Modeling Transient Jet Impingement Cooling of Moving Hot Steel Plates

Abstract

Accelerated cooling (ACC) is one of the key processing steps in the production of Advanced High Performance Steels. In order to obtain thermo-mechanically controlled processed (TMCP) steel products with desired microstructures and mechanical properties, it is necessary to properly adjust the processing parameters of the cooling facility, and therefore it is critically important to quantify the physical process of heat removal by applying water jets on the hot surface of steel. In the present study we propose a mechanistic model for top jet cooling of a moving plate with circular and planar nozzles. The simulation model has been developed based on the extensive experimental database generated with pilot scale runout table tests, and it provides a potentially powerful tool for simulation of cooling of steel strips and plates over the entire length of the cooling facility.

1. Introduction

The increasing demand for advanced high performance steels with improved properties in terms of strength, fracture toughness, ductility and weldability has necessitated to develop novel and/or improved processing routes. In particular, the global market development in the construction and energy sectors has put additional pressure on steel suppliers to produce advanced hot rolled steel plates through improved Thermo-Mechanically Controlled Processing (TMCP). The austenite-ferrite transformation is the key metallurgical tool to engineer the complex microstructures required in most of these advanced steel grades and this can be achieved by carefully tailoring the runout table cooling process in hot mills.

Accelerated cooling technologies (ACC), as an essential part of TMCP, employ lines of water jets impinging on the hot surface of steel strips/plates. Steel typically enters the cooling facility at temperatures in the range of 800–900°C and exits at temperatures as low as 550°C for conventional steel grades, or as low as room temperature for some of the advanced steel grades, e.g. hot-rolled dual-phase steels with lean chemistries. The large range of surface temperatures of steel suggests that when water jets are applied at different positions along the length of the cooling facility very different heat transfer modes may occur. This includes film boiling in the high temperature range, transition and nucleate boiling in the lower temperature range, as well as single phase forced convection and radiation heat transfer. The optimisation of the cooling process, whether through changes in nozzle type, size and configuration or adjustment of process parameters, as well as the development of thermal models as predictive tools, requires fundamental insight into the mechanisms of boiling heat transfer.

In their seminal review paper Wolf et al.¹⁾ gave a comprehensive overview of experimental and modelling research in jet impingement boiling at the turn of the century and provided foundational information for further research activities. Since then, numerous experimental and theoretical studies have been published on the subject of jet impingement boiling with the aim of providing a better quantification of the complex physical phenomena occurring at the contact of the liquid jet with a hot solid surface. Experimental work usually involves tests under steady-state or transient boiling conditions in which heat fluxes and temperature data are collected during the cooling process to build boiling curves. Modelling work is typically coupled with these experimental studies and represents an attempt to develop correlations, whether empirical or phenomenological, as tools to map boiling curves.

The ability to predict the temperature evolution in the steel during jet impingement cooling on the runout table with a particular set of process parameters is critical for hot mill operations. Guedia et al.²⁾ recently reviewed and summarised the modelling efforts with a particular focus on ACC. Most experimental studies carried out with stationary plates suggested that the heat transfer during jet impingement boiling could be significantly larger than concluded using conventional correlations for forced convective boiling curves. This is particularly evident in the transition boiling regime which is one of the key features in runout table modelling. High heat fluxes in the transition boiling regime can be attributed to the breakup of the vapor layer due to the force of the impinging liquid jet and penetration of the liquid. As a result, the liquid-solid contact area is increased such that heat transfer rates increase as well. This pushes the Leidenfrost point, i.e. the initiation of transition boiling, towards higher surface temperatures and makes the change from film to transition boiling very steep with an immediate increase in heat flux by an order of magnitude. Further, a significant region in the boiling curve may be created with almost constant heat flux known as the “shoulder”. This phenomenon, although previously observed by several authors, including Ishigai et al.,³⁾ has been first recorded and analysed in more detail by Robidou et al.⁴⁾ Seiler-Marie et al.⁵⁾ proposed a model of the “shoulder” heat flux at the stagnation point by using the Rayleigh-Taylor instability at the liquid-vapor interface as a mean to quantify the breakup of the vapor layer and liquid penetration volume. Several modifications of the model were made more recently, including those by Karwa et al.⁶⁾ as well as Ahmed and Hamed.⁷⁾

Based on this information and an extensive experimental database for jet impingement of stationary plates by Kashyap et al.,⁸⁾ Guedia et al.⁹⁾ recently proposed a phenomenological model for the entire jet impingement boiling curve in the impingement as well as parallel flow region. The model has been originally developed and validated for cooling with a planar bottom jet impinging on a stationary steel plate. The aim of the present study is to extend the model to top jet cooling with circular and planar nozzles and, more importantly, for moving plates. The data for top jet cooling of stationary plates are taken from Nobari et al.¹⁰⁾ while data of moving plate experiments conducted on a pilot scale facility will be first presented here before the model extension will be described. Similar experimental heat transfer studies have been conducted recently by Wang et al.,¹¹⁾ Lee et al.,¹²⁾ and Fujimoto et al.,¹³⁾ who investigated the heat transfer characteristics of multiple jets using various nozzle arrangements commonly employed on the runout table, as well as by Wang et al.,¹⁴⁾ who investigated heat transfer of a slit jet impingement for ultra-fast cooling. These studies do, however, not contain any substantial modeling analysis such that they were not further considered in the present model development. The proposed model provides a potentially powerful tool for modeling of runout table cooling over the entire length of the cooling facility.

2. Experiments

The runout table pilot scale facility consists of an electric furnace for heating test plates, a cooling section with a pump, headers and nozzles where water is circulated in a closed loop and a hydraulic moving bed for transportation of plates through the cooling section equipped with impinging jets. The specification of the facility and experimental procedures are described in detail elsewhere by Nobari et al.,¹⁰⁾ Prodanovic et al.,¹⁵⁾ and Nobari.¹⁶⁾ In moving plate experiments, the microalloyed low-carbon steel test plates have dimensions of 430 mm in width and 1200 mm in length, and thickness of 6.6 mm. The plates are instrumented with type K thermocouples (Omega INC-K-Mo-1.6 mm) at a depth of 1 mm below the cooled surface, spot-welded inside the plates to make an intrinsic-type connection and minimise the response time. The instrumented plates are heated in the furnace to a desired initial temperature after which they are positioned on the hydraulic bed and driven back and forth through the cooling section to simulate a multi-pass cooling process. The plate speed, water flow rate and water temperature are controlled during experiments. Internal temperatures of steel plates were measured during cooling by using the strategically placed thermocouples. Temperature data were collected at a frequency of 50 Hz, and a 2D finite element Inverse Heat Conduction (IHC) program based on the function specification method combined with the zero-order Tikhonov regularization was subsequently used to calculate surface heat fluxes and surface temperatures. The embedded future information in the sequential-in-time method addressed the time-lag and enhanced the accuracy of calculations.

In case of tests with a circular nozzle, 24 thermocouples were placed along the centre line of the plate, 12 mm apart, and also in the lateral direction spaced 6.35 mm apart from each other, as shown in Fig. 1. This configuration allows to map the temperature variation along the moving as well as radial direction during cooling with a circular nozzle passing along the centreline.

Fig. 1.

The locations of thermocouples on a test plate for circular nozzle experiments.

For planar nozzle experiments only 5 thermocouples were used. They were placed in 50 mm intervals in the moving direction, assuming uniform temperature distribution in the lateral direction. The experimental matrix of the present study is shown in Table 1. A single circular or planar nozzle was used in the tests. In case of circular nozzle experiments, a single nozzle with an inner diameter of 19.2 mm was mounted at a nozzle-to-plate distance of 1.5 m and a test plate moving at a constant speed of 1 m/s was exposed to the jet. Water flow rates were varied between 15 and 45 l/min, and water temperatures between 10 and 40°C. These process parameters are similar to those employed on the runout table of hot mills. For planar nozzle experiments, a nozzle with a slot size of 3 × 300 mm was mounted at a distance of 100 mm from the surface of the plate, and water was supplied at a constant flow rate of 100 l/min with a temperature of 25°C. The aim with circular nozzle experiments was to investigate the effect of water flow rate and subcooling on heat transfer during jet impingement cooling, while the aim of planar nozzle experiments was primarily to quantify the role of plate speed.

Table 1. Experimental matrix.

Nozzle type	Plate speed, m/s	Water flow rate, L/min	Jet impingement velocity, m/s	Water temperature, °C	Nozzle to plate distance, m
Circular	1.0	15	5.5	25	1.5
Circular	1.0	30	5.7	25	1.5
Circular	1.0	45	6.0	25	1.5
Circular	1.0	15	5.5	10	1.5
Circular	1.0	15	5.5	40	1.5
Planar	0.5	100	2.3	25	0.1
Planar	1.0	100	2.3	25	0.1
Planar	1.6	100	2.3	25	0.1

Typical experimental data obtained during a single test are shown in Fig. 2. Each test consists of a series of passes through the cooling section, and each pass is identified by a distinct temperature drop as illustrated in Fig. 2(a). A closer look at the temperature change (Fig. 2(b)) shows the extent of the temperature drop as well as the temperature recovery when the plate leaves the cooling section. The entry temperature, T_entry, denotes the temperature at which the plate enters the cooling section during each pass. Significant variations of the temperature drop and corresponding heat fluxes (Figs. 2(c) and 2(d)) were observed for different entry temperatures which essentially determine the dominant boiling mode for each pass.

Fig. 2.

Example of experimental results for a circular nozzle test with a plate speed of 1 m/s, flow rate of 15 l/min and water temperature of 40°C: (a) surface temperature vs time, (b) surface temperature vs. time for cooling pass 1, (c) heat flux vs. time for cooling pass 1 with an entry temperature of 710°C, (d) heat flux vs. time for a low entry temperature of 150°C (cooling pass 21).

As shown in Fig. 2(c), it is convenient to identify the Peak Heat Flux (PHF) which represents the maximum heat flux obtained during a cooling pass. The sequence of PHF plotted vs surface temperature shows a striking similarity with a boiling curve, and hence can be used in a similar way to identify different boiling regimes, including the Leidenfrost point and the Maximum Heat Flux (MHF) point. As shown in Fig. 3, there can be significant variations in the PHF as a function of temperature and position with respect to the jet. In this particular test (circular jet, flow rate 15 l/min, and water temperature 40°C), the PHF values at the location under the jet (black line in Fig. 3) effectively double from the minimum which is observed at the temperature of 450°C (Leidenfrost point) to the maximum at around 200°C (MHF). The red and green curves show the PHF sequence for the same test obtained at distances of 25 mm and 50 mm, respectively, in lateral direction from the stagnation line. The differences in the PHF values for these three positions show the decrease of heat transfer rate with increasing distance from the jet. This information can be used to identify the effective jet impingement cooling area associated with a single jet.

Fig. 3.

Effect of lateral distance on PHF in a circular nozzle test with a plate speed of 1 m/s, flow rate of 15 l/min and water temperature of 40°C. (Online version in color.)

An important aspect is to quantify the effect of plate speed on PHF. For this purpose a set of planar jet experiments with three different plate speeds was carried out and the results were compared with stationary plate experiments performed under otherwise the same conditions.¹²⁾ As shown in Fig. 4, increasing the plate speed decreases the PHF in the film and transient boiling regions but does not affect the nucleate boiling regime. These findings are consistent with the observation of Gradeck et al.¹⁷⁾

Fig. 4.

Effect of plate speed on peak heat flux for planar jet tests with a flow rate of 100 l/min and a water temperature of 25°C. Stationary data are taken from Nobari et al.¹⁰⁾ (Online version in color.)

3. Modeling

3.1. Stationary Plate Model

3.1.1. Hydrodynamics of Jet Impingement

Guedia et al.⁹⁾ developed a mechanistic heat transfer model for cooling of a stationary plate with planar bottom jets by considering both impingement and parallel flow regions. In order to appreciate the model extension to top jet cooling, it is useful to characterize the differences in the boiling patterns between top and bottom cooling.

In the impingement region, depending on the surface temperature, the heat transfer modes are transition and nucleate boiling. For the applied range of flow rates and water temperatures (subcooling), film boiling is typically not present in the impingement zone of stationary plates. There is however an initial cooling period that is related to the development of the water flow on the plate surface and associated hydrodynamic and temperature profiles. Boiling curves show a steep increase as the heat transfer changes from air cooling to water cooling in the transition boiling regime. Transition boiling has two stages. In the high temperature range a prolonged region of nearly constant heat flux is observed which is known as “shoulder”. In the lower temperature range the heat flux increases further with decreasing temperature until the MHF is reached. Further decrease of surface temperature leads to heat transfer being characterized by nucleate boiling.

In the nucleate boiling regime we consider two distinct regions. One at lower surface temperature (i.e. lower superheat), where the heat flux is only dependent on surface temperature (called the full nucleate boiling, FNB), and one at higher superheat, starting at the point known as departure from nucleate boiling (DNB), where heat flux is also dependent on jet velocity and subcooling.

For the parallel flow region the hydrodynamic model by Mozumder et al.¹⁸⁾ has been adopted which was initially developed for a surface quenched with a circular bottom jet of water. The model maps the heat transfer due to the progression of the wetting front on the bottom surface, and assumes four distinct but dynamic cooling zones: (i) The wet zone which is central to the impingement point where the surface temperature has already dropped below saturation and the boiling process stopped, (ii) the boiling zone characterized by a high temperature gradient where most heat transfer occurs, (iii) the precursory cooling zone and (iv) the unaffected zone of mostly air cooling. Guedia et al.⁹⁾ have developed a comprehensive set of equations for bottom jet impingement boiling in all distinct heat transfer regions for both impingement and parallel flow zones. The model has been benchmarked with experimental data of Kashyap⁸⁾ for bottom cooling with a planar jet.

The principal difference between bottom and top cooling of stationary plates is related to the buildup of a water layer on the top surface, which greatly affects heat transfer in the parallel flow region. The presence of water which quickly evaporates in contact with the hot top surface, and the subsequent buildup of a vapor layer enclosed between the plate and the cooling water effectively develops conditions for film boiling. The film boiling region ends with a breakdown of the vapor layer at the Leidenfrost point, leading to transition and nucleate boiling. Schematic boiling curves are shown in Fig. 5 for the impingement and parallel flow zones adopted for top jet impingement.

Fig. 5.

Schematic of boiling curves for top jet cooling of stationary plates in the impingement (solid line) and parallel flow regions (dashed line).

In order to apply the bottom jet model of Guedia et al.⁹⁾ to top cooling, the jet impingement velocity, jet impingement diameter, stagnation pressure and total acceleration have to be adapted for a jet impinging vertically downward on an upper facing surface. The adapted equations are listed in Table 2.

Table 2. Hydrodynamic parameters.

Impingement velocity, Vj	$V_j={\left(V_n^2+2gH\right)}^{0.5}$
Impingement diameter/width, ϕj	${\varphi }_j=\psi \varphi \frac{V_n}{V_j}$, ψ = −0.18∙Tl + 16 for planar nozzle ψ = 6 for circular nozzle
Jet acceleration, γjet	${\gamma }_{jet,stag}=\frac{V_j^2}{2{\delta }_l}$
Stagnation pressure, Pstag	$P_{stag}=P_0+\frac{1}{2}{\rho }_l{V}_j^2$
Normalised pressure, Pnorm (x)	$P_{norm}\left(x\right)=\frac{P-P_0}{P_{stag}-P_0}=1-{\left(\frac{u_l}{V_j}\right)}^2$
Streamwise velocity, u	$\frac{u_l}{V_j}=\text{tanh}\left(\frac{x}{{\varphi }_j}\right)$
Total acceleration, γtot	γtot = γjet + g γjet (x) = γjet,stag∙Pnorm (x)

3.1.2. Nucleate Boiling

The conceptual approach to modelling nucleate boiling in the impingement and parallel flow zones for top cooling remains the same as in the original model for bottom cooling introduced by Guedia et al.⁹⁾ Small adjustments were made for selected adjustable parameters, in order to provide a better agreement with experimental data. For low superheat, the fully developed nucleate boiling (FNB) regime is considered. It depends on the surface temperature, T_surface (in °C), and the surface heat flux, $q_{FNB}^{″}$ (in MWm⁻²) such that⁹⁾   

$$q_{FNB}^{″}={\left(T_{surface}-130\right)}^{0.75}$$(1)

For higher superheat the departure from the nucleate boiling (DNB) regime has to be considered where, in addition to surface temperature, the surface heat flux, $q_{DNB}^{″}$ (in MWm⁻²), is also dependent on jet velocity (in m/s) and subcooling (in °C), i.e.   

$$q_{DNB}^{″}=c_0{\left(T_{surface}-T_0\right)}^{0.42}$$(2)

where   

$$c_0=1.08{\left(q_0^{″}\right)}^{0.25}$$(3)

and   

$$T_0=130+{\left(q_0^{″}\right)}^{4/3}-{\left(\frac{q_0^{″}}{c_0}\right)}^{2.38}$$(4)

The change from one nucleate boiling model to the other occurs at the temperature T₀ which corresponds to the heat flux $q_0^{″}$. An empirical correlation was previously developed for $q_0^{″}$ by Guedia et al.⁹⁾ for bottom cooling, and here an adjusted version, accounting for top cooling conditions is adopted such that   

$$q_0^{″}=\left(V_j-0.0675T_l+12\right)$$(5)

where $q_0^{″}$ is in MWm⁻².

The adjustments of the original model, made in Eqs. (3), (4), (5) for top cooling conditions, are based on experimental data of Nobari et al.¹⁰⁾ In the parallel flow zone the departure from nucleate boiling is not observed and the nucleate boiling heat flux reaches its maximum point at the MHF, $q_{max}^{″}$. The description of the maximum heat flux consists of two terms quantifying the two major heat transfer processes which periodically take place in the high heat flux region. The cycle consists of the heating of subcooled water and evaporation of saturated liquid which subsequently escapes through the vapor layer. The cycle is determined by the eviction velocity of the vapor, as explained in detail by Guedia et al.⁹⁾ The expression for the maximum heat flux is given by   

$$q_{max}^{″}=c_1{\rho }_v\left(0.46\frac{{\rho }_l}{{\rho }_v}{C}_{p,l}\mathrm{\Delta }{T}_{sub}+h_{lv}\right){\left(\frac{1}{{\delta }_v}\right)}^{1/2}{\left(\frac{\sigma }{{\rho }_v}\right)}^{1/2}$$(6)

where ${\delta }_v=c_2\sqrt{{\displaystyle \frac{2{\mu }_v\varphi {\displaystyle \frac{V_n}{V_j}}}{{\rho }_l{V}_j}}}$

3.1.3. Transition Boiling

The mechanistic transition boiling model proposed by Guedia et al.⁹⁾ has been applied in this study for the impingement zone. The principles of transition boiling are described by two contributing mechanisms, each one becoming dominant in their respective surface temperature range. The two mechanisms have been originally proposed by Ahmed and Hamed.⁷⁾ Both mechanisms are based on the transient conduction to the liquid which comes in contact with the solid surface, but with different frequency of vapor growth and break-up cycles and associated instabilities of the vapor-liquid interface which are contributing to the dynamics of the process. The rate of heat transfer by transient conduction to the liquid is dependent on the ability of the liquid to protrude through the vapor layer and wet the surface. In the lower surface temperature range, following a growing disturbance at the vapor-liquid interface, the liquid rushes in a predictable cycle through the vapor and contacts the surface providing an area for transient conduction. This process results in relatively high heat transfer rates and it is called liquid quenching transient conduction, $q_q^{″}$. This ability of the liquid to penetrate through the vapor layer diminishes with increasing surface temperature. At higher surface temperatures the dominant mechanism is liquid intrusion transient conduction, $q_i^{″}$. The overall transition boiling surface heat flux, $q_{TB}^{″}$, is calculated by using   

$$q_{TB}^{″}=A{q}_i^{″}+\left(1-A\right)q_q^{″}$$(7)

where A is the transitional function given by   

$$A=\frac{1}{1+exp\left\{-0.021\left(T_{surface}-T_1\right)\right\}}$$(8)

The transitional function has values between 0 and 1 and represents the fraction of the liquid intrusion heat flux. In Eq. (8), T₁ indicates the surface temperature when liquid intrusion and quenching modes are equally present. Values of T₁ were determined from the experimental data and vary for different nozzle types, as shown in Table 3.

Table 3. Parameters for the top cooling stationary plate model.

Parameter	Circular nozzle	Planar nozzle
c1	0.003	0.024
c2	90	140
c3	0.01·$P_{norm}^{0.27}$	0.01·$P_{norm}^{1.5}$
c4	200000	360000
c5	0.022+8·10−5VjTl−3.5·10−3Vj−1.4·10−4Tl	0.017+6·10−5VjTl−2.6·10−3Vj−1.3·10−4Tl
c6	0.012	0.03
c7	0.86	2.55
c8	0.95	1.76
c9	0.91	0.4
T1	530	450

Guedia et al.⁹⁾ provide a more detailed description of the model, including the application of the Kelvin-Helmholtz instability at the vapor-liquid interface to calculate the rate of intrusion of the liquid, i.e. the growth rate of the most dangerous wavelength, ω_d, and the wetting frequency, f. In summary, the intrusion heat flux is given by   

$$q_i^{″}=c_3{\rho }_l{C}_{p,l}\mathrm{\Delta }{T}_{sub}{\lambda }_d{\omega }_d$$(9)

where   

$${\omega }_d^2=\frac{{\rho }_l^{″}{\rho }_v^{″}{\left(u_l-u_v\right)}^2{\eta }^2+\left[-\sigma {\eta }^3+\left({\rho }_l-{\rho }_v\right){\gamma }_{tot}\eta \right]\left({\rho }_l^{″}+{\rho }_v^{″}\right)}{{\left({\rho }_l^{″}+{\rho }_v^{″}\right)}^2}$$(10)

Based on the original analysis by Ahmed and Hamed,⁷⁾ Guedia et al.,⁹⁾ described the quenching surface heat flux by   

$$q_q^{″}=k_l\left(T_{surface}-T_l\right)\sqrt{f/\pi {\alpha }_l}$$(11)

where   

$$f=f_{stag}\cdot \overline{P}$$(12)

with   

$$f_{stag}=c_4\cdot \text{exp}\left(-c_5\cdot \frac{T_{surface}}{T_{sat}}\right)$$(13)

The parameters c_3–5 are adapted for top jet impingement based on the experimental data, as given in Table 3.

3.1.4. Film Boiling

The mechanistic transition boiling model is directly applied for the stagnation point. In order to construct boiling curves at distances from the stagnation point (in the parallel flow and in the intermediate region) additional steps must be considered. First, an extended region of film boiling has been observed in the parallel flow zone in all top cooling experiments. For this region, a modified form of the model by Filipovic et al.,¹⁹⁾ i.e.   

$$q_{FB}^{″}=c_6{\lambda }_v\frac{\mathrm{\Delta }{T}_{sub}}{\overline{x}}\left(\frac{{\mu }_l}{{\mu }_v}\right)\left(\frac{P{r}_v{C}_{p,l}}{C_{p,v}}\right){\left(2u_s+7\right)}^{0.2}R{e}^{0.8}P{r}_l{}^{-0.67}{u}_s{}^{0.6}$$(14)

is adopted by adjusting the empirical coefficient c₆ (see Table 3) to match the experimental data. The dimensionless velocity, u_s, of the liquid at the vapor-liquid interface is a function of subcooling such that   

$$u_s=\frac{1}{1+\beta \cdot P{r}_l^{1/3}}$$(15)

with the subcooling parameter   

$$\beta =\frac{P{r}_v{C}_{p,l}\mathrm{\Delta }{T}_{sub}}{P{r}_l{C}_{p,v}\mathrm{\Delta }{T}_{sup}}$$(16)

The stable vapor layer which governs heat transfer in film boiling will collapse at the Leidenfrost point. The mathematical expressions for the Leidenfrost superheat can be written such that   

$$\mathrm{\Delta }{T}_{sat,Leidenfrost}=c_7\cdot e^{\left(-c_8\cdot x\right)}\cdot {\left(\frac{c_{p,l}\mathrm{\Delta }{T}_{sub}}{h_{lv}}\right)}^{-c_9}\cdot \mathrm{\Delta }{T}_{sub}$$(17)

where the parameters c_i (i = 7,8,9) depend on nozzle type, as listed in Table 3. Past the Leidenfrost point, transition boiling will occur. The shoulder heat flux was not observed in the parallel flow zone and transition boiling has to be addressed in a different way than in the impingement zone. For this purpose a simple empirical linear variation of heat flux with surface temperature is proposed for the region between the Leidenfrost point and the MHF point. Based on the experimental data the slope of the line for the circular nozzle is −0.025 MWm⁻²°C⁻¹, and for the planar nozzle −0.044 MW m⁻²°C⁻¹.

Similar to Guedia et al.,⁹⁾ the boiling curves in the intermediate region (locations between the impingement and parallel flow zone) are calculated as the weighted sum of the heat fluxes at the boundary of the impingement and parallel flow zones:   

$$\begin{array}{l}{q}_{inter}^{″}={\displaystyle \frac{P_{norm}\left(x\right)-P_{norm}\left(x_{para}\right)}{P_{norm}\left({\varphi }_j\right)-P_{norm}\left(x_{para}\right)}}\cdot q_{{\varphi }_j}^{″}\\ \mathrm{\hspace{1em}\hspace{1em}}+{\displaystyle \frac{P_{norm}\left({\varphi }_j\right)-P_{norm}\left(x\right)}{P_{norm}\left({\varphi }_j\right)-P_{norm}\left(x_{para}\right)}}\cdot q_{x_{para}}^{″}\end{array}$$(18)

where, $q_{{\varphi }_j}^{″}$ is the heat flux from the boiling curve at the edge of the impingement zone and $q_{x_{para}}^{″}$ the heat flux from the boiling curve at the beginning of the parallel flow zone. As discussed by Guedia et al.,⁹⁾ experimental heat transfer in the intermediate zone is characterised by large fluctuations of the heat flux. The stochastic nature of this process makes it challenging to model, but Eq. (18) provides a reasonable approach for the construction of boiling curves.

3.2. Comparison of Model with Experimental Data for Stationary Plates

The proposed top cooling model of a stationary plate with a single circular nozzle has been validated by using experimental data of Nobari et al.¹⁰⁾ Satisfactory agreement within the experimental errors has been reached for all tests, as illustrated in Fig. 6 for two representative cooling conditions. The figure shows the experimental data and model boiling curves for the impingement (x = 0 mm), the parallel (x = 120 mm) and the intermittent (x = 80 mm) flow zones, for experiments conducted with a flow rate of 15 l/min and different water temperatures.

Fig. 6.

Experimental and calculated boiling curves for stationary plates: (a) Water flow: 15 l/min, water temperature: 25°C, nozzle to plate distance: 0.1 m, nozzle diameter: 19 mm; (b) Water flow: 15 l/min, water temperature: 10°C, nozzle to plate distance: 0.1 m, nozzle diameter: 19 mm.

3.3. Moving Plate

For the extension of the proposed model to moving plates, it is important to note that the stationary plate model provides the upper limit for the heat flux represented by a boiling curve at a given set of conditions (subcooling, flow rate, and jet impingement velocity). This maximal heat flux at a given temperature is reached when the plate speed is approaching zero, i.e. the surface is exposed to the cooling fluid for sufficiently long times such that steady state heat transfer conditions are met. If the plate is moving, the exposure time is shorter and it is a function of plate speed. Thus, scaling the stationary plate boiling curves with plate speed is used to generate a family of boiling curves for moving plate jet impingement.

All segments of the boiling curve, except for the nucleate boiling correlation, Eq. (1), are dependent on plate speed. Linear scaling functions ϕ_v,1-6 have been developed using the planar jet experiments (see Table 1), and further adjusted for circular jet conditions, as listed in Table 4. It has to be emphasized that these correlations are only valid for the experimental plate speed range of up to 1.6 m/s. Non-linear extrapolations to higher plate speeds would need to be considered that must also account for the asymptotic limit of zero heat flux for plate speeds approaching infinity.

Table 4. Parameters for the top cooling moving plate model.

Parameter	Circular nozzle	Planar nozzle
ϕv,1	(1−0.62Vp)·$P_{norm}^{0.6}$	1.5·(1−0.34Vp)·$P_{norm}^{0.5}$
ϕv,2	0.74·(1−0.55Vp)	(1−0.32Vp)
ϕv,3	0.75·(1−0.73Vp)	(1−0.027Vp)
ϕv,4	0.3·(1−0.36Vp)	0.5·(1−0.36Vp)
ϕv,5	(1−0.049Vp)	0.22865·exp(−Vp/0.27094)+0.771
ϕv,6	(1−0.32Vp)·0.2·${\left(\frac{P_{norm}}{P_{norm,para}} \right)}^{0.5}$	(1−0.32Vp)·0.85·${\left(\frac{P_{norm}}{P_{norm,para}} \right)}^{0.5}$

The moving plate model for the entire surface is developed by multiplying the stationary plate heat fluxes for each segment of the boiling curve with the plate speed scaling functions, and for circular jets lateral direction scaling factors, S_lat,i, have to be considered as well. For planar jets the latter are equal to one. In detail the following equations are used to obtain a family of boiling curves and quantify heat flux over the entire top surface of the moving plate. Here, the subscripts mov and stat refer to moving and stationary plates, respectively.   

$$q_{DNB,\mathrm{\hspace{0.17em} }mov}^{″}=q_{DNB,stat}^{″}\cdot {\phi }_{v,1}\cdot s_{lat,1}$$(19)

  

$$q_{max,\mathrm{\hspace{0.17em} }mov}^{″}=q_{max,stat}^{″}\cdot {\phi }_{v,6}\cdot s_{lat,1}$$(20)

  

$$q_{TB,mov}^{″}=q_{TB,\mathrm{\hspace{0.17em} }stat}^{″}\cdot {\phi }_{v,2}\cdot s_{lat,2}$$(21)

  

$$q_{FB,mov}^{″}=q_{FB,stat}^{″}\cdot {\phi }_{v,4}\cdot s_{lat,2}$$(22)

  

$$T_{Leid,\mathrm{\hspace{0.17em} }mov}=T_{Leid,\mathrm{\hspace{0.17em} }stat}\cdot {\phi }_{v,5}\cdot s_{lat,3}$$(23)

  

$$T_{1,mov}=T_{1,stat}\cdot {\phi }_{v,3}\cdot s_{lat,3}$$(24)

For circular jets the entire boiling curve can be scaled with respect to the distance from the stagnation point. The scaling has been done by analyzing the variation of the maximum and minimum peak heat flux, as well as the temperature of the minimum peak heat flux. Figure 7 represents the variation of the maximum peak heat flux (PHF_max) with distance from the stagnation point for all circular nozzle experiments with moving plates (see Table 1). The variation is presented in a normalized format, i.e. PHF_max has been normalized with respect to the peak heat flux at stagnation (PHF_max,y=0), taking the quantity of 1 at stagnation and asymptotically approaching 0 far from the jet, and the distance has been normalized with the jet diameter (d_j). The impingement zone, parallel flow zone as well as the intermediate region are clearly observed on the graph. The peak heat flux gradually changes within the impingement zone until it reaches its edge, at which point it abruptly changes to less than 50 percent of its initial value. As the distance from the stagnation further increases the slope becomes again more gradual indicating the presence of the parallel flow zone. It can also be observed that there is very little variation of PHF_max with respect to flow rate and water temperature, and therefore a unique expression for the variation of heat flux in the lateral direction has been derived for all conditions within the experimental matrix. The proposed scaling factor, S_lat,1 can be written as   

$$s_{lat,1}=0.41+\frac{0.58}{1+exp\left(\left(\frac{y}{d_j}-2.76\right)/0.78\right)}$$(25)

Fig. 7.

Normalized PHF_max profile in lateral direction. (Online version in color.)

A similar approach has been used to develop scaling factors for PHF_min and the corresponding temperature T_PHF,min. These scaling factors are used to scale transition and film boiling in the lateral direction, the Leidenfrost point and the transitional temperature T₁. The normalized graph for PHF_min is shown in Fig. 8 and the scaling factor is obtained to be   

$$s_{lat,2}=0.072+\frac{1.02}{1+exp\left(\left(\frac{y}{d_j}-2.32\right)/1.03\right)}$$(26)

The change of the normalized temperature of PHF_min in the lateral direction is presented in Fig. 9 and the corresponding scaling factor is   

$$s_{lat,3}=0.56+\frac{0.49}{1+exp\left(\left(\frac{y}{d_j}-3.79\right)/1.63\right)}$$(27)

Fig. 8.

Normalized PHF_min profile in lateral direction. (Online version in color.)

Fig. 9.

Normalized T_PHF,min profile in lateral direction. (Online version in color.)

3.4. Model Comparison

The simulation for moving plates has been carried out by using a 1D finite difference through thickness temperature model to calculate the temperature profiles in the steel using the proposed boiling model to determine the surface heat flux boundary conditions. The application of a 1D model reflects that the through-thickness temperature gradients are much larger than those in lateral directions which, therefore, can be neglected in a first approximation. The simulation results, i.e. temperature evolution and surface heat fluxes, have been compared with the experimental results. The simulation domain is 2000 mm long, starting 1000 mm upstream from the nozzle position, and ending 1000 mm downstream from the nozzle. The temperatures were calculated at 1240 equally spaced nodes through-thickness, and the time step was 0.001 s. The simulation domain includes two air cooling sections upstream and downstream respectively, and the boiling region in between for the top surface, and air cooling on the bottom surface. The simulation domain with the pre-determined cooling regions is shown in Fig. 10.

Fig. 10.

Simulation domains for a moving plate under a single circular jet. (Online version in color.)

Air cooling (sections L₁ and L₅ in Fig. 10) is calculated by assuming convection and radiation heat transfer from the plate using   

$$\begin{array}{l}{q}_{air}^{″}=12\cdot \left(T_{surface}-298\right)+\\ \hspace{1em}\left[\left({\displaystyle \frac{T_{surface}}{1\mathrm{\hspace{0.17em} }000}}\right)\left(0.125{\displaystyle \frac{T_{surface}}{1\mathrm{\hspace{0.17em} }000}}-0.38\right)+1.1\right]\cdot 5.67\times {10}^8\\ \hspace{1em}\left(T_{surface}{}^4-{298}^4\right)\end{array}$$(28)

Here, a heat transfer coefficient for air of 12 W/m²K accurately represent the forced convective heat transfer of steel plates traveling at low speed. The heat fluxes obtained in these two sections are substantially lower than those in the boiling sections. Section L₂ starts at the wetting front upstream from the nozzle. The wetting front appears because of the opposing directions of the moving plate and liquid spreading on its top surface, and its location is a function of the liquid flow rate, nozzle diameter and plate speed. The distance L₂ from the stagnation point can be estimated using the correlation proposed by Seraj and Gadala,²⁰⁾ i.e.   

$$L_2=6.5d_n{\left(\frac{\dot{Q}}{d_n{\nu }_l}\right)}^{0.39}{\left(\frac{d_n{V}_p}{{\nu }_l}\right)}^{-0.36}$$(29)

In this equation $\dot{Q}$(m³/s) is the volumetric flow rate of the jet, d_n(m) is the nozzle diameter, and ν_l(m²/s) is the liquid kinematic viscosity. The wetting front in this simulation, i.e. plate moving at a speed of 1 m/s, and cooled by a single circular jet from a nozzle with an inner diameter of 19.2 mm and flow rates ranging from 15 l/min and 45 l/min, is between 145 mm and 225 mm, respectively, upstream from the nozzle.

Section L₃ is immediately downstream from the nozzle, and it is chosen to be equal to half of the length of the plate, i.e. 600 mm. Section L₄ has been introduced because of the water pooling effect downstream from the nozzle, but its size depends on the surface temperature. For higher temperatures the water layer downstream from L₃ will subside very quickly and the dry surface will immediately become exposed to air cooling while at lower temperatures a prolonged section is observed with water remaining on the top surface where parallel flow boiling is adopted to determine the heat transfer rate. Based on the experimental evidence, the length of L₄ can be correlated to the entry temperature of the plate such that:

if T_entry>250°C, L₄ = 0, but if T_entry<250°C, $L_4=L_3\left(\frac{V_p}{V_j-V_p}\right)$

Once the lengths of the sections L₁–L₅ have been established, a family of limiting model boiling curves (i.e. in the limit of zero velocity) can be constructed for different locations with respect to the stagnation point. As the plate is moving through the cooling section, the amount of heat removed from a point on the surface and the associated temperature change can be calculated for each time increment which also determines the change of location of the point with respect to the nozzle. The calculated and measured heat flux and temperature evolution is depicted in Fig. 11 for one particular cooling pass (T_entry = 385°C, flow rate = 45 l/min, water temperature = 40°C) and a plate location that passes directly under the nozzle. Within the errors of measurements, agreement can be noted between experiment and model. Figure 11(c) shows the heat flux evolution with respect to the limiting boiling curves that are displayed for different locations in section L₃. Here, the heat fluxes close the nozzle position are approaching the limiting heat flux concluded for stationary plate cooling for otherwise the same conditions.

Fig. 11.

Comparison of calculated and experimental data for the circular nozzle moving plate test with a flow rate of 15 l/min and a water temperature of 40°C at an entry temperature of 385°C for a plate position that will pass through the nozzle location. (a) heat flux vs. time, (b) surface temperature vs. time, (c) family of boiling curves for stationary plates at various distances from the stagnation point, and model predictions for the change of surface heat flux during the cooling pass. (Online version in color.)

Similarly, satisfactory agreement between experiments and model exists for all other passes, as shown in Fig. 12. The agreement is particularly good in the high temperature region and some discrepancies are seen in the low temperature region (e.g. T_entry = 185°C). The overall accuracy of the model calculations is within +/− 10% for all experimental data with a single circular nozzle, as demonstrated by comparing in Fig. 13 the experimental and model temperatures at the distance 500 mm downstream from the jet. A distance of 500 mm is relevant as it represents the approximate spacing between neighboring jet lines in conventional laminar runout table cooling sections.

Fig. 12.

Comparison of calculated and experimental surface heat fluxes and surface temperatures for cooling passes with different entry temperatures for the stagnation line in the circular nozzle test with a flow rate of 15 l/min and a water temperature of 40°C. (Online version in color.)

Fig. 13.

Comparison of experimental and model temperatures 500 mm downstream from the circular nozzle.

The accuracy of model calculations for planar jets is shown in Fig. 14. It can be observed that, while the accuracy is somewhat lower than for a single circular jet, it is still within +/− 15% and, thus, satisfactory with respect to experimental errors. Similar to the single circular nozzle cooling, the accuracy of model calculations is less in the low temperature range (i.e. below 200°C) which is likely due to the model approximations related to water pooling. Further investigations are required to develop improved relationships to account for water pooling.

Fig. 14.

Comparison of experimental and model temperatures 500 mm downstream of planar nozzle.

3.5. Model Application to a Jet Line with Three Nozzles

Industrial runout tables consist of multiple jet lines with either planar or circular nozzles. In the latter case, each jet line may be represented by a row of equally spaced circular nozzles. The application of the model to multiple nozzles becomes more complicated as it may be affected by jet spacing in rolling and lateral directions and the effect of jet interaction between neighboring nozzles. In order to provide some insight into the applicability of the present model to runout table cooling, a model simulation was conducted for a series of experiments with a moving plate cooled by three equally spaced nozzles placed in a single row. Here, the nozzles with an inner diameter of 10.2 mm are spaced 30 mm apart and placed 1600 mm above the surface of the plate moving at a speed of 1 m/s. Multi-pass cooling tests were conducted as described in Section 2 using a flow rate of 17.4 l/min per nozzle and a water temperature of 40°C.

Figure 15 shows the experimental and simulated heat flux and surface temperature patterns on the top surface of the plate during a single pass with an entry temperature of 600°C. The contour plots representing simulation results have been constructed by running the model for three different lateral locations with respect to the central jet (0 mm, 7.5 mm and 15 mm from the stagnation line), which provide temperature and heat flux variations from the stagnation line of the central jet to the mid-point between two neighboring jets in either direction. This pattern can then be replicated across the width of the plate. The localized high heat transfer areas in the vicinity of the impinging jets, and corresponding temperature variations are accurately predicted by the model. Further evidence of good agreement between the experimental data and model predictions can be observed by comparing the measured and simulated temperatures 500 mm downstream from the jet line, at the thermocouple location 1 mm below the top surface, as shown in Fig. 16. Each data point represents a temperature at the end of a single pass, for a total of 19 passes carried out during the experiment. The model predictions are well within +/− 10%, except for the last two passes in the low temperature range, which can be attributed to the uncertainties related to water pooling.

Fig. 15.

Contour plots for cooling with a jet line of three nozzles and an entry temperature of 600°C: (a) experimental surface heat flux, (b) model predicted surface heat flux, (c) experimental surface temperature, (d) model predicted surface temperature.

Fig. 16.

Comparison of measured and predicted temperatures 500 mm downstream: 3 nozzles, flow rate per nozzle 17.4 l/min, water temperature 40°C.

4. Conclusion

A mechanistic boiling model has been developed for the prediction of temperature distribution on the surface of moving hot steel plates during runout table cooling with circular and planar jets, respectively. The model represents an extension of the previously developed boiling model for bottom jet impingement cooling of stationary plates. After adapting the model to top jet cooling of stationary plates, it has been extended for moving plates by incorporating scaling factors to account for plate speed.

The proposed boiling model has been combined with a 1D through-thickness temperature model. The combined model has been benchmarked with experimental data for top cooling of moving plates with a single circular and planar nozzle, respectively. Measured temperatures can be replicated within the accuracy of measurements as long as the entry temperature is above 200°C, i.e. as long as water pooling does not significantly affect the overall heat extraction rates. Further, the model has been applied to predict cooling with a jet line of three circular nozzles having different diameters as compared to the benchmark singular nozzle case. Satisfactory temperature predictions at the exit from the cooling section have been obtained with an accuracy of +/−10%.

The proposed boiling model provides boundary conditions necessary for a full runout table cooling model. The model is coupled with transient conduction to calculate surface heat fluxes, surface temperatures and through-thickness temperature distributions by following a given steel position when water jets are applied for cooling of steel. When coupled with phase transformation models through thickness of steel, it represents a powerful tool for the simulations of microstructure evolution. Implementing appropriate structure-property relations will then facilitate to predict the mechanical properties of hot rolled steel products.

Acknowledgements

The authors acknowledge the financial support of ArcelorMittal Dofasco and the Natural Science and Engineering Research Council of Canada. Further, they wish to thank Mr. Gary Lockhart, Mr. Ross McLeod, Mr. David Torok and Mr. Carl Ng for their technical contributions and assistance towards the experimental work of this study.

Nomenclature

A: Transitional function

c_{i (i = 0 to 9)}: Empirical parameters

C_p, Jkg⁻¹K⁻¹: Specific heat capacity

d_n, m: Nozzle diameter

f, s⁻¹: Wetting frequency

g, ms⁻²: Gravitational acceleration

h, Wm⁻²K⁻¹: Heat transfer coefficient

H, m: Nozzle to plate distance

h_lv, Jkg⁻¹: Latent heat of vaporization

k, Wm⁻¹K⁻¹: Thermal conductivity

L_i (i = 1…5): Length of cooling sections

P, Pa: Pressure

P₀, Pa: Ambient Pressure

P_norm: Normalized pressure

Pr: Prandtl number

PHF, MWm⁻²: Peak heat flux

$\dot{Q}$, m³s⁻¹: Volumetric flow rate

q^″, MWm⁻²: Heat flux

$q_0^{″}$, MWm⁻²: Heat flux at DNB

Re: Reynolds number

S: Scaling factor in lateral direction

T, °C: Temperature

T_entry, °C: Entry temperature

T_surface, °C: Plate surface temperature

T₀, °C: Temperature at DNB

T₁, °C: Intrusion-quenching transition temperature

u, ms⁻¹: Streamwise velocity

u_s: Interfacial velocity

V_j, ms⁻¹: Impingement velocity

V_n, ms⁻¹: Jet velocity at nozzle

x, m: Position from stagnation

y, m: Distance in lateral direction

α, m²s⁻¹: Thermal diffusivity

β: Subcooling parameter

γ, ms⁻²: Acceleration

δ_v, m: Vapor layer thickness

δ_l, m: Liquid layer thickness

ΔT, °C: Temperature difference

ΔT_sub, °C: Subcooling

η, m⁻¹: Wavenumber, $\eta =\frac{2\pi }{\lambda }$

λ, m: Wavelength

μ, Pas: Dynamic viscosity

ρ, kgm⁻³: Density

ρ^″, kgm⁻³: Modified density

σ, Nm⁻¹: Surface tension

ϕ, m: Jet width/diameter

ϕ_j, m: Impingement width/diameter

ψ: Parameter for stream profile

ϕ_{v,i (i = 1−6)}: Scaling functions for moving plates

ω, s⁻¹: Growth rate of wavelength

Subscripts

air: Air

d: Most dangerous

DNB: Departure from nucleate boiling

FB: Film boiling

FNB: Full nucleate boiling

i: Intrusion

inter: Intermediate

j: Jet impingement

jet: Jet

l: Liquid (water)

lat: Lateral

leid: Leidenfrost point

max: Maximum

min: Minimum

mov: Moving plate

p: Plate

para: Parallel flow region

q: Quenching

sat: Saturation

stag: Stagnation point

stat: Stationary

sub: Subcooling

sup: Superheat

v: Vapor

TB: Transition boiling

tot: Total

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