# Numerical Model of Simultaneous Multi-Regime Boiling Quenching of Metals

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

_{s}is the thermal conductivity of the solid workpiece, n is the normal direction to the wall, h

_{w}and T

_{w}are the heat transfer coefficient and wall temperature, respectively, and T

_{sat}is the saturation or boiling temperature. In their work, the heat transfer coefficient was estimated for each boiling regime: stable vapor film and transition boiling (the authors call it transition boiling, but it should be nucleation boiling, as explained above). The limit between these regimes is the Leidenfrost temperature, which was computed using empirical correlations implemented in the AVL FIRE

^{®}CFD program and used to solve the corresponding differential equations. To improve the results, the authors proposed a variable Leidenfrost temperature, which may depend on the pool temperature, dipping velocity of the heated workpiece, immersion route, geometry of the workpiece, and wall surface roughness. Particularly, the authors derived an expression for the Leidenfrost temperature as a function of vertical position along the surface of the workpiece. The rates of vaporization/condensation of water were computed using semi-empirical equations that assume that the heat flux is proportional to the mass flux (between vapor and liquid). The model validation included a comparison between computed and measured cooling curves obtained from sub-superficial thermocouples installed in the workpiece. Sometime later, Zhang et al. [3] presented an application of the previous model for immersion quenching of a cylinder head. Although there was a good agreement between computed and measured cooling curves, their results and conclusions reflect uncertainty in predicting the Leidenfrost temperature. Srinivasan et al. [4] presented a numerical study of the immersion quench cooling process of a metallic trapezoidal block using the same Eulerian multi-fluid approach implemented in the AVL-FIRE

^{®}v8.6 software. Their assumptions and the applied empirical and semi-empirical equations were very similar to those used by Kopun et al. [2]. However, two differences should be pointed out. First, the solution method for coupling heat conduction in the solid with convection boiling. The authors used the AVL-Code-Couple-Interface (ACCI), which basically solves first the convection boiling under an isothermal solid wall and then the heat conduction in the solid under a constant heat transfer coefficient. This calculation process is repeated a few times to update both the wall temperature and heat transfer coefficient before moving on to the next step. This procedure has the advantage of accepting mutually independent meshes in solids and fluids. In contrast, Kopun et al. [2] used the Multi-Material Approach (MMAT), which considers both solid and fluid as a single domain. Therefore, a single, whole solution is computed at every step. A second difference between these works is the Leidenfrost temperature. Srinivasan et al. [4] considered that Leidenfrost temperature did not depend on position on the workpiece surface. Model validation was carried out by comparing computed and measured cooling curves. Another numerical simulation of flow-boiling heat transfer was presented by Krause et al. [5]. Their model also belongs to the interpenetrated phases category; however, there are some differences with respect to the previously described works. First, they used the mixed model, which solves the continuity and momentum equations for the mixture of vapor and liquid rather than solving these equations individually for each phase. Further, average properties are weighted on volume fractions and phase densities rather than exclusively volume fractions. Moreover, the vaporization/condensation rates are computed from the mass source/sink terms in the continuity equation using the bubble crowding model. This model only establishes that the rate of phase transformation is proportional to the super-heating or under-heating of the fluid with respect to the boiling temperature, T

_{sat}. The coefficient of proportionality depends on the local mass transfer coefficient, which in turn is a function of the boiling regime, the temperature at the critical heat flux, T

_{crit}, and the Leidenfrost temperature, T

_{L}. Therefore, there is no need for a priori knowledge of nucleation site density or a vapor-liquid interphase to start boiling. The authors supported this assumption by recalling that workpiece surfaces are technically rough and liquids are not pure, resulting in a great number of nucleation/condensation sites. The authors implemented their model in the CFD Fluent 6.3 code to compute iteratively the mass transfer coefficient by assuming a starting value of 1 m/s and minimizing the difference between their computed heat transfer coefficient and the corresponding measured values. Model validation was based on comparing calculations with experimental results for two cases: a steady-state case using an isothermal metallic cylinder immersed in a vertical tube where water was flowing at an average velocity of 0.3 m/s. In this case, validation was based on a qualitative comparison between computed and observed bubble fractions at the half height of the vertical cylinder that was held at 400, 500, and 1000 K. The second case was a transient heat transfer during the quenching of the same cylinder. Validation was based on a comparison between computed and experimentally determined heat transfer coefficients. Notice that the mass transfer coefficient at the vapor-liquid interphase was determined using this experimental heat transfer coefficient. Stark et al. [6] presented a numerical study of the quenching process using a confined water jet. The liquid velocities at the nozzle were 1 or 3 m/s, and the distances from the nozzle to the austenitic steel wall were 50 or 100 mm, respectively. They simulated steady-state convection boiling, assuming an isothermal wall. The wall temperatures had different values between 300 and 900 °C; therefore, all the boiling regimes were analyzed. The computed heat transfer coefficients as a function of wall position and temperature were used in a separate calculation to solve the heat conduction equation for the solid plate. Therefore, they decoupled heat conduction in the solid from convective boiling. Regarding convective boiling, it was computed using a mixed model, like Krause et al.’s model, and the interaction between liquid and vapor “bubbles” was defined using the Schiller–Naumann correlation for momentum exchange and the Ranz-Marshall correlation for heat transfer. The required bubble diameter for these correlations was a linear function of the vapor fraction. Turbulent flow was considered using the k-w-SST (Shear Stress Transport) model. Sinks and sources, included in the conservation equations, expressed the rate of phase change from boiling/condensation. The model was implemented in the CFD package, Ansys Fluent

^{®}13. The authors claimed that their single model approach allows for the investigation of all occurring boiling phases during the quenching process, avoiding specifying in advance the temperature range where each boiling regime is present. However, the authors did not present a model validation by comparing their numerical results against experimental measurements. Passarella et al. [7] developed another multiphase mixed model to simulate the quenching of a low-carbon steel cylinder using mineral oil flowing in a vertical tube at 1 m/s. The momentum equation was solved under turbulent conditions using the k-e model implemented in the Comsol

^{®}Multiphysics code.

_{sat}to avoid solving the energy equation for this phase. The proportion of heat for each contribution determined the boiling regime that was present. This proportion was controlled by damping functions: one depending on wall temperature, f

_{T}, and another on vapor fraction, f

_{v}. Different empirical and semi-empirical expressions were used for the heat flux under the stable vapor film regime and for the nucleation boiling regime. The authors showed the computed evolution of the vapor fraction and solid temperature. The vapor-rich layer starts in two zones: the upper area and the lower area of the cylinder surface. Then, it moves towards the mid-height of the cylinder. Despite the authors claiming the correctness of these results, no experimental evidence was shown in the article to support their calculations on the vapor fraction distribution. Nevertheless, the authors obtained computed heat transfer coefficient values with the same order of magnitude as previous experimentally determined values. In a recent study, Petrovic and Stevanovic [8] reported a coupled two-fluid flow and wall heat conduction modeling of nucleate pool boiling. Their model is also in the category of Eulerian multi-fluid with interpenetrated phases, solving numerically the mass and momentum differential equations for each phase: liquid and vapor. The energy equation for the liquid was coupled with the heat conduction equation for the solid workpiece. The interaction between liquid and vapor was given by an interfacial drag force per unit volume for momentum transfer and the rates of evaporation/condensation for mass transfer. The finite volume method [9] was used to solve the conservation differential equations. In contrast with the previous works, the total wall heat flux was divided into two contributions: a convective heat flux and a vaporization heat flux. The former represents the heat flowing from the wall to a liquid film, while the second one represents the heat flowing to growing bubbles. This approach took several results from previous works regarding the dynamics of bubble growth, which required input data like nucleation site density and the wetting contact angle. The authors found that the computed boiling curve strongly depended on these input data. Model validation was carried out using data from steady-state experiments, where a heater was used to hold a time-independent temperature distribution in a steel plate immersed in still water. The authors compared the obtained relationship between computed wall heat flux and nucleation site density with the respective observed relationship reported by other authors. Also, the authors showed comparisons between observed and computed vapor fraction distributions and heat transfer coefficients along the height of the heater. Finally, using transient heat flow results, they compared their computed temperature profiles across a cold spot at the location of a bubble growth with the corresponding measured values reported previously. In this study, wall temperatures were always below 130 °C; therefore, a stable film boiling regime was not present.

^{+}was set equal to 10 to properly calculate the turbulent viscosity. The nozzle was 3 mm in diameter and was located 100 mm from the plate surface. The circular plate was 100 mm in diameter and 60 mm thick. The water left the nozzle at 20 °C at a velocity such that the Reynolds number was 15,000. The authors compared the measured cooling curves for a location under the stagnation point with the corresponding model predictions, and they also compared the respective boiling curves. The authors computed a radial component of the temperature gradient in the plate that was larger than the axial component of the temperature gradient. This was attributed to the radial distribution of different boiling regimes on the wall, which led to quite different heat fluxes along the metallic surface. Single-phase convection was present at the stagnation point, and a stable vapor layer prevailed at some distance from the stagnation point. Zhang et al. [12] studied the influence of wall thickness and thermal diffusivity on dynamics and heat transfer in the nucleate pool boiling of a single bubble. They used the Ghost Fluid Method for sharp interface representation. Two Level Set functions were used to capture the liquid-vapor and liquid-wall interfaces. The computational domain included an axisymmetric solid and a fluid on its upper face. The fluid region was divided into micro- and macro-regions. The momentum, energy, and continuity equations were solved considering laminar flow and constant physical properties. The heat conduction equation was solved for the solid wall, assuming that its lower face was isothermal. The authors found that the bubble growth time decreases with the wall thickness, but the departure bubble diameter increases. A local and periodic expanding-receding low-temperature region was produced inside the wall under the bubble base because of movement of the contact line (vapor-liquid-wall line) due to evaporation of the liquid microlayer. An increase in wall thermal diffusivity lags the movement of the local low-temperature region. The authors did not show any measurements or observations to validate their results. It is evident that this interface-capturing method offers the capability to compute, from first principles, macroscopic parameters used by multi-fluid interpenetrated models, for example, nucleation site density. A recent work by Cukrov et al. [13] shows a model aimed at the stable vapor film regime using the VOF method. They considered the immersion quenching of a short cylinder in saturated still water at 600 °C. The authors used the energy jump mass transfer model to determine the evaporation rate. They claimed that this model does not require empirical input to estimate the mass transfer rate across the interface. The calculations agree with a previously measured cooling curve within the stable vapor film regime. The authors claim that this agreement shows an accurate calculation of the temperature gradient at the vapor-liquid interface in the mass transfer model and adequate turbulence modeling, although turbulence kinetic energy was considered uniform and constant. The authors, Ilic et al. [14], presented a review and future prospects of boiling heat transfer modeling. They pointed out that, besides interpenetrated (macro-scale boiling) and interface-capturing (micro/meso-scale boiling) models, molecular dynamics (nano-scale boiling) methods represent an alternative to improve our understanding of boiling heat transfer mechanisms. These methods are based on computing the trajectories of individual molecules, which move according to the interacting forces. The author’s opinion is that the most important advantage of these methods is that the bubble nucleation site can be detected for different geometrical configurations and wettability conditions of the wall. According to the authors, multi-scale modeling is the future prospectus for an improved predictive methodology of boiling heat transfer phenomena. Multi-scale modeling is a combination of macro-scale, micro/meso-scale, and nano-scale models. However, the authors accept that there is no definite answer to the question of how these scales should be coupled with each other.

Author, Year | Model Validation | Remarks |
---|---|---|

Ramezanzadeh et al. (2017) [10] | Cooling curves in the workpiece | Volume Of Fluid method. No radiation, T_{w} < 627 °C (900 K), laminar flow (Re ≤ 1770). Calculation of simultaneous different boiling regimes on the wall. |

Moon et al. (2022) [11] | Cooling curves and boiling curves | Volume Of Fluid method. No radiation; T_{w} ≤ 900 °C. Turbulent flow, Re_{nozzle} = 15,000, k-w SST model. Calculation of simultaneous different boiling regimes on the wall. |

Zhang et al. (2015) [12] | No validation with experimental results | Level Set method was applied twice: for the vapor-liquid interface and for the liquid-wall interface. Gosh Fluid Method for a computed sharp interface. No radiation; T_{w} ≤ 106 °C. Pool boiling, laminar flow. |

Cukrov et al. (2023) [13] | Cooling curve and qualitative shape of the vapor bubble column | Volume Of Fluid method. No radiation, T_{w} ≤ 600 °C. Turbulent flow, k = 0.25 m^{2}/s^{2}. Stable vapor film regime |

## 2. Materials and Methods

#### 2.1. Quenching Zone and Probe Geometry

#### 2.2. Experimental Conditions

_{water}and T

_{water}, respectively. The probe was removed from the muffle and moved downward at a manually controlled velocity, V

_{imm}. The probe tip touched the water-free surface in the open basin just at the time t

_{imm}= 0, when the probe temperature was T

_{imm}. The recorded temperatures indicated that after 25 seconds of immersion, the metal reached essentially the water temperature.

## 3. Mathematical Model

#### 3.1. Governing Differential Equations

#### 3.1.1. Mixture Model

^{−6}, which means that the vapor bubbles follow the water streamlines. The bubble loading is defined as the mass density ratio of vapor bubbles to that of the liquid water. Estimated orders of magnitude values considering vapor volume fractions of 0.05 and 0.95 are 10

^{−5}and 10

^{−2}, respectively. This is a very low loading, so the coupling effect between phases is one-way, which means the water influences bubbles via drag and turbulence, but the bubbles have no effect on the water flow. On the other hand, the Eulerian interpenetrated phases models do not compute explicitly the interphase boundary, as opposed, for example, to the Level Set or the Volume of Fluid methods, which do track the actual position of the interphase boundary. However, in the mixture model, it is possible to specify an interphase position by conventionally setting a volume fraction value. For example, a vapor-liquid boundary can be defined as those points in space where the volume fraction of vapor is equal to 0.5. Complementary, notice that for pure water, the liquid volume fraction is one, and analogously for pure vapor. Therefore, the considered differential equations also apply for the single-phase regions.

_{k}is the sensible enthalpy for phase k, k

_{eff}is the effective thermal conductivity, which is defined by the following equation:

_{k}the material thermal conductivity of phase k, and k

_{t}the turbulent thermal conductivity defined according to the used turbulence model. The heat source, S

_{h}, includes two terms: (1) the latent heat for evaporation/condensation, which was computed from the Lee model combined with the semi-mechanistic boiling model, and (2) the thermal radiation absorption, which was evaluated from the Discrete Ordinates (DO) radiation model. Both models are explained below.

#### 3.1.2. Evaporation/Condensation Model

_{evap}and c

_{cond}are the evaporation and condensation coefficients, respectively, which are interpreted as the inverse of relaxation times, which must be fine-tuned to match experimental data for each system, and T

_{1}and T

_{2}are the local temperatures for liquid and vapor phases, respectively.

_{sp}and M

_{nb}are the heat flux multipliers for the single phase and nucleate boiling, respectively. The first term on the right side of this equation represents the single-phase contribution, and the corresponding heat flux, q

_{sp}, was calculated from the basic relationship for heat transfer by convection: q

_{sp}= h

_{sp}ΔT, where the single-phase heat transfer coefficient, h

_{sp}, was obtained from the following equation.

_{1}and h

_{2}single-phase heat transfer coefficients for liquid and vapor, respectively, f is the area fraction of wall surface wetted by the liquid, and ΔT is the difference between the wall and boundary cell temperatures (T

_{w}− T

_{c}). The factor F is strictly a fluid flow parameter and is expressed as follows [22]:

_{tt}, which is used to determine the influence of two-phase presence on convection, and it is calculated from the following expression:

_{nb}, was computed using the general equation q

_{nb}= h

_{nb}ΔT

_{sup}. The corresponding heat transfer coefficient, h

_{nb}, was calculated using the Foster and Zuber correlation, written as follows [23],

_{sat,Tw}and P

_{sat,Tsat}the saturation pressures corresponding to wall temperature and saturation temperature, respectively; k

_{1}, c

_{p}

_{1}, ρ

_{1}, and μ

_{1}are the thermal conductivity, specific heat, density, and viscosity of the liquid, respectively; ρ

_{2}is the vapor density; H

_{12}is the latent heat of vaporization; and γ is the surface tension of water. The superheat, ΔT

_{sup}, is equal to the difference T

_{w}− T

_{sat}.

_{sub}accounts for the subcooled effects, and it is calculated from the expression proposed by Steiner et al. [24].

_{ref}a reference temperature. S

_{fc}represents the suppression factor due to forced convection, which can be found elsewhere [22] and depends on the two-phase Reynolds number, according to the following expression.

#### 3.1.3. Thermal Radiation Model

#### 3.1.4. Turbulence Model

#### 3.1.5. Heat Conduction

#### 3.2. Initial, Boundary, and Internal Conditions

^{+}> 30 that grants inclusion of the viscous sublayer and the buffer layer. Notice that quenching modeling starts with the fully immersed probe that still corresponds to point “F” since t = 0. The duration of this velocity increment was equal to the actual immersion time. After this short period, the probe is at point “A”. Table 4 summarizes the initial conditions for the solved differential equations. The table shows that the initial water velocity field was computed previously in an independent calculation, assuming that the probe is already in its final position inside the vertical tube and that it is at water temperature. Therefore, the continuity, momentum, and turbulence equations were solved under steady-state and isothermal conditions. The initial pressure and turbulent quantities, k and ω, correspond to this previous calculation. Finally, the probe temperature is uniform since the Biot number is smaller than 0.1 during air-cooling when transporting the probe from the muffle to the quenching position.

_{atm}, was specified. A zero-velocity wetting condition was specified over the tube surface and on the probe surface, and a near-wall treatment was used to specify turbulent quantities on those wall surfaces.

^{+}, which is defined by the following equation:

_{t}, is defined as

_{w}the shear stress on the wall. A value y

^{+}< 30 represents a region that includes the viscous sublayer and the buffer layer. In the present approach, semi-empirical equations, called “wall functions”, were used to bridge the variables between the viscous sublayer and the fully turbulent region. Wall functions are used in meshes with y

^{+}> 30. Therefore, the first control volume next to the wall will include the first two regions, viscous and buffer. This avoids the appearance of unbounded errors in computing wall shear stress and heat flux.

#### 3.3. Materials Properties

^{3}for the solid metal, water, and water-vapor phases, respectively. The parameters and constants used in the semi-mechanistic boiling model, in the interfacial mass exchange equations, and in the turbulence and radiation models are shown in Table 5. The studied cases are V1, with an initial probe temperature of 930 °C and a water velocity of 0.6 m/s, and case V2, with an initial steel temperature of 850 °C and a water velocity of 0.2 m/s. These values were chosen for the purpose of model validation, not to carry out a parametric study. They are intended to represent the limits of actual quenching conditions. This table also shows additional parameters and constants related to the four models used in the simulation that were not indicated previously. In the case of the Wall Boiling Model, y* is the reference distance from the wall and n is the power law superposition constant. For the Interfacial Mass Transfer Model, D

_{b}is the bubble diameter. For the Turbulence Model, k is the turbulent kinetic energy value. On the other hand, it was recorded that after full immersion of the probe, some incubation period was needed for the wetting front to start to move, and t

_{B}is the corresponding time, point “B”. Finally, in the Radiation Model, ε

_{w}is the probe surface emissivity, a

_{p,water}, and a

_{p,vapor}are the absorptivity coefficients of liquid and vapor water, respectively, σ

_{s}and C are the scattering and anisotropy coefficients, respectively, and n

_{w}and n

_{v}are the refraction indexes of liquid and vapor water, respectively.

#### 3.4. Solution Method

#### 3.4.1. Meshing

^{+}parameter, defined as the instantaneous fluid film responsible for the thermal diffusion of the heat delivered by the solid and given by Equation (25). The semi-mechanistic model requires that this parameter remain above a value of 30, or in the worst case, equal to or greater than 12, during calculation. This implies that the height of the first element on the solid side must have at least a normal value of 1.4 mm. On the other hand, the fluid region on the Plexiglas wall side does not involve boiling convection, and therefore it requires a regular fine mesh for the wall function. In the remainder of the bulk region, water velocity is nearly uniform, and a relatively coarse mesh was used. Having set the mesh characteristics for the fluid region, the solid probe mesh was assessed by testing several mesh sizes. Figure 4 shows the chosen mesh, which consists of 3567 elements in the whole computational domain with an average orthogonal quality of 0.98.

#### 3.4.2. Numerical Solution

^{®}Fluent code. Simulations were conducted for the cases shown in Table 3 and included 25 s of the quenching process using a time step of 0.001 s. Convergence was achieved with a residual value of 10

^{−3}for continuity and momentum equations, while a value of 10

^{−6}was applied for the energy equation.

## 4. Results and Discussion

#### 4.1. Wetting Front Position

^{2}, along the main surface of the probe. This is characteristic of the stable vapor film regime. No wetting front appears clearly defined yet. Figure 6b shows the probe after 3.15 s, which is the time at which the wetting front reaches the base of the inverted cone of the probe, point “C”. The respective photo shows this front as a gray ring, which represents the nucleation boiling regime since bubbles are rising from this position. The conical region shows a dark color, which indicates a lower temperature, leading to the single-phase convection regime. Above the wetting front position, the stable vapor layer can be observed, and the model predicts its thickness value of ~3 mm. Notice that all three regimes are present: stable vapor film, nucleation boiling, and single-phase convection.

#### 4.2. Cooling and Cooling Rate Curves

#### 4.3. Heat Flux at the Wall

_{z}/q

_{r}, as a function of time at the thermocouple z-positions.

_{z}/q

_{r}, for the cases studied. Figure 9a shows that during the initial quenching, this ratio is << 1, which is a result of no temperature variation in the z-direction at the thermocouple positions. However, once the wetting front reaches these positions, both heat flux components increase. Interestingly, the axial component exceeds, by a factor > 5, the radial component of the heat flux. Radial heat flux increases because a nucleation boiling regime appears, but axial heat flux increases because different boiling regimes are present in the wall along the z-direction. After the wetting front passes over the thermocouples, the wall cools down under the one-phase convection regime, and the axial heat flux component decreases. The three curves are mutually lagged, consistent with the z-position of the thermocouples in the probe. Figure 9b shows similar curves when water flows at a velocity of 0.2 m/s. However, the heat flux ratio reaches a value of ~10. This is attributed to the lower radial heat flux resulting from the lower water velocity. In contrast, the axial heat flux is maintained high because of the distribution of different boiling regimes along the wall. The main lesson to learn from these results is that quenching under multi-regime boiling convection is undesirable. Wall heat flux changes abruptly with position, and therefore solid-phase transformation along the workpiece should lag, especially in a metal layer near the surface. Stresses generated from this lagging may cause deformation or even cracks. Optimal immersion routes should be considered when quenching workpieces [29,30], and they may be related to this multi-regime boiling phenomenon.

_{L}, the temperature at the Critical Heat Flux, T

_{CHF}, and the Nucleate Boiling start temperature, T

_{NB}, are the respective limits and are also included. The computed results are mutually consistent, as explained below.

_{VF}, are also relatively lower than the rest of the values reported in this table and were obtained by averaging the local wall heat flux in the temperature range, T

_{L}< T < T

_{s}.

_{CHF}< T < T

_{L}. For case V1, the table shows that the temperature at the critical heat flux is essentially constant, at 254 ± 3 °C and a constant critical heat flux of 5.8 MW/m

^{2}. Therefore, the average transition boiling heat flux is also constant (2.1 MW/m

^{2}). The results for case V2 are similar except for the TC3 thermocouple. At this position, T

_{CHF}and q

_{CHF}are higher than the values for thermocouples TC2 and TC1. This is attributed to the combination of a higher wall temperature when the wetting front reaches that position and the low water velocity.

_{NB}< T < T

_{CHF}. The table shows that temperature T

_{NB}is essentially constant but is higher for case V1 as compared with the corresponding values for case V2. A similar trend is observed for the average heat flux q

_{NB}. Finally, a single-phase convection regime occurs for temperatures T < T

_{NB}, and the computed heat flux values are consistent with the expected results.

## 5. Summary and Conclusions

_{z}/q

_{r}, at the wall at the thermocouple z-positions. It was found that the axial component can be up to 5 times larger than the radial component for case V1, and up to 10 times larger than the radial component for case V2. This is a result of having all three boiling regimes simultaneously distributed along the wall. This multi-regime boiling phenomenon is undesirable since it leads to non-uniform wall heat flux and potentially poor workpiece quality. Finally, one-dimensional IHCP analysis is not appropriate to determine the wall heat flux when multi-regime boiling is present. Two-dimensional IHCP analysis and using thermocouples embedded along radial and axial directions should be preferred.

## Author Contributions

^{®}; validation, M.A.G.-M., B.H.-M. and O.A.R.-R.; formal analysis, M.A.G.-M. and F.A.A.-G.; investigation, M.A.G.-M. and F.A.A.-G.; resources, F.A.A.-G.; data curation, M.A.G.-M.; writing—original draft preparation, M.A.G.-M.; writing—review and editing, F.A.A.-G.; visualization, M.A.G.-M.; supervision, F.A.A.-G.; project administration, F.A.A.-G.; funding acquisition, F.A.A.-G. All authors have read and agreed to the published version of the manuscript.

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) A schematic representation of experimental set-up used to measure the sample thermal history and the wetting front kinematics. (

**b**) Photograph of conical-end cylindrical probe immersed in quenching zone.

**Figure 2.**(

**a**) Photograph of probe indicating thermocouple axial positions, z. (

**b**) Top view of a probe showing thermocouple radial positions, r. Lengths are in mm.

**Figure 3.**Schematic representation of axisymmetric domain indicating the boundary conditions applied to the numerical solution of the momentum and energy equations.

**Figure 4.**Discretization of the computational domain, pointing out the first adjacent cell length of 1.4 mm, which satisfies the y

^{+}parameter requirement.

**Figure 5.**Mesh sensitivity results. (

**a**) Computed maximum cooling rate using three different numbers of cells in the solid probe. (

**b**) Computed evolution of the cooling rate at two thermocouple locations, using the meshes indicated in (

**a**). (

**c**) Computed displacement of the wetting front using the meshes indicated in (

**a**).

**Figure 6.**Sequence of recorded photographs of the cylindrical probe, side “Photo”, paired with the corresponding model predictions, side “Model”, of immersion quenching of a conical-end cylindrical probe from 950 °C in water flowing upward at an average velocity of 0.6 m/s, case V1. Each probe is accompanied by its computed profiles of wall heat flux and vapor film thickness along the probe surface. (

**a**) point “B”, after 0.7 s of probe immersion, (

**b**) point “C”, after 3.15 s, (

**c**) point “D”, after 8.3 s, and (

**d**) point “E”, after 11.9 s.

**Figure 7.**Computed and observed evolution of the wetting front position. (

**a**) Conditions for case V1 and (

**b**) conditions for case V2. Points “A” to “E” refer to the wetting front positions at the corresponding times previously described in Section 3.2 and illustrated in Figure 6a–d.

**Figure 8.**Computed (discontinuous lines) and experimental (continuous lines) cooling curves and their respective cooling rates. (

**a**) Cooling curves for case V1, and (

**b**) their corresponding cooling rate curves. (

**c**) Cooling curves for case V2, and (

**d**) their corresponding cooling rate curves. Points “A” to “F” refer to the times previously described in Section 3.2 and illustrated in Figure 6a–d.

**Figure 9.**Computed ratio of wall heat flux components, q

_{z}/q

_{r}, at the positions of the thermocouples for the study cases: (

**a**) V1 and (

**b**) V2.

Author, Year | Model Validation | Remarks |
---|---|---|

Srinivasan et al. (2010) [4] | Cooling curves in the solid body | Pool boiling, laminar flow, no radiation. Predetermination of a uniform Leidenfrost temperature to know which boiling regime is present and apply the proper heat transfer coefficient. |

Krause et al. (2010) [5] | A qualitative comparison was computed and observed for the bubble fraction on the wall. Heat transfer coefficient as a function of T_{w}. | Pool or convective flow boiling, laminar flow 10^{4} < Re < 2 × 10^{5}, no radiation T_{w} < 1000 K (727 °C). Mass transfer coefficients are computed from experimentally determined heat transfer coefficients during the quenching of a workpiece. |

Stark et al. (2012) [6] | No validation with experimental results | Steady-state convection boils at wall temperatures in the range of 300 to 900 °C. Decoupling heat conduction in the solid by using the computed heat transfer coefficient h (T_{w}, location) in a separate calculation of solid temperature evolution. |

Passarella et al. (2012) [7] | Heat transfer coefficient in the whole temperature range. | Convective flow boiling, turbulent flow, gray medium radiation, solid initial temperature = 850 °C. There was no discussion on the determination of the damping functions. |

Petrovic and Stevanovic (2021) [8] | Wall heat flux as a function of nucleation site density | Nucleate pool boiling, laminar flow, no radiation, and no stable vapor film regime, (T_{w} < 130 °C). |

Case | Immersion Temperature ${\mathit{T}}_{\mathit{i}\mathit{m}\mathit{m}}$ (°C) | Water Temperature ${\mathit{T}}_{\mathit{w}\mathit{a}\mathit{t}\mathit{e}\mathit{r}}$ (°C) | Water Average Velocity ${\mathit{v}}_{\mathit{w}\mathit{a}\mathit{t}\mathit{e}\mathit{r}}$ (m/s) | Immersion Velocity * ${\mathit{v}}_{\mathit{i}\mathit{m}\mathit{m}}$ (m/s) |
---|---|---|---|---|

V1 | 930 | 60 | 0.6 | 0.28 |

V2 | 850 | 0.2 |

Equation | Initial Condition | Comments |
---|---|---|

Continuity, Equation (2) | α_{1} = 1, α_{2} = 0, ${\overrightarrow{v}}_{m}={\overrightarrow{v}}_{0}(r,z)$ | There is only liquid water, which flows upward at a previously computed steady velocity field. |

Momentum, Equation (5) | p = p_{0}(r,z) | Previously computed steady pressure field |

Energy, Equation (9) | T = T_{water} | Measured uniform temperature in water |

Turbulence, k-ω SST | k = k_{0}(r,z)ω = ω _{0}(r,z) | Previously computed steady turbulent field quantities |

Heat conduction, Equation (24) | T = T_{s} | Measured temperature in a solid probe |

**Table 5.**Summary of the parameters and constants for the models used in the simulation of both study cases (V1 and V2).

Study Case | Semi-Mechanistic Boiling Model | Interfacial Mass Exchange | Turbulence Model | Radiation Model | |
---|---|---|---|---|---|

V1: v_{water} = 0.6 ms^{−1}, T_{s} = 930 °C | y* = 250 h _{sp} (estándar)h _{nb} (Foster/Zuber) [23]F (Chen) [21] S (Chen-Steiner) [24] n = 1 | M_{sp} = 5M _{nb} = 1h _{factor} = 0.3 | C_{evap} = 30 s^{−1}C _{cond} = 0.2 s^{−1}D _{b} = 10^{−4} m | k_{c0c1} = 5 m^{2} s^{−2} | ε_{w} = 0.75a _{p,water} = 1.678a _{p,vapor} = 0.25σ _{s} = 0C = 0 n _{w} = 1.333, n_{v} = 1 |

V2: v_{water} = 0.2 ms^{−1}, T_{s} = 850 °C | M_{sp} = 4M _{nb} = 4h _{factor} = 0.8 | C_{evap} = 25 s^{−1}C _{cond} = 0.2 s^{−1}D _{b} = 10^{−4} m | k_{c0c1} = 3 m^{2} s^{−2} |

**Table 6.**Computed average wall heat flux by regime. The values are computed for wall z-positions of thermocouples TC1, TC2, and TC3, and for cases V1 and V2.

V1: T_{s} = 930 °C, v_{z} = 0.6 m/s | TC1 | TC2 | TC3 |
---|---|---|---|

Vapor Film (VF) | |||

Leidenfrost, T_{L} (°C) | 716 | 746 | 770 |

q_{VF} (MW/m^{2}) | 0.268 | 0.275 | 0.297 |

Transition Boiling (TB) | |||

q_{TB} (MW/m^{2}) | 2.096 | 2.144 | 2.149 |

Critical Heat Flux (CHF) | |||

T_{CHF} (°C) | 251 | 254 | 256 |

q_{CHF} (MW/m^{2}) | 5.746 | 5.800 | 5.836 |

Nucleate Boiling (NB) | |||

T_{NB} (°C) | 141 | 142 | 143 |

q_{NB} (MW/m^{2}) | 3.461 | 3.524 | 3.586 |

Single-Phase Convection (SP) | |||

q_{SP} (MW/m^{2}) | 0.396 | 0.374 | 0.344 |

V2: T_{s} = 850 °C, v_{z} = 0.2 m/s | TC1 | TC2 | TC3 |

Vapor Film (VF) | |||

Leidenfrost, T_{L} (°C) | 639 | 668 | 695 |

q_{VF} (MW/m^{2}) | 0.217 | 0.226 | 0.234 |

Transition Boiling (TB) | |||

q_{TB} (MW/m^{2}) | 0.78 | 0.872 | 1.542 |

Critical Heat Flux (CHF) | |||

T_{CHF} (°C) | 252 | 251 | 263 |

q_{CHF} (MW/m^{2}) | 4.622 | 4.822 | 5.797 |

Nucleate Boiling (NB) | |||

T_{NB} (°C) | 119 | 120 | 121 |

q_{NB} (MW/m^{2}) | 2.247 | 2.406 | 2.334 |

Single-Phase Convection (SP) | |||

q_{SP} (MW/m^{2}) | 0.245 | 0.229 | 0.210 |

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## Share and Cite

**MDPI and ACS Style**

González-Melo, M.A.; Rodríguez-Rodríguez, O.A.; Hernández-Morales, B.; Acosta-González, F.A.
Numerical Model of Simultaneous Multi-Regime Boiling Quenching of Metals. *J. Manuf. Mater. Process.* **2024**, *8*, 31.
https://doi.org/10.3390/jmmp8010031

**AMA Style**

González-Melo MA, Rodríguez-Rodríguez OA, Hernández-Morales B, Acosta-González FA.
Numerical Model of Simultaneous Multi-Regime Boiling Quenching of Metals. *Journal of Manufacturing and Materials Processing*. 2024; 8(1):31.
https://doi.org/10.3390/jmmp8010031

**Chicago/Turabian Style**

González-Melo, Marco Antonio, Omar Alonso Rodríguez-Rodríguez, Bernardo Hernández-Morales, and Francisco Andrés Acosta-González.
2024. "Numerical Model of Simultaneous Multi-Regime Boiling Quenching of Metals" *Journal of Manufacturing and Materials Processing* 8, no. 1: 31.
https://doi.org/10.3390/jmmp8010031