Journal of Applied Mechanics and Technical Physics, Vol. 52, No. 1, pp. 96–106, 2011

ENHANCEMENT OF TURBULENT HEAT TRANSFER DURING INTERACTION OF AN IMPINGING AXISYMMETRIC MIST JET WITH A TARGET c M. A. Pakhomov and V. I. Terekhov 

UDC 536.24:532.529

The flow structure and heat transfer of a mist jet with a low mass concentration of droplets (within 1%) impinging onto a flat surface aligned normal to the jet are studied numerically. The mathematical model is based on solving a system of Reynolds-averaged Navier–Stokes equations for a two-phase flow with the kinetic equation of the probability density function for coordinates, velocity, and temperature of particles. Addition of droplets is demonstrated to enhance heat transfer substantially, as compared with an impinging single-phase air jet in the region directly adjacent to the stagnation point of the jet.

Key words: impinging jet, heat and mass transfer, evaporation of droplets.

Introduction. Studying the flow structure and heat and mass transfer in the region of interaction of an impinging jet with a target located normal to the jet is of interest for numerous applications (deposition of coatings, quenching of metallic articles, cooling of electronic equipment, turbine blades, etc.). The main specific feature of such flows is the co-existence of three flow domains (Fig. 1): domain of a free submerged jet (I), domain of a gradient flow in the stagnation point region and turning of the jet (II), and domain of a near-wall (radial or plane) jet (III). The flow is characterized by high intensity of heat and mass transfer in the region of deceleration of the impinging jet and by substantial deformation of the turbulent flow structure. Specific features of the flow in the above-mentioned domains were considered in [1–5]. Based on numerous experimental [6–13] and numerical studies with the use of turbulence models of different levels [14–19], it was found that the influence of the target on hydrodynamic parameters of the jet impinging onto a solid surface starts to manifest itself at a distance from the target surface equal to (2.4–3.0)R (R is the nozzle radius). All relations typical for a free jet are valid in domain I at large distances [20]. Of greatest interest is the gradient flow domain (domain II), which involves essential reconstruction of the flow and substantial enhancement of heat and mass transfer, as well as friction. An effective tool for studying the flow structure and heat transfer in domain III is the theory of near-wall submerged jets [3, 20, 21]. Impinging jet flows are of interest for development and testing of modern turbulence models. The majority of the existing two-parameter models of turbulence were tested with flows parallel to the walls; therefore, such models cannot ensure sufficient accuracy in modeling more complicated spatial flows, for instance, impinging jets where the streamlines are not parallel to the confining surfaces. Such flows are usually described with the use of Reynolds-averaged Navier–Stokes (RANS) equations closed by various modifications of linear k–ε eddy-viscosity models (LEVMs) constructed on the basis of the renormalized group (RNG) theory. The studies performed in [14–19] revealed the drawbacks of this approach. The turbulent kinetic energy at the stagnation point predicted by the “standard” k–ε model isoverestimated approximately by 75%, which leads to a 100% increase in the Nusselt number. The degree of jet expansion and the temperature on the jet axis are not correctly predicted either; the flow velocity is underpredicted near the wall and overpredicted in the external flow region (see the data given

Kutateladze Institute of Thermophysics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090; [email protected]. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 52, No. 1, pp. 119–131, January–February, 2011. Original article submitted April 29, 2009; revision submitted December 16, 2009. 96

0021-8944/11/5201-0096

U1, UL1, T1, TL1, ML1,

2R r

H I

x

II

III

0 Fig. 1. Impinging flow: domain of a free submerged jet (I), domain of a gradient flow in the stagnation point region and jet turning (II), and domain of development of a near-wall jet (III).

in [14, 18]). Various corrections are used for better accounting for the curvature of streamlines in the stagnation region and for calculating the generation of the turbulent kinetic energy, for instance, the Kato–Launder correction [14] (introduction of additional source terms into the equation for k) or the Durbin correction [22] (restriction of the time turbulent scale and modification of the equation for the dissipation rate ε). As was argued in [18], the Durbin correction ensures significant refinement of the heat-transfer mechanism in impinging flows. Sufficiently precise results are provided by using the Reynolds stress models [14], the multi-parameter eddy-viscosity models [15], and the nonlinear eddy-viscosity models [19] modified for impinging flows. One possible method that allows several-fold enhancement of heat and mass transfer between the wall and the impinging jet is the use of mist flows as a coolant [23–27]. Li et al. [24] performed an experimental study of an impinging plane steam-droplet jet and considered the case of low mass concentrations (≈ 2%) of small-size (d1 = 10 μm) droplets; the degree of heat-transfer enhancement at the stagnation point reached α0 /αV 0  4 (αV 0 is the heat-transfer coefficient in the single-phase water-steam flow). Garbero et al. [25] used the Fluent commercial CFD system to study an impinging circular gas-droplet jet numerically. The gas phase was described by RANS equations with allowance for the existence of two phases; the dynamics of droplets was described by the Lagrangian approach. Several turbulence models were used to close these equations: “standard” k–ε model, Menter’s k–ω model, and k–ε model constructed with the theory of renormalized groups included into the Fluent system. In the present work, we chose the k–ε RNG model, which ensures the most accurate description of jet dynamics. Using the “volume of fluid” (VOF) method, we studied the effect of the droplet shape on heat transfer during the droplet–surface contact. The influence of the basic thermogasdynamic parameters, such as the concentration and size of droplets, two-phase flow velocity, nozzle diameter, and distance between the nozzle and the plate, on near-wall heat transfer was also considered. Here we studied only heat transfer between the gas and the wall in the case where the temperatures of the plate surface and particles were identical, and the difference in temperatures of the gas-droplet mixture and the wall was 20◦ C. With such a small difference between the wall and gas-flow temperatures, the domain of applicability of the numerical model is bounded, but thermophysical properties of the gas may be considered as constant. The Nusselt number in a gas-droplet flow was shown to increase approximately by a factor of 1.5, as compared with a single-phase flow. In the case of high mass concentrations of droplets described by the VOF method, heat transfer during the droplet-wall contact should be taken into account. Note, in the case of low concentrations of the disperse phase, satisfactory results can be obtained by taking into account only heat-transfer enhancement due to evaporation of droplets in the gas flow. The experimental study [26] was performed under the following conditions: nozzle diameter 2R = 51 mm, Reynolds number Re = U1 2R/ν = 5 · 104 , density of the heat flux on the plate wall qW = 1.4 kW/m2 , mass concentration of water droplets ML1 = 0–0.5%, initial temperature T1 = 293 K, and initial droplet diameter d1 = 14 μm. The impinging flow was directed from bottom to top. 97

Thus, impinging two-phase flows were modeled in the publications considered above with the use of RANS equations closed by various modifications of LEVMs. The disperse phase motion was modeled within the framework of the Lagrangian approach. The influence of the concentration and size of mist droplets on near-wall heat transfer was practically omitted from considerations. There are practically no data on the structure of turbulence of the impinging gas-droplet flow. The objective of the present work was a numerical study of the influence of droplet evaporation on enhancement of heat transfer in an impinging mist flow with variations of the basic thermogasdynamic parameters of the two-phase flow (diameter of droplets and their concentration). System of Equations for the Gas Phase. Despite all drawbacks of the approach based on solving RANS equations with linear k–ε turbulence models, it is still the basic method for many engineering applications. This method is used in the present work for the case of propagation of an impinging gas-droplet jet. To describe the dynamics and heat and mass transfer of a steady axisymmetric gas flow, we present the system of RANS equations in the form 6J ∂Uj Φ, = ρ ∂xj d ρ

 ∂U ∂ (Ui Uj ) ∂ (P + 2k/3) ∂  ∂Uj  i (μ + μT ) =− + + ∂xi ∂xi ∂xj ∂xj ∂xi

− (Ui − ULi )

 ΦρL  1 ∂Φ CD ρ |U − UL | + J + ρL τ gu ui uj  , τρ 8 ∂xj

ρ

μT  ∂T ∂ (Ui T ) ∂ μ 6Φ + [α(T − TL ) + JL] = − ∂xi ∂xi Pr PrT ∂xi d

+

 ∂K ∂T  C ρ τ g ρDT ∂Φ V PL L ut + (CP V − CP A ) uj t , CP ∂xi ∂xi CP ∂xj

μT  ∂KV P ∂ (Ui KV ) ∂ μ 6JΦ + , ρ= ¯ . = + ∂xi ∂xi Sc ScT ∂xi d RT Here, CD is the drag coefficient of the evaporating droplet, written with allowance for deviation from the Stokes law, CP is the specific heat of the gas, d is the droplet diameter, gu and gut are the coefficients of involvement of the disperse phase into the fluctuating motion of the gas phase, taken from [28], J is the mass flux of vapor from the surface of the evaporating droplet, L is the evaporation heat, KV is the mass concentration of vapor in ¯ is the specific gas constant, T is the binary vapor–gas mixture, k is the turbulence energy, P is the pressure, R the temperature, Ui are the components of the averaged velocity, xi are the projections onto the coordinate axes, Φ is the volume concentration of the disperse phase, α is the heat-transfer coefficient, μ is the dynamic viscosity, and τ is the time of dynamic relaxation of droplets. The subscripts refer to air (A), disperse phase (L), turbulent parameter (T ), and water vapor (V ). The values of the turbulent Prandtl and Schmidt numbers were taken to be PrT = ScT = 0.85; in addition, the relation from [29] was used to calculate the turbulent Prandtl number. The difference in the Nusselt numbers calculated for a single-phase flow with PrT = 0.85 and by the dependence [29] was smaller than 3%. Two-Parameter Turbulence Model. We use the model [30] with the Durbin correction [22] as a turbulence model. The equations for the kinetic energy k and its dissipation rate ε˜, modified to the case with a disperse phase, have the form μT  ∂k  1 ∂  k ∂ ε˜  ∂  ∂ (Uj k) μ+ − μ + ρΠ − ρε + Sk , = ρ ∂xj ∂xj σk ∂xj 2 ∂xj ε ∂xj ρ

ρ

ε μT  ∂ ε˜  ∂  ε˜ ∂k  ρ˜ ∂ (Uj ε˜) ∂  − (Cε1 f1 Π + Cε2 ε˜f2 ) + Sε ; μ+ − μ = ∂xj ∂xj σε ∂xj ∂xj k ∂r kΛ μT =

98

k2 ρCμ fμ . 1 + (Π/˜ ε − 1 − Ak /˜ ε)/E ε˜

(1)

(2)

Here, Λ is the Durbin correction [22] for a two-dimensional impinging flow and Sij is the local strain tensor of flow velocity: 1  ∂Uj ∂Ui  1.2  ; , Sij = + Λ= 2 ∂xi ∂xj 3fμ Cμ 2Sij Sij E = 2 is the constant of the Rotta approximation for correlations of pressure fluctuations and the strain rate, σk , σε , f1 , and f2 are constants and damping functions, which have the same form as in [29]. The terms Sk and Sε characterize additional dissipation of gas-phase turbulence owing to the presence of fine particles and the influence of inhomogeneity of the disperse-phase concentration profile; these terms have the form similar to the corresponding expressions in [31]. Relation (2) is derived by expanding the system of implicit algebraic equations for Reynolds stresses with respect to the averaged velocity gradient. With this method of derivation of Eq. (2), the equations of the turbulence model remain unchanged. Justification of the choice of Eq. (2) in this form for describing the turbulent dynamic viscosity of the gas can be found in [28]. Disperse-Phase Model. The disperse-phase model involves the Eulerian description based on solving the equation of the probability density function for coordinates, velocity, and temperatures of the disperse phase [28, 29]. The system of averaged equations that describe the transport processes in a disperse medium has the form 6JΦ ∂ULj , =− ∂xj d ρL

∂ (ΦULj ULi ) ∂ (ΦuLi uLj ) ρL 1 ∂ (DLij Φ) + ΦρL g − + ρL = Φ(Ui − ULi ) , ∂xj ∂xj τ τ ∂xj ρL

(3)

Θ Φ) ∂ (ΦULj TLi ) ∂ ρL ρL ∂ (DLij + ρL (ΦθuLj ) = Φ(Ti − TLi ) − , ∂xj ∂xj τΘ τΘ ∂xj

Θ = τΘ uLj tL  + τ guΘ uj t are the tensors of turbulent diffusion and where DLij = τ (uLi uLj  + gu ui uj ) and DLij turbulent heat transfer by particles [28], τθ is the time of thermal relaxation of droplets, and guΘ is the coefficient of involvement of particles into gas-phase temperature fluctuations. To calculate the Reynolds stresses, turbulent heat flux, velocity fluctuations, and disperse-phase temperature fluctuations, we used equations derived by the approach [28] in system (3). Note that two components of velocity fluctuations and the Reynolds stresses of the disperse phase were calculated in this work. The system of equations took into account convective transfer of velocity fluctuations, diffusion, generation of fluctuations due to the gradient of averaged motion of the disperse phase, and interaction between the phases. The model used for calculating the heat and mass transfer of a single droplet and heat transfer during a direct contact between the droplet and the wall was described in detail in [31]. Numerical Implementation and Boundary Conditions. The method of the numerical solution used in this work can be found in [31, 32]. The computational grid was nonuniform in both axial and radial directions. The computational nodes were condensed in the vicinity of the jet axis, the nozzle exit, and the wall. All computations were performed on a grid containing 200 × 250 control volumes for the case x/(2R) = 10. For smaller distances, the number of grid nodes along the x coordinate was reduced in a manner to obtain an acceptable ratio of the lengths of the control volume edges. Computations on a grid containing 300 × 400 control volumes were additionally performed. The difference in the Nusselt numbers calculated for a two-phase flow and velocities of the carrier medium and particles was smaller than 2%. The conditions of symmetry were set on the jet axis for the gas and disperse phases. No-slip and nopermeability conditions were imposed on the wall for the gas phase. The boundary conditions on the wall surface for velocity, disperse-phase temperature, and their fluctuations corresponded to an “absorbing surface” [29], i.e., to the case where the droplets do not return to the flow after their contact with the wall. Zero values of the derivatives of parameters in the streamwise direction were set in the output section of the computational domain. Testing of the Model for the Case of an Impinging Single-Phase Jet. To perform a comparative analysis in the case of a single-phase air jet, we used experimental data on hydrodynamics [10] and heat transfer of an impinging air jet [7, 8] and results calculated by using the multi-parameter turbulence model [15] and large eddy simulations [16].

99

x/(2R) 0.5

a

b

c

d

0.4

0.3

0.2

0.1

0

0.5

1.0 0

0.5

1.0 0

0.5

1.0 0

1.0 (U 2+V 2)1/2/U1m

0.5

Fig. 2. Profiles of gas velocity in an impinging single-phase jet at Re = 23,000, H/(2R) = 2, and different distances from the stagnation point of the jet: r/(2R) = 0 (2), 0.5 (b), 1.0 (c), and 2.5 (d); the points are the experimental data [10], the dashed curves are the results calculated in [15], and the solid curves are the calculations performed in the present work.

TABLE 1 Distributions of the Maximum Gas Velocity Vmax in a Single-Phase Near-Wall Jet at Different Distances from the Stagnation Point Vmax /U1m r/(2R)

Data [33]

Data [34]

Data of this work

2.0 2.5 3.0 4.0

0.66 0.53 0.44 0.33

0.64 0.50 0.41 0.30

0.61 0.49 0.40 0.28

Figure 2 shows the experimental [10] and calculated profiles of the total velocity of the gas at different distances from the stagnation point of the jet. For comparison, Fig. 2 also shows the numerical results calculated in [15]. In the vicinity of the stagnation point (see Figs. 2a and 2b), the profiles obtained in the present work are seen to agree well with the data [10, 15]. The model with the turbulent time-scale correction [22] used here slightly underpredicts the total velocity in the region of near-wall jet formation, where boundary-layer development and flow acceleration occur (see Figs. 2c and 2d) near the wall and slightly overpredicts the total velocity in the external part of the jet. This is typical for all linear turbulence models used to describe propagation of a near-wall jet formed from the flow in the vicinity of the stagnation point (see, e.g., [14–19]). Table 1 gives the maximum velocity of the near-wall jet Vmax obtained in this work and with the use of the dependence that follows from the momentum conservation condition [33] Vmax /U1m = 1.34[r/(2R)]−1 , and the semi-empirical formula [34] Vmax /U1m = 1.4[r/(2R)]−1.12 . Here, U1m is the mean-mass velocity of the gas flow on the nozzle exit. It follows from Table 1 that the calculated distributions of the maximum velocity in the near-wall jet are in good agreement with the results obtained by using these dependences. 100

TABLE 2 Distributions of the Turbulent Kinetic Energy at the Stagnation Point of an Impinging Jet at Different Distances from the Nozzle Exit to the Plate k

H/(2R)

2.0 6.0

Data [15]

Data [19]

“Standard” k–ε model

Results calculated in this work

0.035 0.050

0.04 0.05

0.085 0.090

0.055 0.060

Nu 1 2

5

120

4

3

100

80

40

0

1

2

3

4

5 r/(2R)

Fig. 3. Distributions of the Nusselt number over the target radius at H/(2R) = 2 and Re = 23,000: the points are the experimental data of [9] (1) and [10] (2); the curves are the calculated results of [15] (3), [16] (4) (LES), and this work (5).

Table 2 shows the distributions of the turbulent kinetic energy at the stagnation point of the flow at distances between the nozzle exit and the plate equal to H/(2R) = 2 and 6, which were obtained in the present work and in [15, 19]. It is seen that the values of turbulence predicted by the k–˜ ε model [30] are 30% higher than the data [15, 19], but still in better agreement with them than the data obtained by using the “standard” k–ε model. The turbulence maximum is located in the vicinity of the stagnation point, whereas the data [15, 19] predict that the coordinate of this peak is shifted away from the stagnation point to a distance approximately equal to the nozzle diameter. Note that the maximum value of the Nusselt number for a single-phase jet is located in its stagnation region, which agrees with data of other investigations. The distributions of the Nusselt number (Nu) over the plate radius for the case of impingement of a singlephase air flow, which were obtained in the present work, in experiments [7, 8], and in calculations [15, 16], are shown in Fig. 3 for the distance between the nozzle exit and the plate H/(2R) = 2. The results of the present work are seen to provide a fairly accurate description of the experimental distributions of the heat-transfer coefficient. A typical feature of the data obtained in [7, 8, 15, 16] is the presence of a second local maximum in the Nusselt number distributions. This maximum is less pronounced in our calculations, but it is determined within the framework of the model [30]. The minimum value of heat transfer is observed in the domain of flow turning at r/(2R) ≈ 1, which is consistent with the data of other publications. Thus, the use of RANS equations and the turbulence model [30] with the correction [22] ensures an accurate description of the complicated features of heat transfer in impinging jet flows, which, in turn, offers a possibility of calculating a more complicated mist flow interacting with a flat target. Calculated Results and Discussion. The calculations were performed for a monodisperse mixture of water droplets and air at atmospheric pressure. The nozzle diameter was 2R = 20 mm. Profiles of gas-phase parameters obtained by prior calculations of a single-phase flow in a circular pipe with a length of 150R were set in the input section (nozzle exit). The mean-mass velocity of the gas flow on the nozzle exit was U1m = 20 m/sec; the Reynolds number for the gas phase was Re = 2RU1m /ν = 26,600. The disperse phase was defined in the 101

a

x/(2R) 0.5

b

c

0.4

0.3 1

0.2 2

1 1

0.1

0

0.2

0.4

0.6

0

2

2

0.2 0.4 0.6 0.8

0

0.2

0.4

0.6 (U 2+V 2)1/2/U1m, (UL2+VL2)1/2/U1m

Fig. 4. Profiles of the radial velocity component of the phases in the near-wall jet: r/(2R) = 0.5 (a), 1.0 (b), and 2.0 (c); the solid curves refer to the gas phase in a single-phase flow with ML1 = 0 (1) and in a two-phase flow with ML1 = 1% (2); the dashed curves refer to the droplets.

ML /ML1 1.0

1 2

0.8 3

0.6 0.4 0.2

0

4

1

2

3 r/(2R)

Fig. 5. Profiles of the mass concentration of the disperse phase in the jet at different distances from the nozzle exit at Re = 26,600, H/(2R) = 2, ML1 = 1%, and d1 = 50 µm: x/(2R) = 0 (1), 0.5 (2), 1.35 (3), and 1.9 (4).

form of uniform distributions of parameters over the nozzle cross section. The initial velocity of the disperse phase was UL1 = 0.8U1m . The initial size of droplets varied in the interval d1 = 0–100 μm, and their mass concentration was ML1 = 0–1%. The wall temperature was TW = 373 K, and the temperature of droplets and gas was T1 = TL1 = 293 K. The distance between the nozzle exit and the target was H/(2R) = 1–10. The distributions of the radial velocity component for impinging one- and two-phase jets are shown in Fig. 4. The radial velocities of the gas and particles increase in the downstream direction, especially in the vicinity of the wall at x/(2R) = 1.9, where the radial velocities of the phases drastically increase as a result of jet turning and entrainment of droplets by the gas flow. 102

a

Nu

Nu

4

5

300

b

3 4

6

300 3

2

200

2

200 1

1

100

100 0

1

2

3

4

5 r/(2R)

0

1

2

3

4

5 r/(2R)

Fig. 6. Distributions of the Nusselt number over the target radius at Re = 26,600 and H/(2R) = 2: (a) d1 = 50 µm and ML1 = 0 (single-phase flow) (1), 0.2 (2), 0.5 (3), and 1.0% (4); (b) ML1 = 1.0% and d1 = 2 (2), 5 (3), 10 (4), 50 (5), and 100 µm (6); curve 1 refers to ML1 = 0 (single-phase flow).

The profiles of the mass concentration of the disperse phase along the axis from the nozzle exit are shown in Fig. 5. A uniform concentration profile is set on the nozzle exit (curve 1). In the downstream direction, the mass concentration of droplets decreases because of jet expansion and phase transitions (curves 2 and 3). The profile of the mass concentration of particles in the near-wall region (curve 4) changes substantially (ML /ML1 ≈ 0.5), which is caused by significant expansion of the jet in the vicinity of the stagnation point and by intense evaporation of droplets in the near-wall region. Figure 6 illustrates the heat-transfer intensity as a function of the mass concentration and initial diameter of droplets. It is seen that an increase in the mass concentration of droplets enhances heat transfer between the jet and the target owing to evaporation processes. The most significant (two-fold) increase in heat-transfer intensity is observed in the region x/(2R) < 2. Further downstream, the heat transfer approximately corresponds to a similar value for an impinging single-phase flow because of the decrease in the disperse-phase concentration due to flow expansion and evaporation of particles in the vicinity of the wall. An increase in the droplet size exerts a more complicated effect on heat transfer (see Fig. 6b). As for a near-wall gas-droplet sheet in a pipe [31], a decrease in the particle diameter leads to heat-transfer enhancement, especially in the vicinity of the stagnation point (curves 2–4) because of significant growth of the interface area between the phases. Note that the heat-transfer intensity is lower in the case of particles with the smallest diameter (curve 2) than in the case of coarser particles (curves 3–5), because fine droplets evaporate far from the wall and increase the near-wall heat transfer to a smaller extent. The Nusselt number for droplets with the greatest diameter d1 = 100 μm (curve 6) in the vicinity of the stagnation point is slightly smaller than the corresponding value for particles with the diameter d1 = 50 μm (curve 5). This is explained by the fact that inertial droplets are not intensely entrained into the fluctuating motion of the gas phase, and the majority of these droplets falls down onto the plate surface in the vicinity of the stagnation point. These conclusions are in qualitative agreement with the data calculated in [25]. The behavior of the heat-transfer coefficient at the stagnation point with addition of droplets is shown in Fig. 7. Significant enhancement of heat transfer is observed in the range of small sizes of the disperse phase. A clearly expressed peak is located in the interval of the initial diameters d1 = 5–10 μm. In the case with finer droplets (d1 = 2–5 μm), the heat-transfer intensity decreases, because most particles evaporate before they reach the impingement surface. After that, a gradual decrease in the Nusselt number is observed owing to slower evaporation of coarse droplets. In the case considered, the increase in the particle size does not induce such a drastic decrease in heat-transfer intensity, as it happens in the case of a gas-droplet flow in a pipe [35]. Comparison of the Calculated Results with Experimental Data for an Impinging Gas-Droplet Jet. Figure 8 shows the distributions of the Nusselt number for different mass concentrations of droplets. The use of an impinging gas-droplet mist flow as a coolant leads to significant enhancement of heat transfer (more than two-fold, as compared with an impinging single-phase jet) (curves 3 and 4) owing to the latent heat of vapor 103

Nu 400

Nu 800 1 2 3 4

4

3

300

600 2 3 2

1

200

400

1

100

0

50

100 Fig. 7

150

d1, mm

200

0

0.5

1.0

1.5

r/(2R)

Fig. 8

Fig. 7. Nusselt number at the stagnation point of a gas-droplet flow versus the initial droplet size for different mass concentrations: ML1 = 0.2 (1), 0.5 (2), and 1% (3); the remaining parameters are the same as in Fig. 6a. Fig. 8. Distributions of the Nusselt number in an impinging gas-droplet flow over the target radius at Re = 5 · 104 , qW = 1.4 kW/m2 , d1 = 14 µm, ML1 = 0.15%, H/(2R) = 8, and different mass concentrations of droplets: the points are the experimental data [26]; the curves are the results calculated in this work for ML1 = 0 (single-phase flow) (1), 0.045 (2), 0.1 (3), and 0.15% (4).

formation during evaporation of droplets. Note that the most significant enhancement of heat transfer is observed in the gradient zone, both in the experiments [25] and in our calculations. This fact confirms the validity of the results obtained (see Fig. 6). The presence of only one peak in the distribution of the local heat-transfer coefficient is explained by a large distance between the cooled surface and the nozzle. Conclusions. Based on the Eulerian approach, a mathematical model is developed for calculating the processes of hydrodynamics and heat and mass transfer in an impinging two-phase jet. It is demonstrated that the use of the linear turbulence model in the modification [30] with the correction [22] provides a qualitatively reliable description of the complicated heat-transfer processes during interaction of the impinging jet with a flat target aligned normal to it. An increase in the mass concentration of droplets leads to substantial (several-fold) enhancement of heat transfer, as compared with an impinging single-phase air jet owing to the latent heat of the phase transition during evaporation of droplets. This work was supported by the Russian Foundation for Basic Research (Grant No. 09-08-00197), by the Council on Grants of the President of the Russian Federation for supporting young Russian researchers — candidates of sciences (Grant No. MK-504.2010.8), and by Program No. 11 of the Department of Power Engineering, Machine Building, Mechanics, and Control Processes of the Russian Academy of Sciences.

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