3D Transient Spray Cooling Heat Transfer Simulation for Metallic Slabs of Various Alloys

These last years, the high heat flux applications have been widely studied as a promising solution for spray cooling.

Spray cooling of surfaces at very high temperatures is a very complex process. It involves the formation of many droplets from a nozzle, their continuous fragmentation with complicated dynamics, the nucleation of several vapor bubbles from the very hot surface, droplet-vapor bubble interactions and, heat transfer which results. This process became an increasingly required technology, found in several industrial applications with high heat flux. It is characterized by intense heat transfer and uniformity of heat dissipation.

Water spray cooling is widely used in metallurgical processes and heat treatment to control the temperature of the process. It is used statically, with fixed nozzles relatively to the cooled target, or dynamically, with moving. The static regime is typical for quenching systems intended for the heat treatment of fixed sheets and the dynamic regime is used in metal processing such as finish rolling and other applications.

Water spray cooling heat transfer coefficient of continuous casting for a heated steel plate, sprayed by water have been investigated [1]. The temperature of the surface of the plate was measured during its variation in the investigation. Experimental measurement data was used to calculate the heat transfer coefficient.

The lowest surface temperature possible far the being of spray evaporative cooling is determined experimentally to be a linear function of the impinging spray mass flux [2]. A conduction-controlled analytical model of droplet evaporation gives quite good agreement with experimental measurements at ambient conditions.

Various cooling methods investigated the heat transfer coefficients of copper, aluminum, and nickel billets heated, then cooled by water spray [3]. Measured temperature close to the surface was used to compute the heat transfer coefficient. Results allow analytical relation among the heat transfer coefficient and other operational spray cooling parameters.

An experimental investigation of hot metallic surfaces to study water spray heat transfer was performed in different regimes of the quench curve [4]. Heat transfer measure and hydrodynamic properties were made locally in the spray field area. Correlations established to serve up a universal approach to develop quench curves for industrial sprays.

An analytical spray cooling study of a hot surface was performed [5]. The radiation from the hot surfaces to the environment is the first study, the second is the heat transfer by conduction to the droplets of the sprayed water. Spray cooling heat transfer data of the proposed procedure was predicted within the experimental range mentioned.

A method has been developed for the general correlation of effective heat transfer for sprays impacting vertically downward on a high-temperature surface [6]. Heat flux correlation data in the film boiling regime was greatly improved by dimensional analysis. Heat transfer performance depends on spray mass flux and droplet diameter.

A three-dimensional numerical temperature field into a steel billet was analyzed with enthalpy formulation which depends on only two non-linear material parameters, thermal conductivity, and enthalpy [7]. The numerical simulation by finite element method aid to compute the temperature field and thickness of the solid shell using commercial software.

Cooling of hot surfaces using full cone sprays investigated to link spray properties at heat flux removed from the heated surface [8]. The influence of the mass flow on spray cooling heat flux and cooling efficiency is in qualitative agreement with other studies performed in similar conditions. Best performances are reached for spray having highest mass flux.

The formulation of a model for heat transfer by finite volume to predict the temperature field in a continuous casting process under different conditions has been carried out [9]. The results of modeling were verified by the measured slab surface temperatures and a good agreement was achieved.

A great discord over the fundamental mechanisms of spray cooling heat transfer has been confronted due to complexity of this process [10]. Reliable correlations based on spray parameters are needed and dividing energy between single- and two-phase mechanisms at higher wall temperatures and mechanisms by which CHF come also needs further studies.

Water spray cooling is an efficient technology used in high-temperature surfaces in the metal production and processing industry [11]. Below a specific surface temperature, the heat transfer coefficient shows a strong dependence on temperature. The results give a correlation of the heat transfer coefficient which totally depends on temperature difference.

Experimental studies and numerical simulations have been carried on multiphase sprays for cooling applications [12]. Studies divulge that the heat transfer of multiphase sprays and efficiency of impact to be highly dependent on spray droplet size, air-to-liquid loading, and nozzle distance from target surface. In the view of heat transfer, if smaller is the droplet size, better is spray heat transfer efficiency.

The duty cycle is the major parameter of more accurate control of the cooling process [13]. Small duty cycles upgrade heat removal by phase-change. The formation of a thin liquid film is affected by larger duty cycles which evolve to continuous spray improving thermal response of cooling but hit the performances of the system. Intermittent spray cooling enables the design of thermal cooling management systems.

A predictive model of heat and mass transfer allows a favorable comparison which has been validated by the experimental results [14]. Parameters sensitivity were analyzed to get the influence of them on the surface temperature distribution. The heat transfer produced by droplet–film impact and film-surface convection are dominant in the condition that the surface is not superheated, but when the surface is superheated, the boiling effect increases rapidly.

Often in the literature, the process of spray cooling is a very basic level considering the dynamic phenomena involved in the impact of each drop of spray [15]. However, the interaction between drops modifies these phenomena and does not allow the spray to be qualified as the sum of individual effects. These phenomena include partial deposition of incident mass; the appearance of secondary droplets due to fragmentation mechanisms; and extraction and dissipation of the heat.

Spray cooling is much used in several domains and in hot surfaces cooling enclosed hot strip mill [16]. It plays an important role in metal production and industrial process, for very high-temperature steel strip casting and final microstructure optimization after hot rolling. A jet of water drops carried by gas is sprayed toward a hot surface to be cooled.

Control parameters such as spray cooling of a metal plate for different alloys, speed of development, flow rate have the same overall influence on the heat transfer defined by "boiling curves". But, by using a flat spray ensuring homogeneous atomization and high drop rates, the overall transfer is significantly improved which allows faster and more stable cooling kinetics over a large temperature range [17].

The displacement of the target, the variation of the cooling section, the position of the nozzles, and the heterogeneity of distribution of droplets leads to abnormal and non-uniform cooling conditions. It is therefore important to define the effect of these parameters on the intensity of heat transfer during the cooling process of steel sheets [18].

A study reviews two-part of spray cooling. The first study the relatively high-flux with low-temperature mechanisms, the second examine the relatively high-temperature transition boiling and film boiling regimes. Conflicting results point to a need for future test work that must be achieved systematically using many fluids with very different thermophysical properties and wide ranges of operating conditions [19].

A critical review of spray cooling by two levels is presented; by spray and by droplets [20]. The critic of the level of spray is center on the spray cooling performance which is a function of fluid properties, flow and surface conditions, and nozzle position. The droplet level study is based on the impact of droplet flow, on film flow which is the key to the spray cooling mechanism.

For determining how cooling effect of copper alloy changes with roughness, type of water, and type of nozzle, several experiments were performed on this heated alloy plates with various combination of parameters [21]. Results of the cooling rate were studied on the unidimensional model. The results were controlled for the two types of nozzles by experiments.

High-speed visualization and image processing of a heated target is studied to see the dynamic behavior of the liquid film produced by spray impact [22]. If the spray characteristics remain constant, the efficiency of spray cooling decreases. On the other hand, cooling is more efficient of the heated surface if the jet is modified by larger sizes and speeds of drops.

A conduction heat transfer simulation in a Titanium alloy slab was achieved [23]. The slab sprayed by water on its upside, must cool to reach the saturation temperature of the water and is in thermal interaction with the surrounding medium of its all sides by convection and radiation. The simulation is based on numerical solving in three dimensions of the nonlinear transient heat equation, using commercial software to obtain a temperature distribution inside the slab.

Spray cooling is a very efficient technology, surpassing all other conventional cooling methods. This process is extremely complex, thus most design rules to date are highly empirical in type. The review has the goal of sum up some of these recent advances and to establish a scope for the future development of more reliable and universal physics-based correlations to describe quantities involved in spray cooling [24].

Numerical simulation is a powerful tool to study physical problems. In our case, we are studying the spray cooling heat transfer for some materials.

A calculation program was written by a computer language running on a computer that translates from a mathematical model that describes a physical phenomenon is called a digital simulation. Numerical simulation is essential to study the behavior of physical phenomena whose mathematical models are very complex to arrive at analytical solutions, which is the case of many nonlinear problems.

Many secondary phenomena are the consequence of spray cooling which is very complicated to model and the complexity increases when all these models are simultaneously simulated. Boiling in spray cooling complicates modeling and very little literature is available on this subject. The experimentation on which the simulation is based is difficult due to the problem of measuring the velocity and diameter of the droplets and the experimental capture of the wall film thickness.

COMSOL Multiphysics v3.5a software was used for its powerful finite element partial differential equation solver, also used to solve some physical problems. From our side, we chose it for its efficiency to solve complex thermal problems.

3.1 Concept

The use of COMSOL Multiphysics software in this study is for its specificity to make it possible the coupling of different phenomena, so as to describe multiphase phenomena.

The “Heat Transfer” module is an optional module that extends the possibilities of COMSOL with customized user interfaces and optimized functionality for heat transmission analysis. They have been developed for a wide audience, including researchers, developers, teachers, and students.

Heat transfer is involved in almost all types of physical processes and can be the limiting factor for many of them. Its study is often of vital importance and requires powerful analytical tools. In addition, heat transmission often appears coupled with other physical phenomena, which can be addressed in COMSOL. The thermal modulus supports all the fundamental mechanisms of heat transfer conduction, convection, and radiation. We have most often neglected the radiation, thermal equations solved by the simulator are then:

$\text{ }\nabla \cdot \left( -\lambda \cdot \nabla T \right)=Q$     (2)

$-n\cdot \left( -\lambda \cdot \nabla T \right)={{q}_{0}}+h\cdot \left( {{T}_{inf}}-T \right)$     (3)

3.2 Initial and boundary conditions

The heat equation accepts two types of boundary conditions: a specified temperature and a specified heat flux. The first prescribes the temperature at the entire body:

$\text{ T=}{{\text{T}}_{0}}$    (4)

While the second specifies a heat flux:

$q_{0}^{\prime \prime}=n \cdot q^{\prime \prime}$      (5)

where, q" is the total heat flux vector:

$\text{ q''}=-\lambda \cdot \nabla T$     (6)

n is a normal vector at the border and q₀" is the heat flux, normal at the boundary too.

A thermal simulation of spray cooling process using a numerical resolution of the transient nonlinear three-dimensional heat equation from the initial condition and specific boundary conditions for a titanium alloy Ti6Al4V metal plate initially brought to high temperature using simulation software to predict the desired thermal parameters in terms of temperature and heat flux [25].

Note that when heat transfer by convection is active, the condition at the boundary of heat flux is of the mixed type, or Robin, rather than a condition at the pure Neumann boundary. The heat flux q₀" is normally a sum of the contributions of the different heat transfer processes. It is often convenient to divide the heat flux boundary condition into:

$\text{ }-\text{n(}\lambda \cdot \nabla \text{T)}=q{{''}_{0}}$     (7)

It is the amount of waste heat dissipated as a heat flow.

Initially, the temperature at any point of the (uniform) domain, represented by the function T(x,y,z,t), is fixed at time t = 0 s, defines the initial condition:

$\text{ T(x,y,z,0)}={{\text{T}}_{\text{0}}}\text{(x,y,z)}={{\text{T}}_{\text{ini}}}$    (8)

This temperature is obtained from the melting temperature of each material, reduced by a certain specific temperature to prepare the cooling process by natural convection and by radiation to the sides faces and the lower face of the outside of the plate when it leaves the furnace to the spray cooling zone.

The area of the plate is parallelepiped in shape with six faces, each of which is subject to the condition at the appropriate boundary.

We designate:

1) Front face,                   2) Right side face,

3) Under face,                  4) Upper face,

5) Left side face,             6) Back face.

The boundary conditions applied in our problem are of the heat flux type specified. We have two types of imposed flux: the first on the upper face and the second on the other surfaces.

For the condition at the boundary of the upper face (4) for z = 100 mm, we find the boiling convection with phase change characterized by a water film in boiling at the temperature 100℃, the convection of the air and radiation which are governed by the equation:

$q{{''}_{0}}=q{{''}_{spra}}+q{{''}_{conv}}+q{{''}_{radi}}$     (9)

${{\left. -\lambda \left( T \right)\cdot \frac{\partial T}{\partial z} \right|}_{z=z,up}}=$${{h}_{D}}\cdot \left( T-{{T}_{ebu}} \right)+{{h}_{a}}\cdot \left( T-{{T}_{amb}} \right)+\sigma \cdot \varepsilon \cdot \left( {{T}^{4}}-T_{_{amb}}^{4} \right)$    (10)

$\sigma$ = 5,67·10^-8W/(m²·K⁴) and h_D is the two-phase heat transfer coefficient obtained from the previous hydrodynamic calculations (it is assumed to be uniform over the entire sprayed surface):

${{h}_{D}}=\frac{q{{''}_{spra}}}{{{T}_{sur}}-{{T}_{ebu}}}$     (11)

q"_spray is the heat flux in water spray cooling and can be evaluated using the Bratuta and Ivanovsky equation [26]:

$q{{''}_{spra}}=4.6\cdot {{10}^{6}}\cdot {{\left( \Delta P \right)}^{0.1}}{{\left[ {{g}_{\max }}\left( x,y \right) \right]}^{0.4}}$     (12)

After calculations of hydrodynamic parameters [25], the necessary results of the coefficient h_D are obtained. The plate emits radiation to the surrounding ambient medium at a temperature of 25℃ with nonlinear exchange property, knowing that h_a the heat transfer coefficient by natural air convection depends also on the temperature of the slab.

$h_{a}=f(T)$   (13)

the emissivity e depends on the temperature for each material.

$\varepsilon=f(T)$    (14)

3.3 Selected study points

Four points on each face of the simulation slab were selected. These points are represented in Table 1 as shown in Figure 1.

Table 1. Coordinates of selected points

1.png

Figure 1. Selected study points: P1 to P4 for the upper side and P5 to P8 for the down side

In order to give a referential aspect to our simulation results and to give them a credit, we take works applying similar experimental or numerical conditions of investigation or which are close to the conditions of our study and we try to establish a critical comparison of our curves versus those we referenced to make our results valid.

A brief review of some works in the literature will allow us to clearly see the relevance of our results and to confirm their validity. Many references have been indicated, among which we have selected a few, others remain without citation despite their relevance and their close link with our results.

History of measured transient temperature for a full cone of a round spray impinges on stationary hot steel surface was used to compute surface heat flux [27]. The cooling curve shows an illustration of transient temperature profile. These profiles are fully like our temperature histories in Figure 7.

Increasing spray cooling using different coolants which improve the rate of heat removal by creating a high heat transfer zone on heating plate [28]. The curves of the variation of surface temperature over time are very similar to ours.

Hot surface spray-cooling rate experiments of a small metal specimen show both the fundamental validity and limitation of the theory [29]. A typical temperature-time curves are provided which illustrates the relation between temperature and time observed during the cooling process.

Generic history of the midplane temperature of a hot plate in aluminium cooled by water sprays, registered and analyzed in order to obtain the wall temperature curves [30] is shown schematically look the same as those in Figure 7 for temperature and Figures 8 and 9 for the heat flux.

The effect of the orientation of a hot test plate on heat transfer during spray cooling was studied. An example of curves was computed from surface temperatures [31] and some classical curves are traced as in our case.

The temperature field is shown by figure map in the cross-section at the straightening point using the new nozzle and a specific water flow rate nozzle [32]. This map is very similar to our temperature field maps (Figure 2 to 5) by bottom faces.

The temperature histories at selected points were compared with the temperature distributions obtained from the inverse solution [33]. Results are in good agreement with the temperatures measured during the cooling. Our approach, also adopted in Figure 1, allowed us to obtain comparable results.

A comparison of the measured temperatures with the predicted temperatures at the measured locations is shown by plotted curves of the prediction [34] which are analogous in shape to our curves of the same kind.

Experimented and computed temperature profile above heated flat disk up the center of the plate [35] shows the rapid decrease in temperature as those in Figure 6 with a distance from the heated surface but not with time as in our case.

Comparison of the predicted and measured temperatures was determined with an acceptable accuracy of the prediction [36]. Most of the curves obtained are comparable to ours in condition and look.

Comparison of temperature curves of different measures for a smooth surface and rough/real surface is shown. The influence of the initial wall temperature on the dependence of the surface temperature on time for a higher mass flux density is plotted [37] and its profile is broadly like ours.

Several surface temperatures versus time curves showing the results of the transient spray cooling confirm the excellent agreement between experimental results and theoretical predictions [38]. Comparing some of these curves to ours, there clearly seen similarity, thus reinforce the validity of our results.

Take from, for example, Figure 9 of the history curves for aluminum and titanium alloys, we see the same aspect of the descending curves indicating an analog behavior of cooling. However, major deviations are observed, due to the facts of the investigative conditions which are not quite the same.

We record a good agreement between the temperature and heat flux curves of our simulation results compared to those of the references cited in the literature that we have reviewed. The comparison is visibly clear, thus providing a certain validity of our results.