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Interfacial stress in non-Newtonian flow through packed bed
Powder Technology 211 (2011) 127–134
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Powder Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p ow t e c
Interfacial stress in non-Newtonian flow through packed bed Suresh Kumar Patel, Subrata Kumar Majumder ⁎ Department of Chemical Engineering, Indian Institute of Technology Guwahati, Guwahati-781039, Assam, India
a r t i c l e
i n f o
Article history: Received 7 February 2011 Received in revised form 2 April 2011 Accepted 13 April 2011 Available online 20 April 2011 Keywords: Packed bed Interfacial area Non-Newtonian liquid Wettability Shear stress
a b s t r a c t This study investigates the pressure drop characteristics, shear stress in packed bed with shear thinning power law type non-Newtonian liquid. A mechanistic model has also been developed to analyze the pressure drop and interfacial stress in packed bed with non-Newtonian liquid by considering the loss of energy due to wettability. The Ergun's and Foscolo's equations were used for comparison with the experimental data. The Ergun equation was modified to account for the effect of flow behavior index of non-Newtonian fluid in the column. The intensity factor of shear stress and the friction factor were analyzed based on energy loss due to wettability effect of liquid on the solid surface. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Packed bed reactors are most commonly employed in the chemical process industries among the several possible types of multiphase catalytic reactors. Their popularity halts from their effectiveness in terms of performance as well as low capital and operating costs. The non-Newtonian fluid flow through particulate bed system is important in a variety of chemical and biochemical processes [1]. Various examples of applications of the particulate system have been described by many authors [1–4]. Studies on the flow of fluids through porous media were restricted mostly to Newtonian fluids. Recently, the flow of non-Newtonian fluids through packed beds and porous media has received considerable attention because of its importance in various industrial applications. Considerable research efforts have been expended in exploring and further understanding of the basic phenomena of momentum, heat and mass transfer processes with and without chemical reactions in particulate system.
1.1. Previous work Voluminous literature available on the flow of a variety of nonNewtonian materials through packed beds has been critically reviewed previously [5,6]. Wu and Pruess [6] described the non-Newtonian flow behavior in packed bed including beds of uniform size and of multi-size particles. Some other different studies related to Newtonian and non-
⁎ Corresponding author. Tel.: + 91 3612582265; fax: + 91 3612582291. E-mail addresses: [email protected], [email protected] (S.K. Majumder). 0032-5910/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2011.04.010
Newtonian flow behavior in packed bed has been thoroughly reported in literature [7]. From the studies it is concluded that each study has its some uniqueness in its morphology which is contributing in some measure to the complexity of the problem. Similarly, often inadequate rheological characterization also adds to the complexity of such systems. While it is usual for most non-Newtonian materials to exhibit shear thinning behavior, many other features including time-dependency, viscoelasticity, yield stress, etc., are also present but not often measured. Certainly, the major research effort has been directed at developing simple and reliable methods of predicting the frictional pressure loss for the flow of non-Newtonian fluids through packed beds. Kozicki et al. [8] generalized the average velocity–pressure gradient relationship for arbitrary time-independent non-Newtonian fluids in porous media, based on the Blake–Kozeny capillary model. Mishra et al. [9] described average shear stress–shear rate relationship to predict the flow behavior of power law as well as non-power law fluids. The wall factor is also another phenomenon to affect the flow behavior in porous media which is a function of both diameter ratio and particle Reynolds number [10]. For Reynolds number below 1.0, this dependency is nearly same for settling in Newtonian and non-Newtonian liquids. In the range of 1 b Re∞ b 200, wall effect can be estimated for the non-Newtonian case from the relation applicable to settling in Newtonian liquids. Zhu and Satish [11] studied the drag phenomena which decrease with a decrease in flow behavior index and with an increase in the characteristic time. They found that both the normal stress difference and the bed voidage have a great influence on the resistance of visco-elastic flow through a packed bed. Rao et al. [12] studied the pressure loss-throughput behavior for the flow of inelastic power law fluids through randomly packed spherical particles and over wide ranges of operating and physical conditions. Sabiri et al. [13] investigation covers a large range of Reynolds number including creeping and inertial flow regimes. They
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S.K. Patel, S.K. Majumder / Powder Technology 211 (2011) 127–134
also studied a large range of Reynolds number including inertial effects of pressure gradient through porous media in the case of purely viscous fluid flow. Vossoughi et al. [14] reported that the pressure drop of a porous media flow is only due to a small extent to the shear force. They also studied effect of matrix polymer as an additive to pressure drop phenomena in the packed bed. Basu et al. [15] developed a model to elucidate the wall effect on pressure drop and mass flux. They found that the pressure drop increases with the increase in flow rate. The pressure drop also increases with the increase in solute concentration for a given flow rate. Chhabra et al. [16] critically analyzed the flow of complex fluids through unconsolidated fixed beds and fluidized beds. They particularly focused on the prediction of macro-scale phenomena of flow regimes, pressure drop in fixed and fluidized beds. Gandhidasan et al. [17] studied the irrigated pressure drop and found that the structured packing has the lower pressure drop and higher capacity compared with random packing. Nemec et al. [18] investigated the wall effect on pressure drop and concluded that the effect is negligible as long as the column-to-particle diameter ratio is above 10. By phenomenological and empirical analyses they upgraded the original Ergun equation. They reported that with the proposed upgraded Ergun equation one is able to predict single-phase pressure drop in a packed bed of any shape of particles. Alopaeus et al. [19] developed a model to analyze the hydrodynamic parameters in packed-bed based on one-dimensional material and momentum balances for gas and liquid phases. Montillet et al. [20] studied the pressure drop through packed beds of spheres. They interpreted the behavior of dense packing which is characterized by porosities in the range 0.36–0.39 (uniform spheres) or even less. Bendova et al. [21] observed the creeping flow behavior of fluids of different rheological behaviors through fixed beds of spherical and nonspherical particles. Yilmaz et al. [22] studied the Newtonian (distilled water) and non-Newtonian (polyacrylamide solutions with concentrations 5 and 10 ppm) flow behaviors in porous medium. They found that the permeability for the distilled water is almost constant. The permeability of the non-Newtonian visco-elastic fluid flow in porous medium significantly depends on the pressure drop in the system. From the literature it is found that voluminous work is available for nonNewtonian fluid flow through packed beds without considering the interfacial stresses. Also there is a lack of studies on pressure drop characteristics based on the wetting effect of the solid–liquid surface during the flow. The objective of the present study is to study the pressure drop characteristics, shear stress phenomena in packed bed with shear thinning power-law type non-Newtonian liquid system and development of mechanistic model to analyze the pressure drop and interfacial stress in packed bed based on the wetting effect.
2. Theoretical background 2.1. Macroscopic model for pressure drop The fluid dynamic aspect of single phase through packed beds has been described in this section using an internal flow model based on analogy with flow through pipes. The dynamic interaction between liquid and solid wall is taken into account in modeling the flow by introducing the rate of energy dissipation. The mechanical energy balance is used to calculate the pressure drop which can be looked upon as either the force per unit area of cross section required to overcome frictional forces or the energy dissipation per unit volume. The model is presented with the following assumptions: (i) the flow is steady and isothermal with the voidage and holdup being uniform. (ii) acceleration effect is negligible due to absence inter-phase mass transfer. (iii) frictional loss is considered as uniform throughout the column for a particular liquid flow rate.
The mechanical energy balance equation for the liquid phase is given by ΔPls Ac Vsl −gΔZAc ρl Vsl −El = 0:
ð1Þ
In Eq. (1) the first term is “energy due to liquid–solid pressure (N.m/s)”, second term is potential energy (N.m/s) and the third term is energy dissipation per unit packed volume due to friction and wettability between liquid and solid. Eq. (1) can be represented as ΔPls −ρl gΔZ = El = Vsl Ac :
ð2Þ
The energy dissipation in solid–liquid surface occurs due to drag force exerted by the fluid on particle [23]. In Eq. (3), the amount of mechanical energy (El) is irreversively converted to thermal energy due to friction. The total energy dissipation can be calculated from the product of the force exerted by the fluid on a single particle, the fluid velocity and the total amount of particle present which can be represented as: 1 21 2 El = Fd ðVsl = εÞNp = Cd π dp ρl ðVsl =εÞ ðVsl = εÞNp 4 2 Np =
ΔZ ð1−εÞAc : ð1 = 6Þπd3p
ð3Þ ð4Þ
Therefore Eq. (2) can be written as: 2
ΔPls −ρl gΔZ = ð3 = 4ÞCD ρl Vsl
1−ε ΔZ : ε3 dp
ð5Þ
From the experiment the total pressure drop can be obtained as summation of frictional pressure drop and the hydrostatic pressure drop as ΔPls = ΔPfl + ρl gΔZ
ð6Þ
where ΔPfl is the frictional pressure loss due to liquid flow. Therefore, Eqs. (5) and (6) give 2
ΔPfl = ð3 = 4ÞCD ρl Vsl
1−ε ΔZ : ε3 dp
ð7Þ
To determine the frictional losses due to liquid flow in the column a model can be formulated on the basis of the following assumptions: (i) the friction factor for liquid phase is a constant multiple, α′ of that if only flow of single liquid phase without packing takes place in the column. (ii) The area of contact of the liquid phase with wall is α″ times to that of only flow of single phase without packing in the column. From these assumptions a simple overall momentum balance for liquid phase can be represented as [12]: ΔPfl ðCross sectional areaÞls ðWall shear stressÞls = ΔPfl0 ðCross sectional areaÞl0 ðWall shear stressÞl0 0:5f ρl Vsl2 ΔPfl Ac ε ð Area of contact with wallÞls ls × ⇒ = ΔPfl0 Ac ð Area of contact with wallÞl0 0:5f ρl Vsl2 l0 2 ð Area of contact with wallÞls fls Vsl ls × = : 2 ð Area of contact with wallÞl0 fl0 Vsl l0 ×
ð8Þ
ð Area of contact with wallÞls 1 α = αl′: 2 :αl″ = 2l ð Area of contact with wallÞl0 ε ε
⇒ΔPfl =
αl ΔPfl0 ε3
ð9Þ
where, the subscripts “l0” refer to liquid single phase, “ls” refers to liquid–solid wall, “sl” refers to liquid superficial, (Vsl)ls = (Vsl)l0/ε,
S.K. Patel, S.K. Majumder / Powder Technology 211 (2011) 127–134
αl = α′l α″l and cross-sectional area for the flow of liquid is equal to εl times the cross-sectional area of the column. α′l is defined as fls/fl0 and α″l is defined as (area of contact)ls/(area of contact)l0. The value of α″l is equal to the specific interfacial area in the packed bed. The parameter αl is an intensity factor which signifies intensity of interfacial shear stress. ΔPfl0 is the frictional pressure drop due to liquid when only liquid phase flows through the column.
The total rate of energy loss due to wettability of liquid with the total surface of solid wall is the summation of the energy loss due to wettability between liquid and surface of column wall and the energy loss due to wettability between liquid and surface of packing. This can be represented as: π dc Vsl σls V σ + sl ls ε ah ε
for Rel b5
ah 0:25 0:1 = 0:85Ch Rel Frl for Rel ≥5 a
for turbulent flow
where Ren is the Reynolds number for non-Newtonian fluid flow which is defined as [1] n n 2−n dc Vsl ρ 4n : n−1 K 3n + 1 8
ð11Þ ð12Þ
where σls, is the interfacial surface tension at the boundaries between liquid (l) and solid (s). Here, σ represents the force needed to stretch an interface by a unit distance. In the present study, it is found that the contact angle is below 90°, which indicates that the solid surface is wetted by the liquid. The range of superficial liquid velocity used in the present study is 0.004 to 0.05 m/s.
h pffiffi i n −0:3185 for transition flow: 0:0112 + Ren fl0 = 0:125 n
αl ΔPfl0 ε3
+ ρl gΔZ:
αl =
ε3 ½δ −1 δfl0 ls
Cd =
4 αl ΔPfl0 dp : 3 ρl Vsl2 ð1−εÞ ΔZ
fls =
αl fl0 αl″:
16 Ren
for laminar flow
ð21Þ
ð22Þ
The value of α″l is equivalent to the hydraulic specific interfacial area in the packed bed which is calculated by Eqs. (11) and (12) for different ranges of Reynolds number. The Blake–Kozeny's hydraulic model assumes that the bed of solid particles consists of irregularly shaped channels provided by the space between the particles in the bed. These channels are considered as large numbers of capillary tubes running parallel to the direction of flow. The tortuous path of the channel, Zc, traversed by the fluid elements is more than the bed length, Z. The average liquid–solid surface interfacial shear stress is expressed as: ΔPfl 1 2 2 fls ρl Vsl = ε = Rh 2 Zc
where Rh is called hydraulic radius which is defined as:
the friction factor is calculated as
ð20Þ
In Eq. (20), except αl, all parameters are known. Using the experimental pressure drop, the corresponding values of αl can be calculated for different variables from Eq. (20). To estimate the values of αl, the experimental data of ΔPls for different operating conditions of non-Newtonian flow in the packed bed were taken from the experimental results. Once the value of αl will be calculated from Eq. (20), the drag coefficient can be calculated from Eq. (21). From the definition of αl, one can calculate the effective friction factor, fls in the packed bed as:
The single liquid phase (without packing) frictional pressure drop was calculated using Fanning's equation: ð14Þ
ð19Þ
where, δls = ΔPls/(ρlgΔz), δfl0 = ΔPfl0/(ρlgΔz). Again from Eqs. (7) and (9), the drag coefficient can be expressed as
τi =
fl0 =
ð18Þ
From Eq. (19) αl can be expressed as
2.3. Determination of model parameters
2
ð17Þ
For transition flow, the friction factor f can be deduced from the generalized pressure loss equation as [26]:
ΔPls =
ð13Þ
ΔPfl0 = 2fl0 ρl Vsl ΔZ = dc
ð16Þ
Substituting the Eq. (9) for ΔPfl into Eq. (6), one gets
where Rel = (Vslρl)/(aμeff), Frl = (V2sla)/g. Values of specific surface area of packing (a) and Ch are characteristic of the particular type and size of packing. The value of Ch represents the shape factor of the particle. In the present system, for ceramic rashing ring of 10 mm size the values of a and Ch were taken as 440 m2/m3 and 0.791 respectively [24]. Wettability is the affinity of the solid matrix for the aqueous phases. It is normally quantified by the value of the contact angle. The contact angle θ b π/2 indicates that the solid is wetted by the liquid, and θ N π indicates non-wetting. The limits θ = 0 and π define complete wetting and complete non-wetting, respectively. The energy loss due to wetting of liquid depends on the dynamic contact angle between liquid and solid wall [25]. The dynamic contact angle (θ) is approximately equal to Ca1/3 [25], where Ca is the capillary number which is defined as, Ca = μeffVsl/εσl [25]. The surface tension between liquid and solid column wall (σls) can be calculated from Young's equation as σl cosθ = σls
2:63
ð10Þ
where ah is the hydraulic or effective specific area of packing. The hydraulic specific area of packing can be calculated from the correlations [24]: ah 0:5 0:1 = Ch Rel Frl a
0:079 n5 ðRen Þ10:5n
Ren =
2.2. Energy loss due to wettability between liquid and solid wall
Ew =
fl0 =
129
Rh =
dp ϕs ε : 6 1−ε
ð23Þ
ð24Þ
Shape factor is defined as: ð15Þ
ϕs =
vp 6 : : ap dp
ð25Þ
130
S.K. Patel, S.K. Majumder / Powder Technology 211 (2011) 127–134
Tortuosity factor, β=
Zc : Z
ð26Þ
Thus, fls =
dp ϕs 3βZ
! ΔPfl ε3 αf = l ″l0 1−ε ρl Vsl2 αl :
ð27Þ
Then the tortuosity factor can be expressed as dp ϕs β= 3Z
! ΔPfl α″l ε3 1 Cd α″l ϕs = : 4 αl fl0 1−ε ρl Vsl2 αl fl0
ð28Þ
If there is a slip at the interface, Eq. (23) can be expressed as τi =
ΔPfl 1 3n + 1 n 2ðVsl −Vis Þ n 2 2 fls ρl Vsl = ε = Rh =K : 2 4n εRh βZ
ð29Þ
approximately full-filled with packing for all trials. The inlet fluid was prepared using cold distilled water by gradually adding the carboxy methyl cellulose (CMC) powder to the water with gentle stirring until the solution becomes homogeneous. The CMC solution is fed into the packed bed by using the centrifugal pump. The volumetric flow rate is measured by rotameter. The CMC solution moved up through the packed bed. The U-tube manometer connected to the packed bed is used to measure the pressure drop across the bed at different flow rates and CMC concentrations. Because of the time needed to prepare the concentrated CMC solution, first the testing was started at low flow rates of CMC to obtain as many trial runs as possible. After the column reached a steady state, the data for pressure drop (before and after the addition of CMC) were recorded. The pressure tap connections to each column were situated in the packing sections at locations 5.0 cm from the top and from the bottom of the bed, to yield a direct pressure drop measurement without the necessity of correction for end effects. Pressure differences were measured with differential carbon tetrachloride (sp. gr. = 1.65) or mercury in glass manometers depending on the pressure drop range. The void fraction of liquid or specific liquid holdup is 0.650. The readings were repeated four times to ensure the reproducibility.
3. Experimental setup and procedure 3.1. Physical properties and effective viscosity of the liquid The schematic diagram of the experimental setup is shown in Fig. 1. A perspex column of inner diameter 0.050 m and of length 0.49 m packed randomly with ceramic rashing rings of 10 mm diameter and 10 mm height is used for the present experiment. The shape factor of the packing is 0.791. The bottom of the packed bed is connected to a pipe to drain liquid from the packed bed after the experiment is over. There is a bypassing arrangement after the pump which returns the liquid back to storage tank. The column is kept
The properties of non-Newtonian fluids cannot be described with Newton's law of viscosity as the viscosity of these fluids proves to be dependent on the rate of shear. Moreover, the viscosity of these fluids can increase or decrease due to the changes of the rate of shear which, again, is subject to the nature of the fluid. The non-Newtonian fluids, unlike the Newtonian, are defined as materials which do not conform to a direct proportionality between shear stress and shear rate. In the present study the carboxy methyle cellulose (CMC) is used as a shearthinning power-law type non-Newtonian liquid. The rheological parameters of the non-Newtonian liquid for its different concentrations are shown in Table 1. The effective viscosity of the liquid flowing through the porous media can be calculated as [27]: n′ −1
μeff = 12
1−n′ = 2 K ′ ð150Kr εÞ :
ð30Þ
The parameters n′and K′ are called Metzner and Reed parameter [28]. For power law fluids, the values of n′and K′ are equal to n and K [(1 + 3n)/4n]n respectively. Substitution of n′ and K′ in Eq. (30) gives K 3 + 9n n ð1−nÞ = 2 ð150Kr εÞ : 12 n
μeff =
ð31Þ
The parameter Kr in Eqs. (30) and (31) is called permeability of the porous medium which can be defined from Blake–Kozeny equation [29] as
Kr =
d2p ε3 150ð1−εÞ2
:
ð32Þ
Table 1 Physical properties of the non-Newtonian liquid.
Fig. 1. Schematic diagram of the experimental setup.
CMC conc. (wt%)
Density (kg/m3)
Consistency (K, Pa.sn)
Flow index properties (n)
1.0 1.5 2.0 2.5 Water
1000.96 1001.13 1001.37 1001.50 998.50
0.00318 0.00419 0.0059 0.00692 –
0.948 0.910 0.871 0.850 –
S.K. Patel, S.K. Majumder / Powder Technology 211 (2011) 127–134
4. Results and discussion
0.12
4.1. Pressure drop characteristics for non-Newtonian fluid flow in packed bed
0.10
Symbol CMC soln. 1.0 wt% 1.5 wt% 2.0 wt% 2.5 wt%
0.08
αl/αl'', [-]
In this present study the frictional pressure drop characteristic has been analyzed by a mechanistic model based on the viscous interaction and wetting of the non-Newtonian liquid. The intensity factor (αl) of the liquid–solid surface interfacial shear stress characterizes the intensity of the pressure drop in the fluid flow. Variation of the frictional pressure drop (ΔPfl/Δz) against Reynolds number based on non-Newtonian flow (Ren) for different CMC concentrations is shown in Fig. 2. It is seen that the pressure drop increases with the increase in liquid flow rate. This is due to the fact that the interfacial shear stress increases with increase in liquid velocity. Also the viscous friction at the wall increases with the increase in liquid velocity, which increases the pressure drop. Further, the pressure drop increases with the increase in CMC concentration for a given flow rate. The increase of CMC concentration leads to the increase in effective viscosity which leads to increase the shear stress and increase of pressure drop in the packed bed. Basu [15] also got the similar trend of pressure drop in non-Newtonian fluid flow through packed bed but with different shapes of packing materials. He reported that the increase in pressure drop with liquid velocity is due to increase in consistency index of the non-Newtonian fluid. This is true because increase in consistency index enhances the effective viscosity of the fluid. A similar behavior was also observed by the other investigators [30–33].
131
0.06
0.04
0.02
0
100
200
300
400
500
600
700
800
Ren, [-] Fig. 3. Variation of intensity factor with non-Newtonian Reynolds number.
depends on the Reynolds number as per Eqs. (11) and (12). Therefore it is concluded that the intensity factor is a function of Reynolds number. Also as the concentration of CMC solution increases, the effective viscosity decreases at constant flow rate and results in decrease of intensity factor with increase in concentration of CMC solution as shown in Fig. 3. The variation of shear stress intensity factor with non-Newtonian liquid Reynolds number can be expressed by developing a correlation (Eq. (33)) as:
4.2. Intensity factor of interfacial shear stress The intensity factor of interfacial shear stress is decreasing with increase in non-Newtonian liquid Reynolds number (Ren) as shown in Fig. 3. The intensity factor depends on the effective viscosity of the liquid. As the flow rate increases the effective viscosity decreases which causes the decrease in viscous force. At the same time the inertial force also increases but the rate of increase in inertial force is higher compared to the rate of decrease in viscous force. Hence increase in Reynolds number leads to increase the intensity factor. This can also be interpreted in terms of hydraulic specific interfacial area. The intensity factor increases with increase in hydraulic specific interfacial area. The intensity factor by definition depends on the ratio of area of contact of liquid with the solid wall with packing to that of without packing. The area of contact of liquid with the solid wall with packing is measured by the hydraulic specific interfacial area which
αl =
3:484α″l n10:98 2:557
Re0:814n n
:
ð33Þ
The parity of goodness of fit of the predicted value of shear stress intensity factor with the value calculated by the equation is shown in Fig. 4. The correlation coefficient of the correlation is 0.992 and standard error 0.00546. The standard error was calculated as per Eq. (34). sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 ½ðx−xÞð y−y2 ∑ð y−yÞ2 − STEYX = ðn−2Þ ∑ðx−xÞ2
ð34Þ
0.12 400 Symbol CMC (wt%) 1.0 1.5 2.0 2.5
ΔPfls/ΔZ, [Pa/m]
300
0.10
αl/α''l-predicted, [-]
350
250 200 150 100
0.08
0.06
0.04
Symbol CMC soln. 1.0 wt% 1.5 wt% 2.0 wt% 2.5 wt%
o
45 line
0.02
50 0 0
0.00 0.00 100
200
300
400
500
600
700
800
0.02
0.04
0.06
0.08
0.10
0.12
αl/α''l-experimental, [-]
Ren, [-] Fig. 2. Frictional pressure drop over a packed bed as a function of Reynolds number.
Fig. 4. Parity plot of experimental and predicted values for intensity factor of shear stress at different CMC solutions.
132
S.K. Patel, S.K. Majumder / Powder Technology 211 (2011) 127–134
100
18.0 Symbol CMC(wt%) 1.0 1.5 2.0 2.5
16.0 14.0 12.0
flsαl'', [-]
flsα''l, [-]
Symbol CMC soln. 1.0 wt% 1.5 wt% 2.0 wt% 2.5 wt% 10
10.0 8.0 6.0 4.0
1
2.0 100
0
300
200
400
600
500
700
0.0 0.0
800
1.0
2.0
3.0
Fig. 5. Friction factor is a function of Reynolds number at different CMC solutions.
where, STEYX is the standard error which is a measure of the amount of error in the prediction of y for an individual x. y's are the dependent data points (predicted values) and x's are the independent data points (experimental values). x and y are the means of total data points of x and y respectively. 4.3. Analysis of friction factor and shear stress The friction factor can be calculated from Eq. (22) by knowing the value of the intensity factor from Eq. (20). Fig. 5 presents some of the typical friction factor–Reynolds number plots for the different concentrations of non-Newtonian liquid (CMC). The friction factor depends on the intensity factor of shear stress and flow behavior index of non-Newtonian liquid. The dependency of friction factor can also be correlated as: 4:844
fls =
4.0
5.0
6.0
7.0
8.0
Ewx104, [N.m/s]
Ren, [-]
900:7n
ð35Þ
0:832 αl″Re1:094n : n
The value of α″l is the value of specific interfacial area of the packed bed. The above correlation is made in the range of 0.1 b Ren b 730 with
Fig. 7. Friction factor is a function of energy loss due to wettability at different CMC solutions.
visco-elastic non-Newtonian fluid. The correlation coefficient and the standard error of the correlation are 0.9943 and 0.62. The parity of goodness of fit is shown in Fig. 6. The present study is beyond the range for Ren b 0.1. The further study will be done in the creeping flow of non-Newtonian fluid in packed bed forRen b 0.1. The friction factor can also be represented by the rate of energy loss due to wettability. The friction factor decreases with the increase in wettability. The energy loss due to wettability increases with the increase in surface tension and decrease in porosity. This is because of capillary effect of flowing of liquid through porous media. The surface tension increases with the increase in concentration of CMC which leads to increase in loss of energy due to wettability. So as the surface tension increases the friction factor increases and hence increase in frictional losses. The variation of friction factor with the loss of energy due to wettability is shown in Fig. 7. A correlation has been made to interpret the fact as a function of energy loss due to wettability effect on friction factor as:
fls =
1:69 × 10−4 n−5:64 1:09n Ew
−0:15
:
ð36Þ
20 18
20.0
16 16.0
12
τix102, [N/m2]
flsα''l-predicted, [-]
o
45 line
14
10 8 Symbol CMC soln. 1.0 wt% 1.5 wt% 2.0 wt% 2.5 wt%
6 4 2 0 0
2
4
6
8
10
12
14
16
18
12.0
8.0 Symbol CMC(wt%) 1.0 1.5 2.0 2.5
4.0
20
flsα''l-experimental, [-] Fig. 6. Parity plot of experimental and predicted values for the friction factor at different CMC solutions.
0.0 0.0
2.0
4.0
K[(3n+1)/4n]
6.0
8.0
10.0
12.0
n[2(V -V )/(εR )]nx104 sl is h
Fig. 8. Variation of shear stress with effective shear rate at different CMC solutions.
S.K. Patel, S.K. Majumder / Powder Technology 211 (2011) 127–134
4.4. Analysis of drag coefficient The drag coefficient includes both hydrodynamic drag forces on particles and the column wall. The drag coefficient was calculated from Eq. (21) by using the values of experimental pressure drop data at above the Reynolds number (Rep) greater than 0.1. The drag coefficient from the present study has been compared with Ergun's equation (Eq. (37)) [34] and Foscolo's equation (Eq. (38)) [35] of drag coefficient which is represented as
Cd =
Cd =
100
Cd-predicted, [-]
The shear stress is a function of the non-Newtonian flow behavior index and the dynamic variables like flow velocity and the wall slip if any. The data for different concentrations of CMC solution for shear stress obtained are shown in Fig. 8. At higher flow rates it deviates from the laminar flow curve showing that inertial effects are beginning or dominating the viscous effects. In the present experimental investigation, slip effects are assumed to be negligible.
133
10
Symbol CMC(wt%) 1.0 1.5 2.0 2.5 1
1
10
! 4 150 + 1:75 3 Rep ð1−εÞ
ðErgunÞ
! 4 17:3 −1:8 + 0:336 ε 3 Rep
ðFoscoloÞ:
ð37Þ
Fig. 10. Parity plot of experimental and predicted values for the drag coefficient at different CMC solutions.
ð38Þ
where k1 = 0:7087n−0:1042
From the graphical representation (Fig. 9), it is seen that the drag that is obtained by the present work is smaller than predicted by Ergun's equation and greater than Foscolo's equation. It is found that the present result is well approximate to the Ergun's result. The deviation may be the non-Newtonian characteristics of the fluid. Since Cd is a function of frictional pressure drop a pressure drop relation such as the Ergun's equation was adapted to develop the model to predict the drag coefficient for the present study with non-Newtonian liquid within the stipulated range of the experimental variables. In this case two parameters k1 and k2 were introduced to signify the effect of flow behavior index of the non-Newtonian liquid. Eq. (42) describes the drag coefficient in the present stipulated range of experimental variables of: 0.004 b usl b 0.05 m/s, CMC concentration 1.0 to 2.5% (wt) which is represented as: " Cd = k1
4 150 + 1:75 3 Rep ð1−εÞ
!#k
2
ð39Þ
40 Symbol CMC (wt%) n 1.0 0.948 1.5 0.910 2.0 0.871 2.5 0.850 As per Ergun eqn. As per Foscolo eqn.
35 30
Cd, [-]
25 20 15 10 5 0
50
100
Cd-experimental, [-]
100
150
200
and k2 = 2:3817−1:2738n:
Fig. 9. Drag coefficient as a function of particle Reynolds number at different CMC solutions.
ð41Þ
It is seen that the present experimental data is well fitted for drag coefficient with predicted values by the correlation (Eq. (39)). The standard error of the predicted values with the experimental data as calculated by Eq. (34) is 1.10. The parity of experimental data with predicted data of drag coefficient is shown in Fig. 10. 5. Conclusions In this present study the frictional pressure drop characteristic has been analyzed by a mechanistic model based on the viscous interaction and wetting of the non-Newtonian liquid. The increase of CMC concentration leads to the increase in effective viscosity which leads to increase the shear stress and increase of pressure drop in the packed bed. The intensity factor is a function of Reynolds number. Also as the concentration of CMC solution increases, the effective viscosity decreases at constant flow rate and results in decrease of intensity factor with increase in concentration of CMC solution. The intensity factor (αl) of the liquid–solid surface interfacial shear stress characterizes the intensity of the pressure drop in the fluid flow. The friction factor decreases with the increase in wettability. The energy loss due to wettability increases with the increase in surface tension and decrease in porosity. This is because of capillary effect of flowing of liquid through porous media. The surface tension increases with the increase in concentration of CMC which leads to increase in loss of energy due to wettability. The drag that is obtained by the present work is smaller than predicted by Ergun's equation and greater than Foscolo's equation. It is found that the present result is well approximate to the Ergun's result. The deviation may be the nonNewtonian characteristics of the fluid. The present study may be useful for further understanding and modeling of specific multiphase reactor in industrial applications.
250
Rep, [-]
ð40Þ
Nomenclature a Specific interfacial area [1/m] Ac Column cross-sectional area [m2] ah Hydraulic specific interfacial area of packing [1/m]
134
ap Cd Ch dc dp El Fd Ew fls flo g Kr K n Np Pls Pfls Pflo Rh STEYX Vis vp Vsl ΔZ
S.K. Patel, S.K. Majumder / Powder Technology 211 (2011) 127–134
Surface area of single particle [m2] Drag coefficient [−] Constant, defined in Eq. (21) [−] Column diameter [m] Particle diameter [m] Energy dispersion per unit packed column [N.m/s] Drag force [N] Energy loss due to wettability [N.m/s] Friction factor of liquid–solid system [−] Friction factor of single liquid system [−] Gravitational acceleration [m/s2] Permeability of porous medium [m2] Consistency of fluid [Pa.sn] Flow behavior index [−] Number of particles [−] Liquid–solid pressure [N/m2] Frictional pressure due to liquid flow [N/m2] Frictional pressure due to single liquid-phase [N/m2] Hydraulic radius [m] Standard error defined in Eq. (34) [−] Interfacial slip velocity [m/s] Volume of single particle [m3] Superficial liquid velocity [m/s] Height of packing [m]
Dimensionless groups Capillary number (=(μeffVsl)/(εσl)) [−] Ca Frl Liquid Froude No. (=V2sl/gdp) [−] Rel Liquid Reynolds No. (=(Vslρl)/(aμeff)) [−] Ren Non-Newtonian liquid Reynolds number ( = n 4n ) [−]
References
dnc Vsl2−n ρ 8n−1 K
3n + 1
Rep
Particle Reynolds number (=(ρlVsldp)/μeff) [−]
Greek letters αl Parameter defined in Eq. (9) [−] β Tortuosity factor [−] δflo Ratio of frictional pressure to hydrostatic pressure [−] δls Ratio of total pressure to hydrostatic pressure [−] ε Porosity [−] μeff Effective dynamic viscosity [kg/m.s] Φs Sphericity [−] ρl Density of liquid [kg/m3] σl Liquid surface tension [N/m] σls Surface tension between liquid and solid [N/m] τi Interfacial shear stress [N/m2] θ Liquid–solid contact angle [radian] Subscripts c column f frictional i interfacial l liquid phase ls liquid–solid 0 Single p particle s superficial, solid w wettability
[1] R.P. Chhabra, J.F. Richardson, Non-Newtonian flow in the process industries, Fundamentals and Engineering Applications, Butterworth-Heinemann, Oxford, ISBN: 0 7506 3770 6, 1999. [2] L.G. Gibilaro, Fluidization-Dynamics, Butterworth Heinemann, 2001. [3] L.S. Fan, Gas–Liquid–Solid Fluidisation Engineering, Butterworths, Stoneham, MA, 1989. [4] D. Siginer, D. De Kee, R.P. Chhabra (Eds.), Advances in the Flow and Rheology of Non-Newtonian Fluids¸ Parts A & B, Elsevier, Amsterdam, 1999. [5] S. Kumar, J. Kishore, P. Lal, S.N. Upadhyay, Non-Newtonian flow through packed beds and porous media, J. Sci. Ind. Res. 40 (1981) 238–245. [6] Y.S. Wu, K. Pruess, Flow of non-Newtonian fluids in porous media, Adv. Porous Media 3 (1996) 87–178. [7] R.P. Chhabra, Bubbles, Drops, and Particles in Non-Newtonian Fluids, CRC Press, Boca Raton, FL, ISBN: 0 8493 6718 2, 1993, Reprinted, 1994. Second edition, 2006. CRC Press, Boca Raton, FL. [8] W. Kozicki, C.J. Hsu, C. Tru, Non-Newtonian flow through packed beds and porous media, Chem. Eng. Sci. 22 (1967) 487–502. [9] P. Mishra, D. Singh, I.M. Mishra, Momentum transfer to Newtonian and nonNewtonian fluids flowing through packed and fluidized beds, Chem. Eng. Sci. 30 (4) (1975) 397–405. [10] A.M. Lali, A.S. Khare, J.B. Joshi, K.D.P. Nigam, Behaviour of solid particles in viscous non-Newtonian solutions: settling velocity, wall effects and bed expansion in solid–liquid fluidized beds, Powder Technol. 57 (1) (1989) 39–50. [11] J. Zhu, M.G. Satish, Non-Newtonian effects on the drag of creeping flow through packed beds, Int. J. Multiphase Flow 18 (5) (1992) 765–777. [12] V.G. Rao, M.S. Ananth, Y.B.G. Varma, Hydrodynamics of two-phase concurrent downflow through packed beds, AIChE J. 29 (3) (1983) 467–473. [13] N.E. Sabiri, J. Comiti, Experimental validation of a model allowing pressure gradient determination for non-Newtonian purely viscous fluid-flow through packed beds, Chem. Eng. Sci. 52 (20) (1997) 3589–3592. [14] S. Vossoughi, Flow of non-Newtonian fluids in porous media, Rheol. Series 8 (2) (1999) 1183–1235. [15] S. Basu, Wall effect in laminar flow of non-Newtonian fluid through a packed bed, Chem. Eng. J. 81 (1–3) (2001) 323–329. [16] R.P. Chhabra, J. Comiti, I. Macha, Flow of non-Newtonian fluids in fixed and fluidized beds, Chem. Eng. Sci. 56 (1) (2001) 1–27. [17] P. Gandhidasa, Prediction of pressure drop in a packed bed dehumidifier operating with liquid desiccant, Appl. Therm. Eng. 22 (2002) 1117–1127. [18] D. Nemec, J. Levec, Flow through packed bed reactors: 1 single-phase flow, Chem. Eng. Sci. 60 (2005) 6947–6957. [19] V. Alopaeus, K. Hynynen, J. Aittamaa, M. Manninen, Modeling of gas–liquid packed-bed reactors with momentum equations and local interaction closures, Ind. Eng. Chem. Res. 45 (2006) 8189–8198. [20] A. Montillet, E. Akkari, J. Comiti, About a correlating equation for predicting pressure drops through packed beds of spheres in a large range of Reynolds numbers, Chem. Eng. Proc. 46 (4) (2007) 329–333. [21] H. Bendova, B. Siska, I. Machac, Pressure drop excess in the flow of viscoelastic liquids through fixed beds of particles, Chem. Eng. Proc. Proc. Intens. 48 (1) (2009) 29–37. [22] N. Yilmaz, A.S. Bakhtiyarov, R.N. Ibragimov, Experimental investigation of Newtonian and non-Newtonian fluid flows in porous media, Mech. Res. Commun. 36 (5) (2009) 638–641. [23] G.D. Mandrusiak, V.P. Carey, Pressure drop characteristics of two-phase flow in a vertical channel with offset strip fins, Exper. Therm. Fluid Sci. 1 (1) (1988) 41–50. [24] J. Benitez, Principles and Modern Applications of Mass Transfer Operations, Willey interscience, 2002. [25] T. Cubaud, C. Ho, Transport of bubbles in square microchannels, Phys. Fluids 16 (12) (2004) 4575–4585. [26] S.E.M. Desouky, N.A. E1-Emam, Program designs for pseudoplastic fluids, Oil and Gas J. 87 (51) (1989) 48–51. [27] A.M. Hemeida, Mathematical model for flow of pseudoplastic fluids in porous media, J. King Saud Univ. Eng. Sci. 7 (1) (1995) 125–137. [28] A.B. Metzner, J.C. Reed, Flow of non-Newtonian fluids—correlation of the laminar, transition and turbulent-flow regions, AIChE J. 1 (1955) 434–440. [29] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, John Wiley & Sons, NY, 1960. [30] R.J. Marshall, A.B. Metzner, Flow of viscoelastic fluids through porous media, Ind. Eng. Chem. Fundamentals 6 (1967) 393–400. [31] F.M. Brea, M.F. Edwards, W.L. Wilkinson, The flow of non-Newtonian slurries through fixed and fluidised beds, Chem. Eng. Sci. 31 (1976) 329–336. [32] D.R. Gregory, R.G. Griskey, Flow of molten polymers through porous media, AIChE J. 13 (1967) 122–125. [33] N.E. Sabiri, J. Comiti, Pressure drop in non-Newtonian purely viscous fluid flow through porous media, Chem. Eng. Sci. 50 (1995) 1193–1201. [34] S. Ergun, Fluid flow through packed columns, Chem. Eng. Prog. 48 (1952) 89–94. [35] P.U. Foscolo, L.G. Gibilaro, S.P. Waldram, A unified model for particulate expansion of fluidised beds and flow in fixed porous media, Chem. Eng. Sci. 38 (1983) 1251–1260.
Contents lists available at ScienceDirect
Powder Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p ow t e c
Interfacial stress in non-Newtonian flow through packed bed Suresh Kumar Patel, Subrata Kumar Majumder ⁎ Department of Chemical Engineering, Indian Institute of Technology Guwahati, Guwahati-781039, Assam, India
a r t i c l e
i n f o
Article history: Received 7 February 2011 Received in revised form 2 April 2011 Accepted 13 April 2011 Available online 20 April 2011 Keywords: Packed bed Interfacial area Non-Newtonian liquid Wettability Shear stress
a b s t r a c t This study investigates the pressure drop characteristics, shear stress in packed bed with shear thinning power law type non-Newtonian liquid. A mechanistic model has also been developed to analyze the pressure drop and interfacial stress in packed bed with non-Newtonian liquid by considering the loss of energy due to wettability. The Ergun's and Foscolo's equations were used for comparison with the experimental data. The Ergun equation was modified to account for the effect of flow behavior index of non-Newtonian fluid in the column. The intensity factor of shear stress and the friction factor were analyzed based on energy loss due to wettability effect of liquid on the solid surface. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Packed bed reactors are most commonly employed in the chemical process industries among the several possible types of multiphase catalytic reactors. Their popularity halts from their effectiveness in terms of performance as well as low capital and operating costs. The non-Newtonian fluid flow through particulate bed system is important in a variety of chemical and biochemical processes [1]. Various examples of applications of the particulate system have been described by many authors [1–4]. Studies on the flow of fluids through porous media were restricted mostly to Newtonian fluids. Recently, the flow of non-Newtonian fluids through packed beds and porous media has received considerable attention because of its importance in various industrial applications. Considerable research efforts have been expended in exploring and further understanding of the basic phenomena of momentum, heat and mass transfer processes with and without chemical reactions in particulate system.
1.1. Previous work Voluminous literature available on the flow of a variety of nonNewtonian materials through packed beds has been critically reviewed previously [5,6]. Wu and Pruess [6] described the non-Newtonian flow behavior in packed bed including beds of uniform size and of multi-size particles. Some other different studies related to Newtonian and non-
⁎ Corresponding author. Tel.: + 91 3612582265; fax: + 91 3612582291. E-mail addresses: [email protected], [email protected] (S.K. Majumder). 0032-5910/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2011.04.010
Newtonian flow behavior in packed bed has been thoroughly reported in literature [7]. From the studies it is concluded that each study has its some uniqueness in its morphology which is contributing in some measure to the complexity of the problem. Similarly, often inadequate rheological characterization also adds to the complexity of such systems. While it is usual for most non-Newtonian materials to exhibit shear thinning behavior, many other features including time-dependency, viscoelasticity, yield stress, etc., are also present but not often measured. Certainly, the major research effort has been directed at developing simple and reliable methods of predicting the frictional pressure loss for the flow of non-Newtonian fluids through packed beds. Kozicki et al. [8] generalized the average velocity–pressure gradient relationship for arbitrary time-independent non-Newtonian fluids in porous media, based on the Blake–Kozeny capillary model. Mishra et al. [9] described average shear stress–shear rate relationship to predict the flow behavior of power law as well as non-power law fluids. The wall factor is also another phenomenon to affect the flow behavior in porous media which is a function of both diameter ratio and particle Reynolds number [10]. For Reynolds number below 1.0, this dependency is nearly same for settling in Newtonian and non-Newtonian liquids. In the range of 1 b Re∞ b 200, wall effect can be estimated for the non-Newtonian case from the relation applicable to settling in Newtonian liquids. Zhu and Satish [11] studied the drag phenomena which decrease with a decrease in flow behavior index and with an increase in the characteristic time. They found that both the normal stress difference and the bed voidage have a great influence on the resistance of visco-elastic flow through a packed bed. Rao et al. [12] studied the pressure loss-throughput behavior for the flow of inelastic power law fluids through randomly packed spherical particles and over wide ranges of operating and physical conditions. Sabiri et al. [13] investigation covers a large range of Reynolds number including creeping and inertial flow regimes. They
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also studied a large range of Reynolds number including inertial effects of pressure gradient through porous media in the case of purely viscous fluid flow. Vossoughi et al. [14] reported that the pressure drop of a porous media flow is only due to a small extent to the shear force. They also studied effect of matrix polymer as an additive to pressure drop phenomena in the packed bed. Basu et al. [15] developed a model to elucidate the wall effect on pressure drop and mass flux. They found that the pressure drop increases with the increase in flow rate. The pressure drop also increases with the increase in solute concentration for a given flow rate. Chhabra et al. [16] critically analyzed the flow of complex fluids through unconsolidated fixed beds and fluidized beds. They particularly focused on the prediction of macro-scale phenomena of flow regimes, pressure drop in fixed and fluidized beds. Gandhidasan et al. [17] studied the irrigated pressure drop and found that the structured packing has the lower pressure drop and higher capacity compared with random packing. Nemec et al. [18] investigated the wall effect on pressure drop and concluded that the effect is negligible as long as the column-to-particle diameter ratio is above 10. By phenomenological and empirical analyses they upgraded the original Ergun equation. They reported that with the proposed upgraded Ergun equation one is able to predict single-phase pressure drop in a packed bed of any shape of particles. Alopaeus et al. [19] developed a model to analyze the hydrodynamic parameters in packed-bed based on one-dimensional material and momentum balances for gas and liquid phases. Montillet et al. [20] studied the pressure drop through packed beds of spheres. They interpreted the behavior of dense packing which is characterized by porosities in the range 0.36–0.39 (uniform spheres) or even less. Bendova et al. [21] observed the creeping flow behavior of fluids of different rheological behaviors through fixed beds of spherical and nonspherical particles. Yilmaz et al. [22] studied the Newtonian (distilled water) and non-Newtonian (polyacrylamide solutions with concentrations 5 and 10 ppm) flow behaviors in porous medium. They found that the permeability for the distilled water is almost constant. The permeability of the non-Newtonian visco-elastic fluid flow in porous medium significantly depends on the pressure drop in the system. From the literature it is found that voluminous work is available for nonNewtonian fluid flow through packed beds without considering the interfacial stresses. Also there is a lack of studies on pressure drop characteristics based on the wetting effect of the solid–liquid surface during the flow. The objective of the present study is to study the pressure drop characteristics, shear stress phenomena in packed bed with shear thinning power-law type non-Newtonian liquid system and development of mechanistic model to analyze the pressure drop and interfacial stress in packed bed based on the wetting effect.
2. Theoretical background 2.1. Macroscopic model for pressure drop The fluid dynamic aspect of single phase through packed beds has been described in this section using an internal flow model based on analogy with flow through pipes. The dynamic interaction between liquid and solid wall is taken into account in modeling the flow by introducing the rate of energy dissipation. The mechanical energy balance is used to calculate the pressure drop which can be looked upon as either the force per unit area of cross section required to overcome frictional forces or the energy dissipation per unit volume. The model is presented with the following assumptions: (i) the flow is steady and isothermal with the voidage and holdup being uniform. (ii) acceleration effect is negligible due to absence inter-phase mass transfer. (iii) frictional loss is considered as uniform throughout the column for a particular liquid flow rate.
The mechanical energy balance equation for the liquid phase is given by ΔPls Ac Vsl −gΔZAc ρl Vsl −El = 0:
ð1Þ
In Eq. (1) the first term is “energy due to liquid–solid pressure (N.m/s)”, second term is potential energy (N.m/s) and the third term is energy dissipation per unit packed volume due to friction and wettability between liquid and solid. Eq. (1) can be represented as ΔPls −ρl gΔZ = El = Vsl Ac :
ð2Þ
The energy dissipation in solid–liquid surface occurs due to drag force exerted by the fluid on particle [23]. In Eq. (3), the amount of mechanical energy (El) is irreversively converted to thermal energy due to friction. The total energy dissipation can be calculated from the product of the force exerted by the fluid on a single particle, the fluid velocity and the total amount of particle present which can be represented as: 1 21 2 El = Fd ðVsl = εÞNp = Cd π dp ρl ðVsl =εÞ ðVsl = εÞNp 4 2 Np =
ΔZ ð1−εÞAc : ð1 = 6Þπd3p
ð3Þ ð4Þ
Therefore Eq. (2) can be written as: 2
ΔPls −ρl gΔZ = ð3 = 4ÞCD ρl Vsl
1−ε ΔZ : ε3 dp
ð5Þ
From the experiment the total pressure drop can be obtained as summation of frictional pressure drop and the hydrostatic pressure drop as ΔPls = ΔPfl + ρl gΔZ
ð6Þ
where ΔPfl is the frictional pressure loss due to liquid flow. Therefore, Eqs. (5) and (6) give 2
ΔPfl = ð3 = 4ÞCD ρl Vsl
1−ε ΔZ : ε3 dp
ð7Þ
To determine the frictional losses due to liquid flow in the column a model can be formulated on the basis of the following assumptions: (i) the friction factor for liquid phase is a constant multiple, α′ of that if only flow of single liquid phase without packing takes place in the column. (ii) The area of contact of the liquid phase with wall is α″ times to that of only flow of single phase without packing in the column. From these assumptions a simple overall momentum balance for liquid phase can be represented as [12]: ΔPfl ðCross sectional areaÞls ðWall shear stressÞls = ΔPfl0 ðCross sectional areaÞl0 ðWall shear stressÞl0 0:5f ρl Vsl2 ΔPfl Ac ε ð Area of contact with wallÞls ls × ⇒ = ΔPfl0 Ac ð Area of contact with wallÞl0 0:5f ρl Vsl2 l0 2 ð Area of contact with wallÞls fls Vsl ls × = : 2 ð Area of contact with wallÞl0 fl0 Vsl l0 ×
ð8Þ
ð Area of contact with wallÞls 1 α = αl′: 2 :αl″ = 2l ð Area of contact with wallÞl0 ε ε
⇒ΔPfl =
αl ΔPfl0 ε3
ð9Þ
where, the subscripts “l0” refer to liquid single phase, “ls” refers to liquid–solid wall, “sl” refers to liquid superficial, (Vsl)ls = (Vsl)l0/ε,
S.K. Patel, S.K. Majumder / Powder Technology 211 (2011) 127–134
αl = α′l α″l and cross-sectional area for the flow of liquid is equal to εl times the cross-sectional area of the column. α′l is defined as fls/fl0 and α″l is defined as (area of contact)ls/(area of contact)l0. The value of α″l is equal to the specific interfacial area in the packed bed. The parameter αl is an intensity factor which signifies intensity of interfacial shear stress. ΔPfl0 is the frictional pressure drop due to liquid when only liquid phase flows through the column.
The total rate of energy loss due to wettability of liquid with the total surface of solid wall is the summation of the energy loss due to wettability between liquid and surface of column wall and the energy loss due to wettability between liquid and surface of packing. This can be represented as: π dc Vsl σls V σ + sl ls ε ah ε
for Rel b5
ah 0:25 0:1 = 0:85Ch Rel Frl for Rel ≥5 a
for turbulent flow
where Ren is the Reynolds number for non-Newtonian fluid flow which is defined as [1] n n 2−n dc Vsl ρ 4n : n−1 K 3n + 1 8
ð11Þ ð12Þ
where σls, is the interfacial surface tension at the boundaries between liquid (l) and solid (s). Here, σ represents the force needed to stretch an interface by a unit distance. In the present study, it is found that the contact angle is below 90°, which indicates that the solid surface is wetted by the liquid. The range of superficial liquid velocity used in the present study is 0.004 to 0.05 m/s.
h pffiffi i n −0:3185 for transition flow: 0:0112 + Ren fl0 = 0:125 n
αl ΔPfl0 ε3
+ ρl gΔZ:
αl =
ε3 ½δ −1 δfl0 ls
Cd =
4 αl ΔPfl0 dp : 3 ρl Vsl2 ð1−εÞ ΔZ
fls =
αl fl0 αl″:
16 Ren
for laminar flow
ð21Þ
ð22Þ
The value of α″l is equivalent to the hydraulic specific interfacial area in the packed bed which is calculated by Eqs. (11) and (12) for different ranges of Reynolds number. The Blake–Kozeny's hydraulic model assumes that the bed of solid particles consists of irregularly shaped channels provided by the space between the particles in the bed. These channels are considered as large numbers of capillary tubes running parallel to the direction of flow. The tortuous path of the channel, Zc, traversed by the fluid elements is more than the bed length, Z. The average liquid–solid surface interfacial shear stress is expressed as: ΔPfl 1 2 2 fls ρl Vsl = ε = Rh 2 Zc
where Rh is called hydraulic radius which is defined as:
the friction factor is calculated as
ð20Þ
In Eq. (20), except αl, all parameters are known. Using the experimental pressure drop, the corresponding values of αl can be calculated for different variables from Eq. (20). To estimate the values of αl, the experimental data of ΔPls for different operating conditions of non-Newtonian flow in the packed bed were taken from the experimental results. Once the value of αl will be calculated from Eq. (20), the drag coefficient can be calculated from Eq. (21). From the definition of αl, one can calculate the effective friction factor, fls in the packed bed as:
The single liquid phase (without packing) frictional pressure drop was calculated using Fanning's equation: ð14Þ
ð19Þ
where, δls = ΔPls/(ρlgΔz), δfl0 = ΔPfl0/(ρlgΔz). Again from Eqs. (7) and (9), the drag coefficient can be expressed as
τi =
fl0 =
ð18Þ
From Eq. (19) αl can be expressed as
2.3. Determination of model parameters
2
ð17Þ
For transition flow, the friction factor f can be deduced from the generalized pressure loss equation as [26]:
ΔPls =
ð13Þ
ΔPfl0 = 2fl0 ρl Vsl ΔZ = dc
ð16Þ
Substituting the Eq. (9) for ΔPfl into Eq. (6), one gets
where Rel = (Vslρl)/(aμeff), Frl = (V2sla)/g. Values of specific surface area of packing (a) and Ch are characteristic of the particular type and size of packing. The value of Ch represents the shape factor of the particle. In the present system, for ceramic rashing ring of 10 mm size the values of a and Ch were taken as 440 m2/m3 and 0.791 respectively [24]. Wettability is the affinity of the solid matrix for the aqueous phases. It is normally quantified by the value of the contact angle. The contact angle θ b π/2 indicates that the solid is wetted by the liquid, and θ N π indicates non-wetting. The limits θ = 0 and π define complete wetting and complete non-wetting, respectively. The energy loss due to wetting of liquid depends on the dynamic contact angle between liquid and solid wall [25]. The dynamic contact angle (θ) is approximately equal to Ca1/3 [25], where Ca is the capillary number which is defined as, Ca = μeffVsl/εσl [25]. The surface tension between liquid and solid column wall (σls) can be calculated from Young's equation as σl cosθ = σls
2:63
ð10Þ
where ah is the hydraulic or effective specific area of packing. The hydraulic specific area of packing can be calculated from the correlations [24]: ah 0:5 0:1 = Ch Rel Frl a
0:079 n5 ðRen Þ10:5n
Ren =
2.2. Energy loss due to wettability between liquid and solid wall
Ew =
fl0 =
129
Rh =
dp ϕs ε : 6 1−ε
ð23Þ
ð24Þ
Shape factor is defined as: ð15Þ
ϕs =
vp 6 : : ap dp
ð25Þ
130
S.K. Patel, S.K. Majumder / Powder Technology 211 (2011) 127–134
Tortuosity factor, β=
Zc : Z
ð26Þ
Thus, fls =
dp ϕs 3βZ
! ΔPfl ε3 αf = l ″l0 1−ε ρl Vsl2 αl :
ð27Þ
Then the tortuosity factor can be expressed as dp ϕs β= 3Z
! ΔPfl α″l ε3 1 Cd α″l ϕs = : 4 αl fl0 1−ε ρl Vsl2 αl fl0
ð28Þ
If there is a slip at the interface, Eq. (23) can be expressed as τi =
ΔPfl 1 3n + 1 n 2ðVsl −Vis Þ n 2 2 fls ρl Vsl = ε = Rh =K : 2 4n εRh βZ
ð29Þ
approximately full-filled with packing for all trials. The inlet fluid was prepared using cold distilled water by gradually adding the carboxy methyl cellulose (CMC) powder to the water with gentle stirring until the solution becomes homogeneous. The CMC solution is fed into the packed bed by using the centrifugal pump. The volumetric flow rate is measured by rotameter. The CMC solution moved up through the packed bed. The U-tube manometer connected to the packed bed is used to measure the pressure drop across the bed at different flow rates and CMC concentrations. Because of the time needed to prepare the concentrated CMC solution, first the testing was started at low flow rates of CMC to obtain as many trial runs as possible. After the column reached a steady state, the data for pressure drop (before and after the addition of CMC) were recorded. The pressure tap connections to each column were situated in the packing sections at locations 5.0 cm from the top and from the bottom of the bed, to yield a direct pressure drop measurement without the necessity of correction for end effects. Pressure differences were measured with differential carbon tetrachloride (sp. gr. = 1.65) or mercury in glass manometers depending on the pressure drop range. The void fraction of liquid or specific liquid holdup is 0.650. The readings were repeated four times to ensure the reproducibility.
3. Experimental setup and procedure 3.1. Physical properties and effective viscosity of the liquid The schematic diagram of the experimental setup is shown in Fig. 1. A perspex column of inner diameter 0.050 m and of length 0.49 m packed randomly with ceramic rashing rings of 10 mm diameter and 10 mm height is used for the present experiment. The shape factor of the packing is 0.791. The bottom of the packed bed is connected to a pipe to drain liquid from the packed bed after the experiment is over. There is a bypassing arrangement after the pump which returns the liquid back to storage tank. The column is kept
The properties of non-Newtonian fluids cannot be described with Newton's law of viscosity as the viscosity of these fluids proves to be dependent on the rate of shear. Moreover, the viscosity of these fluids can increase or decrease due to the changes of the rate of shear which, again, is subject to the nature of the fluid. The non-Newtonian fluids, unlike the Newtonian, are defined as materials which do not conform to a direct proportionality between shear stress and shear rate. In the present study the carboxy methyle cellulose (CMC) is used as a shearthinning power-law type non-Newtonian liquid. The rheological parameters of the non-Newtonian liquid for its different concentrations are shown in Table 1. The effective viscosity of the liquid flowing through the porous media can be calculated as [27]: n′ −1
μeff = 12
1−n′ = 2 K ′ ð150Kr εÞ :
ð30Þ
The parameters n′and K′ are called Metzner and Reed parameter [28]. For power law fluids, the values of n′and K′ are equal to n and K [(1 + 3n)/4n]n respectively. Substitution of n′ and K′ in Eq. (30) gives K 3 + 9n n ð1−nÞ = 2 ð150Kr εÞ : 12 n
μeff =
ð31Þ
The parameter Kr in Eqs. (30) and (31) is called permeability of the porous medium which can be defined from Blake–Kozeny equation [29] as
Kr =
d2p ε3 150ð1−εÞ2
:
ð32Þ
Table 1 Physical properties of the non-Newtonian liquid.
Fig. 1. Schematic diagram of the experimental setup.
CMC conc. (wt%)
Density (kg/m3)
Consistency (K, Pa.sn)
Flow index properties (n)
1.0 1.5 2.0 2.5 Water
1000.96 1001.13 1001.37 1001.50 998.50
0.00318 0.00419 0.0059 0.00692 –
0.948 0.910 0.871 0.850 –
S.K. Patel, S.K. Majumder / Powder Technology 211 (2011) 127–134
4. Results and discussion
0.12
4.1. Pressure drop characteristics for non-Newtonian fluid flow in packed bed
0.10
Symbol CMC soln. 1.0 wt% 1.5 wt% 2.0 wt% 2.5 wt%
0.08
αl/αl'', [-]
In this present study the frictional pressure drop characteristic has been analyzed by a mechanistic model based on the viscous interaction and wetting of the non-Newtonian liquid. The intensity factor (αl) of the liquid–solid surface interfacial shear stress characterizes the intensity of the pressure drop in the fluid flow. Variation of the frictional pressure drop (ΔPfl/Δz) against Reynolds number based on non-Newtonian flow (Ren) for different CMC concentrations is shown in Fig. 2. It is seen that the pressure drop increases with the increase in liquid flow rate. This is due to the fact that the interfacial shear stress increases with increase in liquid velocity. Also the viscous friction at the wall increases with the increase in liquid velocity, which increases the pressure drop. Further, the pressure drop increases with the increase in CMC concentration for a given flow rate. The increase of CMC concentration leads to the increase in effective viscosity which leads to increase the shear stress and increase of pressure drop in the packed bed. Basu [15] also got the similar trend of pressure drop in non-Newtonian fluid flow through packed bed but with different shapes of packing materials. He reported that the increase in pressure drop with liquid velocity is due to increase in consistency index of the non-Newtonian fluid. This is true because increase in consistency index enhances the effective viscosity of the fluid. A similar behavior was also observed by the other investigators [30–33].
131
0.06
0.04
0.02
0
100
200
300
400
500
600
700
800
Ren, [-] Fig. 3. Variation of intensity factor with non-Newtonian Reynolds number.
depends on the Reynolds number as per Eqs. (11) and (12). Therefore it is concluded that the intensity factor is a function of Reynolds number. Also as the concentration of CMC solution increases, the effective viscosity decreases at constant flow rate and results in decrease of intensity factor with increase in concentration of CMC solution as shown in Fig. 3. The variation of shear stress intensity factor with non-Newtonian liquid Reynolds number can be expressed by developing a correlation (Eq. (33)) as:
4.2. Intensity factor of interfacial shear stress The intensity factor of interfacial shear stress is decreasing with increase in non-Newtonian liquid Reynolds number (Ren) as shown in Fig. 3. The intensity factor depends on the effective viscosity of the liquid. As the flow rate increases the effective viscosity decreases which causes the decrease in viscous force. At the same time the inertial force also increases but the rate of increase in inertial force is higher compared to the rate of decrease in viscous force. Hence increase in Reynolds number leads to increase the intensity factor. This can also be interpreted in terms of hydraulic specific interfacial area. The intensity factor increases with increase in hydraulic specific interfacial area. The intensity factor by definition depends on the ratio of area of contact of liquid with the solid wall with packing to that of without packing. The area of contact of liquid with the solid wall with packing is measured by the hydraulic specific interfacial area which
αl =
3:484α″l n10:98 2:557
Re0:814n n
:
ð33Þ
The parity of goodness of fit of the predicted value of shear stress intensity factor with the value calculated by the equation is shown in Fig. 4. The correlation coefficient of the correlation is 0.992 and standard error 0.00546. The standard error was calculated as per Eq. (34). sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 ½ðx−xÞð y−y2 ∑ð y−yÞ2 − STEYX = ðn−2Þ ∑ðx−xÞ2
ð34Þ
0.12 400 Symbol CMC (wt%) 1.0 1.5 2.0 2.5
ΔPfls/ΔZ, [Pa/m]
300
0.10
αl/α''l-predicted, [-]
350
250 200 150 100
0.08
0.06
0.04
Symbol CMC soln. 1.0 wt% 1.5 wt% 2.0 wt% 2.5 wt%
o
45 line
0.02
50 0 0
0.00 0.00 100
200
300
400
500
600
700
800
0.02
0.04
0.06
0.08
0.10
0.12
αl/α''l-experimental, [-]
Ren, [-] Fig. 2. Frictional pressure drop over a packed bed as a function of Reynolds number.
Fig. 4. Parity plot of experimental and predicted values for intensity factor of shear stress at different CMC solutions.
132
S.K. Patel, S.K. Majumder / Powder Technology 211 (2011) 127–134
100
18.0 Symbol CMC(wt%) 1.0 1.5 2.0 2.5
16.0 14.0 12.0
flsαl'', [-]
flsα''l, [-]
Symbol CMC soln. 1.0 wt% 1.5 wt% 2.0 wt% 2.5 wt% 10
10.0 8.0 6.0 4.0
1
2.0 100
0
300
200
400
600
500
700
0.0 0.0
800
1.0
2.0
3.0
Fig. 5. Friction factor is a function of Reynolds number at different CMC solutions.
where, STEYX is the standard error which is a measure of the amount of error in the prediction of y for an individual x. y's are the dependent data points (predicted values) and x's are the independent data points (experimental values). x and y are the means of total data points of x and y respectively. 4.3. Analysis of friction factor and shear stress The friction factor can be calculated from Eq. (22) by knowing the value of the intensity factor from Eq. (20). Fig. 5 presents some of the typical friction factor–Reynolds number plots for the different concentrations of non-Newtonian liquid (CMC). The friction factor depends on the intensity factor of shear stress and flow behavior index of non-Newtonian liquid. The dependency of friction factor can also be correlated as: 4:844
fls =
4.0
5.0
6.0
7.0
8.0
Ewx104, [N.m/s]
Ren, [-]
900:7n
ð35Þ
0:832 αl″Re1:094n : n
The value of α″l is the value of specific interfacial area of the packed bed. The above correlation is made in the range of 0.1 b Ren b 730 with
Fig. 7. Friction factor is a function of energy loss due to wettability at different CMC solutions.
visco-elastic non-Newtonian fluid. The correlation coefficient and the standard error of the correlation are 0.9943 and 0.62. The parity of goodness of fit is shown in Fig. 6. The present study is beyond the range for Ren b 0.1. The further study will be done in the creeping flow of non-Newtonian fluid in packed bed forRen b 0.1. The friction factor can also be represented by the rate of energy loss due to wettability. The friction factor decreases with the increase in wettability. The energy loss due to wettability increases with the increase in surface tension and decrease in porosity. This is because of capillary effect of flowing of liquid through porous media. The surface tension increases with the increase in concentration of CMC which leads to increase in loss of energy due to wettability. So as the surface tension increases the friction factor increases and hence increase in frictional losses. The variation of friction factor with the loss of energy due to wettability is shown in Fig. 7. A correlation has been made to interpret the fact as a function of energy loss due to wettability effect on friction factor as:
fls =
1:69 × 10−4 n−5:64 1:09n Ew
−0:15
:
ð36Þ
20 18
20.0
16 16.0
12
τix102, [N/m2]
flsα''l-predicted, [-]
o
45 line
14
10 8 Symbol CMC soln. 1.0 wt% 1.5 wt% 2.0 wt% 2.5 wt%
6 4 2 0 0
2
4
6
8
10
12
14
16
18
12.0
8.0 Symbol CMC(wt%) 1.0 1.5 2.0 2.5
4.0
20
flsα''l-experimental, [-] Fig. 6. Parity plot of experimental and predicted values for the friction factor at different CMC solutions.
0.0 0.0
2.0
4.0
K[(3n+1)/4n]
6.0
8.0
10.0
12.0
n[2(V -V )/(εR )]nx104 sl is h
Fig. 8. Variation of shear stress with effective shear rate at different CMC solutions.
S.K. Patel, S.K. Majumder / Powder Technology 211 (2011) 127–134
4.4. Analysis of drag coefficient The drag coefficient includes both hydrodynamic drag forces on particles and the column wall. The drag coefficient was calculated from Eq. (21) by using the values of experimental pressure drop data at above the Reynolds number (Rep) greater than 0.1. The drag coefficient from the present study has been compared with Ergun's equation (Eq. (37)) [34] and Foscolo's equation (Eq. (38)) [35] of drag coefficient which is represented as
Cd =
Cd =
100
Cd-predicted, [-]
The shear stress is a function of the non-Newtonian flow behavior index and the dynamic variables like flow velocity and the wall slip if any. The data for different concentrations of CMC solution for shear stress obtained are shown in Fig. 8. At higher flow rates it deviates from the laminar flow curve showing that inertial effects are beginning or dominating the viscous effects. In the present experimental investigation, slip effects are assumed to be negligible.
133
10
Symbol CMC(wt%) 1.0 1.5 2.0 2.5 1
1
10
! 4 150 + 1:75 3 Rep ð1−εÞ
ðErgunÞ
! 4 17:3 −1:8 + 0:336 ε 3 Rep
ðFoscoloÞ:
ð37Þ
Fig. 10. Parity plot of experimental and predicted values for the drag coefficient at different CMC solutions.
ð38Þ
where k1 = 0:7087n−0:1042
From the graphical representation (Fig. 9), it is seen that the drag that is obtained by the present work is smaller than predicted by Ergun's equation and greater than Foscolo's equation. It is found that the present result is well approximate to the Ergun's result. The deviation may be the non-Newtonian characteristics of the fluid. Since Cd is a function of frictional pressure drop a pressure drop relation such as the Ergun's equation was adapted to develop the model to predict the drag coefficient for the present study with non-Newtonian liquid within the stipulated range of the experimental variables. In this case two parameters k1 and k2 were introduced to signify the effect of flow behavior index of the non-Newtonian liquid. Eq. (42) describes the drag coefficient in the present stipulated range of experimental variables of: 0.004 b usl b 0.05 m/s, CMC concentration 1.0 to 2.5% (wt) which is represented as: " Cd = k1
4 150 + 1:75 3 Rep ð1−εÞ
!#k
2
ð39Þ
40 Symbol CMC (wt%) n 1.0 0.948 1.5 0.910 2.0 0.871 2.5 0.850 As per Ergun eqn. As per Foscolo eqn.
35 30
Cd, [-]
25 20 15 10 5 0
50
100
Cd-experimental, [-]
100
150
200
and k2 = 2:3817−1:2738n:
Fig. 9. Drag coefficient as a function of particle Reynolds number at different CMC solutions.
ð41Þ
It is seen that the present experimental data is well fitted for drag coefficient with predicted values by the correlation (Eq. (39)). The standard error of the predicted values with the experimental data as calculated by Eq. (34) is 1.10. The parity of experimental data with predicted data of drag coefficient is shown in Fig. 10. 5. Conclusions In this present study the frictional pressure drop characteristic has been analyzed by a mechanistic model based on the viscous interaction and wetting of the non-Newtonian liquid. The increase of CMC concentration leads to the increase in effective viscosity which leads to increase the shear stress and increase of pressure drop in the packed bed. The intensity factor is a function of Reynolds number. Also as the concentration of CMC solution increases, the effective viscosity decreases at constant flow rate and results in decrease of intensity factor with increase in concentration of CMC solution. The intensity factor (αl) of the liquid–solid surface interfacial shear stress characterizes the intensity of the pressure drop in the fluid flow. The friction factor decreases with the increase in wettability. The energy loss due to wettability increases with the increase in surface tension and decrease in porosity. This is because of capillary effect of flowing of liquid through porous media. The surface tension increases with the increase in concentration of CMC which leads to increase in loss of energy due to wettability. The drag that is obtained by the present work is smaller than predicted by Ergun's equation and greater than Foscolo's equation. It is found that the present result is well approximate to the Ergun's result. The deviation may be the nonNewtonian characteristics of the fluid. The present study may be useful for further understanding and modeling of specific multiphase reactor in industrial applications.
250
Rep, [-]
ð40Þ
Nomenclature a Specific interfacial area [1/m] Ac Column cross-sectional area [m2] ah Hydraulic specific interfacial area of packing [1/m]
134
ap Cd Ch dc dp El Fd Ew fls flo g Kr K n Np Pls Pfls Pflo Rh STEYX Vis vp Vsl ΔZ
S.K. Patel, S.K. Majumder / Powder Technology 211 (2011) 127–134
Surface area of single particle [m2] Drag coefficient [−] Constant, defined in Eq. (21) [−] Column diameter [m] Particle diameter [m] Energy dispersion per unit packed column [N.m/s] Drag force [N] Energy loss due to wettability [N.m/s] Friction factor of liquid–solid system [−] Friction factor of single liquid system [−] Gravitational acceleration [m/s2] Permeability of porous medium [m2] Consistency of fluid [Pa.sn] Flow behavior index [−] Number of particles [−] Liquid–solid pressure [N/m2] Frictional pressure due to liquid flow [N/m2] Frictional pressure due to single liquid-phase [N/m2] Hydraulic radius [m] Standard error defined in Eq. (34) [−] Interfacial slip velocity [m/s] Volume of single particle [m3] Superficial liquid velocity [m/s] Height of packing [m]
Dimensionless groups Capillary number (=(μeffVsl)/(εσl)) [−] Ca Frl Liquid Froude No. (=V2sl/gdp) [−] Rel Liquid Reynolds No. (=(Vslρl)/(aμeff)) [−] Ren Non-Newtonian liquid Reynolds number ( = n 4n ) [−]
References
dnc Vsl2−n ρ 8n−1 K
3n + 1
Rep
Particle Reynolds number (=(ρlVsldp)/μeff) [−]
Greek letters αl Parameter defined in Eq. (9) [−] β Tortuosity factor [−] δflo Ratio of frictional pressure to hydrostatic pressure [−] δls Ratio of total pressure to hydrostatic pressure [−] ε Porosity [−] μeff Effective dynamic viscosity [kg/m.s] Φs Sphericity [−] ρl Density of liquid [kg/m3] σl Liquid surface tension [N/m] σls Surface tension between liquid and solid [N/m] τi Interfacial shear stress [N/m2] θ Liquid–solid contact angle [radian] Subscripts c column f frictional i interfacial l liquid phase ls liquid–solid 0 Single p particle s superficial, solid w wettability
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