# Phase-splitting algorithm for ice slurry heterogeneous-flow pressure drop in straight pipe flow.

INTRODUCTION

As a new kind of phase-change cooling medium, ice slurry is playing a more and more important role in ice storage, air-conditioning systems, heat damage elimination, as well as other central refrigeration systems (Monteiro and Bansal 2010). Despite the advantages that the ice slurry has possessed (like high heat capacity and high value of heat transfer coefficient) (Beata and Wojciech 2006), the research on the complex flow characteristics is still limited. Since the middle of the 19th century, works on the rheological properties of ice slurry have been published successively (Mika 2013). Mellari et al. (2012) gives a description of different existing rheological models and comes up with the conception of critical deposition velocity by comparing the observed data at various fittings and different working conditions.

With the development of numerical calculation, rheological models with yield/shear stresses were presented based on different rheological behavior (Kauffeld et al. 2005). In general, there are four different non-Newtonian rheological models and Kitanovski et al. (2005) gave different rheological properties in various conditions to describe ice slurry rheology, namely, the Bingham, Power Law, Herschel-Bulkley, and Cassion model, respectively (Chhabra and Richardson 2008). Many researchers reported experimental results on ice slurry flow pressure drop and rheological behavior in different kinds of pipe fittings (Illan and Viedma 2009).

For engineering hydraulic calculation of ice slurry pipeline design, however, actual working conditions in a pipeline flow may involve various models, and numerical computation can sometimes be too complicated. So, seeking a new method that can be available for most working conditions is a meaningful task. Molerus and Wellmann (1981) show a new way to determine the pressure drop of solid-particle suspension flow, which can be available in all ranges of concentration, pipe diameters, and particle sizes. According to Molerus and Wellmann (1981), the total pressure drop was split up into two parts, namely the liquid part and the solid part. But this method was developed for slurries for which the density of the particle was between 1270-5250 kg/[m.sub.3] (79.276-327.715 lb/[ft.sub.3]) and cannot be directly applied in ice slurry pressure-drop calculations. This paper will start from the theory of phase-splitting algorithm with an effort to solve the problems that arose when it was applied to ice slurry.

PRESSURE DROP MODEL

Flow Regimes and Terminal Velocity

When pumping slurry ice through pipelines may sometimes lead to an ice jam blocking by reason of ice agglomeration. Therefore, the state of the flow plays an important role to ensure the safe and steady operation of a cooling system. Kitanovski et al. (2005) divided ice slurry flow patterns into four sates: the homogeneous flow, heterogeneous flow, moving bed, and stationary bed. Homogeneous flow requires high flow velocity and relatively low ice fraction. For most actual cooling systems, the ice particle fraction varies in a range of 10%-30%, and heterogeneous flow characteristics dominate the piping process. The existing homogeneous flow studies are mostly derived from heterogeneous flow in practice. Hence, heterogeneous flow is the common case. When the flow velocity decreases to a critical value, the ice particles will float to the top of the pipe and thus lead the system in unstable conditions. Slurry-Ice Based Cooling System Application Guide (Environmental Process Systems Limited) had recommended a safe velocity zone for ice slurry pipe design. Some researchers came up with the concept of terminal velocity (Peker and Helvaci 2008). For horizontal pipe flow, the terminal velocity was also called deposition velocity, which comes from the sand slurry flow. The deposition velocity can be considered as the transition between a flow with a moving bed and heterogeneous flow (Kitanovski and Poredos 2002). Figure 1 shows the terminal velocity at different pipe diameters and ice-particle fractions (Kitanovski et al. 2005).

The deposition velocity can also be obtained from the following empirical correlation (Kauffeld et al. 2005):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [phi] is the ice particle fraction, D the pipe diameter, s the density ratio between phases, and w is the hindered terminal settling velocity (Richardson and Zaki 1997).

For vertical pipe flow, the terminal velocities are educed with the force analysis method. The motion of ice particles is governed by the following forces (Peker and Helvaci 2008):

a. Gravitational force. The gravitational force, [F.sub.g], of an ice particle can be expressed as

[F.sub.g] = [[rho].sub.S.sup.V] = [sub.S]g = [[rho].sub.S] x [1/6] [pi][d.sup.3] x g (2)

where [[rho].sub.S] stands for the concentration of ice particles.

b. Buoyancy force. Buoyancy is caused by the density difference between liquid and ice particles. According to Archimedes' principle, buoyancy force is equal to the weight of the displaced liquid. So, the buoyancy force, [F.sub.b], can be written as

[F.sub.b] = [[rho].sub.L]V[sub.S]g = [[rho].sub.L] x [1/6][pi][d.sup.3] x g (3)

c. Drag force. Drag force is the resistance of the ice particle to motion, which always acts opposite to the direction of the particle's relative velocity to the fluid, and is expressed as

[f.sub.D] = [C.sub.D] x A[1/2][[rho].sub.L] x [v.sup.2.sub.t] = 1/8 [C.sub.D][pi][d.sup.2][[rho].sub.L] [v.sup.2.sub.t] (4)

where A stands for the projection area onto the flow direction; [v.sub.t] is the terminal velocity, which is equal to the relative velocity between the two phases in vertical pipe flow, namely [v.sub.t] = [v.sub.r]; and [C.sub.D] is the drag coefficient, which is a function of the velocity of the particle, shape of the particle, viscosity of the medium, and the roughness of the particle surface. Peker and Helvaci (2008) used the particle Reynolds number, [Re.sub.p], to characterize the drag coefficient's dependence on the influencing factors A, [v.sub.t], and [C.sub.D].

[Re.sub.p] = [[rho].sub.L][v.sub.t]d/[mu] (5)

where [mu] is the viscosity of the medium. There are three regimes for the particle Reynolds number to characterize and thus lead in different forms of drag coefficient expressions.

For laminar regime ([Re.sub.p] < 1), the inertial effects are negligibly small and the flow is considered as Stokes flow, according to the Stokes law:

[C.sub.D] [congruent to] 0.44 (6)

All the conditions that are discussed in this paper are in laminar regime.

Particle motion in a steady-flow pipe is governed by Newton's law, where all the affecting forces sum up to zero, as

[F.sub.b] - [F.sub.g] + [F.sub.D] = 0 (7)

Combining Equations 2-5 into Equation 7, we can get the terminal velocity of ice particles in vertical pipe flow:

[v.sub.t] = ([[rho].sub.L] - [[rho].sub.S])g[d.sup.2]/18[mu] (8)

The Solid Phase Pressure Drop Model

The pressure drop of heterogeneous pipe flow can be regarded as a result of the friction and interaction among ice particles, fluid, and the pipe wall. By comparing the solid-liquid two-phase flow with the single-phase liquid flow, the slip effect between ice particles and fluid can cause a sharp increase in pressure gradient when it is at high ice fractions. This increase, according to Molerus and Wellmann (1981), influenced by the geometric shape and size of ice particles, is regarded as additional energy consumption.

The mean additional power loss per particle can be written as

[bar.P] = [[DELTA][P.sub.S] x [V.sub.sus]]/Z = [DELTA][P.sub.S][[pi][d.sup.3]/6](v - [v.sub.r])/[phi][DELTA]L (9)

where [DELTA][P.sub.S] is the solid pressure drop, [V.sub.sus] is suspension discharge, Z is the particle number, and [v.sub.r] is the mean relative velocity between the solid phase and the liquid one. Together with Equation 4, the dimensional correlation of additional pressure drop per particle can be written as

[bar.P]/[F.sub.D]v = [[DELTA][P.sub.S]/[phi][rho][v.sup.2]] [d/[DELTA]L] [4/3[C.sub.D]] [1 - [[v.sub.r]/v]/[([v.sub.r]/v).sup.2]] (10)

The power loss of the drag force transfers from the fluid to the particle and hereby requires that Equation 10 equals 1. Together with Equation 5, we can get solid phase expression. Lee et al. (2002) discusses the nondimensional pressure drop on fractional volumetric concentration of solid particles:

When 0 < [phi] < 0.25:

[DELTA][P.sub.S] = [[v.sup.2.sub.r] /[1 - [v.sub.r]]][[v.sup.2]/[v.sup.2.sub.t]][phi] x [[rho].sub.L](s - 1)gL (11)

When [phi] > 0.25:

[DELTA][P.sub.S] = [v.sup.2]/[v.sup.2.sub.t][[rho].sub.L](s - 1)g[L.sub.[phi]][[[v.sup.2.sub.r]/[1 - [v.sub.r]]] + 0.1[N.sup.2.sub.FRT]([phi] - 0.25)] (12)

where [N.sup.2.sub.Frt] is the pipe Froude number, which will be explained in further detail in the section Relative Velocity Between Phases.

Pressure Drop of Ice Slurry

Based on the solid phase pressure drop model, for heterogeneous ice slurry flow, the total pressure drop in a pipe can be separated into two parts: the liquid part and the solid one. With given ice fraction and pipe flow conditions, the relative velocity between phases could be fixed and substituted into Equation 11 or 12, and the additional pressure drop, namely the solid pressure drop, could be determined.

As for the fluid part, the liquid pressure drop is determined as any Newtonian fluid in a pipe:

[DELTA][P.sub.L] = [[lambda].sub.(Re)][[DELTA]L/D] x [[[rho].sub.L]/2][v.sup.2] (13)

where [lambda] is friction loss factor. For the laminar region,

[[lambda].sub.(Re)] = 64/Re (14)

and for turbulent flow, the Blasius model represents the friction factor quite well (Peker and Helvaci 2008)

[[lambda].sub.(Re)] = 0.3164/[Re.sup.0.25] (15)

The pressure drop of the pipe flow can be obtained by the sum of the two parts

[DELTA]P = [DELTA][P.sub.L] + [DELTA][P.sub.S] (16)

Relative Velocity Between Phases

During the process of ice slurry pressure drop calculation, finding how to acquire the relative velocity is the crux of this algorithm. Darby et al. (2001) summarized a relative velocity diagram on the basis of a large number of a pneumatic, hydraulic mortar experiments. However, there are still no ready existing charts for ice slurry to determine the relative velocity. In order to obtain the crucial velocity, a dimensional analysis of the particle motion is carried out along with energy dissipation considerations. The final dimensionless groups can be written as follows.

The particle Froude number:

[N.sup.2.sub.Frp] = [v.sup.2.sub.L]/(s - 1)dg

The pipe Froude number:

[N.sup.2.sub.Frt] = [v.sup.2.sub.t]/(s - 1)Dg (18)

[v.sub.r]/[square root of s] = f([square root of s][N.sub.Frp], [N.sup.2.sub.Frt]) (19)

According to (Molerus and Wellmann 1981), the relative velocity is a function of the nondimensional numbers [N.sup.2.sub.Frp] and [N.sup.2.sub.Frt]. Because the diagram cannot be applied in ice slurry calculation, Table 1 shows the process of drawing the diagram of horizontal relative velocity.

As is shown in Table 1, the solid pressure drop can be obtained by subtracting the liquid pressure drop from the measured pressure drop. Then the relative velocity can be determined according to Equation 11 or 12.

The relative velocity in vertical pipe flow can be acquired in the same way, but because of a lack of experimental data, the relative velocity of vertical pipe flow that was presented in Figure 6 is obtained by numerical computation software.

Figure 2 depicts the relative velocity in horizontal pipe flow. This chart is developed according to the method shown in Table 1 and can be valid for those conditions when the nondimensional number [N.sup.2.sub.Frp] is in a range of 0.02-0.2. With the further development of experimental data in more working conditions, the scope of application will be expanded in future work.

EXAMPLE VALIDATIONS AND ANALYSIS

Example Conditions

To further verify the accuracy of the algorithm, the authors select several experimental cases on ice slurry horizontal and vertical pipe flow that were reported in Lee et al. (2002) and Grozdek et al. (2009). However, because of the economy of existing measurement techniques, it is difficult to find complete experiment data in published literature. So we choose an ice slurry transportation system with a vertical test pipe (D = 0.0224 m) at a mixed-delivery speed in v = 0.1-2.25 m x [s.sup.-1], which is reported in Lee et al. (2002). The horizontal test pipe experimental data is from Grozdek et al. (2009) with a diameter D = 0.009 m (0.0295 ft) and the mixed delivery speed v = 0.1 - 2.2 m/s (0.33-7.22 ft/s). The basic physical parameters are shown in Table 2.

Results and Analysis

Figures 3-5 compare the computational results with the experiment value. As is shown in Figure 3, with the content of the ice particles ranging from 0%-30%, the error for calculating pressure drop in a horizontal pipe flow by using this algorithm is within 15%. For vertical pipe, pressure drop varies with flow direction, as shown in Figures 4 and 5, and the error for calculation is within 20%.

Figure 6 shows relative velocity between liquid and solid phases for ice slurry vertical pipe flow. Differing from horizontal pipe flow, the relative velocity in a vertical pipe is not always positive, which means ice particles can sometimes move faster than fluid. This phenomenon, reflecting on the pressure drop, means a decrease in contrast to fluid flow. So the relative velocity is further expanded as follows: when [V.sub.r] > 0, solid velocity is less than the fluid, ice particles slide backward relative to fluid, and thus lead to an energy consumption and a pressure drop increase (shown as Figure 6a). When [V.sub.r] < 0, the solid velocity is larger than the fluid velocity, therefore the ice slurry flow is enhanced. This is more obvious in vertical upflow by reason of smaller density of ice particles whose buoyancy can further promote the slurry motion.

Figure 7 demonstrates the proportion of pressure drop between phases on various particle concentrations and flow velocity in horizontal pipe flow. Since the solid pressure drop is actually caused mainly by particle fluid friction, the proportion of eachphase can illustrate the particle's effect on solid-liquid two-phase flow.

As is shown in Figure 7, the pressure drop ratio between particles and fluid is increasing along with velocity decrease while the concentration keeps the same value. This is because the carrying capacity of the medium decreases with the velocity becoming slower and thus leads to an increase of the consumption aroused by the particle fluid friction. When the flow velocity reaches a critical value (namely, the deposition velocity), the slurry will step into the moving bed stage, with a remarkable increase on pressure drop and large-size ice block appearance. On the other hand, when the velocity keeps a fixed value, the pressure-drop ratio rises up along with the concentration increasing. This is also a validation on the view that was come up with in Ashley et al. (2010).

In summary, discarding the hypothesis of one-dimensional homogeneous flow, which is based on the single-phase flow principle, the ice slurry phase-splitting algorithm takes full account of slip effect. This method has no restriction on rheology in range of application and can further provide a change rule among velocity, particle fraction, and solid pressure drop ratio.

CONCLUSION

Based on the kinetic theory of two-phase flow, this paper proposes anew algorithm which is applicable on pressure drop calculation of heterogeneous ice slurry pipe flow. Due to the physical properties, dissipation of energy is fully considered to characterize additional pressure drop in various working conditions. Through the example validation, with a content of the ice particles ranging from 0%-30%, the error for the calculation is within 20%.

Terminal velocity plays an important role during the algorithm, as well as in ice slurry transportation. Due to the relationship between force and movement direction, this paper shows a different way to determine terminal velocities in horizontal and vertical pipe flow.

The acquisition of relative velocity is the pivotal issue of the algorithm. Due to lack of experimental pressure drop data, relative velocities in vertical pipe are acquired by the aid of numerical software. Figure 2, showing relative velocity in horizontal pipe, is available for those conditions for which the Froude number is in the range of 0.02-0.2. With further completion of the experiment data, extension of the application scope will be concentrated in further work.

During the horizontal flow process, the proportion of pressure drop between two phases is demonstrated during the calculation. The ratio increases along with velocity decrease, while the concentration keeps a same value. On the other hand, when the velocity keeps a fixed value, the ratio rises up along with concentration increase. This phenomenon is also applicable to vertical flow process.

From a view of practicability, the phase-splitting algorithm makes up for the complexity that numerical computation has possessed. By balancing the relationship between computing efficiency and the prediction precision, this algorithm can fully ensure convenience and reliability in engineering applications.
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NOMENCLATURE

[C.sub.D]     = drag coefficient
d             = ice particle diameter, m (ft)
D             = pipe diameter, m (ft)
F             = force, N ([lb.sub.y])
g             = gravitational acceleration, m/s2 (ft/[s.sup.2])
[N.sub.Frt]   = pipe Froude number
[N.sub.Frp]   = particle Froude number
P             = pressure drop, Pa (psf)
Re            = Reynold number
s             = density ratio between phases
V             = velocity, m/s (ft/s)
V             = volume, [m.sup.3] ([ft.sup.3])
w             = hindered terminal settling velocity, m/s (ft/s)
Z             = particle number

Greek Symbols

[rho]         = density, kg/[m.sup.3] (lb/[ft.sup.3])
[pi]          = pi number
[mu]          = viscosity, kg/m-s (lb/ft x s)
[phi]         = ice particle fraction
[lambda]      = friction loss factor

Subscripts

b             = buoyancy
g             = gravity
dep           = deposition
D             = drag
L             = liquid phase
P             = particle
sus           = suspension
S             = solid phase
t             = terminal
r             = relative
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REFERENCES

Beata, N.Z., and Z. Wojciech. 2006. Momentum transfer of ice slurry flows in tubes, experimental investigations. International Journal of Refrigeration 29(3): 418-28.

Chhabra, R.P., and J.F. Richardson. 2008. Non-Newtonian flow and applied rheology, 2nd ed. Oxford, UK: Butterworth-Heinemann.

Darby, R. 2001. Chemical engineering fluid mechanics. Basel: Marcel Dekker, Inc.

Environmental Process Systems Limited. Slurry-ice based cooling system application guide. www.epsltd.co.uk/files/slurryice_manual.pdf

Grozdek, M., R. Khodabandeh, and P. Lundqvist. 2009. Experimental investigation of ice slurry flow pressure drop in horizontal tubes. Experimental Thermal and Fluid Science 33(2): 357-70.

Illan, F., and A.Viedma. 2009. Experimental study on pressure drop and heat transfer in pipelines for brine based ice slurry Part II: Dimensional analysis and rheological model. International Journal of Refrigeration 32(5):1024-31.

Kauffeld, M., M. Kawaji, and P.W. Egolf. 2005. Handbook on ice slurries: Fundamentals and engineering. Paris: International Institute of Refrigeration.

Kitanovski, A., and A. Poredos. 2002. Concentration distribution and viscosity of ice-slurry in heterogeneous flow. International Journal of Refrigeration 25(6): 827-35.

Kitanovski, A., D. Vuarnoz, D. Ata-Caesar, P.W. Egolf, T.M. Hansen, and C. Doetsch. 2005. The fluid dynamics of ice slurry. International Journal of Refrigeration 28(1): 37-50.

Lee, D.W., C.I. Yoon, E.S. Yoon, and M.C. Joo. 2002. Experimental study on flow and pressure drop of ice slurry for various pipes. Fifth Workshop on Ice-Slurries of the International Institute of Refrigeration. Stockholm, Sweden.

Mellari, S., M. Boumaza, P.W. Egolf. 2012. Physical modeling, numerical simulations and experimental investigations of Non-Newtonian ice slurry flows. International Journal of Refrigeration 35(5): 1284-91.

Mika, L. 2013. Pressure loss coefficients of ice slurry in horizontally installed flow dividers. Experimental Thermal and Fluid Science 45: 249-58.

Molerus, O., and P. Wellmann. 1981. A new concept for the calculation of pressure drop with hydraulic transport of solids in horizontal pipes. Chemical Engineering Science 37(10): 1623-32.

Monteiro, A.C.S., and P.K. Bansal. 2010. Pressure drop characteristics and rheological modeling of ice slurry flow in pipes. International Journal of Refrigeration 33(8): 1523-32.

Peker, S.M., and S. Helvaci. 2008. Solid-liquid two phase flow. Oxford: Elsevier Science.

Richardson, J., and W. Zaki. 1997. Sedimentation and fluidization: PART 1. Chemical Engineering Research and Design 75 (Supplement):S82-S100.

Shugang Wang, PhD, Yanjie Lv, JihongWang, PhD, Tengfei Zhang, PhD, Member ASHRAE

Shugang Wang is a professor, Yanjie Lv is a graduate student, Jihong Wang is a PhD student, and Tengfei Zhang is an associate professor in the Department of Infrastructure Engineering, Dalian University of Technology, Dailin, Liaoning, China.
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Table 1. Process of Calculating the Relative Velocity
between the Liquid and Solid Phase

Mean Velocity,       Experimental       Liquid Phase     Solid Phase
m/s (ft/s)          Pressure Drop,     Pressure Drop,   Pressure Drop,
kPa/m (psf/ft) (1)   kPa/m (psf/ft)   kPa/m (psf/ft)

2.07 (6.79)          4.42 (9.36)        1.30 (2.75)      3.12 (6.60)
1.84 (6.04)          4.03 (8.53)        1.16 (2.46)      2.87 (6.07)
1.58 (5.18)          3.53 (7.47)        0.99 (6.88)      2.53 (5.38)
1.29 (4.23)          3.03 (6.41)        0.81 (5.63)      2.21 (4.70)
1.06 (3.48)          2.63 (5.57)        0.67 (4.65)      1.96 (4.15)
0.77 (2.53)          2.13 (4.51)        0.48 (3.33)      1.65 (3.49)
0.52 (1.71)          1.61 (3.41)        0.33 (2.29)      1.28 (2.71)

Mean Velocity,    [N.sub.Frp]    [N.sub.Frt    [v.sub.r]
m/s (ft/s)        [s.sup.1/2]      .sup.2]

2.07 (6.79)          111.73         0.171        0.015
1.84 (6.04)          99.31          0.171         0.02
1.58 (5.18)          85.28          0.171        0.025
1.29 (4.23)          69.63          0.171         0.04
1.06 (3.48)          57.21          0.171        0.045
0.77 (2.53)          41.56          0.171        0.065
0.52 (1.71)          28.07          0.171         0.11

Mean Velocity,     Calculated      Relative      Relative
m/s (ft/s)         Pressure       Error for      Error for
Drop, kPa/m    Solid Phase    Ice Slurry
(psf/ft)       Pressure      Pressure
Drop, %        Drop, %

2.07 (6.79)       2.80 (5.93)        10.1           7.1
1.84 (6.04)       2.69 (5.69)        6.1            4.3
1.58 (5.18)       2.33 (4.93)        7.9            5.7
1.29 (4.23)       2.01 (4.25)        9.4            6.2
1.06 (3.48)       1.84 (3.89)        6.5            4.8
0.77 (2.53)       1.60 (3.39)        2.6            2.0
0.52 (1.71)       1.24 (2.62)        2.5            2.0

(1) pounds per square foot (PSF).

Table 2. Basic Physical Properties for Liquid
and Solid Phase

Phase                    Density,       Particle Diameter,
kg/[m.sup.3]           mm (ft)
(lb/[ft.sup.3])

Vertical solid         917 (57.24)         0.27 (0.0009)

Vertical liquid        1008 (62.92)             --

Horizontal solid       917 (57.24)         0.5 (0.0016)

Horizontal liquid      986 (61.55)              --

Phase                Concentration             Viscosity,
Fraction, %         kg/m x s (lb/ft x s)

Vertical solid           5%-30%              0.00277-0.00732
(0.00186-0.00492)

Vertical liquid         95%-70%             0.0024 (0.00161)

Horizontal solid         5%-30%       0.1557-0.04168 (0.105-0.028)

Horizontal liquid       95%-70%            0.005032 (0.00338)

Source: Lee et al. (2002); Grozdek et al. (2009)
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