Numerical simulation on flow of ice slurry in horizontal straight tube

Abstract

Numerical simulation on flow of ice slurry in horizontal straight tubes was conducted in this paper to improve its transportation characteristics and application. This paper determined the influence of the diameter and length of tubes, the ice packing factors (IPF) and the flow velocity of ice slurry on pressure loss by using numerical simulation, based on two-phase flow and the granular dynamic theory. Furthermore, it was found that the deviation between the simulation results and experimental data could be reduced from 20% to 5% by adjusting the viscosity which was reflected by velocity. This confirmed the reliability of the simulation model. Thus, two mathematical correlations between viscosity and flow velocity were developed eventually. It could also be concluded that future rheological model of ice slurry should be considered in three sections clarified by the flow velocity, which determined the fundamental difference from single-phase fluid.

Keywords

ice slurry / horizontal tubes / numerical simulation / pressure drop / viscosity model

Introduction

The ice slurry, as a cool agent, has shown great advantages in cooling capacity, flow characteristics and pump ability, compared to the conventional ice-thermal storage technology. It is made up of ice crystals, water and certain depressant additives. Large latent heat is absorbed by ice slurry due to the presence of ice crystals, which makes it possible to design the heat exchanger in a much more compact way. In recent years, the ice slurry has been increasingly used in practical application, such as the HVAC systems (heating ventilation air conditioning), the food cooling area, the medical and health systems, and the shipping industry, etc [¹].

It is believed that three major factors hinder the study on flow behavior of ice slurry in pipeline systems through which the ice slurry is transported to target places for a second refrigerant. First of all, the ice slurry, a mixture of liquid-solid two phases, is different from single-phase liquid in terms of the flow characteristics [²]. Secondly, the heat transfer between two phases occurs because of the melting process, which simultaneously causes the mass transfer. Finally, the ice slurry with certain concentration has granularity effects due to the presence of ice particles. In general, most published literature only take account of the above factors partly in describing the flow characteristics in specific conditions.

Ayel et al. [³] have reviewed four rheological models, namely Bingham, Casson, Power and Herschele Bulkley, which can predict the behavior of ice slurry in tubes on specific conditions. Kumano et al. [⁴] have investigated flow characteristics of ice slurry in narrow tubes experimentally and made theoretical analysis on relationships between the coefficients of pipe friction and Reynolds number. It is found that the flow characteristics of ice slurry can be treated as pseudoplastic fluid and clarified by laminar and turbulent flow. Illán and Viedma [⁵] have found that the increase in pressure drop properties for corrugated pipes is in the same range as that found for smooth pipes, and the heat transfer increases in smooth pipes with ice concentration increment while corrugated pipes show the opposite behavior. Kalaiselvam et al. [²] have numerically analyzed the heat transfer and pressure drop characteristics of a tube-fin heat exchanger in an ice slurry HVAC system and determined optimization parameters like louver angle, fin pitch, and ice slurry flow velocity. Wang et al. [⁶] have made a numerical investigation on ice slurry flowing through the horizontal 90° elbow and found a clear secondary flow phenomenon which enhances the mixing between ice particles and carrier fluid. Shi et al.[⁷] has developed an Euler-Euler two-fluid model incorporating the kinetic theory of granular flow to describe the steady-state two-phase flow in a tubular loop propylene polymerization reactor composed of loop and axial flow pump.

The diameters of the ice crystals and the IPFs of transported ice slurry may differ from the initial inlet value due to the phase change in the tubes. To determine the effects of two-phase and solid ice particles on the pressure drop of the mixture, this paper has omitted the transformation of two phases as well as the heat exchange occurring in the process. According to the theory of two-phase flow and granular dynamics, numerical simulation analyses have been conducted to study the relationship between pressure drop and such parameters as the IPFs, the diameters of tubes and the length of tubes. The results of numerical simulation have an error below 20% compared to the previous experimental data. Indeed, Kousksou et al. [⁸] believes that the value of viscosity is a function of IPF which has been validated and accepted by other researchers. This paper has eventually developed two correlations between the viscosity and flow velocity. It is found that deviation between simulation results and experimental data can be reduced to below 5% using the new modified viscosity model.

Theoretical analyses

To study the ice slurry, it is assumed in this paper that IPF is between 5% and 30%, the diameters of tubes ranges from 5 mm to 10 mm, and the diameter of ice crystals is approximately 100\mathrm{\mu m}. Under this circumstance, the aqueous fluid has a great granularity effect on the flow behavior [⁸]. Shi et al. [⁷] has made analyses on liquid-solid two-phase flow in a tubular loop polymerization reactor numerically, also based on the kinetic theory of granular flow. According to the theory of granular dynamics, the carrier fluid and particles in the mixture are believed to move under three forces [⁹], the forces unrelated to the flow such as gravity and inertia forces etc., the forces along the flow direction such as tangential friction force and virtual mass force, and the forces perpendicular to the flow direction such as buoyancy lift.

Conservation of mass

The Euler-Euler model of multiphase mixture has been adopted to describe the solid-liquid two-phase ice slurry flow behavior. The volume fraction of either of the two phases, called phase q, is calculated using Eq. (1).

(1)V_q={\displaystyle {\int }_V{\alpha }_q}dV,

where

(2){\displaystyle \sum _{q=\mathrm{1}}^n{\alpha }_q}=\mathrm{1.}

The effective density of phase q is calculated by using Eq. (3).

(3){\widehat{\rho }}_q={\alpha }_q{\rho }_q,

where {\rho }_q represents the physical density of phase q.

The continuity equation for phase q is shown as

(4){\displaystyle \frac{\partial }{\partial t}}({\alpha }_q{\rho }_q)+\nabla \cdot \left({\alpha }_q{\rho }_q{\stackrel{\rightharpoonup }{v}}_q\right)=-{\displaystyle \sum _{p=\mathrm{1}}^n{\dot{m}}_{pq}},

where {\stackrel{\rightharpoonup }{v}}_q represents the velocity of phase q, {\dot{m}}_{pq} represents the mass transfer from phase q to the other phase, called phase p, and {\alpha }_q represents the volume fraction of phase q.

Equations (5) and (6) can be derived from the equation of mass conservation,

(5){\dot{m}}_{pq}=-{\dot{m}}_{qp},

(6){\dot{m}}_{pp}=\mathrm{0.}

Conservation of momentum

The ice slurry is regarded as two-phase liquid containing particles which show granularity. Therefore, the momentum equation of liquid phase is shown as

(7)\begin{array}{l}{\displaystyle \frac{\partial }{\partial t}}\left({\alpha }_q{\rho }_q{\stackrel{\rightharpoonup }{v}}_q\right)+\nabla \cdot \left({\alpha }_q{\rho }_q{\stackrel{\rightharpoonup }{v}}_q{\stackrel{\rightharpoonup }{v}}_q\right)=-{\alpha }_q\nabla p+\nabla \cdot \overline{\overline{{\tau }_q}}+{\alpha }_q{\rho }_q\stackrel{\rightharpoonup }{g}\\ \hspace{1em}\hspace{1em}\hspace{1em}+{\alpha }_q{\rho }_q({\stackrel{\rightharpoonup }{F}}_q+{\stackrel{\rightharpoonup }{F}}_{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t},q}+{\stackrel{\rightharpoonup }{F}}_{\mathrm{v}\mathrm{m},q})+{\displaystyle \sum _{p=\mathrm{1}}^n\left(K_{pq}({\stackrel{\rightharpoonup }{v}}_p-{\stackrel{\rightharpoonup }{v}}_q)+{\dot{m}}_{pq}{\stackrel{\rightharpoonup }{v}}_{pq}\right)},\end{array}

where \overline{\stackrel{‾}{{\tau }_q}} represents the stress-strain tensor of phase q, {\stackrel{\rightharpoonup }{F}}_q represents the external body force, {\stackrel{\rightharpoonup }{F}}_{\mathrm{lift},q} represents the lift power, {\stackrel{\rightharpoonup }{F}}_{\mathrm{vm},q} represents the virtual mass force, {\stackrel{\rightharpoonup }{R}}_{pq} represents the interaction forces between phases, and p is the mutual forces applying on all phases.

{\stackrel{\rightharpoonup }{v}}_{pq} represents the relative velocity between phases, which is determined by Eqs. (8) and (9).

(8){\stackrel{\rightharpoonup }{v}}_{pq}={\stackrel{\rightharpoonup }{v}}_p,\mathrm{if}\hspace{0.17em}{\dot{m}}_{pq}>\mathrm{0},

(9)

The equation of momentum applying to solid particle phase is shown as

(10)\begin{array}{l}{\displaystyle \frac{\partial }{\partial t}}\left({\alpha }_s{\rho }_s{\stackrel{\rightharpoonup }{v}}_s\right)+\nabla \cdot \left({\alpha }_s{\rho }_s{\stackrel{\rightharpoonup }{v}}_s{\stackrel{\rightharpoonup }{v}}_s\right)=-{\alpha }_s\nabla p-\nabla p_s+\nabla \cdot \overline{\stackrel{‾}{{\tau }_s}}+{\alpha }_s{\rho }_s\stackrel{\rightharpoonup }{g}\\ \hspace{1em}+{\alpha }_s{\rho }_s({\stackrel{\rightharpoonup }{F}}_s+{\stackrel{\rightharpoonup }{F}}_{\mathrm{lift,s}}+{\stackrel{\rightharpoonup }{F}}_{\mathrm{vm,s}})+{\displaystyle \sum _{l=\mathrm{1}}^n\left(K_{\mathrm{ls}}({\stackrel{\rightharpoonup }{v}}_l-{\stackrel{\rightharpoonup }{v}}_s)+{\dot{m}}_{\mathrm{ls}}{\stackrel{\rightharpoonup }{v}}_{\mathrm{ls}}\right),}\end{array}

where p_s represents the solid pressure,K_{\mathrm{ls}}(=K_{\mathrm{sl}}) represents the momentum exchange coefficient fluid phase (l) and solid phase (s), n is the total number of phases, and {\stackrel{\rightharpoonup }{F}}_q,\hspace{0.17em}{\stackrel{\rightharpoonup }{F}}_{\mathrm{lift},q}and {\stackrel{\rightharpoonup }{F}}_{\mathrm{vm},q} are the external volume force, the lift force, and the virtual mass force, respectively.

Gidaspow et al. have proposed the theoretical model which can be used to explain the interaction forces between the liquid phase and the granular phase [¹⁰]. Interchange coefficient K_sl, the key parameter in the model, can be determined by using Eqs. (9) and (10).

When

(11){\alpha }_{\mathrm{1}}>\mathrm{0.8},\hspace{1em}\hspace{1em}{{\displaystyle K}}_{\mathrm{sl}}={\displaystyle \frac{\mathrm{3}}{\mathrm{4}}}{{\displaystyle C}}_{\mathrm{D}}{\displaystyle \frac{{\alpha }_s{\alpha }_{\mathrm{1}}{{\displaystyle \rho }}_{\mathrm{1}}\left|\overrightarrow{{{\displaystyle v}}_s}-\overrightarrow{{{\displaystyle v}}_{\mathrm{1}}}\right|}{{{\displaystyle d}}_{\mathrm{s}}}}{\alpha }_{\mathrm{1}}^{-\mathrm{2.65}},

where

(12)C_{\mathrm{D}}={\displaystyle \frac{\mathrm{24}}{{\alpha }_{l}R{e}_{\mathrm{s}}}}[\mathrm{1}+\mathrm{0.15}{({\alpha }_{l}R{e}_{\mathrm{s}})}^{\mathrm{0.687}}].

When

(13){\alpha }_{\mathrm{1}}\le \mathrm{0.8},\hspace{1em}\hspace{1em}{{\displaystyle K}}_{\mathrm{s}\mathrm{l}}=\mathrm{150}{\displaystyle \frac{{\alpha }_{\mathrm{s}}(\mathrm{1}-{\alpha }_{\mathrm{1}}){{\displaystyle \mu }}_{\mathrm{1}}}{{\alpha }_{\mathrm{1}}{{\displaystyle d}}_s^{\mathrm{2}}}}+\mathrm{1.75}{\displaystyle \frac{{{\displaystyle \rho }}_{\mathrm{1}}{\alpha }_s\left|\overrightarrow{{{\displaystyle v}}_{\mathrm{s}}}-\overrightarrow{{{\displaystyle v}}_{\mathrm{1}}}\right|}{{{\displaystyle d}}_{\mathrm{s}}}},

where Re represents the relative Reynolds number, and the relative Reynolds number of phases p and q is expressed as

(14)Re={\displaystyle \frac{{{\displaystyle \rho }}_q\left|\overrightarrow{{{\displaystyle v}}_p}-\overrightarrow{{{\displaystyle v}}_q}\right|{{\displaystyle d}}_p}{{{\displaystyle \mu }}_q}}.

Results and discussion

The ice slurry for numerical simulation consists of liquid sodium chloride solution and ice crystals with a diameter of 100\mathrm{\mu m}. The straight tubes in which ice slurry flows have diameters of 9 mm and 15 mm. For boundary conditions, the inlet has been set to a velocity boundary from 0 to 2.6 m/s and the outlet has been set to outflow. The energy option is selected and gravity accounted. In addition, the RNG mixture option under the k-x model is selected to describe the turbulent flow. To evaluate the results calculated under the above conditions, the experimental data derived by Grozdek et al. [¹¹] are used for comparison.

This study has used the ICEM to build the geometric model and the grids to conduct the simulation in the commercial code CFD. The grids of straight tubes include 62426 hexahedral cells and the quality is as high as 0.8 which is helpful to improve the accuracy of the numerical simulation.

Error analysis of numerical results

Figure 1 displays the contour of the pressure in the longitudinal section of the horizontal tube. It is observed clearly that the pressure drops gradually along the straight tube.

The errors between numerical simulation results and experimental data [¹¹] for IPF= 10% and IPF= 20% are less than 20%, which confirms the reliability of the numerical model, as shown in Figs. 2 and 3. What is special is that the simulation results are obtained using different viscosities, as the legends in Figs. 2 and 3 indicate.

Effects of flow parameters on pressure loss

Analyses have been made on the effects of the flow velocity, the IPFs and the diameters of tubes on pressure drop when several parameters are fixed, namely the size of ice crystals and the whole length of tubes. Flow velocities are varied from 0 to 2 m/s and IPFs are respectively set to 10%, 20% and 30%.

Figures 4 and 5 respectively show the comparison of the numerical and experimental results on pressure drop at IPF= 10% and IPF= 20%, presenting the numerical data calculated using different viscosity values. It is conspicuously observed that the pressure drop increases with the increasing velocity both in simulation and experimental results. The experiment data at IPF= 10% are in good agreement with the simulation results calculated at viscosity= 0.0012 when the liquid velocity ranges from 1 m/s to 1.5 m/s, as depicted in Fig. 4 while the experiment data at IPF= 20% are in good agreement with the simulation results calculated at viscosity= 0.0013 when the liquid velocity ranges from 0.8 m/s to 1.7 m/s, as illustrated in Fig. 5. It can be seen that the simulation results of pressure drop at a higher or lower velocity is a little lower than the experimental data in both Figs. 4 and 5. Hence, a new modification model of viscosity needs to be developed to improve the accuracy of the simulation which is based on the kinetic theory of granular, which will be detailed in Section 3.3.

Figures 6 and 7 demonstrate the relationships between the pressure of cross-section and the distance from the cross-section to the inlet. It can be concluded that the pressure drop varies with the distance to the inlet in an almost linear fashion. This paper verifies the feasibility of using Dp/L to represent the pressure drop in any per length of tubes, which has been applied in many previous works [¹², ¹³].

Figure 8 indicates that the pressure loss increases with the decreasing diameters of the tubes when IPF and the flow velocity are kept constant, which is mainly caused by two possible reasons, i.e., the pressure is inversely proportional to the diameter of pipes, and the pressure loss is a function of ratio between diameter of ice crystals and diameter of tubes [¹⁴]. Anyway, the granularity of the ice crystals, being more evident with decrease of tube diameters, boosts interaction between the liquid and particles and increases loss in momentum. This has been proved in previous research [¹⁵].

It can be seen from Fig. 9 that the simulation values of pressure drop appear much higher than the experimental data at IPF= 30%. The reason for this is that the IPF level is so high that solid particles, as a group, have a great effect on the flow characteristics of the solution. Comparing Fig. 9 with Figs. 4 and Fig. 5, it can be deduced that the pressure drop increases with the increasing velocity. More attention is paid to the fact that the growth rate of the pressure loss keeps decreasing with the increasing velocity at IPF= 30%. When the growth rate descends and shows an opposite trend at IPF= 10% and IPF= 20%, it agrees with the kinetic theory of granular. Therefore, the flow characteristics of ice slurry with high an IPF may not apply to the granular dynamic theoretical model.

Above all, the numerical model which combines the two-phase flow with the kinetic theory of granular dynamics only applies to the ice slurry with low IPFs. Furthermore, to improve the simulation accuracy, the viscosity value needs to be modified according to different velocities, especially at a speed higher than 1.6 m/s or lower than 0.9 m/s. Viscosity values show little dependence on velocities when the speed remain between 1 m/s and 1.6 m/s.

Relationships between viscosity and velocity

Kousksou et al. [⁸] have concluded that viscosity is depended on IPF for two-phase and particles flow. The equation is expressed as

(15)\mu ={{\displaystyle \mu }}_{\mathrm{1}}[\mathrm{1}+\mathrm{2.5}{{\displaystyle \phi }}_i+\mathrm{10.05}{{\displaystyle \phi }}_i^{\mathrm{2}}+\mathrm{0.00273}({\mathrm{e}}^{\mathrm{16.6}{{\displaystyle \phi }}_i}-\mathrm{1})].

Ming and Chen [¹⁶] have found that the flow resistance of ice slurry in horizontal tubes increases first, then, decreases when IPF increases. In addition, the flow parameters seem to be volatile conspicuously at an extremely low velocity. The presence of solid particles causes the reduction of the flow resistance by depressing the turbulent effect. On the other hand, the agglomeration of particles results in a higher viscosity of the whole solution, which leads to a larger pressure loss compared to that of pure water. It can also be inferred that when IPF keeps constant, the ice slurry of different velocities which caused solid particles have two opposite effects on the viscosity value and flow resistance to some degree.

The viscosity values have been adjusted according to different flow velocities. Obviously, Figs. 10 and 11 indicate that the viscosity decreases before increasing with increasing velocity when IPF is set to 10% and 20%. This paper has developed a viscosity model calculated by corresponding velocities at IPF= 10% and IPF= 20%. In the modified viscosity model, the error of the numerical results of pressure drop is below 5%. Notably, the fact that viscosity is calculated by velocity does not mean dependence of the viscosity on velocity directly, but it indicates that the viscosity is determined by parameters related to the flow speeds indirectly. Correlations between the viscosity and velocity are derived by fitting the initial curves as

(16)\upsilon =\mathrm{0.0043}-\mathrm{0.0096}v+\mathrm{0.0085}{v}^{\mathrm{2}}-\mathrm{0.0022}{v}^{\mathrm{3}}(\mathrm{IPF=10\%,}\hspace{0.17em}D=\mathrm{0.015}\mathrm{m}),

(17) \upsilon =\mathrm{0.009}-\mathrm{0.0153}v+\mathrm{0.00935}{v}^{\mathrm{2}}-\mathrm{0.00177}{v}^{\mathrm{3}}(\mathrm{IPF=20\%,}\hspace{0.17em}D=\mathrm{0.015}\mathrm{m}).

It can be concluded that three regions of velocity, i.e., the lower velocity region, the transition region, and the higher velocity region should be clarified to study the pressure drop, the viscosity and the flow resistance. Different velocity regions would be used to describe the flow characteristics or to define new calculation models. As is known to all, the flow of single-phase fluid is separated from two regions, i.e., the laminar flow and the turbulent flow. However, it is the interaction forces between the particle phase and liquid phase that mainly makes the differences in the ice slurry flow, compared to the single-phase liquid. Though Doron and Barnea [¹⁷] have proposed three flow patterns in their research which has been applied by Liu [¹⁸] to develop new expressions fit for suspension bed; most of other scholars do research only by laminar and turbulent flow, which can only explain the single-phase liquid flow. Thus, by making numerical simulation analyses in this paper, it can be drawn that future study on the flow of ice slurry in horizontal straight tubes should be considered in three conditions.

Conclusions

1) Numerical simulation results of ice slurry flowing in horizontal tubes based on kinetic theory of granular have errors below 20% compared to the experimental data, which confirms the reliability of the numerical model at IPF= 10% and IPF= 20%.

2) For IPF= 10% and IPF= 20%, the experimental data are in good agreement with the simulation results calculated at a relatively low viscosity when the liquid velocity ranges from 0.8 m/s to 1.7 m/s while the numerical results of pressure drop calculated at the same low viscosity looks a little higher than the experimental data when the velocity varies. Hence, a new modification model of viscosity needs to be developed to improve the accuracy of the simulation based on the kinetic theory of granular.

3) The flow characteristics of ice slurry with high IPF may not apply to the granular dynamic theory.

4) The granularity effects of the ice slurry due to the presence of the ice crystals become stronger with decreasing tube diameter, which causes greater interaction between the liquid and particles and much more loss in momentum.

5) The feasibility of using Dp/L to represent the pressure drop in any per length of tubes, which is applied in many previous works, has been verified.

6) A viscosity model calculated by corresponding velocities has been developed at IPF= 10% and IPF= 20%. It has been found that viscosity values decrease at first and then increases with the increase of velocities. Two correlations between the viscosity values and the velocities derived from fitting curves are given.

7) The numerical values of pressure drop bear an error below 5%, using the modified viscosity model. Thus, according to the numerical simulation analyses in this paper, it can be concluded that future study on flow of ice slurry in horizontal straight tubes should be considered under three conditions clarified by the flow velocities.

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