Reconsideration of the hydrodynamic behavior of

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beds has not been extensively investigated. Some authors argue ... with experimental data. .... However, some authors report that non-equilibrium effects already begin at .... the ratio of Eqs. (19a) and (19b) for round particles (ϕ N 0.8). ..... Technology in the Chemical Industry, Wiley-VCH Verlag GmbH, Weinheim, 2015. 176.
Powder Technology 287 (2016) 169–176

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Reconsideration of the hydrodynamic behavior of fluidized beds operated under reduced pressure Sayali Zarekar a,⁎, Andreas Bück a, Michael Jacob b, Evangelos Tsotsas a a b

Thermal Process Engineering, Otto von Guericke University Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany Glatt Ingenieurtechnik GmbH, Nordstrasse 12, 99427 Weimar, Germany

a r t i c l e

i n f o

Article history: Received 15 March 2015 Received in revised form 28 July 2015 Accepted 22 September 2015 Available online 25 September 2015 Keywords: Fluidized bed Gas–solid flow Vacuum Slip flow Minimum fluidization velocity

a b s t r a c t Fluidized beds operated at sub-atmospheric pressure can be employed for granulation and drying of thermosensitive materials in the food and pharmaceutical industries. However, the hydrodynamics of vacuum fluidized beds has not been extensively investigated. Some authors argue that at low pressures, the slip flow of gas is the major factor influencing the hydrodynamic behavior. The influence of change in gas properties on the hydrodynamics due to reduction in pressure has not been clearly distinguished. In this contribution, the individual effects of gas properties and slip flow on the hydrodynamic behavior, particularly on the minimum fluidization velocity, of vacuum fluidized beds are quantified. This has been achieved by expanding the classical minimum fluidization velocity correlation, valid under atmospheric pressure, to include the slip flow term. The results obtained describe a critical Knudsen number which indicates when the slip term begins to significantly influence the flow behavior. The derived correlation is compared with correlations reported in literature as well as validated with experimental data. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Fluidized bed granulation and drying processes are increasingly used for the production and treatment of granular solid materials as they offer high heat and mass transfer rates. In the production of bioactive materials in the pharmaceutical or food industry, the problem of deactivation is often encountered. While the deactivation can be lowered by reducing the drying temperature, this can also lead to a low process throughput due to the reduced drying potential of the gas medium. Other low temperature methods such as freeze-drying can be used, but they are costly and time consuming for bulk production as compared to high temperature drying processes. An alternate approach is to operate the fluidized bed at moderate vacuum conditions. Thus the gas will continue to be used as a heat carrier and a considerable reduction in the product temperature and equipment cost can be achieved. Vacuum fluid-bed dryers show a reduction of the vaporization temperature and shortening of drying time [1]. There are relatively few studies of fluidized beds operated under reduced pressure. Wraith and Harris [2] considered the application of vacuum fluidized beds in metallurgy for metal extraction. They observed large velocity gradients which resulted in an intermediate fluidization regime characterized by a fluidization front for deep beds. Tatemoto et al. [3] studied the drying characteristics of heat-sensitive material immersed in a fluidized bed of inert particles under reduced pressure. They ⁎ Corresponding author. E-mail address: [email protected] (S. Zarekar).

http://dx.doi.org/10.1016/j.powtec.2015.09.027 0032-5910/© 2015 Elsevier B.V. All rights reserved.

observed that a high drying rate can be achieved even at a relatively low mass velocity of the drying gas, when the chamber pressure is low. The hydrodynamic characteristics of fluidized beds operated at elevated temperatures and pressures, such as minimum fluidization velocity, bed voidage and elutriation, have been rigorously analyzed [4]. In fluidized beds operated under elevated pressures (101–1600 kPa), the minimum fluidization velocity depends on particle size. It is unaffected by pressure for fine Geldart A powders and decreases with increase in pressure for coarse materials. The dependence of minimum fluidization velocity on pressure is due to the effect of pressure on the physical properties of gas, namely, the density and viscosity [5]. In comparison, the hydrodynamics of fluidized beds operated under low pressure has not been extensively studied. Some authors report that the minimum fluidization velocity derived for atmospheric pressure conditions cannot be used to predict the velocity at reduced pressure. The proposed argument is that at low pressure, the hydrodynamic flow regime is no longer laminar but is better described by assuming slip flow [6,7]. They derived correlations for predicting the minimum fluidization velocity which takes into account the slip flow regime. The correlation from Llop et al. [6] has been developed for high as well as sub-atmospheric pressures. Kozanoglu et al. [8,9] have also proposed new equations by modifying the constants used in the Llop correlation. These correlations account for the slip flow via the dimensionless Knudsen number (Kn), which is the ratio of the mean free path of the gas molecules to the characteristic length. However, there are three key aspects in this argument that are not discussed by any of the authors: (1) the proposed correlation is not compared for the limiting

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case of Kn → 0 to existing correlations derived for atmospheric conditions; (2) the range of validity of the slip flow regime is not analyzed and (3) as in the case of fluidized beds operated under elevated pressure, the influence of gas properties on minimum fluidization has not been quantified for low pressure conditions. This analysis is essential because if, indeed, the slip term dictates the flow behavior at sub-atmospheric pressures, then not only the minimum fluidization velocity correlation will have to be modified, but other hydrodynamic parameters such as elutriation velocity, bed expansion, and bubble formation will also have to be remodeled. The noncontinuum effects due to slip flow on the drag force experienced by the fluidized particles will also have to be accounted for. In the case of particles suspended in air, the drag force is calculated using Stokes' law, which is derived from equations of continuum mechanics. But as the particle diameter approaches the same order of magnitude as the mean free path of air molecules, non-continuum effects are accounted for in the Stokes' law by introducing a slip correction factor. The slip correction factor is generally applied for particle diameters less than 10 μm as in the case of pollutant particles suspended in air [10]. Along with the hydrodynamics, the slip flow will also affect the heat and mass transfer characteristics. Hence the Nusselt and Sherwood number correlations would also have to be modified. In the field of micro-reactors, where the characteristic dimension of the channel is of the order 50–200 μm, the slip flow regime and rarefaction effects are taken into account. The heat and mass transfer characteristics are different from macro-reactors as rarefaction effects also cause jumps (discontinuities) in temperature and concentration at the walls of the reactors [11]. In the case of fluidized beds operated under reduced pressure, Kozanoglu et al. [12] have developed a correlation for Sherwood number in terms of Reynolds and Knudsen number. But here as well the range of applicability of the slip effect has not been discussed in terms of the particle diameter and operating pressures. The following correlation proposed by Richardson for the minimum fluidization Reynolds number, Remf, is widely used: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   Rem f dp ; ρp ; P ¼ −25:7 þ 25:72 þ 0:0365Ar:

ð1Þ

This correlation was based on data obtained only at atmospheric pressure conditions [13] and expresses the minimum fluidization Reynolds number as a function of the Archimedes number. The Archimedes number is a function of the particle diameter (dp) and density (ρp), gas density (ρp) and kinematic viscosity (vg). Of these, gas density and kinematic viscosity vary with pressure. As the pressure is reduced, the change of gas properties is reflected in Remf. However, for a fixed particle diameter the operating pressure (P) at which the change in gas properties can no longer accurately predict the Remf is the critical pressure below which the non-continuum effects also become significant. This critical point analysis can be done in terms of the dimensionless Knudsen number, which distinguishes between these two effects. The critical point analysis is essential because only after the identification of this critical value it is possible to conclude whether slip flow is the main phenomenon responsible for the change in Remf at a particular particle diameter and operating pressure. In layering granulation or drying processes, where the typical particle size is above 200 μm, it is essential to evaluate if slip flow affects the motion of particles in this size range. In this study, a new correlation is developed for the minimum fluidization Reynolds number applicable at atmospheric as well as sub-atmospheric operating pressures in the fluidized bed. The correlation is compared to the classical Richardson equation (derived for atmospheric conditions) as well as to correlations reported in literature which account for the reduced pressure. The aim of this study is to quantify the individual effect of change in gas properties with pressure and that of slip flow on the minimum fluidization Reynolds number. A critical Knudsen number is identified which distinguishes between

these effects. The need to apply a slip flow correction in the minimum fluidization Reynolds number is reevaluated and the conditions of pressure and particle diameter at which slip flow becomes relevant are identified. In other words, the range of validity of the classical correlation for minimum fluidization Reynolds number is determined for operating pressure smaller than atmospheric pressure. This paper is organized as follows: The theory of rarefied gas flow in the fields of microfluidics and porous media is discussed in Section 2. A new correlation for the minimum fluidization Reynolds number which accounts for the slip flow is derived in Section 3. The critical Knudsen number which distinguishes between the influence of change in gas properties and slip flow is determined in Section 4. In the same section, the results are compared to correlations from literature as well as to experimental data. Pressure dependency is further elaborated in Sections 4 and 5, before coming to the conclusion. 2. Theory of rarefied flow The slip flow regime in hydrodynamics becomes relevant when the characteristic geometric dimension is comparable to the mean free path of the gas molecules or for gas flows at low pressures. In the field of microfluidics, gas flow through the devices is usually in the slip and transition flow regimes [14]. In micro- and nanodevices, the gas exhibits non-continuum dynamics and the traditional Euler or Navier–Stokes equations fail in predicting the flow [15]. In porous media such as tight sands, with micro- or nanoscale pores, the gaseous flow through them also falls into slip or transition regimes [16]. The range of hydrodynamic regimes for rarefied gas flow is expressed in terms of the dimensionless Knudsen number (Kn). This is the ratio of the mean free path of gas molecules to the characteristic dimension: Kn ¼

λ ; dp

ð2aÞ

where the modified mean free path λ can be obtained according to [17] from: λ¼2

2−γ γ

rffiffiffiffiffiffiffiffiffiffiffiffi kg 2πRT  ; M P 2cp;g −R=M

ð2bÞ

and the accommodation coefficient γ can be calculated by means of the correlation:  log

 1 ð1000K=T Þ þ 1 −1 ¼ 0:6− : γ C

ð2cÞ

In these equations, kg is the thermal conductivity of the gas, T is the absolute temperature and C depends on the molar mass of the gas. For air, C = 2.8. As there is a strong analogy between low-pressure flows and microflows, the following classification of the flow regimes as a function of Kn is generally used [18,14]: • Kn b 0.01: continuum flow regime. This is modeled using compressible Navier–Stokes equations and no-slip boundary conditions. However, some authors report that non-equilibrium effects already begin at lower Knudsen numbers (Kn N 10−3) [19]. • 0.01 b Kn b 0.1: slip flow regime. The Navier–Stokes equations are still applicable, but only with slip velocity and temperature jump boundary conditions. • 0.1 b Kn b 10: transition flow regime. Rarefaction effects dominate and continuum Navier–Stokes equations are invalid. • Kn N 10: molecular flow regime. Intermolecular collisions are negligible as compared to collisions between molecules and surface. Slip models (of first and second order) are used to improve the predictions of continuum methods for the slip and marginally transitional

S. Zarekar et al. / Powder Technology 287 (2016) 169–176

flow regimes [14]. For porous media, the Ergun equation [20], which only accounts for the frictional and inertial flow regimes, is generally used to obtain the pressure drop over the length (L) as: 2 1:75ρg ð1−ϵÞ 2 ΔP 150μ g ð1−ϵÞ ¼ us þ us : 2 3 L dp ϵ3 d ϵ

ð3Þ

p

Here dp is the equivalent sphere diameter of the packing and us is the superficial velocity. Tang et al. [16] modeled the permeability of porous media considering the compressibility and rarefaction effects in the micropores. They introduced a first order slip term in the Ergun equation for moderate Knudsen number flows. On similar lines, this paper proposes a correlation for the minimum fluidization velocity by also introducing higher order slip terms in the Ergun equation, as is discussed in the following section. 3. Derivation of new correlation In this section, a new correlation is developed for the minimum fluidization Reynolds number which estimates the influence of slip flow on the hydrodynamic behavior in a fluidized bed. The packed bed or porous medium is approximated as a bundle of capillaries with uniform circular cross-section. For a rarefied gaseous flow in an ideal porous medium composed of a bundle of N straight and parallel circular tubes of equal diameter dt, the flow rate q as a function of the Knudsen number can be obtained from [16]:

In order to include the turbulent (inertial) flow regime, the term for pressure drop at high velocities ð 1:75ð1−ϵÞ ϵ3

ρg u2s ϕdp

Þ is added to Eq. (7) to ob-

tain the pressure drop valid across all three flow regimes (slip, laminar and turbulent flow): 72μ g 1:75ρg 1−ϵ 2 dP ð1−ϵÞ2 ¼ 2 2 us þ u : dx ϕ d ð1 þ 8C 1 Kn þ 16C 2 Kn2 Þ ϵ3 ϕdp ϵ3 s p

ð8Þ

For a packed bed of particles the onset of fluidization occurs when the drag force by the upward moving gas is equal to the weight of the particles. In other words, from the momentum balance:  i  h  ΔPA ¼ AΔx 1−ϵm f ρp −ρg g :

ð9aÞ

By rearrangement, the pressure drop for minimum fluidization conditions is obtained as:   dP  ¼ 1−ϵm f ρp −ρg g: dx

ð9bÞ

From Eqs. (8) and (9b) we obtain: 

1−ϵm f

  ρp −ρg g ¼



 dP Nπdt  ; 1 þ 8C 1 Kn þ 16C 2 Kn2 dx 128μ g

171



72μ g

1−ϵm f 2 2 2 ϵ3m f ϕ dp ð1 þ 8C 1 Kn þ 16C 2 Kn Þ 1:75ρg 1−ϵm f 2 þ us : ϕdp ϵ3m f

2 us ð10aÞ

4

q¼ 

ð4Þ

C1 and C2 are coefficients for the first and second-order slip models, respectively. Cercignani et al. [15] used the hard-sphere model as an approximation to the real gas behavior for isothermal flows. The values obtained for the first and second-order coefficients from their slip model were C1 = 1.11 and C2 = 0.31 respectively. Their model is in excellent agreement with numerical solutions of the linearized Boltzmann equation for hard spheres, up to Kn b 0.4 [21]. Using continuum methods to obtain the solution of flow problems is significantly more efficient as compared to molecular-based approaches, hence accurate first and second order slip coefficients are highly desirable. At Kn = 0, Eq. (4) gives the flow rate through a circular tube for the laminar or viscous flow regime. The flow rate can also be written as a function of mean velocity in the tube ut: 2

q¼ 

Nπdt ut : 4

ð5Þ

3

dp 2 g νg

Multiplying both sides of Eq. (10a) with ρ Ar ¼

  72 1−ϵm f ϕ2 ϵ3m f ð1 þ 8C 1 Kn þ 16C 2 Kn2 Þ

Rem f þ

Ar ¼

  3 gdp ρp −ρg ρg ν2g

Rem f ¼

;

ð10cÞ

us dp : νg

ð10dÞ

Eq. (10b) can be rewritten as:

ð6aÞ

K1 ¼

dt ¼

2 ϵ ϕdp ; 3 1−ϵ

ð6bÞ

and

ð7Þ

ð11aÞ

where

us ; ϵ

dP 72μ g ð1−ϵÞ2 us ¼ : dx ϕ2 d2 ϵ3 1 þ 8C 1 Kn þ 16C 2 Kn2 p

ð10bÞ

and the Reynolds number is defined as:

ut ¼

where ϕ is the sphericity of the particles. Substituting Eqs. (5), (6a) and (6b) in Eq. (4) gives the pressure drop across the porous medium for the laminar (viscous) and slip flow regimes:

1:75 2 Re ; ϕϵ3m f m f

where the Archimedes number is defined as:

K 1 Re2m f þ K 2 Rem f ¼ Ar; The superficial velocity (us) for the porous medium and the equivalent particle diameter (dp) can be obtained from:

and rearranging yields:

K2 ¼

1:75 ; ϕϵ3m f

ð11bÞ

  72 1−ϵm f ϕ2 ϵ3m f ð1 þ 8C 1 Kn þ 16C 2 Kn2 Þ

:

ð11cÞ

Solving the quadratic Eq. (11a) gives the general equation for Remf:

Rem f

K2 ¼− þ 2K 1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   K 2 2 Ar þ : K1 2K 1

ð12Þ

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S. Zarekar et al. / Powder Technology 287 (2016) 169–176

Substituting the expressions for K1 and K2 in Eq. (12) and for porosity of ϵmf = 0.4, the new correlation for Remf is obtained as: 12:3 ϕ 1 þ 8C ð Kn þ 16C 2 Kn2 Þ 1 ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 12:3 þ þ 0:0365ϕAr : ϕð1 þ 8C 1 Kn þ 16C 2 Kn2 Þ

Rem f ¼ −

ð13Þ

To compare the derived correlation with existing correlations, we first only look at the continuum flow regime. For continuum flow, it is Kn ≪ 1 and there is no effect of slip. Hence the term in the denominator of K 2 (Eq. (11c)) becomes approximately unity, and we get K 2 ¼

72ð1−ϵm f Þ ϕ2 ϵ3m f

. For continuum flow, the correlation pro-

posed by Richardson (Eq. (1)) is generally applied. The constants K2 proposed by Richardson are: K11 ¼ 0:0365 and 2K ¼ 25:7. For particle 1

4. Determination of critical Knudsen number and comparison with literature In this section, the critical Knudsen number is derived from Eq. (17) as well as from the correlation developed by Llop et al. [6]. The critical Knudsen number indicates when the slip flow term begins to significantly influence the minimum fluidization Reynolds number. It differentiates between the individual effects of gas properties and slip flow on the minimum fluidization Reynolds number. To obtain Kncr, the Reynolds number correlation which considers both the slip and continuum flow regimes is compared with the same correlation but with the Knudsen number set as Kn = 0 (only continuum flow regime). This comparison is expressed in the form of the ratio ζ as: ζ¼

Rem f ðAr; KnÞ : Rem f ðAr; Kn ¼ 0Þ

ð18Þ

sphericity of ϕ = 1 and bed porosity of ϵmf = 0.4, from Eqs. (11b) and (11c) we get

1 K1

¼ 0:0365 and

K2 2K 1

¼ 12:3. Hence the term K 2

needs to be modified such that the derived correlation gives the same result as the Richardson equation for continuum flow. The modified term is obtained as: K 02 ¼

  150 1−ϵm f ϕ2 ϵ3m f

:

ð14Þ

Including the Knudsen number term in K2′, and substituting K1 and K2′ in the general equation (Eq. (12)), the modified correlation valid for continuum and slip flow regimes is obtained as:

Rem f

  42:85 1−ϵm f ¼− ϕðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 8C 1 Kn þ 16C 2 Kn2 Þ s ffi    2 42:85 1−ϵm f 3 þ 0:571ϕϵm f Ar: þ ϕð1 þ 8C 1 Kn þ 16C 2 Kn2 Þ

ð15Þ

Rem f ¼ −

ð16Þ

As the minimum fluidization porosity varies for different particle systems, if information on ϵmf and ϕ of the particles is available, the general form of the correlation (Eq. (15)) should be used. The values for coefficients of the slip model are obtained from [15] as C1 = 1.1 and C2 = 0.31. The second-order slip coefficient is usually employed for flow regimes beyond slip flow (Kn N 0.1), which is well into the transition regime [19,14]. For the current analysis, we only consider Knudsen numbers in the slip flow regime and hence the second-order term can be neglected. Neglecting C2 in Eq. (16), the final correlation is obtained as:

Rem f

25:7 þ ¼− ϕð1 þ 8:8KnÞ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 25:7 þ 0:0365ϕAr : ϕð1 þ 8:8KnÞ

Rem f ;Llop ðAr; KnÞ ¼ −

29:4

1þ 32:4Kn ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s  2 29:4 þ þ 0:0357Ar; 1 þ 32:4Kn

Rem f ;Llop ðAr; Kn ¼ 0Þ ¼ −29:4 þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 29:42 þ 0:0357Ar :

ð19aÞ

ð19bÞ

For sharp particles (0.5 b ϕ ≤ 0.8):

A random packing of spheres usually has a porosity of 0.38–0.42. If we approximate the minimum fluidization porosity to be the porosity of a packed bed of spheres, i.e. ϵmf = 0.4, the modified equation can be written as: 25:7 ϕ 1 þ 8C ð Kn þ 16C 2 Kn2 Þ 1 ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 25:7 þ þ 0:0365ϕAr : ϕð1 þ 8C 1 Kn þ 16C 2 Kn2 Þ

For a particular particle diameter and density, ζ is obtained as a ratio of Eqs. (17) and (1). Whereas for the Llop correlation [6], ζ is obtained as the ratio of Eqs. (19a) and (19b) for round particles (ϕ N 0.8). For sharp particles (0.5b ϕ ≤ 0.8), the ratio of Eqs. (20a) and (20b) is considered. For round particles (ϕ N 0.8):

ð17Þ

Eq. (17) will be used in the following sections for further analysis.

Rem f ;Llop ðAr; KnÞ ¼ −

38:6

1þ 20:3Kn ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s  2 38:6 þ þ 0:0571Ar; 1 þ 20:3Kn

Rem f ;Llop ðAr; Kn ¼ 0Þ ¼ −38:6 þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 38:62 þ 0:0571Ar :

ð20aÞ

ð20bÞ

The critical Knudsen number can be obtained from that value of ζ for which slip flow can be considered to significantly affect the minimum fluidization Reynolds number. Since deviations of 10% or 20% from reality are usually considered acceptable in engineering calculations for practical purposes, a value of ζ = 1.1 or ζ = 1.2 can be used to derive Kncr. Using, for example, ζ = 1.1 means that the effect of slip flow is regarded to be significant if its calculated influence on Remf is more than 10%. 4.1. Effect of particle size The critical Knudsen number is derived from Eq. (17) and from the Llop correlation for particles with different diameters, keeping the particle density and sphericity constant. In Fig. 1 ζ is plotted against Knudsen number for Eq. (17) and the Llop correlation. For ζ = 1.1 (i.e. at 10% influence of slip flow) the corresponding Knudsen number is obtained for both correlations. This is taken as the critical Knudsen number (Kncr) and is shown in Fig. 1 by the round marks for each of the correlations. For both the particle sizes, Llop correlation gives Kncr = 0.0034 whereas Eq. (17) gives Kncr = 0.0122. For a somewhat higher slip influence of 20% (i.e. ζ = 1.2), the critical Knudsen number is obtained from Fig. 1 as 0.023 and 0.0065 for Eq. (17) and Llop correlation, respectively. For both the correlations, Kncr is nearly independent of the particle size.

S. Zarekar et al. / Powder Technology 287 (2016) 169–176

173

Fig. 1. Critical Knudsen number derivation for different particle densities and constant particle diameter of 728 μm using Llop correlation [6] and Eq. (17). The critical point is taken at a difference of 10% between the Reynolds numbers with and without the contribution of slip.

Fig. 3. Critical Knudsen number derivation for different particle diameters, constant sphericity of 0.6 and constant particle density of 2500 kg·m−3 using Llop correlation [6] and Eq. (17). The critical point is taken at a difference of 10% between the Reynolds numbers with and without the contribution of slip.

4.2. Effect of particle density

to Eq. (17). Table 1 summarizes the results obtained from the analysis of Eq. (17) and Llop equations for ζ = 1.1 and 1.2. To determine the influence of gas properties on the critical Knudsen number, a similar analysis as presented above was performed using superheated steam as the gas in a dryer instead of air. The critical Knudsen number obtained with steam as gaseous medium was the same as that obtained with air for both, Eq. (17) and the Llop correlation. Thus, the critical Knudsen number is essentially unaffected by the gas properties.

To determine the dependency of the critical Knudsen number on the particle density, Kncr is derived from both the correlations for varying particle densities but keeping the particle size (dp = 728 μm) and sphericity (ϕ = 1) constant. In Fig. 2, the critical Knudsen number is taken as the Knudsen number at which ζ = 1.1 for the Llop correlation and Eq. (17). Kncr shown by the two round marks in Fig. 2 does not change for particles with different densities and is similar to that obtained for varying particle size for each of the correlations. Thus, the critical Knudsen number is essentially unaffected by the particle density. 4.3. Effect of particle sphericity The effect of particle sphericity on Kncr is investigated keeping the particle diameter and density constant. For sphericity of 0.6, Fig. 3 shows the values of Kncr derived from Eqs. (17) and (20a). While the value of Kncr derived from Eq. (17) is not significantly affected by the particle sphericity (Kncr = 0.0118 instead of 0.0122 for ϕ = 1), the Llop equation gives Kncr = 0.0052 as compared to a value of 0.0034 obtained for ϕ = 1. For the higher difference of 20%, Kncr is obtained as 0.023 and 0.010 for Eq. (17) and Llop equation, respectively. An effect of slip flow larger than 10% is seen for Kn N 0.012, irrespectively of the particle diameter, density and sphericity, according

4.4. Comparison with experimental data Experimental data for minimum fluidization velocity from Tatemoto et al. [22] is compared with Eq. (17) and the Llop correlation (Eq. (19a)) in Fig. 4. The experimental data was obtained from fluidization of glass beads (0.12 mm in diameter) using superheated steam as the gas in a dryer. The operating pressure of the drying chamber was varied between 100 and 1000 mbar, at a fixed temperature of 423 K. A good agreement of experimental data with Eq. (17) is obtained as compared with the Llop correlation which overestimates the minimum fluidization Reynolds number. For Kncr = 0.012, the critical pressure for the glass beads of 0.12 mm can be calculated as 247 mbar, at 423 K. Above 247 mbar, slip flow does not influence the hydrodynamics of the fluidized bed significantly. This is also seen from Fig. 4, where for Kn b 0.012, the value of ζ according to the experiments is smaller than 1.1. In other words, the Richardson equation (Eq. (1)) satisfactorily predicts the minimum fluidization velocity for Kn b 0.012. Eq. (15) is further verified with experimental data reported by Kusakabe et al. [7] for the minimum fluidization velocity at reduced operating pressures. The experimental data was obtained from fluidization of spherical glass beads of various particle diameters using air as the gas in the fluidized bed. The operating pressure was varied between 10 and 1000 mbar at a temperature of around 298 K. The minimum fluidization porosity for the glass beads was given to be 0.38. The experimental data showed that the minimum fluidization velocity increased as the pressure was reduced. The comparison of experimental data with Eq. (15) and Llop correlation is represented using a parity plot in Fig. 5. The experimental data is Table 1 Values of critical Knudsen number obtained from Eq. (17) and Llop correlations at 10 and 20% influence of slip flow on minimum fluidization velocity.

Fig. 2. Critical Knudsen number derivation for different particle densities and constant particle diameter of 728 μm using Llop correlation [6] and Eq. (17). The critical point is taken at a difference of 10% between the Reynolds numbers with and without the contribution of slip.

ζ Kncr (Eq. (17)) Kncr (Llop Eq. (19a), ϕ = 1) Kncr (Llop Eq. (20a), ϕ = 0.6)

1.1 0.012 0.0034 0.0052

1.2 0.023 0.0065 0.010

174

S. Zarekar et al. / Powder Technology 287 (2016) 169–176 Table 2 Minimum fluidization Reynolds number and velocity calculated using Eq. (17) for spherical particles of diameters 213 μm and 728 μm and density 2500 kg·m−3, fluidized with air at 293 K. Pressure [mbar]

1000 500 300 100 60 30 20

213 μm

728 μm

Remf [–]

umf [m·s

0.62 0.31 0.19 0.068 0.044 – –

0.044 0.044 0.045 0.048 0.052 – –

−1

]

Remf [–]

umf [m·s−1]

18.4 10.3 6.6 2.4 1.51 0.80 0.56

0.38 0.43 0.46 0.50 0.52 0.56 0.59

regime. Beyond this regime, the higher-order coefficients would have to be included in the derivation. Fig. 4. Comparison of experimental data for minimum fluidization Reynolds number from Tatemoto et al. [22] with Eq. (17) and correlation from Llop et al. [6]; (drying gas: steam, dp = 0.12 mm and ρp = 2500 kg·m−3). For the experimental data, ζ has been obtained by dividing measured Remf by Remf according to the Richardson equation.

plotted on the X axis and minimum fluidization velocity obtained from the correlations is plotted on the Y axis, for different particle diameters. As the minimum fluidization porosity for the particles is specified, the generalized correlation (Eq. (15)) is used for comparison instead of Eq. (17). The particle sphericity is taken as ϕ = 1. A good agreement of experimental data with Eq. (15) is obtained as seen in Fig. 5. The Llop correlation is in good agreement with experimental data at higher pressures (i.e. at smaller umf values for each of the particle sizes). At lower pressures, however, the Llop correlation overestimates the minimum fluidization velocity as seen by the points at higher values of umf for all the particles sizes. As the derived correlation (Eq. (15)) is a semi-empirical equation, the limitations of its application can be due some of the assumptions made during the derivation. The Richardson equation is selected as the reference equation to calculate the minimum fluidization Reynolds number for atmospheric pressure conditions. However, several authors have proposed values for K1 and K2 (in Eq. (12)) for calculating the minimum fluidization Reynolds number at normal pressure conditions [13]. Depending on the application, an alternate reference equation may be selected. In this study, the influence of slip flow above 10% (i.e. ζ = 1.1) is taken as the criteria for determining the critical Knudsen number. The correlation only takes into account the first order slip coefficient and hence is applicable only in the slip flow

5. Pressure dependency of minimum fluidization Reynolds number and velocity As the pressure in the fluidized bed is reduced, the operating gas becomes less dense as its density decreases. The kinematic viscosity of the gas increases. Both these changes in the gas properties lead to a decrease in Archimedes number (Eq. (10c)) as the pressure decreases. For very small Knudsen number (Kn ≪ 1), the minimum fluidization Reynolds number decreases with pressure only due to the decrease in Archimedes number, as can be seen from the Richardson equation (Eq. (1)). The minimum fluidization velocity, however, shows different pressure dependency for large and small particles. For small particle diameters the influence of pressure on the minimum fluidization velocity (umf) is negligible, whereas for large particles, umf is inversely proportional to the square root of pressure. A detailed explanation for this behavior is provided in the Appendix A. The minimum fluidization Reynolds number calculated from Eq. (17) depends on the Archimedes number as well as the Knudsen number. As seen from an example in Table 2, Remf decreases with pressure for both particle diameters. The decrease in Remf with pressure is due to the Archimedes number (via the gas density and viscosity) up to the critical Knudsen number, above which the slip flow also contributes to the change in the value of Remf. The critical pressures (Pcr) corresponding to Kncr = 0.012 for 213 μm and 728 μm particles fluidized in air, at 293 K, are obtained as 103.9 mbar and 30.4 mbar, respectively. The minimum fluidization velocity for 213 μm particles is nearly constant with pressure up to the critical pressure (103.9 mbar). This trend is similar to that obtained from the Richardson equation. Below Pcr, there is a small increase in umf. Above Pcr, for 728 μm particles, umf increases as the pressure is reduced, similar to the umf obtained from the Richardson equation. This trend is also observed for pressures below Pcr (30.4 mbar). Below Pcr, the increase in umf with reduction in pressure, for both large and small particles can be attributed to the slip flow of gas. 6. Discussion The operating pressures below which the slip flow significantly influences the hydrodynamics are listed for two particle diameters in Table 3 for air and steam used as the gas in a dryer. Smaller particles Table 3 Critical Knudsen numbers and pressures (ζ = 1.1) for spherical particles of different sizes. Particle size [μm]

Fig. 5. Comparison of experimental data for minimum fluidization velocity from Kusakabe et al. [7] with Eq. (15) and correlation from Llop et al. [6]; (gas: air, ϵmf = 0.38 and ρp = 2500 kg·m−3, T = 298 K).

213 728

Kncr [–]

Pcr (air at 293 K) [mbar]

Pcr (steam at 423 K) [mbar]

Eq. (17)

Llop Eq. (19a)

Eq. (17)

Llop Eq. (19a)

Eq. (17)

Llop Eq. (19a)

0.012 0.012

0.0034 0.0034

103.9 30.4

366.6 107.3

139.4 40.8

491.9 143.9

S. Zarekar et al. / Powder Technology 287 (2016) 169–176

Fig. 6. Relationship between umfP versus pressure for particles with diameters 213 and 728 μm. The shaded zone is the region in which the Richardson equation is sufficiently accurate for pressures less than 1 bar; (drying gas: air, ϕ = 1, ρp = 2500 kg·m−3, T = 293 K).

show a critical pressure higher than larger particles. Critical pressure of particles fluidized in steam is higher than that in air. In granulation processes, where particles are generally larger than 200 μm, the slip flow would be relevant only when the operating pressure is below 100 mbar (in air). For particle diameter dp, operating pressure P and a constant temperature T, the Knudsen number is inversely proportional to the operating pressure and particle size, Kn 

1 : Pdp

ð21Þ

For particle sizes such that Kn b 0.012, the slip flow term does not affect the minimum fluidization velocity by more than 10%. In Fig. 6, Eq. (17) and the Llop correlation (Eq. (19a)) are compared with the Richardson equation for two different particle sizes, using air as drying medium at 293 K. For operating pressure less than Pcr the Richardson equation underestimates significantly the minimum fluidization velocity due to the slip flow not being considered. The region of interest is highlighted in Fig. 6, where although the operating pressure is below 1 bar, it is larger than Pcr such that the Richardson equation is still valid for minimum fluidization velocity calculations. In this operating region, the advantage of reduced pressure on the granulation and drying processes can be availed without having to remodel the hydrodynamics and heat and mass transfer characteristics of the fluidized bed to include the slip flow. For Pcr = 1 bar, the particle diameter corresponding to Kncr = 0.012 can be calculated as 26 μm, in air at 293 K. At 1 bar, hydrodynamics of particles with diameter ≤ 26 μm will already be influenced by slip flow. For granulation processes, where particles are much larger, slip flow would not be relevant at these conditions. However, primary particles in agglomeration processes, which can be below 26 μm, would already experience slip at atmospheric conditions. Particles smaller than 26 μm could be processed at elevated pressure (above 1 bar) in order to avoid the effect of slip flow on the bed hydrodynamics. 7. Conclusion A new correlation (Eq. (17)) for predicting the minimum fluidization Reynolds number in fluidized beds operated under atmospheric as well as sub-atmospheric pressures has been developed. The correlation was derived based on the Ergun equation and includes the first order slip coefficient to account for the slip flow regime. A critical Knudsen number (Kncr) was determined to distinguish between the influence of gas properties and slip flow on the minimum fluidization velocity.

175

A value of 0.012 for the critical Knudsen number was obtained from Eq. (17). For Kn b Kncr, the influence of slip flow of gas on the fluidized bed hydrodynamics is negligible (≤10%). Kncr showed hardly any dependence on the particle size, density and sphericity as well as the gas properties. It was found to be independent of the Archimedes number. The value of Kncr = 0.012 obtained from the derived correlation matches well with the reported values [23] of Knudsen number below which the gas is in the continuum flow regime. The value of Kncr obtained from the Llop correlation is smaller than that obtained from Eq. (17) by almost one order of magnitude. Eq. (17) was also validated using experimental data and predicted the minimum fluidization velocity well, whereas the Llop correlation overestimated this velocity. Below Kncr, the classical Richardson equation can be applied to obtain the minimum fluidization velocity, which accounts for the effect of pressure due to the change in gas properties. For Kn N 0.012 the slip flow regime has been accounted for by including the first order slip flow term in the new correlation. The minimum fluidization Reynolds number was found to decrease as the pressure was reduced for both large and small particles. The minimum fluidization velocity, on the other hand, showed different pressure dependencies for different particle sizes. It was found to be independent of pressure for small particles (b200 μm) in the continuum flow regime. In the slip flow regime, it showed a small increase with reduction in pressure. The minimum fluidization velocity for large particles was found to increase as the pressure was reduced in the continuum as well as the slip flow regimes. For drying and granulation fluidized bed processes, the standard approach that accounts for the influence of pressure on gas properties (density and kinematic viscosity) is sufficient until a Knudsen number of 0.012. Above this value, slip flow would have to be accounted for not only in the minimum fluidization velocity but also in other hydrodynamic and heat and mass transfer characteristics of fluidized beds. Nomenclature A area, [m2] Ar Archimedes number, [–] cp specific heat capacity at constant pressure, [J·kg−1·K−1] d diameter, [m] g acceleration of gravity, [m·s−2] k thermal conductivity, [W·m−1·K−1] Kn Knudsen number, [–] L length, [m] M molar mass, [kg·mol−1] N number of capillary tubes, [–] P absolute pressure, [Pa] q volumetric flow rate, [m3·s−1] R gas constant, [J·mol−1·K−1] Re Reynolds number, [–] T temperature, [K] u velocity, [m·s−1] x axial coordinate, [m] 

Greek symbols γ ϵ ζ λ μ ν ρ ϕ

accommodation coefficient, [–] porosity, [–] ratio of Remf with and without slip flow, [–] modified mean free path, [m] dynamic viscosity, [Pa·s] kinematic viscosity, [m2·s−1] density, [kg·m−3] sphericity, [–]

Subscripts cr g

critical gas

176

S. Zarekar et al. / Powder Technology 287 (2016) 169–176

2. For large particle diameters (i.e. Ar → ∞), Eq. (22) can be written as:

minimum fluidization particle superficial tube

mf p s t

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi Rem f ¼ 25:72 0:000053Ar ¼ 0:1872 Ar :

ð29Þ

As the pressure is reduced:

Acknowledgments

pffiffiffi um f dp 1 P ⇒um f ≅ pffiffiffi : ¼ ⇒um f ≅ P νg P

The authors gratefully acknowledge funding of this work by the German Federal Ministry of Science and Education (BMBF) within the “Unternehmen Region” project WIGRATEC+ (O3WKCI5A, O3WKCI5B).

Rem f

Appendix A

For large particles, the minimum fluidization velocity is inversely proportional to the square root of pressure.

In the continuum region (Kn ≪ 1), the Richardson equation derived for atmospheric pressure conditions and used for obtaining the minimum fluidization Reynolds number can be written as: Rem f

hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i ¼ 25:72 1 þ 0:0000553Ar −1 :

ð22Þ

The density of gas (ρg), calculated using the ideal gas law, is directly proportional to pressure (P). The dynamic viscosity (μg) is not affected μ

by pressure. Hence, the kinematic viscosity (ν g ¼ ρg) is inversely proporg

tional to pressure. The pressure dependency of gas properties can be expressed as: ρg  P; νg 

ð23aÞ

1 1 ⇒ν g  : ρg P

ð23bÞ

The pressure dependency of Archimedes number can be expressed as:

Ar ¼

  3 gdp ρp −ρg ν 2g ρg

3

≅

gdp ρp ν2g ρg

;

ð24aÞ

with Ar 

P2 ⇒Ar ≅ P: P

ð24bÞ

Two cases are considered for large and small particle sizes. 1. For small particle diameters (i.e. Ar → 0), Eq. (22) can be written as: Rem f ¼ 25:72  0:0000553  Ar ¼ 0:000711  Ar;

ð25Þ

using the relation:  lim ½ 1 þ xÞ0:5 ¼ 1 þ 0:5x:

x→0

ð26Þ

For small particles, the minimum fluidization Reynolds number is directly proportional to pressure: Rem f ≅ P:

ð27Þ

As the pressure is reduced: Rem f ¼

um f dp P ⇒um f ≅ ⇒um f ≅ const: P νg

ð28Þ

For small particles, the minimum fluidization velocity (umf) is nearly independent of the pressure.

ð30Þ

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