ 6=ðj Dm Þ

(3.13)

where j is sphericity defined by Wadell (1935). Table 4 lists definitions of the composite variables, derived from Equations (3.2), (3.4) and (3.11), 3.12, and 3.13. These definitions contrast with

Table 4 Definitions of composite variables in the GCM listed for clarity.

Fig. 1. Archie (1942) curve gives good fit to CFD numerical data.

Common variable

Applied to porous bed

L U Dx Lx

Lb/E7/12 Ub/E19/12 j Dm E/(6(1  E)) (p/6)1/3 Dm E1/2/(1  E)1/3

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G. Hoyland / Water Research 111 (2017) 163e176 Table 5 Range in exponent value from Equation (4.3) is similar to experimental range.

Table 7 Agreement between GCM and Blake/Kozeny models for fixed beds depends on flow regime.

E range

Expansion exponent, y

Flow

GCM

Blake/Kozeny

0.50e0.90 0.55e0.90 0.60e0.90

4.52 4.63 4.76

Viscous Inertial

5.00 (Equation (4.1)) 5.75 (Equation (4.2))

5.00 (Equation (1.4)) 4.00 (Equation (3.1))

those given in Table 3, apart from the definition of Dx, which remains effectively unchanged. 4. Testing of composite variables This chapter tests the validity of the GCM's composite variables. The test method involves deriving the viscous and inertial equations of the fixed and fluidised components of the GCM, and then comparing the E functions of these equations against historical experimental data. The viscous and inertial flow equations in the GCM are obtained by substituting the definitions listed in Table 4 into Equations (2.1) and (2.2), giving:

Dp=Lb ¼ Kl m Ub ð1eEÞ4=3

.

j D2m E11=3

.

Dp=Lb ¼ Kt r Ub2 ð1eEÞ

j Dm E19=4



(4.1)



(4.2)

Whilst these two equations do not contain the wall-effect factors described later in Section 5.1, this omission does not affect test legitimacy. Specifically for fluidised beds, Equations (4.1) and (4.2) are simplified by eliminating Dp/Lb using Equation (1.11), giving:

Ub ¼ ðrm erÞg j D2m E11=3

.  Kl mð1eEÞ1=3

(4.3)

. Ub2 ¼ ðrm erÞg j Dm E19=4 ðKl rÞ:

(4.4)

To obtain theoretical E functions compatible with the experimental data, Equations (4.1) and (4.2) are transformed to the following:

Dp is proportional to Ey

(4.5)

exponent values obtained depend on the range of E values in the input data. For example, Table 5 shows the range in the estimated values of the exponent obtained from Equation (4.3). Remarkably, the range of exponent values from 4.52 to 4.76 reflects the range found generally in previous experimental works; for example, see the results of Chong et al. (1979). Rumpf and Gupte (1971) experimentally investigated the sensitivity of Dp to E in fixed beds, over a range of Rem values of from 1 to 100. They constructed eight fixed beds, comprising polystyrene beads having diameters of 0.67 ± 0.13 mm, giving a range of E values of from 0.37 to 0.64. Paraffin and air were used as experimental fluids. Rumpf and Gupte found that their exponent values were equal to 5.5 ± 0.5 and independent of Rem. For consistency with the Rumpf and Gupte experiments, the exponent values estimated from Equations (4.1) and (4.2) have been determined for the same E range. All the R Squared values from the regression analyses were 0.9965, indicating that the simulated power equations are practically indistinguishable from the original E functions defined in Equations (4.1) to (4.3). Table 6 shows the results of the comparison between the theoretical (GCM) values and experimental exponent values. The agreement is good in all four cases, giving credibility to the validity of the composite variables in Table 4. For comparison, exponent values were similarly estimated from the Blake-Kozeny equation (Equation (1.4)) and the Blake inertial equation (Equation (3.1)). The comparison in Table 7 shows that Equations (4.1) and (1.4) yield the almost identical exponent value of 5.00, implying that the GCM shares the predictive success of the Blake-Kozeny equation for viscous flow. However, there is a significant difference in exponent values between Equation (4.2) and Equation (3.1) for inertial flow, which suggests that historical fixedbed models based on Equation (3.1) do not accurately predict the sensitivity of pressure loss to E in the turbulent regime.

where variables, apart from Dp and E, are constant. Similarly, Equations (4.3) and (4.4) are transformed to:

5. Construction of GCM

Ub is proportional to Ey :

This chapter constructs the GCM and estimates the empirical constants for the fixed and fluidised components. The wallcorrection factors applied in the GCM are described first.

(4.6)

In total, there are four values of the y exponent, covering two flow regimes and two bed types. Equation (4.4) is a power curve, which clearly has a value for the y exponent of 19/8. The other three exponents are estimated using regression analysis to fit power curves to data points generated from Equations (4.1) to (4.3). The

5.1. Wall-effect factor Values of Rd in the experimental data used herein to derive the

Table 6 Excellent agreement for exponent values between GCM and historical experiments. Bed

Flow

GCM

Historical experiments

Reference

Fixeda

Viscous Inertial Viscous Inertial

5.00 5.75 4.63c 2.375 (exactly)

5.5 ± 0.5b

Rumpf and Gupte (1971)

4.65 2.39

Lewis et al. (1949) Richardson and Zaki (1954)

Fluidised Notes: a E range is 0.37e0.64. b Rem range is 1e100. c E range is 0.55e0.90.

G. Hoyland / Water Research 111 (2017) 163e176

171

Table 8 Vessel wall affects fixed beds in at least three different ways. Physical effect on bed

Effect on pressure loss

Implications for modelling

A B C

Increase Reduce Uncertain

Decrease value of Dx or F using wall-effect factor as denominator Use overall value of E in model Make T function of Rd

Increased surface area Increased E value near wall and overall Change in packing pattern near wall

coefficients in the fixed-bed component GCM are in the range of from 0.002 to 0.158; the higher values in this range have the potential to affect pressure loss significantly. Based on theoretical considerations, vessel walls containing fixed beds can affect pressure loss in at least three ways, as indicated in Table 8. Effect C in Table 8 is probably secondary and is not considered further herein. With regard to effect B, the GCM, like most historical models, defines E as the overall value of the bed. Zou and Yu (1995) have empirically related the overall E of a fixed bed of spheres to E of the central bulk of spheres and to Rd. With regard to effect A, Mehta and Hawley (1969) made a notable development. They modified the definition of Dx by adding Carman's (1937) wall effect factor, u, as a denominator. They then inserted the modified Dx into the Ergun model, in which Dx appears twice, once in qd and again in Red, giving the model listed in Table 1. Many other workers have followed the same approach to derive similar models. Also, the models listed in Table 1 indicate other approaches to wall effect. As indicated in Chapter 4, the sensitivity of pressure loss to E for turbulent flow differs between the GCM and other models. Thus, wall effects developed for other models will generally affect pressure loss differently if applied directly to the GCM. The GCM uses u as the wall-effect factor but the factor appears only once, as a denominator in F. This approach, which is considered theoretically sound, is tested for the viscous and turbulent regimes in Section 5.3.1. The definition of u for the GCM is an adaption of Carman's definition given in Equation (1.6), modified for near-spherical particles. The factor, ua, for a circular fixed bed is defined by:

ua ¼ Sb =ðSm ð1eEÞÞ ¼ 1 þ 4=ðDv Sm ð1eEÞÞ

(5.1)

where Sb is the total specific surface area of bed including circular wall, and Dv is diameter of vessel. Eliminating Sm from Equation (5.1) using Equation (3.13), gives the following expression for ua:

ua ¼ 1 þ 2 Rd j=ð3ð1eEÞ Þ

(5.2)

The factor, ua, is considered applicable to fixed beds when Rd
0.2 and over the full range of hydrodynamic conditions. However, ua is inappropriate for fluidised beds because the value of ua may become unreasonably large at E values greater than, say, 0.80. For fluidised beds, Equation (5.2) is subjectively modified to:

ub ¼ 1 þ 8 j Rd E=3:

(5.3)

Equations (5.2) and (5.3) yield the same value for ua and ub when E ¼ 0.5, giving a smooth transition in the factor at the transition between fixed and fluidised modes. As indicated by the results of the regression analysis in Section 5.3, the two factors significantly improve the fit of the GCM to experimental data. The two factors have different functions. Factor, ua, increases pressure loss across fixed beds in proportion to its value, whereas factor, ub, reduces value of Ub for any particular value of E. 5.2. Construction of GCM The core hydrodynamic equation of the GCM is Equation (2.9),

which is now reproduced:



Fm ¼ ðKl ReÞm þ Kt Re2

m

(2.9)

The dimensionless groups, F and Re are defined by substituting the composite variables from Table 4 into Equations (2.6) and (2.7) and inserting the appropriate wall correction factor into definition of F. The dimensionless groups for fixed beds are as follows:

F ¼ Dh j Ga E31=12

.  Lb ua ð1eEÞ5=3

(5.4)

and

.  Re ¼ Dm Ub r m E13=12 ð1eEÞ1=3

(5.5)

where Ga is Galilei number, defined by:

. Ga ¼ D3m r2 g m2 :

(5.6)

The definition of F, specifically for fluidised beds, is obtained by substituting Equation (1.11) into Equation (5.4) and changing the wall-effect factor, giving:

F ¼ j Ar E31=12

.

ub ð1eEÞ2=3



(5.7)

where Ar is Archimedes number, defined by:

. Ar ¼ D3m ðrm erÞr g m2 :

(5.8)

The definition of Reynolds number, Equation (5.5), is the same for both fixed and fluidised beds. Equation (1.11) transforms to:

Dh=Lb ¼ ð1eEÞDe:

(5.9)

where

De ¼ ðrm erÞ=r ¼ Ar=Ga

(5.10)

A final equation, necessary for determining depth of fluidised beds is as follows:

Lb =L0 ¼ ð1eE0 Þ=ð1eEÞ:

(5.11)

The GCM comprises Equations 2.9 and 5.2e5.11. Table 9 lists the dimensionless terms used by the model, separated into variables

Table 9 Model outline, showing variables and constants. Bed

Empirical constants

Independent variables

Dependent variables Main

Supplementary

Fixed and Fluidised beds Additional for fluidised beds

Kl, Kt m

Ga, E0, j, Rd

Dh/Lb, Re

Rem, Res

e

De

E, Lb/L0

e

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G. Hoyland / Water Research 111 (2017) 163e176

Table 10 Wide range of validated experimental data used for evaluation of GCM coefficients. Source

Wilhem and Kwark (1948) Lewis et al. (1949) Loeffler (1953) Kang (2010) Schriever (1930) Coulson (1949)

Nr of runs

15 12 3 4 9 4 5

Bed type

Fixed Fluid. Fluid. Fluid. Fixed Fixed Fixed

Granules

Fluid

Sand, beads, shot, & rock Glass beads Glass spheres Glass spheres Round sand Ball bearings

and constants. The three empirical constants have different values for fixed and fluidised beds. In contrast to most historical models for fluidised beds, the GCM does not use Uf (or Ref) as an independent variable.

Air/water Water Water Glycol/water Air/water Oil Oil

The three constants, Kl, Kt and m, for each bed type were estimated from experimental data obtained from the six previous major works listed in Table 10. Additional experimental details are available in the references to the previous works. Run numbers used in this work are the original run numbers used by the respective workers. A preliminary exercise, which included regression analyses, was performed on the experimental data obtained from the sources to identify rogue or inappropriate data. This exercise uncovered the following:  Kang Run T22 was performed at Rd value of 0.274. It was found that the data have a significant deviation from the GCM, indicating that the wall correction in the GCM is not valid at this Rd value. This run was not used for estimating constants.  Wilhem and Kwark Run 55 is a run performed in fixed and fluidised modes with water. Data from this run deviated grossly from the GCM for both operational modes. Similarly, Runs 37, 41 and 51, which are fixed mode runs performed with air, had inexplicable high deviations. These four runs were also not used for estimating constants.  Fluidised runs with air, performed by Lewis and co-workers and Wilhem and Kwark, were excluded at the outset to avoid potential inaccuracies caused by slugging. Thus, the applicability of the fluidised-bed component of the GCM to air fluidisation is not tested in this work. Thus, five runs were excluded from the regression analysis. However, the 52 runs (including the Coulson and Schriever runs) listed in Table 10 contributed to the analysis. Values of Ga and Ar (or De) were calculated for this work from the parameter values given in the specified references. Values of j for the sand and crushed rock granules were assumed equal to 0.90 and 0.65 respectively, based on values published by Wadell (1935) for typical granule shapes and types. All other granules were allocated a j value of 1.0. The experimental results published by Schriever (1930) and Coulson (1949) were used exclusively to derive the Kl value for viscous flow in fixed beds. Carman (1937) also used these data to determine his Kl value of 5, given in Equation (1.5). Coulson and Shriever did not present their raw data fully, but the data reported contains sufficient detail to enable reliable calculation of wallcorrected Kl values. Rd values in the experimental data ranged from 0.002 to 0.158; the highest value is in the Kang (2010) data.

Re range

Min.

Max. 103

Min.

Max.

5.72 314 49.4 1.67 10257 4.39 N/A

1379 1870 1.09 2290 107528 0.0404 N/A

1.61

6845

0.208 0.014 533 N/A N/A

27.7 4398 43653 e e

5.3.1. Fixed beds It follows from Equations (2.9), (5.4) and (5.5) that, for viscous flow:

Kl ¼ Dp j D2m E11=3 5.3. Estimation of constants

Ga or Ar range

.  Lb m ua ð1eEÞ4=3

(5.12)

Fig. 2 shows values of Kl, obtained by calculation from the Coulson and Schriever data, plotted against Rd. The regression line, based on Coulson's data, has a gradient of practically zero, indicating the validity of the wall-effect factor, ua, when applied to viscous flow. The average values of Kl from the Coulson data and the Shriever data were both equal to 69.4. Thus, this particular value serves as the Kl value for fixed beds. The equivalent Kl values obtained from the Coulson and Schriever data for the Blake-Kozeny equation are 4.98 and 5.07 respectively, supporting the approximate value of 5 determined by Carman (1937). Fig. 3 shows all qualified values of F plotted against Re. The Coulson and Shriever data are represented in the figure by a single straight line, because individual points for this figure could not be determined from the data reported. Non-linear regression was used to fit Equation (2.9) to the experimental data, giving the following best-fit equation:



F8=9 ¼ ð69:4 ReÞ8=9 þ 0:23 Re2

8=9

(5.13)

In this case, Equation (2.9) has two degrees of freedom, namely Kt and m, since Kl was predetermined from the Coulson and Shriever data, as described. Table 11 compares regression characteristics of two scenarios labelled A and B performed to determine the significance of the

Fig. 2. Coulson (1949) and Schriever (1930) viscous flow data are independent of Rd after wall effect applied.

G. Hoyland / Water Research 111 (2017) 163e176

Fig. 3. GCM gives good fit to fixed-bed experimental data from four sources.

wall-effect factor on the fit of the experimental data to the GCM. In Scenario B, the wall-effect factor was made equal to 1. The bottom row of the table indicates the 95-percentile, multiplicative, confidence-interval, CI95, for each scenario. The regression results indicate that inclusion of the wall-effect factor significantly reduced Cl95 from 1.32 to 1.22. Most of the experimental data in this particular analysis relate to the turbulent regime. 5.3.2. Fluidised bed Experimental data relating to fluidised runs were confined to an E range of from 0.50 to 0.90. Larger E values exceed the valid range of the GCM, and smaller values are potentially in the transition zone between fixed and fluidised beds, as indicated by Richardson and Zaki (1954). Fig. 4 shows all relevant values of F plotted against Re. The best fit of Equation (2.9) to the experimental data gives:



F2=3 ¼ ð43 ReÞ2=3 þ 0:41 Re2

2=3

(5.14)

In this case, Equation 2.10 had three degrees of freedom, since Kl was determined simultaneously with Kt and m. As for fixed beds, Table 12 lists regression results for two scenarios labelled A and B, performed to determine the significance of the wall-effect factor, ub. CI95 has a significantly lower value of 1.26, when ub is included. In both the fixed bed and fluidised regression analyses, residual deviations were roughly evenly distributed throughout the experimental data. 5.4. Comparisons with previous models This section compares GCM predictions against those of previous models, for both fixed and fluidised beds. Table 11 GCM gives best fit to data for fixed beds when ua is included. Item

Regression characteristics

Scenario Wall factor, ua, included Number of points R Squared Multiplicative CI95 for F

A Yes 173 0.9994 1.22

B No 173 0.9989 1.32

173

Fig. 4. GCM gives good fit to fluidised-bed experimental data from three sources.

Table 12 GCM gives best fit to data for fluidised beds when ub is included. Item

Regression characteristics

Scenario Wall factor included Number of points R Squared Multiplicative CI95 for F

A Yes 189 0.9992 1.26

B No 189 0.9984 1.39

5.4.1. Fixed bed For fixed beds, predictions of the relationship between Dh/Lb and Rem are compared. The GCM is compared against three fullrange models previously described in Section 1.2.1; these are the models of Montillet et al. (2007), Erdim et al. (2015) and Ergun (1952). Values of Dh/Lb predicted using the three selected models are expressed as deviations (%) from corresponding GCM predictions. A negative deviation indicates a larger Dh/Lb value predicted by the GCM. Five scenarios, comprising E values of 0.37, 0.4 and 0.43 and Rd values of 0.05 and 0.15, are investigated. Values of Ga and j are fixed at 104 and 1 respectively. Fig. 5A shows line plots of the predicted pressure loss deviations against Rem, for the Erdim and Ergun models. Predictions cover the Rem ranges given in Table 1 for each model. Agreement between the GCM and the Erdim model is reasonable; for an E value of 0.38 and Rd values of 0.03 and 0.10, the deviation varies from 20% to þ4% over more than four orders of magnitude variation in Rem. The change in the size of the deviation with Rd value is, of course, entirely due to the GCM. Agreement with the Ergun model is poor when Rem > 500, as also discovered by Hicks (1970). Fig. 5Bshows similar line plots for the Montillet and Foscolo models. The Montillet line plots are less curved than the Erdim plots but wider apart, owing to the model's inverse wall correction. The Foscolo plots are similar to the Ergun plot given in Fig. 5B because the Foscolo model defaults to the Ergun model when E ¼ 0.4. However, it is notable that the Foscolo plots are very similar for E values of 0.38 and 0.42, which indicates that the sensitivities of pressure loss to E in the Foscolo and GCM are similar. 5.4.2. Fluidised beds For fluidised beds, the comparison between models is made on the basis of fluidisation curves, relating Rem to E. The GCM is compared against the Foscolo et al. (1983) model, the Loeffler and Ruth (1959) model for viscous regime and the classic Richardson

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G. Hoyland / Water Research 111 (2017) 163e176

Fig. 5. A: Comparison between CGM and Erdim et al. (2015) and Ergun (1952) models for different fluid fractions and wall corrections. The Erdim and Ergun models do not contain wall corrections. Agreement between GCM and Erdim model is reasonable, over specified values of Rd. B: Comparison between GCM and the Montillet et al. (2007) model and the Foscolo et al. (1983) fixed-beds models for fluid fractions and wall corrections. Sensitivities of pressure loss to fluid fraction are very similar in Foscolo model and GCM.

and Zaki (1954) model, as re-developed by Khan and Richardson (1989), which is called herein the RKZ model. Fig. 6A compares predicted fluidisation curves obtained from the GCM and RKZ models, for Ar values of 10, 103 and 105 and for Rd values of 0 and 0.2. Agreement between the models is reasonable, but the wall correction is clearly much stronger in the RKZ model, particularly at lower Rem values. The GCM expansion curves for different Rd values and any particular Ar value are effectively parallel. Also, the GCM expansion curves are closer together than the RKZ curves. These two characteristics of the GCM curves agree with the Felice and Kehlenbeck (2000) experimental findings from settling experiments. The original Richardson and Zaki (1954) model has a much poorer fit to the GCM, owing to exaggerated wall effect. Fig. 6Bcompares predictions made using the GCM and the Foscolo and Loeffler models. Agreement between the GCM and the Foscolo model is reasonable for Rem values greater than about 10 and in the viscous region but poor in the transition region. The Foscolo model, which was derived using the Richardson and Zaki

Fig. 6. A: Reasonable agreement between GCM and Khan and Richardson (1989) fluidisation model. Khan model has larger wall correction. B: Agreement of GCM with Foscolo et al. (1983) fluidisation model is reasonable when 10 < Rem < 1. Agreement with Loeffler and Ruth (1959) fluidisation model for viscous flow is good.

(1954) model as a template, gives similar predictions to this model (and the RKZ model) when Rd ¼ 0. By comparison, the GCM has excellent agreement with the Loeffler model for the viscous regime.

6. Comparison between individual runs This section compares experimental data from runs performed by Wilhem and Kwark (1948) against GCM predictions. The runs are special because they comprise extensive data for both fixed and fluidised bed operation in the same run. The comparison investigates the goodness of fit of the GCM to raw experimental data within the transition zone from fixed to fluidised operation. As indicated in Table 13, five runs have been selected from the Wilhem and Kwark data set to cover wide ranges of experimental conditions and values of independent variables. These five runs are a subset of the 52 runs used for the regression analysis. Fig. 7A and B compare predicted and experimental plots of Dh/Lb and E against Rem. Fig. 7A shows the data for Runs 3, 10 and 16, and

G. Hoyland / Water Research 111 (2017) 163e176 Table 13 Wide ranges of values for independent variables in simulation of Wilhem and Kwark (1948) data. Run

Granules

j

u

Ga

Ar

E0

3 10 14 16 58

Sea sand Glass beads Lead shot Crushed rock Porous beads

0.90 1.00 1.00 0.65 1.00

1.007 1.075 1.018 1.015 1.032

1682 1379408 20621 27669 843488

2765 1870080 202365 45413 511333

0.408 0.385 0.375 0.447 0.368

Fig. 7B for Runs 14 and 58. GCM predictions are presented as lines. The following observations are made from the comparisons:  Predicted fixed bed curves, relating Dh/Lb to Rem, are generally good fits to the experimental data.  Predicted fluidisation curves, relating E to Rem, give excellent fits to the experimental data for E values up to 0.90.  The agreement between predicted and experimental fluidisation curves is also generally good at E values down to the value, E0, of the fixed beds. Run 3 is an exception, showing a discernible transition zone, when E0 < E < 0.5.  Generally, the intercepts of the GCM and experimental fixed and fluidised pressure gradient curves occur approximately at E0,

175

indicating beds are fluidised at this point. Richardson and Zaki (1954) explain that between E values of E0 and approximately 0.5, the granules teeter, held in position by the expanding contiguous structure of the bed.

7. Conclusions A General Combined Model applicable to granular fixed beds and fluidised beds has been derived from fundamental hydrodynamic principles controlling pressure loss and expansion. The model gives an excellent fit to experimental data for both bed types over a wide range of granule properties and hydrodynamic conditions from the viscous to inertial regimes. For fixed beds, air as well as water was used as the operating fluid. With regard to fixed beds, the GCM has excellent agreement with the Blake-Kozeny equation, which is applicable to the viscous regime. However, The GCM has a significantly higher sensitivity of pressure loss to E in the turbulent regime compared with historical models generally. The fluidised component of the GCM is validated for E values up to 0.90. The GCM deviates significantly from the classical Richardson fluidisation models especially when the granule to vessel diameter ratio is comparatively high. The GCM's theoretical basis and wide applicability is offered as a means of rationalising historical development and focusing future development in this field. Acknowledgement This article was prepared with the encouragement of Bluewater Bio Ltd, and Richard Haddon, the Chairman, has given his permission for publication of the article in Water Research. Also, I show much appreciation to Prof Gerald Noone, Newcastle University, for his help in editing the article. References

Fig. 7. A and B: Good fits of GCM, fixed and fluidised predictions to experimental data for individual runs taken from Wilhem and Kwark (1948) data. A shows Runs 3, 10 and 16 and B shows Runs 14 and 58.

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