1 Introduction

Impinging jet cooling is a widely used technique in heat treatment processes owing to the wide range of controllability of the cooling speed. When a single circular jet impinges on a solid surface, the fluid spreads around the wall. The flow can be divided into stagnation region, acceleration region, and wall jet region. The stagnation point coincides with the jet point and exhibits the maximum pressure [1]. In the accelerating region, the flow velocity increases gradually along the radial direction until it approaches the jet velocity. In the wall jet region, the fluid flows along the radial direction with a decreased velocity caused by the viscous force.

When the low-temperature fluid contacts the high-temperature wall, a vapor film will form, which limits the heat transfer between fluid and wall. After the collapse of the vapor film, the fluid can contact the hot surface again, which is known as a rewetting phenomenon [2]. More effective surface cooling occurs when the hot surface is rewetted. The cooling process is gradually carried out along the radial direction from the jet point. When the surface temperature downstream reaches the rewetting temperature, the rewetting will extend to the downstream [3]. Different cooling times result in different boiling structures on the surface at the same time, including nucleate boiling, transition boiling, and film boiling. The wetting front is located between the dry and wet regions [4], and the whole transition boiling region is located at a point that can be considered as being part of the wetting front [5]. In the dry region outside the wetting front, since the surface temperature is higher than the minimum temperature for film boiling, a continuous vapor film is formed, and the fluid flows over it. The wet region is inside the wetting front, where nucleate boiling occurs.

The peak of the heat flux curve during cooling is referred to as the maximum heat flux, which is often used to characterize the surface quenching ability [6]. It is traditionally believed that the maximum heat flux occurs at the boundary between nucleate boiling and transition boiling. When the surface is in a nucleate boiling state, a higher heat transfer efficiency can occur under a relatively low wall-fluid temperature difference [7]. Therefore, one approach to improve the heat transfer efficiency is to increase the rewetting speed in order to induce the wall to enter the nucleate boiling state at an earlier time. At present, the movement mechanism of the wetting front is still not clear [8]. Woodfield et al. [9] reported that the increase of the radial temperature gradient causes a decreased wetting front width. Karwa et al. [10] found that the ability for the fluid to sweep bubbles at the wetting front might determine the moving speed of the wetting front. Hatta et al. [11] found that, when the wetting region is spread to 80 mm, the wetting front width increases to 20 mm, and they claimed that the decreasing flow velocity would lead to an increasing width. Several studies have analyzed the single-jet impingement heat transfer and found that the jet diameter, jet angle, jet velocity, and other parameters can affect the heat transfer ability [12,13,14]. In addition, several researchers have proposed correlations of the maximum heat flux for different experimental conditions. Agrawal et al. [15] and Paul et al. [16] found correlations between the jet height and heat flux. According to the flow velocity, supercooling, and superheat, Wang et al. [17] proposed a correlation to predict the surface heat flux.

In this study, double water-jet cooling experiments were performed using jet exit velocities of 2.95, 5.90, 8.06, and 11.80 m/s and jet heights of 50, 150, 250, 350, and 450 mm on a 50 mm thick AISI 304 austenitic stainless-steel plate. The stagnation and acceleration regions are here denoted as the jet region, the region where the two jets interact is referred to as the confluence jet region, and the outside flow region without interference is denoted as the wall jet region. The rewetting phenomenon and heat exchange process for the different flow regions were discussed, and the results provide data and basis for optimizing the cooling technology of thick plates.

2 Experimental materials and methods

The chosen stainless steel was of grade 304, with specifications of 50 (thickness) × 80 (width) × 150 (length) mm3. The density ρs of the experimental material was 7930 kg·m− 3. The specific heat cs and thermal conductivity λs were calculated according to Eqs. (1) and (2) [18]. Temperature-measured holes with a diameter of 3.2 mm and depth of 40 mm were drilled at h = 3, 25, and 47 mm to install thermocouples, as shown in Fig. 1. h is the distance to the cooled surface, while r represents the horizontal distance between the surface position and jet point. Symbols “-” and “+” indicate that the position is located in the confluence jet region or wall jet region, respectively. The embedded thermocouples were inserted into the holes parallel to the surface, and the gap was filled with high-temperature glue to further secure the thermocouples. Pure water with a constant temperature of 12.8 °C was selected as the working fluid.

$$ {c}_s=\left(-0.2274\times {10}^{-\frac{T}{788}}+0.62\right)\times 1000 $$
(1)
$$ {\lambda}_s=-31.1\times {10}^{-\frac{T}{1703}}+47.7 $$
(2)
Fig. 1
figure 1

Experimental layout

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The jet-cooling system consisted of heating, cooling, and data recording units. The heating equipment consisted of a resistance heating furnace, while the cooling device included mainly a valve, flowmeter, and nozzle. The jet velocity could be adjusted via the flowmeter. Two circular nozzles with a length of 90 mm, inner diameter of 3 mm, and spacing of 60 mm were fixed above the target plate. A 20 (thickness) × 80 (width) × 150 (length) mm3high-alumina refractory brick was placed under the target plate to maximize the heat insulation for the lower surface. The jet height z and flow Q used in this study are shown in Table 1, while the jet exit velocity u was calculated through the flow. During the experiment, the plate was heated to an average temperature of 938.2 °C in the furnace and then transferred to the refractory brick. At the beginning of the cooling process, an iron box was used to collect water; such box was removed when the water flow was stable. The jet stagnation point was located on the upper surface, above P1-3, and the water curtain caused by the intersection of the two jets was located on the upper surface, above P1-1. The data-recording module was composed of thermocouples, a multi-channel temperature-data collector, and a computer. The temperature-data collector was used to record the various temperatures at different points by converting the electrical signal into a temperature signal and storing them in the connected computer.

Table 1 Experimental parameters
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The error of the K-type thermocouple was approximately ± 1.0 %. The jet flow was controlled within ± 3 L/h, and the jet height error was within ± 3 mm. The deviation distance of the thermocouples and jet point was within ± 0.5 and ± 1 mm, respectively. For the different experimental conditions, the initial cooling temperature error did not exceed 0.9 %.

A nine-point moving average method was used to reduce the temperature noise. Assuming that the average noise for all temperature values was zero, the true temperature could be calculated according to:

$$ {Y}_{k,M}=\frac{\sum_{i=1}^n\left({x}_{M-i}+{x}_{M+i}\right)+{x}_M}{2n+1} $$
(3)

where, \( {Y}_{k,M} \)is the filtered temperature at point K and time M, \( {x}_{M-i} \) is the measured temperature at time M-i, and n represents the radius of the sliding window, here set to four.

The ability of the fluid-cooling plate was evaluated via the heat flux. Based on the temperature history, the surface heat flux could be calculated through the inverse heat conduction. The solution method used in this work has been provided in a previous study [18]. According to the calculation results, the relative error did not exceed 0.1 %, 1.5 %, and 3.7 % at h = 3, 25, and 47 mm, respectively, and the largest errors were found to occur at points far away from the coolant surface. Therefore, it was possible to infer that the estimated surface heat flux has a sufficiently high calculation accuracy.

3 Experimental results and discussions

In order to simplify the discussion, several points and terms are here defined. When the boiling structure changed from film boiling to transition boiling, the wall was rewetted [19, 20] (the wall color changed from red to gray, and the heat flux began to increase explosively), and the occurrence time and surface temperature were denoted as the rewetting time tmin and rewetting temperature Tmin. When the boiling structure changed from transition boiling to nucleate boiling, the wall was completely wetted (the wall color changed from gray to dark, and the heat flux reached its maximum value qmax), and the occurrence time and surface temperature were denoted as the complete-wetting time tmax and complete-wetting temperature Tmax. The point on the upper surface above P1-1 to P1-5 was marked as P0-1 to P0-5, respectively, and P0-3 was rewetted at t = 0 s.

3.1 Double‐jet heat transfer characteristics

The heat transfer characteristics for the double-jet experiments were investigated at z = 150 mm and u = 2.95 m/s. Figure 2 shows that the wetting front can be clearly distinguished and that the rewetting and complete-wetting occur on its outer and inner edges, respectively. As times passes, a varying wetting front width can be observed. When t = 0.72 s, the wetting front width rapidly increases to approximately 17.5 mm; according to Fig. 2, this phenomenon appears to be related to the deflection fluid of the jet along the wall. Afterwards, the width decreases to 1.82 mm at t = 0.90 s. Woodfield et al. [9] reported that such decrease is caused by the increased radial temperature gradient. At t = 1.19 s, the width increases again to 5.4 mm. This might be due to the decreased flow velocity, which reduces the ability for the fluid to sweep bubbles at the wetting front, thus resulting in an increasing width.

Fig. 2
figure 2

Photographs of the wetting front at increasing times

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The measured temperature curves are shown in Fig. 3a. The slow cooling stage before jet cooling is attributed to the transfer of the target plate. Upon starting the experiment, it can be seen that the temperature at P1-1 to P1-5 rapidly decreases to below 200 °C within 48.4 s on average, and the average temperature at h = 25 and 47 mm is 705.3 and 817.9 °C, respectively. The temperature increases with the increased distance between the point and cooled surface until the end of the cooling process. The cooling speed (temperature gradient with respect to time) and temperature gradient as a function of time are shown in Fig. 3b-c. For h = 3 mm, the cooling speed and temperature gradient begin to increase rapidly at 38.4 s (approximately the time at which the plate is transferred), while for h = 25 mm a significant increase occurs at 43.3 s, which explains the hysteresis of the cooling process away from the surface. Grad3mm→25mm (temperature gradient between h = 3 mm and h = 25 mm) keeps growing until the cooling speed at h = 3 mm becomes equal to that at h = 25 mm, and Grad25 mm→47 mm was found to have a similar change to Grad3 mm→25 mm. The average maximum cooling speed at h = 3, 25, and 47 mm was measured to be 71.87, 5.86, and 4.07 °C/s, respectively. The above results prove that the temperature gradient determines the cooling speed field.

Fig. 3
figure 3

a Temperature curve, b cooling speed curve, and c temperature gradient curve

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Figure 4a shows the tmin and Tmin at z = 150 mm and u = 2.95 m/s. Due to the frictional resistance and vapor film, it takes a certain amount of time for the fluid to rewet the surface, and tmin increases along both sides from P0-3. Most researchers have found that, within a certain range of jet diameters, the fluid is deflected away from the surface due to the deflection force caused by bubbles at the wetting front [19,20,21]. In this study, the deflected fluid can be observed in Fig. 2, but the deflection has no effect on the rewetting phenomenon due to the splashing out of the test area. As momentum decreases and fluid temperature increases, it can be seen that Tmin decreases with increasing the radial distance (for r = -15, + 15, and + 30 mm). At r = -30 mm, Tmin is slightly higher than at r = -15 mm, since the accumulated water curtain increases the fluid supply before rewetting. In addition, the fluid layer in the confluence jet region prolongs the film boiling duration, and a lower Tmin is thus observed compared with that in the wall jet region.

tmax and Tmax are shown in Fig. 4b. P0-3 is impacted by the jet and completely wetted first. tmax is the lowest at P0-3 and gradually increases along both sides. It should be noted that a local minimum for Tmax is observed at P0-3. This might be due to the high-pressure fluid at the jet region reducing the bubble circulation frequency and enhancing heat convection in the transition boiling, resulting in a lower surface temperature. At r = ± 15 mm, Tmax was found to be higher than at r = 0, due to the increased bubble circulation frequency, with an average increase of 7.86 °C. Afterwards, the decreased flow velocity and increased fluid temperature lead to a low heat transfer, with the minimum Tmax occurring at r = ± 30 mm.

Fig. 4
figure 4

a Rewetting time and temperature; b complete-wetting time and temperature

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3.2 Influence of the jet parameters on the rewetting and complete‐wetting phenomena

The rewetting time and rewetting temperature for different values of u and z are shown Fig. 5. As u increases from 2.95 to 11.80 m/s, the impinging stream accumulates into a higher turbulence zone, which results in an accelerations of the rewetting speed. tmin was found to be reduced by 0.67 and 0.88 s at r = -30 and + 30 mm, respectively, while Tmin was observed to increase by 9.85 °C on average. According to Fig. 5a, it can be inferred that u has a lower influence on the confluence jet region, and a possible reason for this phenomenon might be that the accumulated fluid layer weakens the u influence. When changing z, the interference and interaction between the two jets are also changed. However, the same variation of the rewetting parameter as a function of z can be observed for the wall jet region and confluence jet region, as shown in Fig. 5b. With the increase of z, tmin firstly decreased and then increased, Tmin followed the opposite trend. The lower fluid splash volume and faster fluid velocity lead to a smaller tmin and higher Tmin at z = 150 mm.

Fig. 5
figure 5

Characteristic values for the rewetting phenomenon at z = 50–450 mm and u = 2.95–11.80 m/s: a rewetting time and b rewetting temperature

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The complete-wetting time and complete-wetting temperature for different values of u and z are shown Fig. 6. As u increases, the high-momentum fluid accelerates the bubble detachment rate from the surface and prevents bubbles from accumulating in the wetting front [17], which promotes the structure changing from transition boiling to nucleate boiling. A decreased tmax and increased Tmax are observed. As the converging fluid provides a stronger turbulence while reducing the fluid momentum, no significant difference for tmax and Tmax can be observed between the confluence jet region and wall jet region. For z = 150 mm, the high kinetic energy of the fluid, which is caused by the lower fluid splash volume and faster fluid velocity, enhances the ability of the fluid to sweep the bubbles, leading to the lowest tmax and highest Tmax. According to Fig. 6a, the impact of z on the confluence jet region is particularly evident. The changing rate of tmax increases radially from r = 0 mm to r = -30 mm, with values of 10.15 and 25.24 % at P0-3 and P0-1, respectively. This indicates that, in the confluence jet region, as the distance from the jet point increases, the offset of the fluid momentum caused by the opposite motion of the jets reduces the ability to sweep the bubbles at the wetting front, and the transition boiling duration is thus extended. By calculating the difference between Tmin and Tmax, the temperature drop at P0-1 was found to be lower than that at P0-5 from tmin to tmax. Therefore, it can be inferred that an excessive amount of the accumulated fluid reduces the heat transfer efficiency in the transition boiling.

Fig. 6
figure 6

Characteristic values for the complete-wetting phenomenon at z = 50–450 mm and u = 2.95–11.80 m/s: a time and b temperature

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3.3 Maximum heat flux and correlation

Figure 7 shows qmax in different regions at u = 2.95–11.80 m/s and z = 50–450 mm. Due to the friction resistance and enthalpy increase, the heat transfer ability between the fluid and wall is slowed down, and qmax in the confluence jet region and wall jet region was found to be lower than in the jet region. In the wall jet region, a larger qmax value was obtained at a high Tmin, which proves that the rewetting temperature affects the maximum heat flux. The radial range from P0-3 to P0-2 also follows this rule. Tmin at P0-1was found to be higher than at P0-2, but a lower qmax was obtained, which might be related to the lower Tmax caused by the confluence fluid. A lower Tmax reduces the bubble boiling circulation frequency and heat convection intensity, thus leading to a low qmax. Therefore, based on the above discussion, it can be concluded that qmax is affected by both Tmin and Tmax, the former being the dominant factor.

Fig. 7
figure 7

Maximum heat flux at z = 50–450 mm and u = 2.95–11.80 m/s

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The actual cooling process was more similar to that in the jet region and confluence jet region. According to the experimental data, a correlation of the maximum heat flux at the jet point (qmax)0 was established, and the equation was extended to other locations in the confluence jet region (qmax)r:

$$ {\left({q}_{\mathrm{max}}\right)}_0={u}^{0.21}\times 2.03\times \left(9.00\times {10}^{-1}+1.17\times {10}^{-3}z-4.71\times {10}^{-6}{z}^2+4.68\times {10}^{-9}{z}^3\right) $$
(4)
$$ {\left({q}_{\mathrm{max}}\right)}_r/{\left({q}_{\mathrm{max}}\right)}_0=3.84\times {10}^2-2.55z-4.75\times 10u+3.2\times {10}^{-1} zu-5.42\times {10}^{-3}r $$
(5)

where r is the distance from the jet point (0 ≤ r ≤ 30) in mm, u is the jet exit velocity (2.95 ≤ u ≤ 11.80) in m/s, and z is the jet height (50 ≤ z ≤ 450) in mm.

qmax was found to be directly proportional to u0.21, rather than to u1/3, as was reported by most previous works [22,23,24]. The reason might be that the temperature change caused by the jet impingement is weakened in thick plates due to the plate heat conduction. Agrawal et al. [15] and Paul et al. [16] respectively proposed that qmax is directly proportional to (z/d)−0.01 and (z/d)0.086, with z/d = 4–16 and z/d = 4–10. However, considering the flatness and deformation of the plate, the above z/d relationship was not applicable for the thick plate. In this study, the jet height range was 16 < z/d ≤ 150, and qmax was found to increase first and then decrease with the increase of z, which is not in agreement with a trend of the form (z/d) n. Therefore, a one-dimensional cubic equation was used. The proposed correlation could well predict the qmax value from P0-3 to P0-1. The maximum error band in Fig. 8 is ± 5 %, and the predicted value falls within the error band within 85 % probability.

Fig. 8
figure 8

Comparison between the predicted and experimental values of qmax

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3.4 Maximum cooling speed and correlation

The maximum cooling speed vmax at different measurement points is shown in Fig. 9. The internal cooling speed is mainly affected by the heat conduction along the thickness and exhibits the same variation trend as the surface heat flux: it increases with the increase of u, and it firstly increases and then decreases with the increase of z. For z = 150 mm and u = 8.06 m/s, the vmax difference along the horizontal direction at h = 3 mm was found to be 24.55 %, while it reduces to 4.26 % at h = 47 mm. The data prove that the horizontal heat conduction plays an important role in the vmax movement process along the thickness. One of the aims of this study was to explore the cooling-speed field along the thickness. Therefore, according to the above results, the average maximum cooling speed along the horizontal vave was used to establish correlations with u, z, and h.

Fig. 9
figure 9

Maximum cooling speed at h = 3, 25, 47 mm: a z = 150 mm and u = 2.95–11.80 m/s and b z = 50–450 mm and u = 8.06 m/s

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The varying trend of vave along h can be predicted using the function model \( {V}_{ave}={A}_0+{A}_1\times {e}^{\left(-h/{A}_2\right)} \), in which A0, A1, and A2 are related to u and z. The power-law relationship of A0 and A1 with u and z is: \( A=a^{\prime}\times {u}^{b\prime}\times \left({c}^{\prime }+{d}^{\prime}\times z+e^{\prime}\times {z}^2\right) \), where the a’, b’, c’, d’, and e’ values are shown in Table 2, and A2 is here assumed to be 6.1. The proposed correlation could be used to predict 93 % of the experimental results within ± 5 % error as shown in Table 3.

Table 2 Parameters used in the power-law relationship
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Table 3 Prediction errors
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4 Conclusions

To understand the heat-transfer characteristics and cooling-speed field for a thick plate with multiple jets, double water-jet cooling experiments were performed on a 50 mm thick plate at jet exit velocities of 2.95–11.80 m/s and jet heights of 50–450 mm. The main results are as follows:

  1. (1)

    The wetting front width is affected by the radial temperature gradient and fluid motion and appears to have a complex trend, with sequential increasing, decreasing, and increasing behaviors upon increasing the wetting region diameter.

  2. (2)

    The accumulated water curtain can increase the fluid supply before rewetting, thus raising the rewetting temperature. However, its effect on the complete-wetting phenomenon was not observed. It is speculated that an excessive amount of the accumulated fluid would reduce the heat transfer efficiency in the transition boiling.

  3. (3)

    The confluence fluid layer between the two jets reduces the influence of different jet velocities, but enhances the complete-wetting parameters difference in the confluence jet region for different jet heights due to the prolonged transition boiling duration.

  4. (4)

    The heat flux can be adjusted via the rewetting temperature and complete-wetting temperature, and the cooling speed field variation along the thickness was provided.