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Concentration distribution and viscosity of ice-slurry in heterogeneous flow
International Journal of Refrigeration 25 (2002) 827–835 www.elsevier.com/locate/ijrefrig
Concentration distribution and viscosity of ice-slurry in heterogeneous flow Andrej Kitanovski*,1, Alojz Poredosˇ 2 Faculty of Mechanical Engineering, Laboratory for Refrigeration, Asˇkercˇeva Ljubljana, Slovenia Received 12 May 2001; received in revised form 12 August 2001; accepted 11 September 2001
Abstract Mathematical modeling of two phase flows, especially liquid–solid flows is very complex. Especially when a distribution of the solid phase in a carrier liquid is not homogenous but heterogeneous or even when a moving or stationary bed occurs. In this case, the rheological characteristics of suspension are changing and affect transport characteristics. Therefore, the slurry flow may present a Newtonian and non-Newtonian fluid as well, depending on the operation characteristics. In this paper the fully suspended ice-slurry flow in a horizontal pipe is analysed. The model allows us to avoid the definition on what kind of fluid ice-slurry is present. For the taken ice-particle diameter, the iceconcentration profiles depending on various average velocities and pipe diameters are shown. The viscosity of the iceslurry is presented, depending on average concentration, velocity, pipe diameter and ice-particle size. The results of the analysis have shown that the ice slurrys can be treated as Newtonian-fluid at higher average velocities, and lower average concentrations as well. As the ice concentration increases and velocity decreases the viscosity depends not only on the ice concentration but also on the average velocity and the pipe diameter. The ice-slurry behaves then as a nonNewtonian fluid. The results show also the area where the safe operation of an ice-slurry-district-cooling system can be performed. # 2002 Elsevier Science Ltd and IIR. All rights reserved. Keywords: Ice slurry; Viscosity; Concentration; Flow; District cooling
Coulis de glace en e´coulement he´te´roge`ne : distribution de la concentration et de la viscosite´ Mots cle´s : Coulis de glace ; Viscosite´ ; Concentration ; E´coulement ; Refroidissement urbain
1. Introduction Many different models have been used for the suspension viscosity determination. Most of them essentially
extend the work of Einstein on spheres and his equation for viscosity (CV <0.01)[1]: ¼ liquid ð1 þ 2:5CÞ
* Corresponding author. Tel.: +386-1-4771-418; fax: +386-1-4771-448. E-mail addresses: [email protected] (A. Kitanovski), [email protected] (A. Poredosˇ ). 1 Member of the Ice-Slurry Working Party. 2 Member of the IIR E2 Commission. 0140-7007/02/$22.00 # 2002 Elsevier Science Ltd and IIR. All rights reserved. PII: S0140-7007(01)00091-3
ð1Þ
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Nomenclature A C CD Cmax CS D dp fi g rh Re s v* v w0 w
Greek " (y)
area (m2) local concentration drag coefficient maximum concentration on the top of the pipe average concentration pipe diameter (m) ice particle diameter (m) friction coefficient gravity constant (m/s2) hydraulic diameter (m) Reynolds number density ratio between the ice and carrier fluid shear velocity (m/s) average velocity (m/s) terminal settling velocity (m/s) hindered terminal settling velocity of cluster of particles(m/s)
diffusion coefficient (m2/s) viscosity (Pas) local viscosity (Pas) average viscosity (Pas)
where C is the concentration of the solid phase in a solid–liquid mixture. In Eq. (1), there is no effect of particle size, nor of particle position, because the theory neglects the effects of other particles. There were several other models developed for the viscosity determination, as shown in Table 1. The most popular determination of the suspension viscosity, which comprises not only the concentration of the solid phase, but also the interaction between the solid particles, is based on the well-known Thomas equation [3]: ¼ liquid 1 þ 2:5C þ 10:05C2 þ 0:00273e16:6C
viscosities. Later, Kauffeld [6] had reported on the ice particle size distribution experiments at DTI. From the rheology on suspensions it is known, that viscosity depends also on particle size and its shape. From the work of Thomas [3] the influence of different particle sizes on viscosity at concentrations C< 20% resulted in a 6% difference of relative viscosity. When the concentration is further increased until it reaches the maximum packing concentration, the influence of particles becomes more important. So, when the flow is heterogeneous, the particle size has an important role, especially at very low average velocities, when the concentration on the top of the pipe (in a case of ice-slurry) is increasing to high values. For concentrations C<25%, Thomas [3] has observed, that the first three terms in his equation accounted over 97.5% of the value of the relative viscosity. For concentrations C> 25%, first three terms in his equation account for very small percentages of the relative viscosity, e.g. 60% at C=40% and 8.7% at C=60%. The interaction forces in the heterogeneous flow become, therefore, more important than in homogeneous, so the local ice-slurry viscosity on the top of the pipe increases rapidly and influence the average value of viscosity. In the past different rheological models have been used for the ice-slurry. One of first tries was performed by Sasaki [7], where he used a dilatant fluid model to describe a rheological behavior of ice-slurry. After that, Egolf et al. [8] proposed the Bingham fluid model, which was taken also by many other authors, such as in Refs. [4,5,9]. All the experimental data on ice-slurry rheological behavior had shown that at a certain low ice concentration the ice-slurry behaves as a Newtonian fluid and at a higher concentrations as a Bingham fluid. To describe both, one of the possibilities is to use the apparent viscosity as it was shown by Guilpart et al. [10]. Another proposal was done by Jensen et al. [11], who had actually modified the Bingham model. Later, a model was proposed by Doetsch [12], where he proposed the Casson model to describe Newtonian and Bingham behavior of ice-slurry. However, none of these models consider the ice-slurry in the heterogeneous flow.
ð2Þ 2. Mathematical model
The model is valid for the concentrations up to C=0.625 and particle size ranging from 0.099 to 435 mm. It considers that the flow is homogenous. The Thomas equation has been widely used almost by all researchers on ice-slurry. However, this equation overpredicts the viscosity of ice-slurry at C> 15%, as was shown by Hansen [4]. Hansen used the Jeffrey’s equation with the constant A=4.5 to get the best fit with experimental results. At the same time, Frei [5] observed the time-dependent behavior of ice-slurry, which resulted in different sizes of ice particles and consequently different
Our model in this paper is based on the Doron‘s two [13] and three [14] layer model, which is based primarily on the solids concentration distribution. The analysis was performed for the ice-particle size dp=1 mm and carrier liquid of 10% ethanol/water mixture for various average velocities v (ranging from 0.25 to 2 m/s), ice concentrations CS (ranging from 0.05 to 0.25) and pipe diameters D (27.2, 100, 200 mm). Let us assume that the ice-slurry flow is fully suspended and homogenous in each tiny volume (Fig. 1).
A. Kitanovski, A. Poredosˇ / International Journal of Refrigeration 25 (2002) 827–835
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Table 1 Expressions for Newtonian suspension viscosity as a function of solids volume fraction CV [2]
1 þ 0:5C 0:5C2 1 2C 9:6C2 2:5C ¼ liquid exp 1 0:609C
Guth and Simha
Predicts ! 1 for C=0.234
¼ liquid
Vand
¼ liquid exp
2:5C þ 2:7C2 1 0:609C
No interparticle forces Includes doublet collisions, but not triplet
2:5C 0.75
Mooney
¼ liquid exp
Simha
Frankel and Acrivos
54 C C ! Cmax ¼ liquid 1 þ 3 5f 1 ðC=Cmax Þ3 1 !1 C 3 ¼ liquid C 1 Cmax ¼ liquid ð1 þ ACÞ 2.5
Jeffrey
K depends upon system
Dilute suspensions
Very concentrated
Concentrated suspensions only
Ellipsoid particles
Then Eq. (2) can be described with local concentration as a function of the y-axis: ðyÞ ¼ liquid 1 þ 2:5CðyÞ þ 10:05C2 ðyÞ þ 0:00273e16:6CðyÞ ð3Þ The concentration profile can be determined using the diffusion equation of the turbulent flow, which was originally determined in the vertical direction by Schmidt and Rouse [15]: "
d2 CðyÞ dCðyÞ ¼0 þw dy2 dy
ð4Þ
Fig. 1. Homogeneous and heterogeneous flow in a pipe. Fig. 1. Homogeneous and heterogeneous flow in a pipe.
where " is the local diffusion coefficient and w is the local hindered terminal velocity of the ice-particle. In our case it was assumed that these are the mean coefficients. The concentration profile can be estimated as [13]: w
CðyÞ ¼ Cmax eð " yÞ
ð5Þ
where Cmax is the maximum concentration on the top of the pipe (Fig. 2). The maximum concentration in the fully suspended flow is not equal to the maximum packing concentration, which is presented later in a paper. Since we know the average concentration, the maximum concentration can be determined as: Cmax ¼ Ð A
CS A w" Þy ð e dAðyÞ
ð6Þ
where CS is the average ice concentration (in situ concentration) and D is the diameter of the pipe. Doron [13] determined the main diffusion coefficient " by equation: " ¼ 0:052v rh
ð7Þ
where rh is the hydraulic radius (in our case rh=D/2) and v* is the shear velocity [13]: rffiffiffi fi v ¼ v 2
ð8Þ
where fi is the friction coefficient, suggested by Doron [13] to be evaluated from the modified Colebrook formula, where the friction coefficient should be multiplied by 2 (modified by Televantos):
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Fig. 2. The concentration profile of ice-slurry flow.
1 dp 2; 51 p ffiffi ffi pffiffiffiffiffiffi ¼ 0:86 ln þ 3:7D Re 2fi 2fi
ð9Þ
where dp is an ice particle diameter and Re is the Reynolds number based on viscosity and density of a carrier liquid. Doron [13] as many other authors [16,17] suggested that the hindered terminal settling velocity of a cluster of particles w can be calculated using the well known Richardson and Zaki [18] correlation: w ¼ w0 ð1 CS Þm
ð10Þ
where parameter for 1 < Rew < 500, m ¼ m=4.45 Re0.1 w d 4:36 þ 17:6 Dp Re0:03 for 0.2
of the maximum packing fractions in slurries, as shown in Table 2. Doron [13,14] has determined the maximum packing concentration as 0.52, which is the concentration of the monodispersed spheres in a simple cube. The values for the maximum packing fractions vary in a range from 0.5 to 0.75 for monodisperse spheres. In our case it was taken, that the maximum ice packing fraction equals 0.52.
3. Concentration distribution in the heterogeneous ice-slurry flow It was assumed in our case, that there is no variation of the concentration distribution in the downstream direction. This can be satisfied by assumption that the horizontal pipe is long enough to have a fully developed flow, and that there is no influence of the heat transfer or pressure drop along the pipe on the concentration profile. Nasr-El-Din [15] has found in his experiment with polystyrene and sand, that there was almost no lateral variation of concentration for the polystyrene particle size of 0.3 mm and CS < 40% (vbulk=2 m/s). At the same bulk-velocity and particle diameter of 1.4 mm the lateral variation of concentration was near to zero for CS < 27%. Since the polystyrene density is close to that of ice, another assumption was made in this paper, that there is no lateral variation of the concentration distribution in the ice-slurry for CS <25%. Calculating Eqs. (4)–(11) one may calculate different concentration profiles, depending on a pipe diameter, ice particle size, average concentration and velocity. At the velocity v=2 m/s the ice slurry flow is near the homogenous distribution of the ice concentration (Fig. 3). Decreasing the velocity below 0.5 m/s, the transition from the heterogeneous flow to that with a moving bed occurs, because the maximum packing fraction on the top of the pipe occurred. Increasing the average concentration of the ice-slurry, at the same pipe diameter, the transition
A. Kitanovski, A. Poredosˇ / International Journal of Refrigeration 25 (2002) 827–835 Table 2 The maximum packing fraction of various arrangements of monodispersed spheres [1] Arrangement
Maximum packing fraction
Simple cube
0.52
Maximum thermodynamically stable configuration
0.548
Hexagonally packed sheets just touching
0.605
Random close packing
0.637
Face-centred cubic/ hexagonal close packed
0.68
Body-centred cubic/ hexagonal close packed
0.74
from the homogenous to heterogeneous flow occurs earlier, as shown in Fig. 4. However, as the ice-particles are very small it is possible that the transition from the heterogeneous to that with a moving bed never occurs [20]. In this case a stationary bed
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occurs and the flow is tending to stop, since more and more ice particles are added.
4. Ice-slurry viscosity As was mentioned before, there are many different models to describe the viscosity of ice-slurry. However, none of them considers ice-slurry as heterogeneous mixture or even flow with a moving bed. When a maximum packing fraction at the top of the pipe occurs, more and more particles are added, the transition from the heterogeneous flow to that with a moving or even stationary bed occurs and tends to ‘‘jam up’’. The viscosity goes in this case towards infinity. There are some models, which describe the so called ‘‘crowding effect’’, such as Ball‘s [1] and Richmond‘s equation for viscosity, Krieger and Dougherty‘s [1] theory and others. Using the Thomas equation we want to show a new approach in determining the viscosity of the heterogeneous ice-slurry flow. There are many other equations or their possible modifications, which could be used in our model. Our model has validation only at parameters when the maximum packing fraction on the top of the pipe is not reached. But it can show the transition from the heterogeneous flow to that with a moving bed, so it can be used to determine ‘‘safe operation’’. Note, that the viscosity presented in Figs. 5–7
Fig. 3. Concentration profiles of ice slurry flow in a horizontal pipe (CS=0.1, D=27.2 mm, dp=1 mm). Fig. 1. Concentration profiles of ice slurry flow in a horizontal pipe (CS=0.1, D=27.2 mm, dp=1 mm)..
Fig. 4. Concentration profiles of ice slurry flow in a horizontal pipe (CS=0.25, D=27.2 mm, dp=1 mm). Fig. 1. Concentration profiles of ice slurry flow in a horizontal pipe (CS=0.25, D=27.2 mm, dp=1 mm)..
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is effective viscosity, so the viscosity variations depending on velocity, particle diameter and pipe diameter are effected almost only by interaction terms in Eq. (3). For very small ice concentrations (e.g. CS < 0.1) the ice-slurry viscosity is almost independent of the average velocities (Fig. 5). When the concentration increases and the velocity decreases the viscosity increases rapidly. The transitional area in Fig. 5 presents the area where the transition from the heterogeneous flow to that with a moving or stationary bed occurs. Increasing the pipe diameter from D=27.2 to D=100 mm (at the same average velocities) the effect of smaller pipe diameter almost does not influence a viscosity at higher velocities, especially at lower ice concentrations (Fig. 6). When the ice concentrations are higher and velocities lower (e.g. v=1 m/s), the viscosity is increasing so it depends on velocity, concentration and the pipe diameter as well. When the pipe diameter is further increased to D=200 mm, the moving bed occurs a bit earlier. At lower average velocities viscosity is much higher than at smaller diameters (Fig. 7). Note, however, that in our analysis the Einstein‘s equation or Jeffrey’s equation for the local viscosity determination in heterogeneous flow cannot be used, more than so because they do not comprise the interaction term between the ice particles, which becomes very important at lower average velocities and higher average concentrations. On the other hand, when the velocity is high and the flow is close to homogeneous concentration distribution, the effect of interaction terms is so low, so Jeffrey´s equation can be used for viscosity determination.
5. Effect of ice-particle size Another calculation was made to describe the effect of ice-particle size on effective viscosity of ice-slurry. When the flow is near to the homogeneous, the effects of iceparticle diameter and pipe diameter on viscosity is very low, almost negligible (Fig. 8). Actually, one must notice, that Figs. 8 and 9 are showing only the effect of ice-particle size on concentration distribution and, therefore, on viscosity. The Thomas´ equation depends not only on the concentration, but also on the constants, which were calculated for his case. That means, that the constants should be calculated again for the ice-slurry at particular ice-particle diameter in order to give new equations for the ice-slurry, which could be also used as a local viscosity equations. When the flow is homogeneous, Jeffrey´s equation could be used, where the parameter A should vary with the ice-particle size or even shape. However, when the flow is heterogeneous, the interaction terms should be involved, similar to how it was done by Thomas. When the velocity is lower, e.g. 0.75 m/s, the maximum concentration on the top of the pipe is very high (Figs. 3 and 4), so the influence of ice-particle size on viscosity is higher than at velocities v=2 m/s (Fig. 9). However, the difference between the viscosity at dp=0.25 mm and dp=1 mm (D=200 mm) is 6.2%, which is still very low, but not negligible. Especially because the constants (which could be determined by experiment for particular ice-particle size) in the Thomas equation could have together with the effect of con-
Fig. 5. Ice slurry viscosity (D=27.2 mm, dp=1 mm, 10% water–ethanol mixture). Fig. 1. Ice slurry viscosity (D=27.2 mm, dp=1 mm, 10% water-ethanol mixture).
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Fig. 6. Ice slurry viscosity (D=100 mm, dp=1 mm, 10% water–ethanol mixture). Fig. 1. Ice slurry viscosity (D=100 mm, dp=1 mm, 10% water–ethanol mixture).
Fig. 7. Ice slurry viscosity (D=200 mm, dp=1 mm, 10% water–ethanol mixture). Fig. 1. Ice slurry viscosity (D=200 mm, dp=1 mm, 10% water–ethanol mixture).
centration distribution (shown in our case) a much higher influence on viscosity at lower velocities. We assume, that in that case, for given dp=0.25 and 1 mm (D=200 mm, v=0.75 m/s), the differencies in viscosities could be higher than 30%.
6. Conclusion The results of this analysis have shown that at higher average velocities and very low ice concentrations (CS < 0.1) the ice slurry viscosity is almost independant
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Fig. 8. Viscosity depending on different pipe and ice-particle diameters (CS=20%, v=2 m/s).
Fig. 9. Viscosity depending on different pipe and ice-particle diameters (CS=20%, v=0.75 m/s).
of velocity as for Newtonian fluids. Similar results were presented by [10] for apparent viscosity. When the ice concentration is further increased, the viscosity depends not only on the ice concentration, but also on the average velocity and pipe diameter as well. The dependence is much stronger at higher ice-concentrations and lower average velocities. At lower average velocities, the viscosity increases with increased pipe diameter. The transitional area in Figs. 5–7 presents the transition from the heterogeneous flow to that with a moving or stationary bed. It enables us to determine the ‘‘safe operation’’ parameters. One must note that since the ice particles are very small, it is quite possible that the transition from the heterogeneous flow to the flow with a stationary bed occurs directly, without a moving bed formation. Snoek [21] has reported on the experimental results on the pressure drop, depending on average velocities and ice concentrations up to CS=0.25 (D=100 mm, l=27 m, 10% glycol–water mixture). From his results one
can determine the critical average velocity. The critical average velocity in his results for the ice concentrations up to CS=0.2 varies between 0 and 0.7 m/s. He also proposed that the average velocities for the operation, where no problems are anticipated, should be maintained at 0.5 m/s or higher (CS < 0.3) which is similar to our results. When the flow is near the homogeneous distribution (v=2 m/s), the viscosity approaches the values calculated by the Thomas equation for =(CS) only. Further analysis was made to determine the effect of ice-particle size on concentration distribution and therefore viscosity of ice-slurry. It shows that in the homogeneous flow, the ice-particle diameter has a very low effect on viscosity of ice-slurry (Fig. 8). When the average velocity is lower, so the concentration on the top of the pipe is high, the effect of ice-particle size (and pipe diameter) on viscosity varies between 3 and 6%, which is still very low, but not negligible (Fig. 9). Especially, because the constants (determined by experiment
A. Kitanovski, A. Poredosˇ / International Journal of Refrigeration 25 (2002) 827–835
for particular ice-particle size) in the Thomas equation, together with the effect of ice-particle size on concentration distribution could strongly affect the viscosity at low average velocities. In future, the equation for the viscosity of ice-slurry has to be determined as a correlation function of the iceconcentration, pipe diameter, average velocity, the iceparticle size and the carrier fluid properties. Such an equation could serve engineers for calculating ice-slurry viscosity at any conditions. The calculation would of course require the modification of constants in Thomas equation, to fit experimental results.
References [1] Barnes HA. An introduction to rheology. Amsterdam: Elsevier, 1989 (chapter 7). [2] Darby R. Hydrodynamics of slurries and suspensions, encyclopedia of fluid mechanics, vol. 5, Slurry flow technology. 1986, 49–92 p. (chapter 2). [3] Thomas DG. Transport characteristics of suspension. Journal of Colloid Science 1965;20. [4] Hansen TM, Kauffeld M. Viscosity of ice-slurry. 2nd Workshop on ice-slurries, IIR, May 2000. [5] Frei B, Egolf PW. Viscometry applied to the Bingham substance ice-slurry. 2nd Workshop on ice-slurries, IIR, May 2000. [6] Kauffeld M. Report on the work of the ice particle size distribution at DTI. 3rd Workshop on ice-slurries, IIR, Lucerne, 2001. [7] Sasaki M, Kawashima T, Takahashi H. Dynamics of snow-water flow in pipelines, Slurry Handling and Pipeline Transport. Hydrotransport 1993;12:533–613.
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[8] Egolf PW, et al. Properties of ice-slurry. IIR conference, Aarhus, 1996. p. 517–26. [9] Sari O, Vuarnoz D, Meili F, Egolf PW. Visualization of ice slurries and ice slurry flow. 2nd Workshop on ice-slurries, IIR, May 2000. [10] Guilpart J, et al. Experimental study and calculation method of transport characteristics of ice-slurries. 1st Workshop on ice-slurries, IIR, May 1999. [11] Jensen EN, et al. Pressure drop and heat transfer with ice slurry. IIR Conference, Pordue-USA, 2000. [12] Doetsch C. Pressure drop and flow pattern of ice slurries. 3rd Workshop on ice-slurries, IIR, May 2001. [13] Doron P, Granica D, Barnea D. Slurry flow in horizontal pipes—experimental and modeling. International Journal of Multiphase Flow 1987;13(4):535–47. [14] Doron P, Barnea D. A three-layer model for solid-liquid flow in horizontal pipes. International Journal of Multiphase Flow 1993;19(6):1029–43. [15] Nasr-El-Din H, Shook CA. The lateral variation of solids concetration in horizontal slurry pipeline flow. International Journal of Multiphase Flow 1987;13(5):661–70. [16] Di Felice R. The voidage function for fluid-particle interaction systems. International Journal of Multiphase Flow 1994;20(1):153–9. [17] Chen RC, Kadambi JR. Experimental and numerical studies of solid-liquid pipe flow. ASME Fluids Engineering Division Summer Meeting, FED 1994;189:123–35. [18] Richardson JF, Zaki WN. Sedimentation and fliudisation: Part 1. Trans Inst Chem Engrs 1954;32:35–53. [19] Bird RB, Stewart WE, Lightfoot EN. Transport phenomena. New York: John Wiley & Sons, 1960 (chapter 6). [20] Jacobs, B.E.A. Design of slurry transport systems. Elsevier Science, 1991. [21] Snoek, C.W. The design and operation of ice-slurry based district cooling systems, IEA District heating, Published by Novem, 1993.
Concentration distribution and viscosity of ice-slurry in heterogeneous flow Andrej Kitanovski*,1, Alojz Poredosˇ 2 Faculty of Mechanical Engineering, Laboratory for Refrigeration, Asˇkercˇeva Ljubljana, Slovenia Received 12 May 2001; received in revised form 12 August 2001; accepted 11 September 2001
Abstract Mathematical modeling of two phase flows, especially liquid–solid flows is very complex. Especially when a distribution of the solid phase in a carrier liquid is not homogenous but heterogeneous or even when a moving or stationary bed occurs. In this case, the rheological characteristics of suspension are changing and affect transport characteristics. Therefore, the slurry flow may present a Newtonian and non-Newtonian fluid as well, depending on the operation characteristics. In this paper the fully suspended ice-slurry flow in a horizontal pipe is analysed. The model allows us to avoid the definition on what kind of fluid ice-slurry is present. For the taken ice-particle diameter, the iceconcentration profiles depending on various average velocities and pipe diameters are shown. The viscosity of the iceslurry is presented, depending on average concentration, velocity, pipe diameter and ice-particle size. The results of the analysis have shown that the ice slurrys can be treated as Newtonian-fluid at higher average velocities, and lower average concentrations as well. As the ice concentration increases and velocity decreases the viscosity depends not only on the ice concentration but also on the average velocity and the pipe diameter. The ice-slurry behaves then as a nonNewtonian fluid. The results show also the area where the safe operation of an ice-slurry-district-cooling system can be performed. # 2002 Elsevier Science Ltd and IIR. All rights reserved. Keywords: Ice slurry; Viscosity; Concentration; Flow; District cooling
Coulis de glace en e´coulement he´te´roge`ne : distribution de la concentration et de la viscosite´ Mots cle´s : Coulis de glace ; Viscosite´ ; Concentration ; E´coulement ; Refroidissement urbain
1. Introduction Many different models have been used for the suspension viscosity determination. Most of them essentially
extend the work of Einstein on spheres and his equation for viscosity (CV <0.01)[1]: ¼ liquid ð1 þ 2:5CÞ
* Corresponding author. Tel.: +386-1-4771-418; fax: +386-1-4771-448. E-mail addresses: [email protected] (A. Kitanovski), [email protected] (A. Poredosˇ ). 1 Member of the Ice-Slurry Working Party. 2 Member of the IIR E2 Commission. 0140-7007/02/$22.00 # 2002 Elsevier Science Ltd and IIR. All rights reserved. PII: S0140-7007(01)00091-3
ð1Þ
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A. Kitanovski, A. Poredosˇ / International Journal of Refrigeration 25 (2002) 827–835
Nomenclature A C CD Cmax CS D dp fi g rh Re s v* v w0 w
Greek " (y)
area (m2) local concentration drag coefficient maximum concentration on the top of the pipe average concentration pipe diameter (m) ice particle diameter (m) friction coefficient gravity constant (m/s2) hydraulic diameter (m) Reynolds number density ratio between the ice and carrier fluid shear velocity (m/s) average velocity (m/s) terminal settling velocity (m/s) hindered terminal settling velocity of cluster of particles(m/s)
diffusion coefficient (m2/s) viscosity (Pas) local viscosity (Pas) average viscosity (Pas)
where C is the concentration of the solid phase in a solid–liquid mixture. In Eq. (1), there is no effect of particle size, nor of particle position, because the theory neglects the effects of other particles. There were several other models developed for the viscosity determination, as shown in Table 1. The most popular determination of the suspension viscosity, which comprises not only the concentration of the solid phase, but also the interaction between the solid particles, is based on the well-known Thomas equation [3]: ¼ liquid 1 þ 2:5C þ 10:05C2 þ 0:00273e16:6C
viscosities. Later, Kauffeld [6] had reported on the ice particle size distribution experiments at DTI. From the rheology on suspensions it is known, that viscosity depends also on particle size and its shape. From the work of Thomas [3] the influence of different particle sizes on viscosity at concentrations C< 20% resulted in a 6% difference of relative viscosity. When the concentration is further increased until it reaches the maximum packing concentration, the influence of particles becomes more important. So, when the flow is heterogeneous, the particle size has an important role, especially at very low average velocities, when the concentration on the top of the pipe (in a case of ice-slurry) is increasing to high values. For concentrations C<25%, Thomas [3] has observed, that the first three terms in his equation accounted over 97.5% of the value of the relative viscosity. For concentrations C> 25%, first three terms in his equation account for very small percentages of the relative viscosity, e.g. 60% at C=40% and 8.7% at C=60%. The interaction forces in the heterogeneous flow become, therefore, more important than in homogeneous, so the local ice-slurry viscosity on the top of the pipe increases rapidly and influence the average value of viscosity. In the past different rheological models have been used for the ice-slurry. One of first tries was performed by Sasaki [7], where he used a dilatant fluid model to describe a rheological behavior of ice-slurry. After that, Egolf et al. [8] proposed the Bingham fluid model, which was taken also by many other authors, such as in Refs. [4,5,9]. All the experimental data on ice-slurry rheological behavior had shown that at a certain low ice concentration the ice-slurry behaves as a Newtonian fluid and at a higher concentrations as a Bingham fluid. To describe both, one of the possibilities is to use the apparent viscosity as it was shown by Guilpart et al. [10]. Another proposal was done by Jensen et al. [11], who had actually modified the Bingham model. Later, a model was proposed by Doetsch [12], where he proposed the Casson model to describe Newtonian and Bingham behavior of ice-slurry. However, none of these models consider the ice-slurry in the heterogeneous flow.
ð2Þ 2. Mathematical model
The model is valid for the concentrations up to C=0.625 and particle size ranging from 0.099 to 435 mm. It considers that the flow is homogenous. The Thomas equation has been widely used almost by all researchers on ice-slurry. However, this equation overpredicts the viscosity of ice-slurry at C> 15%, as was shown by Hansen [4]. Hansen used the Jeffrey’s equation with the constant A=4.5 to get the best fit with experimental results. At the same time, Frei [5] observed the time-dependent behavior of ice-slurry, which resulted in different sizes of ice particles and consequently different
Our model in this paper is based on the Doron‘s two [13] and three [14] layer model, which is based primarily on the solids concentration distribution. The analysis was performed for the ice-particle size dp=1 mm and carrier liquid of 10% ethanol/water mixture for various average velocities v (ranging from 0.25 to 2 m/s), ice concentrations CS (ranging from 0.05 to 0.25) and pipe diameters D (27.2, 100, 200 mm). Let us assume that the ice-slurry flow is fully suspended and homogenous in each tiny volume (Fig. 1).
A. Kitanovski, A. Poredosˇ / International Journal of Refrigeration 25 (2002) 827–835
829
Table 1 Expressions for Newtonian suspension viscosity as a function of solids volume fraction CV [2]
1 þ 0:5C 0:5C2 1 2C 9:6C2 2:5C ¼ liquid exp 1 0:609C
Guth and Simha
Predicts ! 1 for C=0.234
¼ liquid
Vand
¼ liquid exp
2:5C þ 2:7C2 1 0:609C
No interparticle forces Includes doublet collisions, but not triplet
2:5C 0.75
Mooney
¼ liquid exp
Simha
Frankel and Acrivos
54 C C ! Cmax ¼ liquid 1 þ 3 5f 1 ðC=Cmax Þ3 1 !1 C 3 ¼ liquid C 1 Cmax ¼ liquid ð1 þ ACÞ 2.5
Jeffrey
K depends upon system
Dilute suspensions
Very concentrated
Concentrated suspensions only
Ellipsoid particles
Then Eq. (2) can be described with local concentration as a function of the y-axis: ðyÞ ¼ liquid 1 þ 2:5CðyÞ þ 10:05C2 ðyÞ þ 0:00273e16:6CðyÞ ð3Þ The concentration profile can be determined using the diffusion equation of the turbulent flow, which was originally determined in the vertical direction by Schmidt and Rouse [15]: "
d2 CðyÞ dCðyÞ ¼0 þw dy2 dy
ð4Þ
Fig. 1. Homogeneous and heterogeneous flow in a pipe. Fig. 1. Homogeneous and heterogeneous flow in a pipe.
where " is the local diffusion coefficient and w is the local hindered terminal velocity of the ice-particle. In our case it was assumed that these are the mean coefficients. The concentration profile can be estimated as [13]: w
CðyÞ ¼ Cmax eð " yÞ
ð5Þ
where Cmax is the maximum concentration on the top of the pipe (Fig. 2). The maximum concentration in the fully suspended flow is not equal to the maximum packing concentration, which is presented later in a paper. Since we know the average concentration, the maximum concentration can be determined as: Cmax ¼ Ð A
CS A w" Þy ð e dAðyÞ
ð6Þ
where CS is the average ice concentration (in situ concentration) and D is the diameter of the pipe. Doron [13] determined the main diffusion coefficient " by equation: " ¼ 0:052v rh
ð7Þ
where rh is the hydraulic radius (in our case rh=D/2) and v* is the shear velocity [13]: rffiffiffi fi v ¼ v 2
ð8Þ
where fi is the friction coefficient, suggested by Doron [13] to be evaluated from the modified Colebrook formula, where the friction coefficient should be multiplied by 2 (modified by Televantos):
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Fig. 2. The concentration profile of ice-slurry flow.
1 dp 2; 51 p ffiffi ffi pffiffiffiffiffiffi ¼ 0:86 ln þ 3:7D Re 2fi 2fi
ð9Þ
where dp is an ice particle diameter and Re is the Reynolds number based on viscosity and density of a carrier liquid. Doron [13] as many other authors [16,17] suggested that the hindered terminal settling velocity of a cluster of particles w can be calculated using the well known Richardson and Zaki [18] correlation: w ¼ w0 ð1 CS Þm
ð10Þ
where parameter for 1 < Rew < 500, m ¼ m=4.45 Re0.1 w d 4:36 þ 17:6 Dp Re0:03 for 0.2
of the maximum packing fractions in slurries, as shown in Table 2. Doron [13,14] has determined the maximum packing concentration as 0.52, which is the concentration of the monodispersed spheres in a simple cube. The values for the maximum packing fractions vary in a range from 0.5 to 0.75 for monodisperse spheres. In our case it was taken, that the maximum ice packing fraction equals 0.52.
3. Concentration distribution in the heterogeneous ice-slurry flow It was assumed in our case, that there is no variation of the concentration distribution in the downstream direction. This can be satisfied by assumption that the horizontal pipe is long enough to have a fully developed flow, and that there is no influence of the heat transfer or pressure drop along the pipe on the concentration profile. Nasr-El-Din [15] has found in his experiment with polystyrene and sand, that there was almost no lateral variation of concentration for the polystyrene particle size of 0.3 mm and CS < 40% (vbulk=2 m/s). At the same bulk-velocity and particle diameter of 1.4 mm the lateral variation of concentration was near to zero for CS < 27%. Since the polystyrene density is close to that of ice, another assumption was made in this paper, that there is no lateral variation of the concentration distribution in the ice-slurry for CS <25%. Calculating Eqs. (4)–(11) one may calculate different concentration profiles, depending on a pipe diameter, ice particle size, average concentration and velocity. At the velocity v=2 m/s the ice slurry flow is near the homogenous distribution of the ice concentration (Fig. 3). Decreasing the velocity below 0.5 m/s, the transition from the heterogeneous flow to that with a moving bed occurs, because the maximum packing fraction on the top of the pipe occurred. Increasing the average concentration of the ice-slurry, at the same pipe diameter, the transition
A. Kitanovski, A. Poredosˇ / International Journal of Refrigeration 25 (2002) 827–835 Table 2 The maximum packing fraction of various arrangements of monodispersed spheres [1] Arrangement
Maximum packing fraction
Simple cube
0.52
Maximum thermodynamically stable configuration
0.548
Hexagonally packed sheets just touching
0.605
Random close packing
0.637
Face-centred cubic/ hexagonal close packed
0.68
Body-centred cubic/ hexagonal close packed
0.74
from the homogenous to heterogeneous flow occurs earlier, as shown in Fig. 4. However, as the ice-particles are very small it is possible that the transition from the heterogeneous to that with a moving bed never occurs [20]. In this case a stationary bed
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occurs and the flow is tending to stop, since more and more ice particles are added.
4. Ice-slurry viscosity As was mentioned before, there are many different models to describe the viscosity of ice-slurry. However, none of them considers ice-slurry as heterogeneous mixture or even flow with a moving bed. When a maximum packing fraction at the top of the pipe occurs, more and more particles are added, the transition from the heterogeneous flow to that with a moving or even stationary bed occurs and tends to ‘‘jam up’’. The viscosity goes in this case towards infinity. There are some models, which describe the so called ‘‘crowding effect’’, such as Ball‘s [1] and Richmond‘s equation for viscosity, Krieger and Dougherty‘s [1] theory and others. Using the Thomas equation we want to show a new approach in determining the viscosity of the heterogeneous ice-slurry flow. There are many other equations or their possible modifications, which could be used in our model. Our model has validation only at parameters when the maximum packing fraction on the top of the pipe is not reached. But it can show the transition from the heterogeneous flow to that with a moving bed, so it can be used to determine ‘‘safe operation’’. Note, that the viscosity presented in Figs. 5–7
Fig. 3. Concentration profiles of ice slurry flow in a horizontal pipe (CS=0.1, D=27.2 mm, dp=1 mm). Fig. 1. Concentration profiles of ice slurry flow in a horizontal pipe (CS=0.1, D=27.2 mm, dp=1 mm)..
Fig. 4. Concentration profiles of ice slurry flow in a horizontal pipe (CS=0.25, D=27.2 mm, dp=1 mm). Fig. 1. Concentration profiles of ice slurry flow in a horizontal pipe (CS=0.25, D=27.2 mm, dp=1 mm)..
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is effective viscosity, so the viscosity variations depending on velocity, particle diameter and pipe diameter are effected almost only by interaction terms in Eq. (3). For very small ice concentrations (e.g. CS < 0.1) the ice-slurry viscosity is almost independent of the average velocities (Fig. 5). When the concentration increases and the velocity decreases the viscosity increases rapidly. The transitional area in Fig. 5 presents the area where the transition from the heterogeneous flow to that with a moving or stationary bed occurs. Increasing the pipe diameter from D=27.2 to D=100 mm (at the same average velocities) the effect of smaller pipe diameter almost does not influence a viscosity at higher velocities, especially at lower ice concentrations (Fig. 6). When the ice concentrations are higher and velocities lower (e.g. v=1 m/s), the viscosity is increasing so it depends on velocity, concentration and the pipe diameter as well. When the pipe diameter is further increased to D=200 mm, the moving bed occurs a bit earlier. At lower average velocities viscosity is much higher than at smaller diameters (Fig. 7). Note, however, that in our analysis the Einstein‘s equation or Jeffrey’s equation for the local viscosity determination in heterogeneous flow cannot be used, more than so because they do not comprise the interaction term between the ice particles, which becomes very important at lower average velocities and higher average concentrations. On the other hand, when the velocity is high and the flow is close to homogeneous concentration distribution, the effect of interaction terms is so low, so Jeffrey´s equation can be used for viscosity determination.
5. Effect of ice-particle size Another calculation was made to describe the effect of ice-particle size on effective viscosity of ice-slurry. When the flow is near to the homogeneous, the effects of iceparticle diameter and pipe diameter on viscosity is very low, almost negligible (Fig. 8). Actually, one must notice, that Figs. 8 and 9 are showing only the effect of ice-particle size on concentration distribution and, therefore, on viscosity. The Thomas´ equation depends not only on the concentration, but also on the constants, which were calculated for his case. That means, that the constants should be calculated again for the ice-slurry at particular ice-particle diameter in order to give new equations for the ice-slurry, which could be also used as a local viscosity equations. When the flow is homogeneous, Jeffrey´s equation could be used, where the parameter A should vary with the ice-particle size or even shape. However, when the flow is heterogeneous, the interaction terms should be involved, similar to how it was done by Thomas. When the velocity is lower, e.g. 0.75 m/s, the maximum concentration on the top of the pipe is very high (Figs. 3 and 4), so the influence of ice-particle size on viscosity is higher than at velocities v=2 m/s (Fig. 9). However, the difference between the viscosity at dp=0.25 mm and dp=1 mm (D=200 mm) is 6.2%, which is still very low, but not negligible. Especially because the constants (which could be determined by experiment for particular ice-particle size) in the Thomas equation could have together with the effect of con-
Fig. 5. Ice slurry viscosity (D=27.2 mm, dp=1 mm, 10% water–ethanol mixture). Fig. 1. Ice slurry viscosity (D=27.2 mm, dp=1 mm, 10% water-ethanol mixture).
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Fig. 6. Ice slurry viscosity (D=100 mm, dp=1 mm, 10% water–ethanol mixture). Fig. 1. Ice slurry viscosity (D=100 mm, dp=1 mm, 10% water–ethanol mixture).
Fig. 7. Ice slurry viscosity (D=200 mm, dp=1 mm, 10% water–ethanol mixture). Fig. 1. Ice slurry viscosity (D=200 mm, dp=1 mm, 10% water–ethanol mixture).
centration distribution (shown in our case) a much higher influence on viscosity at lower velocities. We assume, that in that case, for given dp=0.25 and 1 mm (D=200 mm, v=0.75 m/s), the differencies in viscosities could be higher than 30%.
6. Conclusion The results of this analysis have shown that at higher average velocities and very low ice concentrations (CS < 0.1) the ice slurry viscosity is almost independant
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Fig. 8. Viscosity depending on different pipe and ice-particle diameters (CS=20%, v=2 m/s).
Fig. 9. Viscosity depending on different pipe and ice-particle diameters (CS=20%, v=0.75 m/s).
of velocity as for Newtonian fluids. Similar results were presented by [10] for apparent viscosity. When the ice concentration is further increased, the viscosity depends not only on the ice concentration, but also on the average velocity and pipe diameter as well. The dependence is much stronger at higher ice-concentrations and lower average velocities. At lower average velocities, the viscosity increases with increased pipe diameter. The transitional area in Figs. 5–7 presents the transition from the heterogeneous flow to that with a moving or stationary bed. It enables us to determine the ‘‘safe operation’’ parameters. One must note that since the ice particles are very small, it is quite possible that the transition from the heterogeneous flow to the flow with a stationary bed occurs directly, without a moving bed formation. Snoek [21] has reported on the experimental results on the pressure drop, depending on average velocities and ice concentrations up to CS=0.25 (D=100 mm, l=27 m, 10% glycol–water mixture). From his results one
can determine the critical average velocity. The critical average velocity in his results for the ice concentrations up to CS=0.2 varies between 0 and 0.7 m/s. He also proposed that the average velocities for the operation, where no problems are anticipated, should be maintained at 0.5 m/s or higher (CS < 0.3) which is similar to our results. When the flow is near the homogeneous distribution (v=2 m/s), the viscosity approaches the values calculated by the Thomas equation for =(CS) only. Further analysis was made to determine the effect of ice-particle size on concentration distribution and therefore viscosity of ice-slurry. It shows that in the homogeneous flow, the ice-particle diameter has a very low effect on viscosity of ice-slurry (Fig. 8). When the average velocity is lower, so the concentration on the top of the pipe is high, the effect of ice-particle size (and pipe diameter) on viscosity varies between 3 and 6%, which is still very low, but not negligible (Fig. 9). Especially, because the constants (determined by experiment
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for particular ice-particle size) in the Thomas equation, together with the effect of ice-particle size on concentration distribution could strongly affect the viscosity at low average velocities. In future, the equation for the viscosity of ice-slurry has to be determined as a correlation function of the iceconcentration, pipe diameter, average velocity, the iceparticle size and the carrier fluid properties. Such an equation could serve engineers for calculating ice-slurry viscosity at any conditions. The calculation would of course require the modification of constants in Thomas equation, to fit experimental results.
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[8] Egolf PW, et al. Properties of ice-slurry. IIR conference, Aarhus, 1996. p. 517–26. [9] Sari O, Vuarnoz D, Meili F, Egolf PW. Visualization of ice slurries and ice slurry flow. 2nd Workshop on ice-slurries, IIR, May 2000. [10] Guilpart J, et al. Experimental study and calculation method of transport characteristics of ice-slurries. 1st Workshop on ice-slurries, IIR, May 1999. [11] Jensen EN, et al. Pressure drop and heat transfer with ice slurry. IIR Conference, Pordue-USA, 2000. [12] Doetsch C. Pressure drop and flow pattern of ice slurries. 3rd Workshop on ice-slurries, IIR, May 2001. [13] Doron P, Granica D, Barnea D. Slurry flow in horizontal pipes—experimental and modeling. International Journal of Multiphase Flow 1987;13(4):535–47. [14] Doron P, Barnea D. A three-layer model for solid-liquid flow in horizontal pipes. International Journal of Multiphase Flow 1993;19(6):1029–43. [15] Nasr-El-Din H, Shook CA. The lateral variation of solids concetration in horizontal slurry pipeline flow. International Journal of Multiphase Flow 1987;13(5):661–70. [16] Di Felice R. The voidage function for fluid-particle interaction systems. International Journal of Multiphase Flow 1994;20(1):153–9. [17] Chen RC, Kadambi JR. Experimental and numerical studies of solid-liquid pipe flow. ASME Fluids Engineering Division Summer Meeting, FED 1994;189:123–35. [18] Richardson JF, Zaki WN. Sedimentation and fliudisation: Part 1. Trans Inst Chem Engrs 1954;32:35–53. [19] Bird RB, Stewart WE, Lightfoot EN. Transport phenomena. New York: John Wiley & Sons, 1960 (chapter 6). [20] Jacobs, B.E.A. Design of slurry transport systems. Elsevier Science, 1991. [21] Snoek, C.W. The design and operation of ice-slurry based district cooling systems, IEA District heating, Published by Novem, 1993.