INTRODUCTIONAn important issue of the intensification ofproduction processes and rational use of raw materialsis the possible rapid decrease in the temperature of theliquid system. For example, in processing industries andother sectors of the economy, the cooling of biologicalorigin substances contributes to the maintenance oftheir biological properties, as well as prevents microfloragrowth in the product [1, 2].Currently, the cooling of liquids by using frozensolids is one of the methods for lowering the temperaturein a liquid medium [3–7, 11, 14, 15]. This technology isuseful and in refrigeration engineering, where frozenphysical bodies are balls filled with eutectic solution[8–10, 12]. This is confirmed by results of theoreticaland experimental studies conducted by reserchersof Moscow State University of Food Production andRazumovsky Moscow State University of technologyand management. The results confirm the advantagesof cooling water with frozen balls over other methods,providing a high intensity of the process and reducedenergy consumption.To accelerate the heat transfer process based onwater treatment using the technology of enrichment ofthe working volume of cooling liquid with frozen bodies(cooling agent), it is advisable to carry out this processin the mode of flow of the liquid through a containerwith frozen balls. At the same time, it should be notedthat there are no theoretically based calculation methodsfor predicting and controlling the heat transfer process,including in the flow, when the cooling process developsin a heterogeneous liquid “water-frozen balls” system.Bases on the law of conservation of mass and energy,we presented the results of the analytical and numericalstudies on the of cooling a liquid flow moving througha heat exchanger filled with balls with a frozen eutecticcoolant to justify the intensification of the heat transferprocess between a coolant and liquid.172Slavyanskiy A.А. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 171–176The setting of the problem. Let a liquid flow (forexample, water) be supplied to a certain container filledwith balls with frozen eutectic solution in a stationarymode, with a small productivity Q and a low flow rate v0(Fig. 1). We will consider the selected motion model of atwo-phase liquid system as the filtration flow of a fluidthrough a porous medium formed by balls (coolant) [6].It is assumed that the balls are statistically uniformlydistributed inside a cylinder of length h and radius Rcontaining a liquid (water) and solid (balls) phases.To simplify the process for quantitative analysis ofthe heat removal from the liquid to the balls, we replacedthe pore volume occupied by the liquid (preservingthe values of porosity) with a set of cylindrical tubes(conditionally, capillaries). Each of them had an axisparallel to the axis of the vessel, length h and reducedradius r0 (Fig. 2).For the selected geometric model of the liquidvolume, we introduce the following notations: db is theball diameter and m is the surface porosity of the system(m < 1), numerically equal to the ratio of the volume ofliquid filled pores to the volume of the system [8].Since the volume and surface porosities of theworking volume of the capacitance are quantitatively thesame [8], the approximate ratio, based on the porositydefinition can be written as follows:πr02 /m = πdb2/[4(1 – m)]which yields the relationship:r0 = 0,5d[m/(1 – m)]1/2 (1)henceforward, r0 = r0 (d,m), d = db is the diameter of theball.Thus, as can be seen, the problem of calculationreduces to the quantitative analysis of temperature in theisolated capillary.The solution of the problem. Due to the possibleaxisymmetric nature of heat transfer from the walls ofthe channel to the liquid, the quantitative modelingof this process will be carried out in a cylindricalcoordinate system, in the meridional section of thechannel rOz (Fig. 2).For the selected simulation geometric model, weuse the equation of stationary convective thermalconductivity related to cylindrical coordinates as abasic differential equation describing the heat transferphenomenon in the flow [9]:dкО( ) ( 1 )22rTr ra TzTvz = 1   nii i AJ r r 10 0 ( / ) exp( 0 10 100200300400500Frozen ball LiquidFlow of Flow of cooled uncooledliquidv0 v0hrRvzТ/rrr0hО zr, W/(m2К)Т, (2)where r, z – radial and axial coordinate, respectively;T – the temperature of water; vz – axial componentof fluid flow rate in the capillary; а = λ/(сρ), а – is thecoefficient of thermal diffusivity of water, λ – thermalconductivity coefficient, c – specific conductivitycoefficient, and ρ – is the density of water.In practice, fluid rate vz can be replaced with itsaveraged value over the cross section of the channel,with a small error. Then a simplified form of Eq. (2) canbe written as follows:( 1 ) 22rTr rTzT∂∂+∂∂=∂∂β (3)whereβ = а/vz (4)vz = v0/m, v0 = Q/S – volume rate of fluid flow in thecapacitance (filtration rate [8]), S = πR2 – cross-sectionalarea of the capacitance; β is the specific coefficient ofthermal diffusivity calculated taking into considerationaxial velocity vz.For simplicity, assume inlet temperature to beconstant:Т(r,z) = Т0 at 0 ≤ r ≤ r0, z = 0 (5)The condition of symmetry of the temperaturedistribution along the channel diameter corresponds tothe condition of the maximum temperature in the middlethe capillary walls.∂∂ÒÒT//∂/ ∂rr a==t 0r0 = 0, 0 < z ≤ h (6)Taking into account the fact that heat energydevelops from the liquid to the capillary wall, theboundary condition on the surface of this channel is:Н[Т(r0,z) – Тk] – ∂T(r0,z)/∂r = 0 (0 < z ≤ h) (7)Figure 1 Scheme of liquid cooling with frozen ballsFigure 2 Scheme to calculate the cooling process of liquidwith frozen ballsdкО( ) ( 1 )22rTr ra TzTvz = 1     nii i i AJ r r Fo120 0 ( / ) exp(  ),0 10 20 30100200300400500TFrozen ball LiquidFlow of Flow of cooled liquiduncooledliquidv0 v0hzrRvzТ/rrr0hО zr, W/(m2К)Т, СdкО( ) ( 1 )22rTr ra TzTvz = 1     nii i i AJ r r Fo120 0 ( / ) exp(  ),0 10 20 30100200300400500TFrozen ball LiquidFlow of Flow of cooled liquiduncooledliquidv0 v0hzrRvzТ/rrr0hО zr, W/(m2К)Т, СdкО( ) ( 1 )22rTr ra TzTvz = 1     nii i i AJ r r Fo120 0 ( / ) exp(  ),0 10 20 30100200300400500TFrozen ball LiquidFlow of Flow of cooled liquiduncooledliquidv0 v0hzrRvzТ/rrr0hО zr, W/(m2К)Т, СdкО( ) ( 1 )22rTr ra TzTvz = 1     nii i i AJ r r Fo120 0 ( / ) exp(  ),0 10 20 100200300400500Tv0 v0hzvzТ/rrr0hО zr, W/(m2К)Т, С173Slavyanskiy A.А. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 171–176where Н = α /λ, α i s heat transfer coefficient from theliquid to the capillary wall, Тk is the eutectic temperatureof frozen balls (Fig. 1).According to (5)–(7), in the framework of theterminology adopted in the theory of heat transfer, wehave a problem with boundary conditions of the thirdkind for differential equation (3).From the point of view of quantitative analysis ofthe thermal regime in a steady stream of fluid inside thecapillary, the considered problem is formally equivalent(provided that the diameter of the capillary dk is muchless than the capacitance height h) to the problem of thetemperature distribution over time in an unsteady modein an unlimited cylinder. What is more, in Eqs. (3)–(7)the duration of the heat transfer process is displayed onthe axial coordinate z.Thus, the solution to the problem with an unsteadymode of heat transfer in an unbounded cylinder can beadapted to the boundary value problem (3)–(7) of thetemperature distribution in the convective fluid stream inthe capillary and is formulated as a dependence:dк1 )22rTr rT = 1     nii i i AJ r r Fo120 0 ( / ) exp(  ),0 10 20 30100200300400500TFrozen ball LiquidFlow of cooled liquidv0hzRvzТ/rrr0hz, W/(m2К)Т, С, n → ∞ (8)whereθ = θ(d,m,r,z) = (Т – Т0)/(Тк – Т0) > 0 (9)is specific value reflecting the differential temperature ofthe ball Тk and the initial temperature of the liquid Т0, aswell as the current differential temperature of the liquidТ(d,m,r,z) and the coolant temperature Т0.Аi = 2J1(νi)/{νi[J0(νi)2 + J1(νi)2]} (10)where J0, J1 – Bessel function of the first kind ofzero and first order respectively; positive roots oftranscendental equations.J0(ν)/J1(ν) = ν/Bi (11)Bi = Bi(α,d,m) = αr0/λ – the Biot number (12)Fo* = Fo*(z,d,m) = βz/r02 – the modified Fourier number (13)RESULTS AND DISCUSSIONQuantitative modeling of the heat transfer processwas carried out based on relations (8)–(13) using theMathcad medium.We used the following process parameters: thelength of the capacitance h = 0.5 m; the diameter ofthe capacitance D = 0.1 m; equipment productivity (bywater) Q = 2×10–4 m3/s; kinematic viscosity coefficientν = 10–6 m2/s; ball eutectic temperature Тk = – 10°С;thermal c onductivity c oefficient λ = 0 .58 W /(m·К);thermal diffusivity coefficient a = 13.8×10–8 m2/s; andball diameters d = 0.0375, 0.04, and 0.0425 m. Thecalculation of the current temperature of the liquidwas carried out according to two variants of porosity:m = 0.5 and m = 0.35.In accordance with the selected parameter values,the volume rate of flow (filtration rate) for all calculationoptions was v0 = 4Q/(πD2) = 4×2×10–4/(3.14×0.12) =0.0254 m/s.As a calculated value of the temperatu∫re 02 002 r ( , , , ) , d m r z rdrrθ given overthe radius r inside the capillary, we used its valu∫e 02 002 r ( , , , ) , d m r z rdrrθ av,averaged over the channel cross-sectional area:∫ 02 002 r ( , , , ) , d m r z rdrrθ av(d,m,z) = ∫ 02 002 r ( , , , ) , d m r z rdrrθ (14)The following dependence was used as a calculateddependence for the liquid temperature based on theoperating parameters of the axis coordinate z and thetime of the process τ:T(z) = T0 + (Тк – Т0)θ, (15)To calculate the number of balls N in the capacitance,we used the formula:N(m,d) = 1.5D2h×(1 – m)/d3 (16)where m is the porosity if the liquid system, d isthe diameter of the ball, D is the diameter of thecapacitance, and h is the length of the capacitance.Thus, according to the geometrical parameters,the number of balls of diameter d = 0.04 m was 58 form = 0.5 and 76 for m = 0.35.Previously, to assess the convergence of series (8),we performed test calculations using formula (14).Temperature T0 was 36°С and heat transfer coefficientα was 440 W/(m2·К) ( Fig. 3 ). B ased o n B i = α r0/λ =440×0.02/0.58 ≈ 15, we found partial sums of this series,from the first to the sixth sum inclusively.Since determination of the roots of transcendentalequation (11) when varying the parameters of the Bicriterion involves laborious calculations, we used tabulardata. As in all calculations a slight difference in thevalue of partial sums was noted only starting from thesixth sum (Fig. 4), the sum of six members of this serieswas used in the calculations (8).dкО( ) ( 1 )22rTr ra TzTvz = 1     nii i i AJ r r Fo120 0 ( / ) exp(  ),0 10 20 30100200300400500TFrozen ball LiquidFlow of Flow of cooled liquiduncooledliquidv0 v0hzrRvzТ/rrr0hО zr, W/(m2К)Т, СFigure 3 Heat-transfer coeffici ent as a function of thetemperature of water for the “water-ice” system× ×( ) ( 1 )22rTr ra TzTvz = 1   nii i AJ r r 10 0 ( / ) exp( 0 10 100200300400500, W/(m2К)Т,174Slavyanskiy A.А. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 171–176In addition, to find the dependence of temperaturesin the capillary on the diameter of the balls for each ofthe options, we considered the situation when ballswith a diameter of 0.0375 or 0.0425 m (closed to thediameter of the test ball d = 0.04 m) acted as a coolant.This made it possible, with some approximation, to usetabular data [13] on the roots of equation (11) based onBi = αr0/λ = 15 which corresponded to d = 0.04 m.From relations (8)–(13) and (15), in the range ofvariation of the parameters of water cooled by frozenballs, 7 calculation results were obtained and graphswere plotted (Figs. 5–7). Based on the analysis ofthe graphs, the dependence of the variables of theconsidered problem on the operating parameters wasrevealed.Since the calculated temperature of the coolantreduced along the channel (Fig. 4), the specifictemperature of the liqui∫d 02 002 r ( , , , ) , d m r z rdrrθ increased in the samedirection.In turn, the rate of change in the temperature of theliquid decreased with time (Figs. 5–7). This is becauseof the reduction of the specific surface area of the ball,which is the ratio of the surface area of the ball to itsvolume. This resulted in decreasing heat exchange onthe border between solid and liquid phases in the liquidsystem (for example, in Fig. 5 graph 3 corresponding tothe diameter of the ball d = 0.0425 m is above graph 2for the ball diameter d = 0.04 m).In addition, as it can be seen from Fig. 7, in thecase of water productivity Q fixed for all variants, theincrease in the rate of flow of the coolant along with thedecrease in the porosity of the liquid system naturallyleads to a decrease in the rate of cooling of the liquid.Thus, for example, in Fig. 7 graph 4 corresponding toporosity m = 0.35 and the diameter of the ball d = 0.04 mis above graph 1 for porosity m = 0.35 and the ball withthe same diameter.However, it should be noted that, despite thequalitative consistency of the calculated results withFigure 5 Relation between the outlet temperature of liquid Тand time τ at different values of parameters (z = 0.5 m;m = 0.5; Т0 = 36°С; α = 440 W/(m2·К); Bi = 15: 1 – d = 0.0375m, 2 – d = 0.04 m, 3 – d = 0.0425 m; Т0 = 20°С; α = 230 W/(m2·К); Bi = 8: 4 – d = 0.0375 m, 5 – d = 0.04 m,6 – d = 0.0425 m)0 2 4 6 8010203040T11()T12()T13()T14()T15()T16()Relation between the outlet temperature of liquid Т and time  at different values of parameters (z =Т0 = 36С;  = 440 W/(m2К); Bi 15: 1 – d = 0.0375 m, 2 – d = 0.04 m, 3 – d = 0.0425 m; Т0 =W/(m2К); Bi = 8: 4 – d = 0.0375 m, 5 – d = 0.04 m, 6 – d = 0.0425 m)the calculated temperature of the coolant reduced along the channel (Fig. 4), the specific temperatureincreased in the same direction.the rate of change in the temperature of the liquid decreased with time (Figs. 5–7). This is becauseof the specific surface area of the ball, which is the ratio of the surface area of the ball to its volume.decreasing heat exchange on the border between solid and liquid phases in the liquid system (forFig. 5 graph 3 corresponding to the diameter of the ball d = 0.0425 m is above graph 2 for the ball0.04 m)., s1Т, С234 5 6Figure 6 Relation between the outlet temperature of liquid Тand time τ at different values of parameters (z = 0.5 m; m =0.35; Т0 = 36°С; α = 440 W/(m2·К); Bi = 15: 1 – d = 0.0375 m,2 – d = 0.04 m, 3 – d = 0.0425 m; Т0 = 20°С;α = 230 W/(m2·К); Bi = 8: 4 – d = 0.0375 m, 5 – d = 0.04 m,6 – d = 0.0425 m)of the liquid Ɵ increased in the same direction.In turn, the rate of change in the temperature of the liquid decreased with time (Figs. 5–7). This of the reduction of the specific surface area of the ball, which is the ratio of the surface area of the ball to This resulted in decreasing heat exchange on the border between solid and liquid phases in the liquid example, in Fig. 5 graph 3 corresponding to the diameter of the ball d = 0.0425 m is above graph 2 diameter d = 0.04 m).0 2 4 6 8010203040T11()T12()T13()T14()T15()T16(), s4 5 61 2 3Т, Сdetermination of the roots of transcendental equation (11) when varying the parameters of the Bilaborious calculations, we used tabular data. As in all calculations a slight difference in the value ofnoted only starting from the sixth sum (Fig. 4), the sum of six members of this series was used in the0 0.1 0.2 0.3 0.4 0.500.10.20.31(z)2(z)3(z)4(z)5(z)6(z)zbetween the radius mean specific temperature of liquid and the axial coordinate z for partial sums of0.04 m; m = 0.5; 1, 2, and 3 are numbers of summands from 1 to 3; and 4–6 from 4 to 6.addition, to find the dependence of temperatures in the capillary on the diameter of the balls for each ofconsidered the situation when balls with a diameter of 0.0375 or 0.0425 m (closed to the diameter of0.04 m) acted as a coolant. This made it possible, with some approximation, to use tabular data [13]equation (11) based on Bi = r0/ = 15 which corresponded to d = 0.04 m.relations (8)-(13) and (15), in the range of variation of the parameters of water cooled by frozen balls,results were obtained and graphs were plotted (Figs. 5–7). Based on the analysis of the graphs, thevariables of the considered problem on the operating parameters was revealed.123, 4–6z, mFigure 4 Relation between the radius mean specifictemperature of liquid and the axial coordinate z for partialsums of the series (8). d = 0.04 m; m = 0.5; 1, 2, and 3 arenumbers of summands from 1 to 3; and 4–6 from 4 to 6175Slavyanskiy A.А. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 171–176the physical meaning of the process under study, thequantitative assessment that characterizes the course ofthis process needs additional comments.First of all, this relates to the question of formalizingthe boundary condition at the interface, which is typicalfor many works on the theory of heat conductivity,where, when setting the problem, the temperatureis assumed unchanged, while for the heat exchangebetween phases (for example, between liquid andcoolant) phase temperature tends to level off. This leadsto a decrease in the intensity of heat transfer from theliquid to the ball as naturally as in the studied problem.Therefore, the results of the the cooling rate of waterpresented in Figs. 5–7 on, in fact, are overestimatedcompared to real data.At the same time, despite the simplificationsbased on the theory of heat conduction and used in theformulation of the problem, physical and mathematicalmodeling of processes makes it possible to predict andcontrol their development. To the same extent, this isalso applied to the complex problem of justifying therate of liquid cooling due to the accumulation of frozenballs with a developed heat exchange surface that isanalyzed in this paper.CONCLUSIONTo justify heat transfer from the fluid flow to ballswith eutectic frozen solution, we applied an analyticaltool forecasting the course of this process in theinnovative technology for cooling this liquid.In the quantitative analysis of this problem kineticaspects of filtration fluid motion was used, namely, whenthe working volume occupied by the liquid between theballs was simulated by equivalent plurality of orderedcylindrical capillary channels. This allowed us, fromthe point of view of analytical and numerical analysisof the thermal regime in a steady fluid flow inside theworking volume of the capacitance, to adapt the solutionof this problem to the study of this regime in an isolatedcapillary.The accepted conditions, namely the size of thecapacitance and balls and the volume fraction of balls inthe capacitance created the preconditions for conductingquantitative modeling of the process under study basedon the calculated dependences of the temperaturedistribution in an unlimited cylinder under an unsteadyregime.To assess the efficiency of the cooling processof fluid flow in a heat exchanger with a frozen solidphase and a developed heat exchange surface in thefield of the real values of the process parameters, theobtained temperature dependences were used to carryout a numerical modeling of the cooling process of thismedium.Based on the results of the analytical and numericalstudy of the problem, an acceptable region for varyingthe mechanical and thermotechnical parameters of theseprocesses was determined. This region is of importancefor engineering calculations of the low-temperatureprocessing of raw materials and finished products ofbiological origin.CONTRIBUTIONThe authors were equally involved in writing themanuscript and are equally responsible for plagiarism.CONFLICT OF INTERESTThe authors declare that there is no conflict ofinterest related to the publication of this article.