Study of the convective heat transfer during full melt off of ice slurry in laminar and non-Newtonian flows

This paper presents the experimental results of the heat transfer coefficient of ice slurry in the distribution phase in order to conserve fish and shellfish where the ideal temperature oscillates between 0 °C and 2 °C. The required energy power absorbed during the exchange between the ice slurry and the air is assimilated to the transport capacity (in kW of cooling at full melt off). The secondary fluid was prepared by mixing monopropylene glycol, MPG, and water to obtain an initial concentration varying from 5 to 24% (the initially freezing temperature varying from − 1.84 °C to – 9.39 °C). The heat transfer coefficients for ice slurry in laminar flow conditions increased with velocity and ice concentration but decreased with increasing MPG concentration, according to the results. The heat transfer coefficients for ice slurry in laminar flow conditions increased with velocity and ice concentration but decreased with increasing MPG concentration, according to the results. The heat transfer coefficients of ice slurry are of the order of 7000 W.m−2.K−1.Moreover, the rheological behavior of the ice slurry greatly influenced the results under certain conditions. A new average Nusselt correlation was proposed that accurately predicts the superficial heat transfer coefficients of laminar flow of ice slurry through the horizontal pipe.


Introduction
Several experimental and numerical studies were carried out on laminar flows in a horizontal circular pipe. A decrease in the size of the crystals results in an intensification of heat transfers, as noted by Kasa and Chen [1]. Others have conducted an experimental study on the heat transfer of flowing ice slurries into a horizontal tube. Most observations show that the internal Nusselt number increases with the velocity of flow and the ice fraction. However, there are different relationships in the literature reflecting these heat exchanges between ice slurry and the wall [2][3][4][5][6]. In Sari's experimental survey, they found that wall temperatures are lower when the ice slurry passes through an aluminum pipe in the heat exchanger [7].
Experimental and numerical studies of heat transfers of ice slurry, made up of microcapsules through a plate heat exchanger, confirmed that, regardless of the geometry tested, an increase in the concentration of microcapsules leads to an increase in the local Nusselt number during the laminar flow [8].
Furthermore, and contrary to the previous authors, a numerical study of the heat transfer of ice slurry in the form of a water-salt mixture at different flow concentrations through horizontal pipe has shown that the increase of the initial salt concentration results in a decrease of the mean Nusselt number by [9].
Many recent numerical studies on the heat transfer characteristics of ice slurry in a straight horizontal tube have been conducted. The VOF model in FLUENT 6.3 has been used to simulate ice slurry in laminar and turbulent flow regimes with varying ice fractions. The authors demonstrated that flow velocity and ice mass fraction affect the heat transfer performance of ice slurry. The density of the heat flux has only a small influence on the value of the Nusselt number. A 3D CFD model has been proposed by [10], which treats the ice slurry as a Newtonian and isothermal flow in a horizontal pipe.
The thermal behavior of ethylene glycol flowing through a horizontal heated tube has been investigated where the 3D steady-state model has been used to determine the thermal behavior of ice slurry, which shows the variation of the mean heat transfer. The convective heat transfer coefficient as a function of the ice volume fraction and for the three sets of operating conditions indicates that the present numerical results agree quite well with the experimental data obtained by Grozdek et al. for all ice fractions up to 20% [11]. Table 1 gives some correlations of the Nusselt number in laminar flow.
The present paper aims to use ice slurry as a secondary refrigerant in an indirect circuit to cool a positive cold room at 1 °C, thus making it possible to contribute to the knowledge of the thermal behaviour of the ice slurries in order to obtain a correlation of the Nusselt number in the horizontal tube as a function of the MPG concentrations, ice concentrations, and mass flow.
With regard to the present paper, the work was carried out as follows: one started the calculation with two unknowns, which are the internal exchange coefficient of the ice slurries and the length of the heat exchanger located in the chamber to be cooled, where one was dependent on the other. Then, I used a mathematical analysis on the overall heat transfer coefficient after having determined the convective heat transfer coefficient of the ambient air. Next, one calculates the limit length below which the convective heat transfer coefficients of the ice slurry will be negative in order to remove them. This is the principal contribution of the present work. Finally, a relationship for the average Nusselt number was developed.

Experimental equipment and procedure
The ice particles are formed on the external surface of two evaporator plates situated in a brushed surface heat exchanger Heatcraf model. The condensing unit has a power of 16 kW, a load of 10 kg of R404A and a temperature in the range -15 °C + 110 °C. The measuring devices is Coriolis mass-flow meter ABB/ MC 2000 model calibrated from 0 to 4000 kg h −1 and density from 950 to 1050 kg m −3 , with ± 1% accuracy, the differential pressure meter ROSEMOUNT type; model 3051/3001 ranges of 0 to 40 mbar, with ± 0.1% accuracy and two T type thermocouples with 0.05 K accuracy, conduct of characterization 1 m. Ice slurry is delivered from a storage tank with volume capacity of 2700l equipped with an agitator was used to allow a homogeneous mixture of ice slurry (Fig. 1). Knowing that the cold room is not introduced in the experimental device, it has been assumed. The ice slurry installation has two cooling loops. The primary loop contains a refrigerant which cools the ice slurry circulating in the secondary loop.
A variable speed single stage centrifugal pump Samson type multi-H203, DN 1" (26-33), power 0.55 kW calibrated from 200 to 2000 kg h −1 , maximum functional pressure 10 bars, volume flow rate from 0.5 to 5 m 3 h −1 , allows ice slurry to be circulated from storage tank to exchanger heat It is in the distribution part where the mass flow, density measurements and pressure drop will be carried out. The monopropylene glycol is used for its ability to lower the freezing temperature of water and for its low volatility and low corrosivity when inhibited, it has a high dynamic viscosity and no risk of human contact. Generally, monopropylene glycol is used above -18 °C because its viscosity makes the required pumping power too high.
Regarding the procedure, the concentrations of MPG were modified from 5 to 25% and for each concentration of MPG, the ice concentration was varied between 5% and 25%. For this conditions, the thermal equilibrium corresponding to an average temperature of the ice slurry between − 1.84 °C and -9.39 °C. Then, each ice fraction, the mass flow was varied from 400 to 2000 kg h −1 . The measures taken in each case are temperature, pressure drop, mass flow rate, and density of ice slurry.

Enthalpy and specific heat
h is , h g , and h l are the successive enthalpies of ice slurry, ice, and fluid carrier at temperature T as function of the ice concentration and the initial concentration of MPG. (1) Where L f is the latent heat of fusion of the ice at 0 °C equal to 333.6 kJ kg −1 and h g is the enthalpy of ice calculated by the expression [12]: A simple reasoning of the expressions given by [13], for a eutectic mass fraction of MPG equal to 0.6 gives us the following correlation: With a 1 = 4.2058; a 2 = 1.29583; a 3 = 0.49528, and a 4 = 2.25.10 −3 The apparent specific heat of the ice slurry is obtained easily by

Conductivity
The overall thermal conductivity of ice slurry is estimated by approximation which has been proposed by Jeffrey [14], introducing the interactions between the spherical particles, on the other hand, it is applicable to a wide range of diameters and concentrations of solid particles: λ g is thermal conductivity of ice (W m −1 K −1) calculated by the expression [12].
The thermal conductivity of the carrier fluid phase is determined by the approach of [13] gives us the following correlation:

Rheological properties
Assuming that the ice slurry follows a rheological behavior type Ostwald-de Waele law, the consistency index k = τ w /γ n .
A simple relation between the linear pressure drop (Δp/L) and the wall shear stress [15] gives The shear rate is opposite to the local gradient velocity, γ = − dv z dr . In the case of a laminar flow into a horizontal cylindrical pipe, the shear rate can be determined by using experimental data (flow rate and pressure drop), as well as the Rabinowitch and Mooney [16]: Let us remember the experimental results of Mellari et al., which reveal that the ice slurry behaves as non-Newtonian: thickening flow (n > 1) or shear thinning flow (n < 1) and sometimes as Newtonian fluid (n ≈ 1). Only in particular flow conditions, for the low concentration of the solute (x i = 5%) there is an important increase in the pressure drop, with an increase in ice fraction. This occurs because the size of the particles becomes important. This phenomenon increases the viscosity caused by forces of collision of particles, as well as by the friction forces between the single particles (agglomeration phenomena) and for high concentration of solute (x i = 19% and x g = 25%) and (x i = 24% and x g = 22 and 20%), the pressure drop becomes higher for small velocities where the moving bed is presented [17]. These concepts are very important because one will see later that this directly influences the heat transfer convection coefficient of the ice slurry and gives a most imprecision.

Reynolds and Prandtl numbers
The calculation methodology uses an evaluation of the Reynolds and Prandtl numbers from its non-Newtonian expression over the entire range of ice concentration considered. The rheological properties have been studied previously, where ρ is is the measured density of ice slurry by [17].

Convective heat transfer coefficient calculation
A simple method was sought for predicting the superficial heat transfer coefficient of ice slurry. The convective exchange coefficient between the ice slurry and the internal wall of the air/ice slurry heat exchanger is determined from the experimental knowledge of the overall transfer between the two fluids and the convective transfer between air and the outer wall with certain hypotheses, which are -Isothermal flow (heat flow lost through the insulated negligible); -Flow fully developed; -Flow time independent; -Isothermal surfaces; -Pressure drops of the exchanger neglected; -To maintain a temperature range of 5 to 10 °C between the ambient air and the temperature of the ice slurry, a cold room temperature of 1 °C has been chosen.

Energy balance
Where Q 0 is the required refrigerating power of air/ice slurry exchanger and Q is the cooling energy "at full melt off " of ice slurry given by While pumping power P is neglected so the required refrigerating power is the same as the cooling energy "at full melt off " of ice slurry.
It can also be given by Because the length of the heat exchanger is unknown and the overall heat transfer coefficient has been multiplied by the length as follows: It can also be evaluated from the usual formula Eq. (20). One, it has considered that there is no fooling in the heat exchanger and the thermal resistance of the tube wall is negligible.
Referring to Eq. (20), the convection coefficient on the ice slurry side can be calculated by This requires that the second term must be positive. This suggests that there is a finite limit of the length below which the ice slurry heat transfer coefficients become negative. This means that the wall temperature will be below the temperature of the ice slurry, which contradicts the second law of thermodynamics.
Considering that the heat air/ice slurry exchanger envisaged is placed in staggered rows with forced ventilation, the average Nusselt number given by the following relation was used to determine the convection coefficient on the air side for 10 3 < Re < 2.10 5 by neglecting the (15) corrective term [18]. Where u fr is the air frontal velocity of air, assumed to be in the range of 2.5 to 4 m s −1 .
The two convection heat transfer coefficients of the ice slurry and the air are related to the Nusselt number by the following equation:

Results and discussions
The tests provided 112 measurement points carried out under different conditions of concentration of the solute, the ice fraction, and the mass flow. Figure 2 shows the variation of the superficial heat transfer coefficient of ice slurry and (T w − T is ) versus the length of the exchanger for a 11% MPG concentration.
If the length tends to L ∞ , then the superficial heat transfer coefficient tends to infinite values and (T w − T is ) tends to 0, which means that the ice slurry temperature increases to its maximum to reach the wall temperature.
When the length exceeds L, the superficial heat transfer coefficient of ice slurry decreases and the difference (T w − T is ) becomes very small, starting at 0 °C for xg = 0.05, then 0.1 °C for xg = 0.11 and 0.14, and finally 0.3 °C for xg = 0.24. As can be seen, the wall temperature hardly varies with the velocity of the flow. This can be explained by the very small difference between the wall temperature and the temperature of the two-phase mixture, which is due to the migration of ice particles towards the walls of the tube, which makes new particles replace those already melted. These heat sinks allow a reduction of the temperature at the level of the wall, as demonstrated in Ben Lakhdar's thesis [19]. In addition, particle-particle and particle-wall interactions decrease the thickness of the thermal boundary layer and, consequently, the wall temperature.
When the length tends to infinite values, then the superficial heat transfer coefficient of ice slurry tends to 0 consequently T w tend to T air .
Finally, in the sizing of heat exchangers, a short length with a high superficial heat transfer coefficient is always desirable. For this reason, a length of 20 m in addition to the limit length has been chosen. Figure 3 presents the variety of Reynolds number versus velocity flow in horizontal conduct. The experiment results show that the Reynolds number is generally located between 50 and 2000. That means, of course,  It should be noted that the values of the Reynolds number decrease as the ice concentration increases. This is due to the increase of the slurry viscosity and the decrease of slurry density with ice concentration.
For x i = 5 and 11%, their range increases as the ice concentration increases and for x i = 14, 19, and 24%, their range decreases as the ice concentration increases. This is due to the increase of the slurry viscosity with solute concentration. Figures 4, 5, 6, 7, and 8 show the variation of the superficial heat transfer coefficient of ice slurry versus Reynolds number for five fixed initial MPG concentrations and varying ice fractions.
The convective heat transfer coefficient increases with increasing ice concentration and increasing Reynolds numbers. However, the ice concentration seems to have a greater influence on the heat transfer coefficient than the Reynolds number.
Due to the melting of ice, the convective heat transfer coefficient is higher than that of conventional fluids (500 to 7000 W m −2 K −1 ). Indeed, in laminar flow, the ice particles add a micro-convection effect that increases the thermal conduction, which has already been demonstrated [20]. It has also been observed that the values of the heat transfer coefficient decrease, as the MPG concentration increases. The highest value of the convective heat transfer coefficient (in W m −2 K −1 ) is 7000 at x i = 0.05 then 5500 at x i = 0.11 then 4700 at x i = 0.14 then 2700 at x i = 0.19 to reach 2500 at x i = 0.24.
However, one can see that, for the low solute (x i = 0.05), the values of the convective heat transfer coefficients are high, but the Reynolds number varies slightly due to low velocities at the same time due to the high viscosity of MPG. While for the high concentration of solute x i = 0.19 and x i = 0.24, the heat transfer coefficient increases very much at an ice concentration of 15%, before this concentration, the heat transfer coefficient becomes quasi constant.
On the other hand, ice crystal accumulation in the upper part of the pipe can cause flow blocking phenomena (significant increase in pressure drops with lower velocities, Re 200), while also reducing heat transfer to the lower part of the pipe. This phenomenon has already been observed by [21].
Finally, a Nusselt-Graetz type relation to model the heat transfer of ice slurry in a fully developed laminar flow (long tubes) and isothermal surfaces has been established. With n = 0.33 Valid for R e < 2100 and 40 P r < 900 According to a survey of the results, as can be seen in Fig. 9, the experimental Nusselt number is calculated to An accuracy of ± 5% for 55% of all experimental results; An accuracy of ± 15% for 25% of all experimental results. And an accuracy of − 40% for 20% of all experimental results, As it has already been explained above, these points are These points are not operational, and they were removed, so the values will become an accuracy of ± 5% (28) A = (-1500x i + 890)x g + 80 U d for 68.54% of all experimental results and an accuracy of ± 15% for 30.46% of all experimental results.

Comparison of the results with experimental data of the convective heat transfer coefficient
The experimental results of the heat transfer coefficient of ice slurry are validated and compared with the experimental data applied to laminar flows by some authors.
In the first work of comparison, the authors have experimentally determined the convective heat transfer coefficient of ice slurry flowing into a horizontal pipe when the ice suspension was produced with an aqueous solution of ethanol and with an initial mass of 10%. The test section had an internal diameter of 21.6 mm and a length of 1 m. The pipe is constructed of stainless steel and the flow velocities varied from 0.7 to 2.5 m.s −1 by Christensen-Kauffeld, cited by Ben Lakhdar [19].
In the second study, Guilpart et al. worked with an aqueous ethanol solution of an initial mass of 11% and a diameter of 30 cm to produce the ice slurry [22].
Third, the heat transfer, based on 10.3% of initial alcohol concentration-water mixing through a circular horizontal tube with an ice mass fraction ranging from 0 to 20%, has been examined experimentally by Grozdek et al. [6].
Finally, a study has been conducting an experimental study of the heat transfer coefficient during the flow of ice slurry at different mass fractions in a plate heat exchanger. The ice slurry has been obtained from a water/ ethylene glycol mixture (Gupta and Fraser cited by Snoek cited by Ben Lakhdar [19]). In addition, the results give a good agreement with Christensen-Kauffeld and experimental results of the present paper at 1 ms −1 regardless of the ice fraction but at the same velocity, the values are slightly higher.
On the other hand, with Grozdek et al., the values of heat transfer coefficients are very close to the mass concentration equal to 10%, whether for velocity 1 m.s −1 or 0.6 ms −1 . From this last one begins a deviation of the values.
As one can clearly notice, below 10%, the values of heat transfer coefficients hardly vary with the concentration of ice and vary slightly with the velocity for all authors, because the high concentration has a great effect on microconvection. Figure 11 presents a comparison of heat transfer coefficients as a function of flow rate of the present paper with those of Gupta-Fraser and Christensen-Kauffeld for x i = 11% and 20% of the ice fraction.
The figure shows that the convective heat transfer coefficients of Gupta and Fraser obtained within the plate heat exchanger are greater than those of the present paper and much more at low flow rates. While the values of Guilpart et al. are close to the values of the present paper for xi = 11%, it can be noticed that the values of Guilpart et al. vary slightly with the flow, but with Gupta and Fraser, the evolution of values is the same. These results can be explained by the geometry tested (plate heat exchanger for Gupta-Fraser and cylindrical for both Guilpart et al. and the present paper).  Figure 12 compares the correlation of the present paper (x i = 11%) with other authors' correlations. As previously indicated, the values of the current work and those of Christensen-Kauffeld show a notable agreement with an accuracy of 15%, according to the change in ice concentration. In contrast, Guilpart and Grozdek's Nusselt number values agree with one another but are lower than in the current work, particularly as ice concentration increases. Contrary to Christensen-Kauffeld, as was already established, the convective heat transfer coefficient does not alter significantly with the concentration of ice. Whatever the case, the figures are still lower than the current work, which brings us back to the characteristics of MPG and ethanol.

Conclusion
The main objective of this work consists of an experimental study of the heat convective transfers of ice slurry with an indirect system of refrigeration. While the required refrigerating power is the cooling energy "at full melt off " of ice slurry, varying from 8 to 60 kW, and the velocities varying from 0.25 to 1.2 ms −1 , which are considered reasonable velocities found in a secondary refrigeration system. The overall heat transfer coefficient varies from 40 to 140 Wm −2 K −1 . The various test campaigns were carried out to study the influence of the flow rate of the ice slurry, its particle content, and the concentration of the solute.
The superficial heat transfer coefficients of ice slurry in the fully developed laminar and steady state were determined.
As it has been seen, the heat transfer coefficients are largely influenced by the flow regime.
From the convective heat transfer coefficients, an emperical correlation between the Nusselt number, the Graetz number, solute concentration, ice fraction, and mass flow velocity was established.
The comparison showed that the Nusselt number determined experimentally and that of the correlation are very close. The correlation appears satisfactory. One must note that the lengths of exchangers obtained are in the range of 300−700 m.
The comparison demonstrated that the values given by Christensen-Kauffeld and Guilpart et al. give good agreement with experimental results of present paper regardless of the ice fraction. The deviations of experimental data can be explained by the different operating conditions.
Finally, in the future study, one can expect to see the same results by adding the sensible heat of the aqueous solution to the latent heat, which is sure to improve the heat exchange between the ice slurry and the ambient air.