# Droplet Evaporation in a Gas-Droplet Mist Dilute Turbulent Flow behind a Backward-Facing Step

^{*}

## Abstract

**:**

_{1}= 1–100 μm; they have a mass fraction of M

_{L}

_{1}= 0–0.1. There is almost no influence of a small number of droplets on the mean gas flow and coefficient of wall friction. A substantial heat transfer augmentation in a droplet-laden mist-separated flow is shown. Heat transfer increases both in the recirculating flow and flow relaxation zones for fine, dispersed droplets, and the largest droplets augment heat transfer after the reattachment point. The largest heat transfer enhancement in a droplet-laden flow is obtained for small particles.

## 1. Introduction

_{f}is a non-dimensional parameter defining how the particle or droplet interacts with the mean carrier-phase flow [7,16,17]. Here,$\tau ={\rho}_{L}{d}_{1}^{2}(18\mu W)$ is the particle relaxation time, taking the deviation from the Stokes power law, $W=1+{\mathrm{Re}}_{L}^{2/3}/6$, where ${\mathrm{Re}}_{L}=\left|{\mathbf{U}}_{S}-{\mathbf{U}}_{L}\right|{d}_{1}/\nu $ is the Reynolds number based on the dispersed phase diameter and τ

_{f}= 5H/U

_{1}is the time macroscale (characteristic time of fluid motion) [7,16,17]. ρ

_{L}is the dispersed phase density, d

_{1}is the droplet diameter at the inlet,

**U**

_{S}and

**U**

_{L}are the fluid (gas) velocity seen by the droplet and mean droplet velocity, respectively, and μ and ν are the dynamic and kinematic viscosities of the gas phase, respectively. The tiny particles or droplets (Stk < 1) cause an attenuation of the level of turbulent kinetic energy (TKE) of the gas phase, and they interact well with the motion of the carrier flow [6,7]. The large particles (Stk > 1) cause additional turbulence generation due to the formation of vortices caused by the flow around large particles. The Stokes number Stk

_{K}= τ/τ

_{K}, based on the Kolmogorov time scale τ

_{K}, is another important dimensionless parameter for describing the behavior of the dispersed phase in a two-phase flow [7,17,18].

_{1}= 60 μm, mass concentration M

_{L}

_{1}= 0.04 and Reynolds numbers Re

_{H}= U

_{m}H/ν = (0.5 and 1.1) × 10

^{4}. The study was performed at two heights of the step, H = 10 and 20 mm, and the expansion ratios were ER = (H + h

_{1})/h

_{1}= 1.14 and 1.29, respectively. Numerical simulations were performed using the RANS model with a standard k − ε model. The droplets’ motion was simulated using the Lagrangian stochastic approach. A considerable increase in heat transfer (more than twofold in comparison with a single-phase separated flow) was obtained. It was found in Reference [19] that the heat transfer coefficient increases significantly (almost doubling) when the gas-droplet mist is added at Re

_{H}= (1.25 and 2.5) × 10

^{4}. The mean size of droplets at the inlet was d

_{1}= 10 μm, and the mass fraction was M

_{L}

_{1}= 0.015. The study was performed for two heights of the step, H = 10 and 40 mm, and in two ducts with h = 10 and 60 mm before expansion. The expansion ratios were ER = (H + h

_{1})/h

_{1}= 2 and 1.67.

## 2. Mathematical Model and Method of Numerical Realization

_{1}< 10

^{−4}), and the droplets are fine (d

_{1}< 100 μm). The effect of inter-particle collisions is neglected in the two-phase flow [18]. The droplet–droplet collisions are ignored in such a case, but the effect of the dispersed phase on the gas turbulent flow cannot be ignored [18,28]. The regime of turbulence modification has been identified for Φ = 10

^{−6}−10

^{−3}, and it is called “two-way coupling” [28,29]. The r.m.s. velocities and temperature pulsations and the turbulent heat flux of the dispersed phase are simulated using the model of [26,27].

_{cr}= 7 [30], but for all droplet diameters at the inlet studied in the present paper, We << 1. Here ${U}_{S}=U+\langle {u}_{S}^{\prime}\rangle $ and

**U**

_{L}are the fluid (gas) velocity seen by the droplet [12,18] and mean droplet velocity, where U is mean gas velocity (derived from RANS) and $\langle {u}_{S}^{\prime}\rangle $ is the drift velocity between the fluid and the particles [18]. Breakup and droplet deformation were not observed in the turbulent gas-droplet flow. Droplet fragmentation at its contact with a duct wall was not considered. The predictions are carried out for the water droplets and air (gas-phase) flow at the inlet cross-section at a uniform wall temperature (T

_{W}= 373 K). The temperature inside the droplet radius stays constant since the Biot number is Bi = α

_{L}d

_{1}/λ

_{L}<< 1 [22] and the Fourier number is Fo = τ

_{eq}/τ

_{evap}<< 1 [31], and a droplet evaporates at the saturation temperature. Here, τ

_{eq}is the period of existence of an internal temperature gradient inside a droplet and τ

_{evap}is the droplet lifetime (the time until complete droplet evaporation). The boundary condition on the heated wall for the droplets correlates to the so-called “absorbing surface” [32]. According to this condition, droplets do not return to the flow after making contact with the solid wall. The wall surface is always dry, and droplets deposited from the droplet-laden flow momentarily evaporate and do not form a liquid film or spots on the wall [22,23]. This assumption is valid for the heated surface when the temperature difference between the wall and the droplet is large enough (${T}_{W}-{T}_{L}\ge 40\text{}K$) [31]. The effect of growth of steam bubbles on the wall surfaces was not taken into account. The same assumptions were used in our previous recent numerical simulations [22,23] for droplet-laden mist flow in the pipe with sudden expansion. It may be is important in other thermal boundary conditions on the wall [33].

## 3. Numerical Procedures and Validation

_{+}= yU

_{*}/ν = 0.3–0.5, where y is the distance from the wall, and U

_{*}is the friction velocity of the single-phase flow. A minimum of 10 control volumes were located to resolve the large gradients in the near-wall region, subjected to viscosity (y

_{+}< 10). Grid sensitivity was studied to obtain the optimum grid resolution that provides the grid-independent solution. For all numerical investigations, we used a basic grid with 400 × 100 control volumes along the streamwise and transverse directions. The grid convergence was verified for three grid sizes: “coarse” 200 × 50, “basic” 300 × 150 and “fine” 500 × 150 control volumes. The Reynolds stress components are obtained using the method of [34].

_{j}= a × Δψ

_{j}

_{−1},

_{j}and Δψ

_{j}

_{−1}are the current and previous steps of the grid in the axial or radial directions, respectively, and a = 1.08 (longitudinal direction) and a = 1.05 (transverse direction). At least 10 CVs were generated to ensure the resolution of the mean velocity field and turbulence quantities in the viscosity-affected near-wall region (y

_{+}< 10).

## 4. The Numerical Results and Discussion

_{1}= 20 mm; after expansion, h

_{2}= 40 mm, the step height H = 20 mm and the expansion ratio ER = (h

_{2}/h

_{1}) = 2 (see Figure 1). The mean-mass gas velocity at the inlet is U

_{m}

_{1}= 10 m/s, and the Reynolds number Re

_{H}= HU

_{m}

_{1}/ν ≈ 1.33 × 10

^{4}. We add the droplets to the hydrodynamically fully developed single-phase air flow in the inlet cross-section (the section of sudden expansion) and keep the initial droplet velocity constant across the duct height: U

_{L}

_{1}= 0.8U

_{m}

_{1}. The initial size of droplets in our studies is d

_{1}= 1–100 μm, and the mass concentration of droplets is M

_{L}

_{1}= 0–0.1. The vapor mass fraction at the inlet is M

_{V}

_{1}= 0.005. The temperature of the air and droplets at the inlet is T

_{1}= T

_{L}

_{1}= 293 K, and the wall temperature is T

_{W}= const = 373 K. The mean Stokes number is Stk = τ/τ

_{f}= 0.03–2.9, τ

_{f}= 5H/U

_{1}= 0.01 s, and the Stokes number is ${\mathrm{Stk}}_{K}=\tau /{\tau}_{K}$ = 0.2–19.

#### 4.1. Flow Structure, Wall Friction and Turbulence Quantities

_{R}/H ≈ 5.8 for a single-phase air flow and x

_{R}/H ≈ 5.83 at M

_{L}

_{1}= 0.05, where x

_{R}is the length of the recirculating region. The first three cross-sections are located in the recirculation zone, the fourth cross-section corresponds to the flow reattachment area and the last two correspond to the droplet-laden flow relaxation area. A sharp change in the flow structure is observed downstream of the separation cross-section. The profiles of the streamwise velocities of phases in a two-phase flow correspond to those for a single-phase flow. The modification of the mean flow velocity with such a small addition of the dispersed phase is not observed. This qualitatively agrees with other conclusions for both gas-droplet [17,22] and gas-dispersed [6,7,9,10,11,12] turbulent separated flows. At a large distance from the point of flow reattachment, the two-phase flow takes the form of a fully developed flow in the duct. In the first cross-sections, the gas velocity is higher than the corresponding value for the dispersed phase; this is explained by the initial conditions for the addition of the droplets to the gas phase and their acceleration in the downward direction. Further, the droplet velocity is almost identical to the gas velocity.

_{W}is the shear wall friction and x

_{R}is the recirculation length. The line 1 represents the simulations for the single-phase flow without droplets and with other identical conditions. The addition of a dispersed phase to the turbulent separated single-phase flow has no significant effect on the value of C

_{f}in the flow separation and relaxation (after the reattachment point) regions (see Figure 2b). We can see a slight increase in the wall friction coefficient in a droplet-laden flow.

_{L}

_{1}= 0.05 (3) vs. the Reynolds number Re

_{H}is given in Figure 3. Line 1 is the semiempirical correlation [37] for calculating the wall friction

_{L}

_{1}= 0.05 (the maximal possible steam magnitude and ${M}_{L1}\equiv \Delta {M}_{V}^{evap}$). The maximum values of the steam mass fraction are found close to the wall of the duct, and the smallest values occur in the duct flow core.

_{0,max}is the maximum level of turbulence of the gas phase in a single-phase air flow. The level of gas turbulence in a two-dimensional flow is estimated using the following expression:

_{0}depending on the average Stokes number are shown in Figure 7, where k

_{0}is the turbulence level in the single-phase flow. The carrier-phase turbulence is calculated using relation (11). It is shown that the smallest suppression of gas turbulence is obtained near the wall at the distance y/H = 0.1 (1), where the diameters of droplets are the smallest due to their evaporation (see Figure 4a and Figure 5). The largest value is predicted at the distance y/H = 1 (3). There is almost no evaporation in this region, and the diameters of droplets are maximal. They are roughly equal to their initial size. The different mechanisms of the influence of the average Stokes number on turbulence modification for the cross-sections operate in the recirculation zone (1 and 2) and after flow reattachment (3). There is a sharp bend in the TMR distribution at Stk ≈ 1 for lines (1 and 2). This is caused by the fact that particles at Stk > 1 scarcely penetrate into the separation region, and they are found only in the shear layer and the duct core. Without a dispersed phase, an increase in turbulence was obtained in these two cross-sections for the level k/k

_{0}→ 1 in a single-phase flow. In the cross-section y/H = 1 (3), a decrease in the turbulence level of the gas phase is obtained with growth in the mean Stokes number (initial droplet diameter). For the investigated range of the initial droplet diameters, this cross-section is characterized only by turbulence level suppression, while for more inertial particles, an additional generation of the TKE level can also be obtained.

_{L}

_{, max}is the maximal temperature of the droplets in the corresponding cross-section. The subscripts “W”, “m” and “L” correspond to the wall, mean and droplets terms, respectively, and T

_{m}and ${T}_{L,m}=\frac{2}{{U}_{1}{h}_{2}}{\displaystyle \underset{0}{\overset{{h}_{2}}{\int}}{T}_{L}Udy}$ are the mean-mass gas and droplet temperatures in the corresponding cross-section, respectively. The normalized temperature Θ

_{L}is based on the maximal value of the droplet temperature T

_{L}

_{,max}. It is obvious that the minimal value of the droplet temperature is predicted in the turbulent flow core, and the maximal value of the droplet temperature is determined close to the wall. The gas temperature in the mist flow (2) is lower than that in the single-phase flow (1) due to the vaporization of droplets. The droplet temperature in the flow core is slightly lower than at the inlet due to droplet cooling.

#### 4.2. Heat Transfer

_{W}= const is obtained by the following formula:

_{R}is the length of flow recirculation. The value Nu

_{fd}= h

_{2}U

_{m}

_{2}/ν ≈ 38 (5) is the magnitude of the heat transfer in a fully developed single-phase air flow.

_{1}= 10 μm, Stk = 0.03, 2) evaporate faster and closer to the flow detachment section (see Figure 9b). The latter conclusion confirms the results in Figure 7 showing that droplets are entrained into the separated motion of the gas flow, and they scatter across the recirculating region. The largest droplets at d

_{1}= 100 μm and Stk = 2.9 (4) are badly entrained into the mean motion of the gas phase. The heat transfer enhancement is revealed mainly after the reattachment point due to droplet evaporation. In the recirculation zone, the values of heat transfer are similar to those in the single-phase separated flow. It should be noted that the maximal value of the Nusselt number in the two-phase flow (d

_{1}= 100 μm, Stk = 2.9) is smaller than that determined for finer drops (d

_{1}= 30 μm, Stk = 0.28).

_{max}distribution within the whole range of changes in the mass fraction of droplets studied in this work. In the region of small droplet diameters at the inlet (Stk ≈ 0.2, d

_{1}≈ 20 μm), a slight increase in heat transfer is observed, and for large droplets, a sharp reduction is observed. An intensive evaporation of small droplets leads to a decrease in the rate of their deposition. The large droplets almost are not present in the recirculation region. The increase in the droplet mass fraction at the inlet leads to a significant heat transfer enhancement in the two-phase mist flow as compared to a single-phase flow. The largest augmentation of heat transfer in the gas-droplet flow is observed for small droplets.

_{L}

_{1}= 0.1 as compared to the single-phase air flow).

## 5. Comparison with the Results Obtained for Gas-Dispersed and Droplet-Laden Flows in a Backward-Facing Step

_{1}= 40 mm. The expansion ratio is ER = 1.67. The Reynolds number based on the maximal centerline velocity U

_{1}is Re

_{H}= HU

_{1}/ν = 1.84 × 10

^{4}. The flow is the single-side backward-facing step oriented vertically downward. The transverse profiles of the gas-phase mean streamwise velocities in particle-laden and single-phase air flows in a few stations were presented in [22], and they are not shown here. Symbols are the experimental results of [7] for glass particles; lines represent simulations of the authors.

_{L}phases is calculated according to the following approach [16]: $k=0.5\left({u}^{\prime 2}+{v}^{\prime 2}\right)/{U}_{m1}^{2}$ and ${k}_{L}=0.5\left({u}_{L}{}^{\prime 2}+{v}_{L}{}^{\prime 2}\right)/{U}_{m1}^{2}$. The distributions of turbulence energy for a single-phase flow (1) and the gas (2) and dispersed (3) phases are qualitatively similar. The level of turbulence in a single-phase flow is close to the TKE level in the gas phase of a two-phase gas-dispersed flow. The turbulent energy of droplets is less than the corresponding value for gas (1 and 2). There is a maximum in the turbulence level of the gas and dispersed phases, and it is located in the shear layer. This is characteristic of both the experiments of [16] and our numerical results. The largest values of the turbulent kinetic energy for both phases are observed at the distance x/H = 4–5. In the region of the reattachment point (x

_{R}/H ≈ 6.2), the value of the gas turbulence decreases significantly. The results predicted by the authors and experiments of [16] are in quantitative agreement, and the largest difference is up to 15%.

_{0,max}ratios in the mist flow behind a backward-facing step are shown in Figure 13. Here, St = α/(ρC

_{P}U

_{1}) is the heat transfer rate in a two-phase flow, and St

_{0,max}= α

_{0,max}/(ρ

_{0}C

_{P}

_{0}U

_{1}) is the maximum Stanton number for a single-phase flow behind the BFS, all other parameters being equal. The heat transfer increases by more than 1.5 times in the gas-droplet flow as compared to a single-phase flow, both at the reattachment point and in the flow relaxation zones. Heat transfer enhancement is obtained after the reattachment point at mean Stokes number Stk = 2.2. The value of HTER at Stk = 2.2 (H = 10 mm) in the region of flow relaxation is greater than in the case of a step with Stk = 1.1 (H = 20 mm). The heat transfer enhancement in the recirculation region at Stk = 1.1 is much higher than that at Stk = 2.2. The droplets are well entrained into the recirculation region at a low Stokes number [16]. The local maximum of heat transfer at Stk = 2.2 is significantly below the reattachment point $\left(x-{x}_{R}\right)/{x}_{R}=x/{x}_{R}-1$ > 2, while at Stk = 1.1, the position of the heat transfer maximum almost coincides with the position of the reattachment point of flow.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Nomenclature

${C}_{f}=2{\tau}_{W}/{U}_{1}^{2}$ | wall friction coefficient |

C_{P} | heat capacity |

d | droplet diameter |

ER = (H + h_{1})/h_{1} | expansion ratio |

h_{1} | height of the duct before the sudden expansion |

h_{2} | height of the duct after the sudden expansion |

H | step height |

$2k=\langle {u}_{i}{u}_{i}\rangle $ | turbulent kinetic energy |

L | duct length |

M_{L} | mass fraction |

$\mathrm{Nu}=-{\left(\partial T/\partial y\right)}_{W}H/\left({T}_{W}-{T}_{m}\right)$ | Nusslet number |

Re_{H} = U_{m}_{1}H/ν | the Reynolds number, based on the step height |

Stk = τ/τ_{f} | the mean Stokes number |

T | temperature |

U_{L} | the mean droplet velocity |

U_{m}_{1} | mean-mass flow velocity |

U_{S} | the fluid (gas) velocity seen by the droplet |

U_{*} | friction velocity |

$\mathrm{We}=\rho {\left({\mathbf{U}}_{S}-{\mathbf{U}}_{L}\right)}^{2}/\sigma $ | the Weber number |

x | streamwise coordinate |

x_{R} | position of the flow reattachment point |

x_{Nu_max} | position of the peak of heat transfer rate |

y | distance normal from the wall |

Subscripts | |

0 | single-phase fluid (air) flow |

1 | initial condition |

W | wall |

L | liquid |

m | mean-mass |

Greek | |

Φ | volume fraction |

ε | dissipation of the turbulent kinetic energy |

λ | thermal conductivity |

ρ | density |

μ | the dynamic viscosity |

ν | kinematic viscosity |

τ | the particle relaxation time |

τ_{W} | wall shear stress |

Acronym | |

BFS | backward-facing step |

CV | control volume |

RANS | Reynolds-averaged Navier-Stokes |

SMC | second moment closure |

TKE | turbulent kinetic energy |

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**Figure 2.**The transverse profiles of (

**a**) the mean streamwise phase velocities and distributions of wall friction coefficient along the longitudinal coordinate. d

_{1}= 30 μm. (

**a**): 1 and 2 are single-phase (M

_{L}

_{1}= 0) and gas-droplet (M

_{L}

_{1}= 0.05) flows, respectively; 3 is droplets. (

**b**): 1: M

_{L}

_{1}= 0 (single-phase flow); 2: 0.02; 3: 0.05; 4: 0.1.

**Figure 3.**The value of the wall friction coefficient in the recirculation region as a function of the Reynolds number. 1: semi-empirical correlation of for the single-phase flow [37]; 2: M

_{L}

_{1}= 0 (single-phase flow); 3: 0.05.

**Figure 4.**The transverse dictributions of (

**a**) droplets and (

**b**) steam mass fractions in the droplet-laden separated flow. x/H = 2, M

_{L}

_{1}= 0.05. 1: d

_{1}= 10 μm, Stk = 0.03; 2: 30, 0.28; 3: 100, 2.9.

**Figure 5.**The transverse profiles of normalized droplets diameter in the mist backward-facing step flow. x/H = 2, M

_{L}

_{1}= 0.05. Legends to the figure are the same as in Figure 4.

**Figure 6.**The effect of Stokes numbers on the maximal turbulence kinetic energy modification ratio of the gas-droplet k

_{max}to the single-phase k

_{0,max}phases. x/H = 2, d

_{1}= 0–100 μm. 1: M

_{L}

_{1}= 0; 2: 0.05; 3: 0.1.

**Figure 7.**The TKE modification ratio vs. Stokes numbers at a few stations from the wall surface. x/H = 2, M

_{L}

_{1}= 0.05. 1: y/H = 0.1; 2: 0.5; 3: 1.

**Figure 8.**The transverse profiles of gas and droplet Θ

_{L}temperatures in the droplet-laden mist flow in a backward-facing step. d

_{1}= 30 mm, Stk = 0.28, x/H = 2. 1: single-phase flow (M

_{L}

_{1}= 0); 2: gas phase (M

_{L}

_{1}= 0.05): 3: water droplets.

**Figure 9.**The longitudinal distributions of local Nisselt number for various droplets’ inlet mass fraction (

**a**) and their diameter. (

**a**): d

_{1}= 30 μm, Stk = 0.28. 1: M

_{L}

_{1}= 0 (single-phase flow); 2: 0.02; 3: 0.05; 4: 0.1; 5: the correlation for a fully developed single-phase flow; (

**b**): M

_{L}

_{1}= 0.05. 1: d

_{1}= 0 (single-phase flow); 2: d

_{1}= 10 µm, Stk = 0.03; 3: 30, 0.28; 4: 100, 2.89; 5: the correlation for a fully developed single-phase flow.

**Figure 10.**The effect of mean Stokes numbers (inlet droplets diameter) on (

**a**) the maximal value of heat transfer and (

**b**) averaged heat transfer. 1: M

_{L}

_{1}= 0.02; 2: 0.05; 3: 0.1; 4: single-phase flow.

**Figure 11.**Turbulence modification ratio of the two-phase solid particle-laden flow to the single-phase flow ${u}^{\prime}/{u}_{0}^{\prime}$ longitudinal fluctuating gas-phase velocities. Symbols are the results of experiments of [7] for glass particles, lines are the author’s simulations. d

_{P}= 150 μm, ER = 1.67. 1: M

_{P}= 0.2; 2: 0.4.

**Figure 12.**The profiles of turbulent kinetic energies of gas and dispersed phases. Symbols are the results of experiments of [16], lines are the author’s simulations. ER = 1.3. 1: single-phase air flow ${k}_{0}/{U}_{m1}^{2}$; 2: gas phase of droplet-laden mist flow $k/{U}_{m1}^{2}$; 3: dispersed phase ${k}_{L}/{U}_{m1}^{2}$. d

_{1}= 60 μm, M

_{L}

_{1}= 0.04, H = 20 mm.

**Figure 13.**Heat transfer enhancement ratio St/St

_{0,max}in the gas-droplets flow behind the backward-facing step for H = 10 and 20 mm. Symbols are the results of experiments of [16], lines are the author’s simulations. Um

_{1}= 10 m/s, Re

_{H}= Um

_{1}H/ν = (0.53 and 1.10) × 10

^{4}, d

_{1}= 60 μm, M

_{L}

_{1}= 0.04. 1: T

_{W}= 308 K; 2: 338 K.

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**MDPI and ACS Style**

Pakhomov, M.A.; Terekhov, V.I.
Droplet Evaporation in a Gas-Droplet Mist Dilute Turbulent Flow behind a Backward-Facing Step. *Water* **2021**, *13*, 2333.
https://doi.org/10.3390/w13172333

**AMA Style**

Pakhomov MA, Terekhov VI.
Droplet Evaporation in a Gas-Droplet Mist Dilute Turbulent Flow behind a Backward-Facing Step. *Water*. 2021; 13(17):2333.
https://doi.org/10.3390/w13172333

**Chicago/Turabian Style**

Pakhomov, Maksim A., and Viktor I. Terekhov.
2021. "Droplet Evaporation in a Gas-Droplet Mist Dilute Turbulent Flow behind a Backward-Facing Step" *Water* 13, no. 17: 2333.
https://doi.org/10.3390/w13172333