Modelling of heat and mass transfer processes in phase transformation cycle of sprayed water into gas : 4 . Thermal state analysis of a droplet slipping in humid air flow

a thermal diffusivity, m/s; cp mass specific heat, J/(kg K); Fo Fourier number; g evaporation velocity, kg/s; L latent heat of evaporation, J/kg; m vapour mass flux, kg/(ms); p pressure, Pa; q heat flux, W/m; r radial coordinate, m; Pe Peclet number; T temperature, K; λ thermal conductivity, W/(m K);  molecular mass, kg/kmol;  density, kg/m;  time; w velocity, m/s; subscripts – C droplet centre; co condensation; c convection; e evaporation; f phase change; g gas; i time index in a digital scheme; it number of iteration; I index of control time; j index of radial coordinate; J index of droplet surface; k conduction; k+r conduction and radiation; l liquid; m mass average; η non-dimensional radial coordinate; r radiation; R droplet surface; v vapor; vg gas-vapor mixture; 0 initial state;  far from a droplet; superscripts – + external side of a droplet surface; – internal side of a droplet surface.


Introduction
Hot gases that are produced in organic solid gasification process is effectively cooled down by water injection into transportation systems, conditioned air, combustion process is controlled and eliminated fumes parameters.For mention-above and other water sprayed application cases thermal energy consumptions is important for proceeding phase transformations on the droplet surface and for liquid heating in the droplet.Therefore researches of water droplets heat transfer and evaporation [1][2][3][4][5][6][7] as well vapour condensation on its surface [8][9][10][11][12] is relevant.
The change of gas mixture composition that carry out droplets is caused by ongoing phase transitions on the surface of the droplet and temperature variations are defined by droplet heat transfer processes.Heat flow that is provided by the gas stimulates a liquid evaporation from droplet surface and also warms liquid in a droplet.Liquid warming rate is defined by intensity of heat offtake to droplet.In combined heating case the heat is being provided by convection and radiation: .Heat supplying for droplets by convection and radiation has its own peculiarities that is defined by transfer process nature.A flowing round fluid provides heat for the surface by convection, while at semi-transparent droplet a radiation heat is provided by absorbing electromagnetic waves of light in the infrared spectrum.Therefore, the radiation heat warmth the liquid in the droplet directly, and external convection heat from a droplet surface must be leaded off inside the droplet by internal heat exchange.Therefore, a total heat flow density of a droplet defines a fluid heating intensity.Semi-transparent liquids absorbs spectral radiation by surface poorly [6], so assumption heat flow is defined by droplet circulation and heat conductivity processes.Internal layers of a droplet are heated unevenly, therefore a droplet thermal state is described by variable function T(x, y, z, τ) in time and space.A droplet unsteady temperature field is described by time and radial coordinate function T(r, τ) in spherically symmetric heating assumption case.A thermal state of non-isothermal droplet is defined by mass average temperature function of a droplet: A droplet heating conditions in gas flow [10], has a strong impact for peculiarities of droplet unsteady temperature function T(r, τ).When external heating provides a heat spread by radiation and conductivity, function T(r, τ) is defined according to integral type model [7] that is convenient for numerical modelling.Then heat that is leaded off to the droplet is described by Fourier's low.In case of convective heating droplet slips in gas flow and friction forces rises on droplet surface.On their impact liquid is circulating and intensifies internal heat transfer.In case of forced liquid circulation at the droplet its thermal state is described by energy and Navier-Stokes equation system.The latter analytical solution is not possible and the numerical solution is complicated [12].Therefore a numerical scheme of convection heat flow in a droplet would be susceptible for machine computing and would complicate iterative method application whereby a droplet surface temperature definition is based on.To define unsteady temperature field function Tl(η, τ) ≡ Tl,"c"(η, τ) in case of slipping droplet is quite difficult.
In this work a slipping droplet thermal state defining methodology is developed and slipping intensity impact for water droplet warming in humid air flow is also analysed.

Research method and results
For droplet thermal state definition the onedimensional effective conductivity empirical model is provided [13] and heat transfer by conduction and radiation in the droplet integral models [7].To define a droplet surface temperature time function TR(τ) a balance of fluxes that flows in and off on the surface of a droplet is provided.This is defined by expressions (2) and (3) for condensing and evaporation regimes, respectively. (2) At expressions (2) and (3) a potential radiation flow absorption by droplet surface is denied.External convective heat flow is described by the method [10,13].Phase transformation flow on the surface of the droplet is described by water vapour flow density for convective het flow description that is leaded to droplet a modified Fourier low is applied: At compound heating case by conductivity and radiation an unsteady temperature field gradient is described by infinite integral equations series [7]: Liquid circulation potential input to balance of heat spread in a droplet is evaluated by effective heat conductivity parameter where at expression (6) Peclet number is defined according to liquid flow maximum velocity wl,R: Pel = 2Rρlwl,R / μl [13] on the surface of a droplet.This flow is caused by friction forces.
Iterative numerical scheme is made from expanded (2) and (3) expressions.A droplet surface time function TR(τ) definition of this scheme is applicable for all droplet heating cases.However, it must be remember, that surface temperature function TR(τ) defines only marginal value of unsteady temperature field T(r, Fo) function of a droplet T(r ≡ R, Fo) = TR(Fo).Therefore, a local temperature T(r < R, Fo) according to discussed methodology can be defined only at assumption of stable liquid at the droplet.In humid air slipping water droplet phase transformation cycle is modelled according to boundary conditions Tg = 500 K, p = 0.1 MPa, R0 = 0.000075 m, Tl,0 = 278 K, A point that defines Reynolds number initial value (Fig. 1), also shows phase transformations of condensing regime relative duration in aspect of droplet heating by conduction.It is clearly seen, that condensation regime time increases and convectional heat transformation at the droplet is more prevalent if droplet slips more intensively.Water circulation intensity is sensitive for droplet slipping and dispersity.A droplet growth in condensation regime is suitable factor for circulation in it, while droplet slipping weakening is factor that reduces water circulation in droplet.At the beginning of condensing regime an intensification of convectional heat transfer observes (Fig. 1).This means, that for some time, a droplet growth factor that is favourable for droplet circulation is stronger than slipping weakening factor that repress circulation.At both factors balancing point

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Fo k c  extreme point observes in function graphs (Fig. 1), after which the water circulation consistently suffocates in the droplet.
The droplet initial slipping significantly influences the droplet surface warming process (Fig. 2).The droplet initial slipping significantly influences the droplet surface warming process (Fig. 2).In unsteady phase transformation regime a droplet surface heats up rapidly (Fig. 2, a), however warming rate consistently suffocates until becomes zero at final stage of unsteady evaporation (Fig. 3, a).In equilibrium evaporation regime a temperature of droplet heating by conductivity is steady.A slipping droplet temperature decreases at equilibrium evaporation regime (Fig. 2, b): at the beginning of equilibrium evaporation droplet cooling down even accelerates, however it rapidly reaches maximum cooling down rate and after that is consistently weakening (Fig. 3, b).Therefore, at equilibrium evaporation process when droplet slipping reduces approaching to thermal state of droplet heated by conductivity is getting closer (Fig. 2, b).Deeper inner lay- The latter is difficult to describe due to fact that droplet diameter is changing at phase transformation cycle.Therefore a dimensionless radial coordinate η = r / R(Fo) of droplet is introduced.By its aspect a droplet dimensionless radius is universal, because its unit value remains in all droplet phase transformation cycle Fo ≡ 0 ÷ Foco ÷ Fonf ÷ Fof.It is convenient for droplet unsteady temperature field Tη,''c"(Fo), local gradient gradTη,"c"(Fo) and local convectional heat flow density

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defined by effective thermal conductivity local parameters is existing.Effective thermal heating conductivity theory is applied for convective transfer intensity description inside droplet: Local temperature gradient is described by integral equation: Condition r ≡ R is applied for expression (8) and temperature gradient is defined for leaded off convectional heat flow description at the droplet (4) in expression (Fig. 4).
(Fo) that is caused by circulation and conductive  " " , , c k q  (Fo) that is defined by heat conductivity fluid: Heat conductivity component of a droplet is defined by heat conductivity Fourier low at (9) expression: Convection heat flow component that is caused by liquid circulation in the droplet can be defined according to difference of local convectional heat flow and its conductivity component: It is assumed that at unsteady temperature field in local gradient " " k and " "c heat exchange cases a proportion of the effective thermal conductivity is: Then convectional heat flow component that is caused by liquid circulation in the droplet is defined according to expression: In modelled phase transformation cycle a time change step ΔFoi ≡ FoI / (I-1) is defined by providing a finite number of control points in free chosen Fourier number changing interval . Providing a finite J number of control points a unit radius of a droplet is divided to J-1 interval Δηj ≡ 1 / (J-1).Preserved conditions: When defining temperature field gradient in droplet by temperature difference and radial coordinate change ratio , a droplet local temperature when water circulates in it, can be defined by scheme: A strong and liner receding circulation cases in a droplet is provided for parameter function At case of liner receding water circulation inside the droplet the function  " " , c F  (Fo) is defined according to scheme: A Modelled droplet slipping regimes in air flow is defined by freely chosen Reynolds number Re0 at interval 0 0 100 Re  .At each case I = 21 and J = 41, while Fourier criteria FoI is equated for criterion Foco that defines phase transformation regime duration FoI = Foco(Re0) [17].Up to 225 Fourier time steps ΔFoi is made in phase transformation cycle.
Table 1 Heat flow densitys on the surface of a droplet, kW/m 2  c q ,  f q and  c q at the beginning of phase transformation regime and condensing regime change to evaporation regime as well at equilibrium evaporation beginning moments for different Re0 values is given in Table 1.
Internal convection heat flow  c Q spreads by heat conductivity and is transferred by circulating water in droplet.Therefore internal heat flow intensity is defined by ( 10) and ( 13) expression which is described by conduction and convection heat flow components . Internal convection heat flow density components are given in Table 2. Calculated local heat flow spread in droplet depends from applied model of heat exchange in the droplet (Fig. 5).Heat transfer model influence is brightest in con-densation phase transformation regime, where droplet slipping is the most intensive.
In all modelled slipping droplet cases, heat flow density qη,"k",i calculated according to case "k" is less than heat flow density calculated qη,"c",i according to (7) expression.The difference between them is brighter at more intensive water circulation (Fig. 5, b).Strong and liner receding circulation in the droplet models for small slipping velocities gives close result (Fig. 5, a), while for larger slipping velocities a brighter difference already observes between calculated qη,"c",i heat flows (Fig. 5, b).
In slipping droplet the distribution between spreading local heat flow qη,"c",i convective qη,c,"c",i and conductivity qη,k,"c",i components depends from droplet slipping velocity (Fig. 6).At small slipping velocities conduction heat flow qη,k,"c",i is brighter in a droplet (Fig. 6, a).When slipping velocity increasing a heat flow spread convective component qη,c,"c",i influence also grows (Fig. 6, b).In faster slipping droplet water circulates more intensively and heat is leaded to its central layers more quickly too (Fig. 5).Therefore droplet heats up equally and momentum temperature dissolution in "c" heat transfer case is strong different from "k" heat transfer case (Fig. 7).In strong circulation model case a calculated droplet temperature in central layers is lightly higher than in case of liner receding circulation model.It is observed in low (Fig. 7, a) and strong (Fig. 7, b) slipping droplets.At the initial stage of droplet heating the biggest difference observes between curves 2 and 3 that reflects instantaneous temperature field Tη,i,"c" (Fig. 7).However this difference is less than one and half degrees and in unsteady phase transformation regime is consistently decreasing (Fig. 8).Maximum deviation of local temperature Tη,i,"c" from conductivity heated droplet temperature Tη,i,"k" for slipping droplet reaches 3.5 K and 6.5 K when Re0 = 5 and Re0 = 40 in its central layers, respectively (Fig. 8).
A selected heat transfer model has influence for average droplet mass temperature which is calculated ac according to expression (1) (Fig. 9).Especially bright difference observes between droplet thermal modelling results in heat transfer cases of "c" and "k".Strong and liner receding circulation models give close droplet mass average temperature dynamics (Fig. 9, 2 and 3 curves).For further slipping droplet analysis "c" heat transfer case of strong circulation has been chosen.A calculated droplet surface temperature function TR(Fo) graph do not depends from heat transfer model inside the droplet.However it is strongly affected by droplet slipping intensity in gas (Fig. 10).Droplet average mass temperature function Tm(Fo) graph that describes nonisothermal droplet thermal state change depends from applied heat transfer model for droplet and when droplet slips more intensively, this graph deviates from "k" case graph (Fig. 10).
Influence of applied heat transfer model in droplet for calculated thermal state of droplet changes at time: it is significant in droplet condensing regime (Fig. 9), is weaken at unsteady evaporation regime and in equilibrium evaporation regime "k" and "c" heat transfer models ensures the same calculated droplet thermal state (Fig. 11).Droplet slipping has influence for calculated diameter dynamics (Fig. 12), therefore external convectional heat flow density that describes droplet energy state change in unsteady evaporation closing state is different (Fig. 13).
Water circulation inside the droplet ensures more Fo intensive heat, that is provided to surface, abstraction to central layers and measurable operates droplet layers warming rate (Fig. 14).This impacts droplet unsteady temperature field dynamics (Fig. 7) and droplet layers warms differently (Table 3) in condensing phase transformation regime.When primary slipping velocity in gas increases, then droplet layers average warming rate slowdown in condensing phase transformation regime (Table 4).This leads to duration increasing of condensing phase transformation regime.

Conclusions
Summarizing slipping droplet thermal state modelling results it can be state, that calculated droplet thermal state according to strong circulation model in weak slipping case is close to liner receding circulation modelling results, while for intensive slipping droplet case according to liner receding circulation model calculated droplet thermal state is bright different from strong circulation modelling results.Therefore in all droplets slipping case it is recommended to apply a strong circulation model.
Droplet slipping in humid air flow has an impact for complex heat transfer processes interaction in droplet condensing phase transformation regime and makes preconditions for intensive heat drain to inner layers with circulating water.Water warms evenly in slipping droplet then a surface layers warming rate slows down and condensing phase transformation regime duration increases.This is very important factor for technological processes optimization of heat phase transformation utilization from removable gas.

Fig. 14
Fig. 14 Droplet slipping impact for its layers warming rate in condensing phase transformation regime   

Table 2
Internal convection heat flow density and its conductivity and convectional components, kW/m 2