- Email: [email protected]
Theoretical approach for a pressure drop in two-phase particle-laden flows
International Communications in Heat and Mass Transfer 34 (2007) 153 – 161 www.elsevier.com/locate/ichmt
Theoretical approach for a pressure drop in two-phase particle-laden flows ☆ Seyun Kim a , Kye Bock Lee b,⁎, Chung Gu Lee b , Seong-O Kim a b
a Fluid Engineering Division, Korea Atomic Energy Research Institute, 150 Dukjin-dong, Yuseong-gu, Daejeon, 305-353, Korea School of Mechanical Engineering, Chungbuk National University, 12 Gaeshin-dong, Heungduk-gu, Cheongju, Chungbuk, 361-763, Korea
Available online 15 December 2006
Abstract The purpose of this research is to develop an analytical model for a pressure drop per unit pipe length due to the turbulence modulations of a carrier phase which results from the presence of a dispersed phase in various types of diluted two-phase flows. The wake behind a particle, a particle size, the loading ratio and the density difference between two phases of a particle-laden flow were considered as significant parameters, which have an influence on the turbulence of a particle-laden flow, and the relative velocity of the laden particles was calculated by using a terminal velocity. The frictional pressure drop was formulated by using the force balance in the control volume by considering the shear stresses due to the presence of particles and an analogy of the shear stresses in the solid surfaces. The numerical results show a good agreement with the available experimental data and the model successfully predicted the mechanism of the pressure drop in the particle-laden flows. © 2006 Elsevier Ltd. All rights reserved. Keywords: Particle-laden; Gas-particle; Pressure drop; Turbulence modulation
1. Introduction A discrete flow means the state at which the second phase forms a flow in a continuous first phase flow. Continuous liquid and discrete gas form a bubbly flow, continuous gas and discrete liquid create a droplet flow, and a solid particle flow in a gas composes a particulate flow. In a liquid metal reactor the reaction products of supercritical carbon dioxide and sodium form gas-particle flows. Since a particle-laden shear flow has occurred frequently in industrial processes, many researchers have paid considerable attention to this phenomenon and lots of experimental researches and numerical analyses have been performed. In spite of the many researches, a model that can simulate precisely the particle-laden flow characteristics has not been developed as yet due to the complexity of the physical phenomena. Thus, there are certain peculiarities and inconsistencies in numerical researches about the turbulence aspects of twophase flows. The early experimental studies were mostly focused on a measurement of the pressure drop and the velocity distribution to develop the relevant correlations. Recently, by using up-to-date technologies, i.e. LDV and PIV, studies to analyze the fluid–particle interactions and correlations have been performed by turbulence measurements. ☆
Communicated by Dr. W.J. Minkowycz ⁎ Corresponding author. E-mail address: [email protected] (K.B. Lee).
0735-1933/$ - see front matter © 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2006.11.002
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S. Kim et al. / International Communications in Heat and Mass Transfer 34 (2007) 153–161
These theoretical studies have been conducted to develop a turbulence model to explain the particle-laden effect on the turbulence flow by using experimental results. Several of the analysis models proposed by a few researchers for a particle-laden flow, based on correlations of the experimental data, show limitations in their applications because of an insufficient comprehension of the physical phenomena of the particle-laden flows. To examine the effect of a laden particle size, Gore and Crowe [1] adopted a length scale ratio between the particle diameter, dp and turbulent length scale, l, which can be considered as the eddy size of an energy transportation. They showed a relationship between the ratio and the turbulence by using previous experimental data. When the ratio, dp/l is smaller than 0.1, the turbulence is attenuated. And when the ratio is larger than 0.1, the turbulence is augmented. Hetsroni [2] proposed that a turbulence is increased in the condition where the particle Reynolds number, Rep is about O (0.1), and a turbulence is suppressed when the particle Reynolds number is about O (1000), and the intermediate effect occurs in Rep ∼ O (100), by utilizing the experimental data of Tsuji et al. [3]. The particle Reynolds number is defined by the particle diameter and the relative velocity between the phases. Kenning and Crowe [4] suggested a simple model for a generation and an extinction of a turbulence kinetic energy by considering a turbulence length scale for the space between particles. However, these models need a lot of experimental data. Kim et al. [5] derived source terms of a turbulence kinetic energy based on the physical phenomena and developed, not a correlation of the experimental data but a physical model for turbulence intensity by using the mixing length model and the equilibrium flow conditions. For industrial applications, a pressure loss (or drop) is a significant design parameter. To calculate a pump capacity and to design a pipe system, a precise analysis of a system pressure distribution is essential. A pressure gradient is expressed as a summation of the frictional, gravitational and acceleration components in a single phase flow [6]. These components are attributed to different physical effects respectively. For a two-phase flow, the gravitational and acceleration pressure gradient terms contain a void fraction á through an analysis of the experimental data. For the internal two-phase flow, the frictional pressure loss term is usually dominant when compared to the other terms. To obtain a pressure loss, a simple modification of the correlation coefficient and a regression of the experimental data have been carried out such as in the study of Beattie and Whalley [7]. For this reason, because of the experimental conditions, the reliability of the correlation is decreased. Therefore, instead of merely a curve fitting of the limited experimental data of a particle-laden flow, a theoretical model of a pressure drop based on an investigation of the physical mechanisms of a turbulence variation due to laden particles is required. In the present research, to obtain the pressure drop per unit length in a particle-laden flow, a theoretical basis for the prediction of a pressure drop in two-phase flows has been proposed and shown to be useful in providing the design data for gas-particle flow systems. 2. Theoretical analysis To obtain a pressure drop due to laden particles, the force balance of each phase in a unit volume of a particle-laden flow is arranged as follows. The conceptual drawing is shown in Fig. 1. Dp −aqp g ¼ 0 Dl Dp −ð1−aÞqf g− es w ¼ 0 − es i þð1−aÞ Dl
es i þa
ð1Þ ð2Þ
Where, α is the volume fraction of a particle, and τ˜i is the shear stress between the phases in a unit volume, defined as follows. es i ¼
j j
2Cd aqf Ur Ur dp
ð3Þ
In Eqs. (1) and (2), the shear stresses are eliminated as follows. Dp −aqp g−ð1−aÞqf g þ es w ¼ 0 Dl
ð4Þ
S. Kim et al. / International Communications in Heat and Mass Transfer 34 (2007) 153–161
155
Fig. 1. Particle-laden flow in a pipe.
⁎ can be expressed as The relationship between the wall shear stress per unit area, τw and the wall shear velocity, uw Eq. (5) by a definition. *¼ uw
rffiffiffiffiffi sw qf
ð5Þ
Total shear stress is a summation of the wall shear stress and the shear stress between the phases. However, when the laden particles are diluted, the total shear stress can be approximated by the wall shear stress and the wall shear stress can be expressed by using the turbulent fluctuation velocity in turbulent flows [8]. * cu V uw
ð6Þ
Therefore, sw ¼ qf u V2
ð7Þ
This result is substituted into the equation for the force balance per unit pipe length, Eq. (4) as follows. Dp 4 ¼ aqp g þ ð1−aÞqf g þ Cs qf uV2 Dl D
ð8Þ
A wall shear stress is generated not only in a pipe wall but also on a particle surface where a continuous flow passes over it. Accordingly, by considering the total wall surface area and the particle surface area in a unit volume, the equation can be rearranged. Dp 3 D −1 4 ¼ aqp g þ ð1−aÞqf g þ Cs qf uV2 1 þ a Dl 2 dp D
ð9Þ
where, Cτ is an empirical constant. From Eq. (9), the relationship between the turbulent fluctuation velocity and the pressure drop can be derived. To develop an analytical model for an estimation of the turbulent fluctuation velocity in a particle-laden flow, an approximation of the equilibrium flow is adopted. In a fully developed particle-laden flow in a straight pipe, the lift force of a particle due to a velocity gradient is relatively small in the volume-averaged turbulence kinetic energy equation, and the effect is negligible. The approximation of an equilibrium flow is valid in a core region of a fully developed pipe flow. In this case, the convection term and the diffusion term can be ignored, and the
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S. Kim et al. / International Communications in Heat and Mass Transfer 34 (2007) 153–161
Fig. 2. Schematic of a wake flow.
source term and the sink term should be balanced. Consequently, an equation of the turbulence kinetic energy can be written as
j
f aqp Ui −Vi sr
f j −ð1−aÞq u u AU þ aq ð m m − u m Þ−ð1−aÞq e ¼ 0 s Ax 2
f
P i j
i
j
p
P i i
P i i
r
f
ð10Þ
The first term is a work for the carrier fluid by the drag caused by an acceleration and a turbulence kinetic energy source due to velocity defects in the wakes behind the particles, and this term is expressed as a velocity difference between the particles and the carrier flow [4]. To model the relative velocity between the phases, to calculate the relative velocity, the terminal velocity, which is derived from the kinetic equation of a free-falling particle, is supposed to be an approximated relative velocity between the phases [5]. Ur ¼ VT ¼
gsr qp dp2 g ¼ 18ldf f
ð11Þ
f is the drag factor. When a drag is not a Stokes drag, this factor supplements the differences between a Stokes drag and a particle drag. In the present study, a general use of the Schiller model for the drag factor was adopted [8]. f ¼ 1:0 þ 0:15Re0:687 p
ð12Þ
Accordingly, the first term of Eq. (10) can be modeled as follow. aqp
j
f Ui −Vi sr
j iaq sf V 2
p
r
2 T
ð13Þ
The second term of the left hand side of Eq. (10), which is the source term of the velocity gradient, must contain the effect of the fluctuations of a flow caused by the wake behind the laden particles. The schematic of the wake flow and a notation are depicted in Fig. 2. Velocity profiles in the wake made behind a particle in a uniform flow have a similarity function in the downstream of a particle [9]. The source term due to the velocity gradient can be expressed as Eq. (14) with the mixing length model. − ui uj P
j j
AUi AUi ¼ l2 Axj Axj
AUi 2 Axj
ð14Þ
S. Kim et al. / International Communications in Heat and Mass Transfer 34 (2007) 153–161
157
Fig. 3. Comparisons of the non-dimensional pressure drop with the experimental data.
Where, the velocity gradient is described with a similarity function, Ud −U Ul y ¼ ¼ f ðgÞ; g ¼ Ud −Um Ulm d
ð15Þ
and the function is given by f (η) = 1 − η3 / 2 [10]. AU AU Ag Ulm ¼ ¼ f VðgÞ Ay Ag Ay d
ð16Þ
Therefore 3 AUi U3 3 2 Ulm ½ f VðgÞ3 ¼ l 2 lm ½ f V ð gÞ ¼ b Axj d d3 ¼ Ax−2=3 ; d ¼ Bx1=3
− ui uj
ð17Þ
Ulm
ð18Þ
P
The dimensionless length scale β = l/δ is the ratio of the turbulent length scale for a half width of the wake, which is a function of the streamwise distance. To calculate the average value of the source of the velocity gradient, this term is integrated for the wake length and the wake width Z lw Z d Z lw 3 27 2 Ulm 27 b5 3 4 48 l 3 g3=2 ydy dx ¼ U d ð19Þ dx ¼ A3 Bb2 lw−2=3 lm 3 8 28 7 l d 0 0 0 Now, the generation of a turbulent kinetic energy of a wake per unit volume is l2
3 Ulm 120 A3 b2 −7=3 l ½ f VðgÞ3 ¼ 3 7 B w d
Where, as Yarin and Hetsroni proposed [11], the length of a wake is 2 !1=3 3 q p 5 lw ¼ dp 4X3 qf c
ð20Þ
ð21Þ
In Eq. (21), Ω is the order of unity determined by the experiments. The third term which means a redistribution of the kinetic energy can be neglected because it is small enough near the center of the pipe flow [12]. The last term is the dissipation rate of the kinetic energy and this term can be modeled with the kinetic energy and the
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S. Kim et al. / International Communications in Heat and Mass Transfer 34 (2007) 153–161
Fig. 4. Calculated pressure drops for glass particles of 1 mm and 2 mm. (a) Pressure drop per unit length vs. volume fraction of particles. (b) Pressure drop per unit length vs. volume fraction of particles.
characteristic length scale of a flow when the particles are laden. When particles are introduced into a flow, various length scales are used to characterize a turbulent flow. In single phase flows, the integral length scale and the Kolmogorov scale need to be considered. The wakes produced by the particles yield a length scale in the order of a particle size. If the concentration of the particles introduced into a flow yields an average interparticle spacing smaller than the inherent dissipation length scale, the particles may interfere with the existing eddies by breaking them up, so that the new dissipation length scale is proportional to the average interparticle spacing rather than the geometry of a pipe. Likewise in the particle-laden flow the particle size and the average space among the particles characterize the flow length scale. Namely, to model the dissipation rate of a particle-laden flow, the integral length scale of a single phase flow, l, and the mean distance among the particles which is a function of the concentration of a particle λ are used [5]. When the particles are laden, the energy dissipation depends on the eddies of a small scale being similar to the particle size and the average inter particle spacing for the case where the particle size is smaller than the inherent characteristic length scale of a flow. While for the case where the particle size is larger than the characteristic length scale of a flow, the inherent length scale is dominant. Hence, the length scale, which is corrected for an added particle is expressed with a harmonic mean of the two-length scale.
2lk k 1=3 lh ¼ ; kidp −1 ð22Þ lþk 6a
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159
Dissipation rate term is modeled based on this reasoning. ð1−aÞqf eið1−aÞqf
k 3=2 lh
ð23Þ
By rearranging Eq. (10), aqp
f 2 120 A3 b2 −7=3 k 3=2 lh lw ¼ ð1−aÞqf VT þ ð1−aÞqf sr 7 B lh
The turbulent kinetic energy of a particle-laden flow is " #2=3 3 2 aqp f 120 A b lh lw−7=3 lh VT2 − k¼ 7 B ð1−aÞqf sr
ð24Þ
ð25Þ
This model showed a good agreement with the experimental data [5]. When the kinetic energy is substituted into Eq. (9), the pressure drop per unit pipe length can be obtained. 3. Calculations and interpretation The calculated pressure drop by using Eq. (9) which is the developed pressure drop model for particle-laden flows is compared with the experimental data of Littman et al. [13] for a vertical pipe of 28.45 mm in inner diameter, in which glass particles and rapeseed particles are laden. In the experiments, the particle loading ratio, the particle diameter and the particle concentration are changed for 25 different cases. For the same conditions, the pressure drops of the model are evaluated with the experimental data. The pressure drops nondimensionalized with an inlet dynamic pressure of a carrier phase as expressed in Eq. (26) are calculated and compared with the experimental data. 1 Pin ¼ qf U 2 2
ð26Þ
In Fig. 3, the calculated results of the present model generally agree with the experimental data in various flow conditions. The total pressure drop consists of the body forces of a particle and a carrier fluid and a drag force at the walls as arranged in Eq. (9). Since the wake effect is significant for a relatively large particle size and the model reflects the wake effect well, this model shows a good agreement with the experiments in the case of a relatively large particle size. The dependency of the pressure drop on the volume fraction is presented in Fig. 4. From the experimental data, the pressure drop increases nearly proportionally with an increase of the volume fraction of particles in the ranges studied. The pressure drops per unit pipe length for glass particles of 1 mm and 2 mm were calculated in the experimental conditions. Fig. 4 shows that the analytically calculated pressure drop increases linearly as the volume fraction of a particle increases as predicted and the present model exhibits a satisfactory agreement with the experimental data. 4. Conclusions An analytical model for predicting a pressure drop in a dilute two-phase flow is developed from the force balance of each phase in a unit volume of a particle-laden flow. The model of a turbulence modulation by using the mixing length theory under the assumption of an equilibrium flow is used. The present model is developed based on a theoretical basis and it does not need a lot of experimental information and empirical constants. When the present model is compared to the experimental data, the pressure drop per unit length of a pipe shows a good agreement with the experimental data both qualitatively and quantitatively. Especially, when the particle size is relatively large, the model predicts the pressure drop accurately. The results show that the effects of a particle wake on
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the carrier phase are reflected correctly in the model. The calculated results can be applied to the fundamental data for a design of a gas-particle flow system and the safety analysis of a particle product flow in an advanced energy conversion system of nuclear reactors. And the suggested model can also be used to calculate the pressure drop not only in the particle-laden flow, but also in a droplet flow and a dispersed bubbly flow which have similar physical characteristics and mechanisms. Nomenclature d Mean diameter of dispersed phases f Drag factor g Gravitational acceleration k Turbulent kinetic energy of fluid l Characteristic length scale (integral) lh Hybrid length scale lw Length of wake u, ν Velocity fluctuation uw⁎ Wall shear velocity Cd Drag coefficient Cr Proportional constant Cτ Wall shear constant D Pipe diameter M Turbulence modulation Rer Relative Reynolds number (=Rep) U, V Mean velocity Ur Relative velocity between phases Ul Velocity defect in wake VT Terminal velocity of a falling particle α Volumetric fraction of dispersed phase β Ratio of length scale ε Dissipation rate of k λ Length scale of dispersed phase spacing ρ Density τi Interfacial shear stress τr Relaxation time τw Wall shear stress ΔP Pressure drop Δl Unit length of pipe corresponds to unit volume Subscript f Property of fluid (carrier) phase in two-phase flow p Property of dispersed phase in two-phase flow References [1] [2] [3] [4] [5] [6] [7] [8] [9]
R.A. Gore, C.T. Crowe, Effect of particle size on modulating turbulent intensity, Int. J. Multiph. Flow 15 (1989) 79–285. G. Hetsroni, Partcle–turbulence interaction, Int. J. Multiph. Flow 15 (1989) 735–746. Y. Tsuji, Y. Morikawa, H. Shiomi, LDV measurements of an air–solid two-phase flow in a vertical pipe, J. Fluid Mech. 139 (1984) 417–443. V.M. Kenning, C.T. Crowe, On the effect of particles on carrier phase turbulence in gas-particle flows, Int. J. Multiph. Flow 23 (1997) 403–408. S. Kim, K.B. Lee, C.G. Lee, Theoretical approach on the turbulence intensity of the carrier fluid in dilute two-phase flows, Int. Commun. Heat Mass Transf. 32 (2005) 435–444. P.B. Whalley, Two-Phase Flow and Heat Transfer, Oxford University Press, London, 1996. D.R. Beattie, P.B. Whalley, A simple two-phase frictional pressure drop calculation method, Int. J. Multiph. Flow 8 (1982) 83–87. T.G. Theofanous, J. Sullivan, Turbulence in two-phase dispersed flows, J. Fluid Mech. 116 (1982) 343–362. I. Kataoka, A. Serizawa, Basic equations of turbulence in gas–liquid two-phase flow, Int. J. Multiph. Flow 15 (1989) 843–855.
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[10] F.M. White, Viscous Fluid Flow, 2nd ed. McGraw-Hill, Singapore, 1991. [11] L.P. Yarin, G. Hetsroni, Turbulence intensity in dilute two-phase flows-3, Int. J. Multiph. Flow 20 (1994) 27–44. [12] K. Sengupta, K. Russell, W.J. Minkowycz, F. Mashayek, Numerical simulation data for assessment of particle-laden turbulent flow models, Int. J. Heat Mass Transf. 48 (2005) 3035–3046. [13] H. Littman, M.H. Morgan III, S.Dj. Jovanovic, J.D. Paccione, Z.B. Grbavcic, D.V. Vukovic, Effect of particle diameter, particle density and loading ratio on the effective drag coefficient in steady turbulent gas–solid transport, Powder Technol. 84 (1995) 49–56.
Theoretical approach for a pressure drop in two-phase particle-laden flows ☆ Seyun Kim a , Kye Bock Lee b,⁎, Chung Gu Lee b , Seong-O Kim a b
a Fluid Engineering Division, Korea Atomic Energy Research Institute, 150 Dukjin-dong, Yuseong-gu, Daejeon, 305-353, Korea School of Mechanical Engineering, Chungbuk National University, 12 Gaeshin-dong, Heungduk-gu, Cheongju, Chungbuk, 361-763, Korea
Available online 15 December 2006
Abstract The purpose of this research is to develop an analytical model for a pressure drop per unit pipe length due to the turbulence modulations of a carrier phase which results from the presence of a dispersed phase in various types of diluted two-phase flows. The wake behind a particle, a particle size, the loading ratio and the density difference between two phases of a particle-laden flow were considered as significant parameters, which have an influence on the turbulence of a particle-laden flow, and the relative velocity of the laden particles was calculated by using a terminal velocity. The frictional pressure drop was formulated by using the force balance in the control volume by considering the shear stresses due to the presence of particles and an analogy of the shear stresses in the solid surfaces. The numerical results show a good agreement with the available experimental data and the model successfully predicted the mechanism of the pressure drop in the particle-laden flows. © 2006 Elsevier Ltd. All rights reserved. Keywords: Particle-laden; Gas-particle; Pressure drop; Turbulence modulation
1. Introduction A discrete flow means the state at which the second phase forms a flow in a continuous first phase flow. Continuous liquid and discrete gas form a bubbly flow, continuous gas and discrete liquid create a droplet flow, and a solid particle flow in a gas composes a particulate flow. In a liquid metal reactor the reaction products of supercritical carbon dioxide and sodium form gas-particle flows. Since a particle-laden shear flow has occurred frequently in industrial processes, many researchers have paid considerable attention to this phenomenon and lots of experimental researches and numerical analyses have been performed. In spite of the many researches, a model that can simulate precisely the particle-laden flow characteristics has not been developed as yet due to the complexity of the physical phenomena. Thus, there are certain peculiarities and inconsistencies in numerical researches about the turbulence aspects of twophase flows. The early experimental studies were mostly focused on a measurement of the pressure drop and the velocity distribution to develop the relevant correlations. Recently, by using up-to-date technologies, i.e. LDV and PIV, studies to analyze the fluid–particle interactions and correlations have been performed by turbulence measurements. ☆
Communicated by Dr. W.J. Minkowycz ⁎ Corresponding author. E-mail address: [email protected] (K.B. Lee).
0735-1933/$ - see front matter © 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2006.11.002
154
S. Kim et al. / International Communications in Heat and Mass Transfer 34 (2007) 153–161
These theoretical studies have been conducted to develop a turbulence model to explain the particle-laden effect on the turbulence flow by using experimental results. Several of the analysis models proposed by a few researchers for a particle-laden flow, based on correlations of the experimental data, show limitations in their applications because of an insufficient comprehension of the physical phenomena of the particle-laden flows. To examine the effect of a laden particle size, Gore and Crowe [1] adopted a length scale ratio between the particle diameter, dp and turbulent length scale, l, which can be considered as the eddy size of an energy transportation. They showed a relationship between the ratio and the turbulence by using previous experimental data. When the ratio, dp/l is smaller than 0.1, the turbulence is attenuated. And when the ratio is larger than 0.1, the turbulence is augmented. Hetsroni [2] proposed that a turbulence is increased in the condition where the particle Reynolds number, Rep is about O (0.1), and a turbulence is suppressed when the particle Reynolds number is about O (1000), and the intermediate effect occurs in Rep ∼ O (100), by utilizing the experimental data of Tsuji et al. [3]. The particle Reynolds number is defined by the particle diameter and the relative velocity between the phases. Kenning and Crowe [4] suggested a simple model for a generation and an extinction of a turbulence kinetic energy by considering a turbulence length scale for the space between particles. However, these models need a lot of experimental data. Kim et al. [5] derived source terms of a turbulence kinetic energy based on the physical phenomena and developed, not a correlation of the experimental data but a physical model for turbulence intensity by using the mixing length model and the equilibrium flow conditions. For industrial applications, a pressure loss (or drop) is a significant design parameter. To calculate a pump capacity and to design a pipe system, a precise analysis of a system pressure distribution is essential. A pressure gradient is expressed as a summation of the frictional, gravitational and acceleration components in a single phase flow [6]. These components are attributed to different physical effects respectively. For a two-phase flow, the gravitational and acceleration pressure gradient terms contain a void fraction á through an analysis of the experimental data. For the internal two-phase flow, the frictional pressure loss term is usually dominant when compared to the other terms. To obtain a pressure loss, a simple modification of the correlation coefficient and a regression of the experimental data have been carried out such as in the study of Beattie and Whalley [7]. For this reason, because of the experimental conditions, the reliability of the correlation is decreased. Therefore, instead of merely a curve fitting of the limited experimental data of a particle-laden flow, a theoretical model of a pressure drop based on an investigation of the physical mechanisms of a turbulence variation due to laden particles is required. In the present research, to obtain the pressure drop per unit length in a particle-laden flow, a theoretical basis for the prediction of a pressure drop in two-phase flows has been proposed and shown to be useful in providing the design data for gas-particle flow systems. 2. Theoretical analysis To obtain a pressure drop due to laden particles, the force balance of each phase in a unit volume of a particle-laden flow is arranged as follows. The conceptual drawing is shown in Fig. 1. Dp −aqp g ¼ 0 Dl Dp −ð1−aÞqf g− es w ¼ 0 − es i þð1−aÞ Dl
es i þa
ð1Þ ð2Þ
Where, α is the volume fraction of a particle, and τ˜i is the shear stress between the phases in a unit volume, defined as follows. es i ¼
j j
2Cd aqf Ur Ur dp
ð3Þ
In Eqs. (1) and (2), the shear stresses are eliminated as follows. Dp −aqp g−ð1−aÞqf g þ es w ¼ 0 Dl
ð4Þ
S. Kim et al. / International Communications in Heat and Mass Transfer 34 (2007) 153–161
155
Fig. 1. Particle-laden flow in a pipe.
⁎ can be expressed as The relationship between the wall shear stress per unit area, τw and the wall shear velocity, uw Eq. (5) by a definition. *¼ uw
rffiffiffiffiffi sw qf
ð5Þ
Total shear stress is a summation of the wall shear stress and the shear stress between the phases. However, when the laden particles are diluted, the total shear stress can be approximated by the wall shear stress and the wall shear stress can be expressed by using the turbulent fluctuation velocity in turbulent flows [8]. * cu V uw
ð6Þ
Therefore, sw ¼ qf u V2
ð7Þ
This result is substituted into the equation for the force balance per unit pipe length, Eq. (4) as follows. Dp 4 ¼ aqp g þ ð1−aÞqf g þ Cs qf uV2 Dl D
ð8Þ
A wall shear stress is generated not only in a pipe wall but also on a particle surface where a continuous flow passes over it. Accordingly, by considering the total wall surface area and the particle surface area in a unit volume, the equation can be rearranged. Dp 3 D −1 4 ¼ aqp g þ ð1−aÞqf g þ Cs qf uV2 1 þ a Dl 2 dp D
ð9Þ
where, Cτ is an empirical constant. From Eq. (9), the relationship between the turbulent fluctuation velocity and the pressure drop can be derived. To develop an analytical model for an estimation of the turbulent fluctuation velocity in a particle-laden flow, an approximation of the equilibrium flow is adopted. In a fully developed particle-laden flow in a straight pipe, the lift force of a particle due to a velocity gradient is relatively small in the volume-averaged turbulence kinetic energy equation, and the effect is negligible. The approximation of an equilibrium flow is valid in a core region of a fully developed pipe flow. In this case, the convection term and the diffusion term can be ignored, and the
156
S. Kim et al. / International Communications in Heat and Mass Transfer 34 (2007) 153–161
Fig. 2. Schematic of a wake flow.
source term and the sink term should be balanced. Consequently, an equation of the turbulence kinetic energy can be written as
j
f aqp Ui −Vi sr
f j −ð1−aÞq u u AU þ aq ð m m − u m Þ−ð1−aÞq e ¼ 0 s Ax 2
f
P i j
i
j
p
P i i
P i i
r
f
ð10Þ
The first term is a work for the carrier fluid by the drag caused by an acceleration and a turbulence kinetic energy source due to velocity defects in the wakes behind the particles, and this term is expressed as a velocity difference between the particles and the carrier flow [4]. To model the relative velocity between the phases, to calculate the relative velocity, the terminal velocity, which is derived from the kinetic equation of a free-falling particle, is supposed to be an approximated relative velocity between the phases [5]. Ur ¼ VT ¼
gsr qp dp2 g ¼ 18ldf f
ð11Þ
f is the drag factor. When a drag is not a Stokes drag, this factor supplements the differences between a Stokes drag and a particle drag. In the present study, a general use of the Schiller model for the drag factor was adopted [8]. f ¼ 1:0 þ 0:15Re0:687 p
ð12Þ
Accordingly, the first term of Eq. (10) can be modeled as follow. aqp
j
f Ui −Vi sr
j iaq sf V 2
p
r
2 T
ð13Þ
The second term of the left hand side of Eq. (10), which is the source term of the velocity gradient, must contain the effect of the fluctuations of a flow caused by the wake behind the laden particles. The schematic of the wake flow and a notation are depicted in Fig. 2. Velocity profiles in the wake made behind a particle in a uniform flow have a similarity function in the downstream of a particle [9]. The source term due to the velocity gradient can be expressed as Eq. (14) with the mixing length model. − ui uj P
j j
AUi AUi ¼ l2 Axj Axj
AUi 2 Axj
ð14Þ
S. Kim et al. / International Communications in Heat and Mass Transfer 34 (2007) 153–161
157
Fig. 3. Comparisons of the non-dimensional pressure drop with the experimental data.
Where, the velocity gradient is described with a similarity function, Ud −U Ul y ¼ ¼ f ðgÞ; g ¼ Ud −Um Ulm d
ð15Þ
and the function is given by f (η) = 1 − η3 / 2 [10]. AU AU Ag Ulm ¼ ¼ f VðgÞ Ay Ag Ay d
ð16Þ
Therefore 3 AUi U3 3 2 Ulm ½ f VðgÞ3 ¼ l 2 lm ½ f V ð gÞ ¼ b Axj d d3 ¼ Ax−2=3 ; d ¼ Bx1=3
− ui uj
ð17Þ
Ulm
ð18Þ
P
The dimensionless length scale β = l/δ is the ratio of the turbulent length scale for a half width of the wake, which is a function of the streamwise distance. To calculate the average value of the source of the velocity gradient, this term is integrated for the wake length and the wake width Z lw Z d Z lw 3 27 2 Ulm 27 b5 3 4 48 l 3 g3=2 ydy dx ¼ U d ð19Þ dx ¼ A3 Bb2 lw−2=3 lm 3 8 28 7 l d 0 0 0 Now, the generation of a turbulent kinetic energy of a wake per unit volume is l2
3 Ulm 120 A3 b2 −7=3 l ½ f VðgÞ3 ¼ 3 7 B w d
Where, as Yarin and Hetsroni proposed [11], the length of a wake is 2 !1=3 3 q p 5 lw ¼ dp 4X3 qf c
ð20Þ
ð21Þ
In Eq. (21), Ω is the order of unity determined by the experiments. The third term which means a redistribution of the kinetic energy can be neglected because it is small enough near the center of the pipe flow [12]. The last term is the dissipation rate of the kinetic energy and this term can be modeled with the kinetic energy and the
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Fig. 4. Calculated pressure drops for glass particles of 1 mm and 2 mm. (a) Pressure drop per unit length vs. volume fraction of particles. (b) Pressure drop per unit length vs. volume fraction of particles.
characteristic length scale of a flow when the particles are laden. When particles are introduced into a flow, various length scales are used to characterize a turbulent flow. In single phase flows, the integral length scale and the Kolmogorov scale need to be considered. The wakes produced by the particles yield a length scale in the order of a particle size. If the concentration of the particles introduced into a flow yields an average interparticle spacing smaller than the inherent dissipation length scale, the particles may interfere with the existing eddies by breaking them up, so that the new dissipation length scale is proportional to the average interparticle spacing rather than the geometry of a pipe. Likewise in the particle-laden flow the particle size and the average space among the particles characterize the flow length scale. Namely, to model the dissipation rate of a particle-laden flow, the integral length scale of a single phase flow, l, and the mean distance among the particles which is a function of the concentration of a particle λ are used [5]. When the particles are laden, the energy dissipation depends on the eddies of a small scale being similar to the particle size and the average inter particle spacing for the case where the particle size is smaller than the inherent characteristic length scale of a flow. While for the case where the particle size is larger than the characteristic length scale of a flow, the inherent length scale is dominant. Hence, the length scale, which is corrected for an added particle is expressed with a harmonic mean of the two-length scale.
2lk k 1=3 lh ¼ ; kidp −1 ð22Þ lþk 6a
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Dissipation rate term is modeled based on this reasoning. ð1−aÞqf eið1−aÞqf
k 3=2 lh
ð23Þ
By rearranging Eq. (10), aqp
f 2 120 A3 b2 −7=3 k 3=2 lh lw ¼ ð1−aÞqf VT þ ð1−aÞqf sr 7 B lh
The turbulent kinetic energy of a particle-laden flow is " #2=3 3 2 aqp f 120 A b lh lw−7=3 lh VT2 − k¼ 7 B ð1−aÞqf sr
ð24Þ
ð25Þ
This model showed a good agreement with the experimental data [5]. When the kinetic energy is substituted into Eq. (9), the pressure drop per unit pipe length can be obtained. 3. Calculations and interpretation The calculated pressure drop by using Eq. (9) which is the developed pressure drop model for particle-laden flows is compared with the experimental data of Littman et al. [13] for a vertical pipe of 28.45 mm in inner diameter, in which glass particles and rapeseed particles are laden. In the experiments, the particle loading ratio, the particle diameter and the particle concentration are changed for 25 different cases. For the same conditions, the pressure drops of the model are evaluated with the experimental data. The pressure drops nondimensionalized with an inlet dynamic pressure of a carrier phase as expressed in Eq. (26) are calculated and compared with the experimental data. 1 Pin ¼ qf U 2 2
ð26Þ
In Fig. 3, the calculated results of the present model generally agree with the experimental data in various flow conditions. The total pressure drop consists of the body forces of a particle and a carrier fluid and a drag force at the walls as arranged in Eq. (9). Since the wake effect is significant for a relatively large particle size and the model reflects the wake effect well, this model shows a good agreement with the experiments in the case of a relatively large particle size. The dependency of the pressure drop on the volume fraction is presented in Fig. 4. From the experimental data, the pressure drop increases nearly proportionally with an increase of the volume fraction of particles in the ranges studied. The pressure drops per unit pipe length for glass particles of 1 mm and 2 mm were calculated in the experimental conditions. Fig. 4 shows that the analytically calculated pressure drop increases linearly as the volume fraction of a particle increases as predicted and the present model exhibits a satisfactory agreement with the experimental data. 4. Conclusions An analytical model for predicting a pressure drop in a dilute two-phase flow is developed from the force balance of each phase in a unit volume of a particle-laden flow. The model of a turbulence modulation by using the mixing length theory under the assumption of an equilibrium flow is used. The present model is developed based on a theoretical basis and it does not need a lot of experimental information and empirical constants. When the present model is compared to the experimental data, the pressure drop per unit length of a pipe shows a good agreement with the experimental data both qualitatively and quantitatively. Especially, when the particle size is relatively large, the model predicts the pressure drop accurately. The results show that the effects of a particle wake on
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the carrier phase are reflected correctly in the model. The calculated results can be applied to the fundamental data for a design of a gas-particle flow system and the safety analysis of a particle product flow in an advanced energy conversion system of nuclear reactors. And the suggested model can also be used to calculate the pressure drop not only in the particle-laden flow, but also in a droplet flow and a dispersed bubbly flow which have similar physical characteristics and mechanisms. Nomenclature d Mean diameter of dispersed phases f Drag factor g Gravitational acceleration k Turbulent kinetic energy of fluid l Characteristic length scale (integral) lh Hybrid length scale lw Length of wake u, ν Velocity fluctuation uw⁎ Wall shear velocity Cd Drag coefficient Cr Proportional constant Cτ Wall shear constant D Pipe diameter M Turbulence modulation Rer Relative Reynolds number (=Rep) U, V Mean velocity Ur Relative velocity between phases Ul Velocity defect in wake VT Terminal velocity of a falling particle α Volumetric fraction of dispersed phase β Ratio of length scale ε Dissipation rate of k λ Length scale of dispersed phase spacing ρ Density τi Interfacial shear stress τr Relaxation time τw Wall shear stress ΔP Pressure drop Δl Unit length of pipe corresponds to unit volume Subscript f Property of fluid (carrier) phase in two-phase flow p Property of dispersed phase in two-phase flow References [1] [2] [3] [4] [5] [6] [7] [8] [9]
R.A. Gore, C.T. Crowe, Effect of particle size on modulating turbulent intensity, Int. J. Multiph. Flow 15 (1989) 79–285. G. Hetsroni, Partcle–turbulence interaction, Int. J. Multiph. Flow 15 (1989) 735–746. Y. Tsuji, Y. Morikawa, H. Shiomi, LDV measurements of an air–solid two-phase flow in a vertical pipe, J. Fluid Mech. 139 (1984) 417–443. V.M. Kenning, C.T. Crowe, On the effect of particles on carrier phase turbulence in gas-particle flows, Int. J. Multiph. Flow 23 (1997) 403–408. S. Kim, K.B. Lee, C.G. Lee, Theoretical approach on the turbulence intensity of the carrier fluid in dilute two-phase flows, Int. Commun. Heat Mass Transf. 32 (2005) 435–444. P.B. Whalley, Two-Phase Flow and Heat Transfer, Oxford University Press, London, 1996. D.R. Beattie, P.B. Whalley, A simple two-phase frictional pressure drop calculation method, Int. J. Multiph. Flow 8 (1982) 83–87. T.G. Theofanous, J. Sullivan, Turbulence in two-phase dispersed flows, J. Fluid Mech. 116 (1982) 343–362. I. Kataoka, A. Serizawa, Basic equations of turbulence in gas–liquid two-phase flow, Int. J. Multiph. Flow 15 (1989) 843–855.
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[10] F.M. White, Viscous Fluid Flow, 2nd ed. McGraw-Hill, Singapore, 1991. [11] L.P. Yarin, G. Hetsroni, Turbulence intensity in dilute two-phase flows-3, Int. J. Multiph. Flow 20 (1994) 27–44. [12] K. Sengupta, K. Russell, W.J. Minkowycz, F. Mashayek, Numerical simulation data for assessment of particle-laden turbulent flow models, Int. J. Heat Mass Transf. 48 (2005) 3035–3046. [13] H. Littman, M.H. Morgan III, S.Dj. Jovanovic, J.D. Paccione, Z.B. Grbavcic, D.V. Vukovic, Effect of particle diameter, particle density and loading ratio on the effective drag coefficient in steady turbulent gas–solid transport, Powder Technol. 84 (1995) 49–56.