Heat Transfer Efficiency While Cooling with a Water Spray, Air-Assisted Water Spray and Water Jet Under Boiling and Single-Phase Forced Convection Conditions ^†

Abstract

The main purpose of this paper was to determine and compare the boundary conditions of heat transfer on the cooled surface of a cylindrical sensor made of Inconel 600 alloy while cooling with a water jet, water spray and air-assisted water spray under high-temperature conditions. The inverse method for the heat conduction equation was used to determine the boundary conditions. Experimental tests were carried out, including temperature measurements at several points inside the cylinder while cooling with all the tested systems from a temperature of 900 °C for three values of water pressure: 0.05 MPa, 0.1 MPa and 0.2 MPa. Temperature measurements were used as the input data to identify the heat transfer boundary conditions. The temperature field of the axially symmetric sensor was determined using the finite element method. The boundary conditions were determined as average values of the heat transfer coefficient and heat flux and local values of the heat transfer coefficient. A comparison of the amount of thermal energy dissipating from the sensor surface as a result of boiling and a forced single-phase convection is also presented in the paper. The highest uniformity of cooling was obtained during air-assisted water spray-cooling.

1. Introduction

Water is the most common and widely used cooling medium due to its availability, low cost and lack of negative impact on the environment and humans. While water cooling a surface at a high temperature, a boiling phenomenon occurs which allows for the dissipation of significant amounts of heat from hot surfaces. The heat flux (HF), describing the amount of heat removed from the cooled surface, is conditioned by the type of boiling regimes that occur on the cooled surface. During the cooling of hot materials, the individual stages of the boiling process are opposite to the boiling regimes observed during pool-boiling, described by Nukiyama [1]. The first stage of boiling, while cooling a hot surface at a high temperature, is film-boiling, which is characterized by a low HF. Small values of HF during this boiling regime are caused by the vapor film forming on the cooled surface, which isolates the surface and limits the inflow of liquid. That in turns reduces the heat dissipation from the cooled surface. When the surface temperature decreases, transition boiling occurs in which the liquid intensively wets the cooled surface and the HF increase up to a maximum value called the critical heat flux (CHF). After reaching the maximum, the HF decrease with the decrease in the surface temperature due to the formation of increasingly smaller amounts of vapor bubbles on the cooled surface. It is a nucleate boiling regime. The last stage of the boiling process is a single-phase convection. The most popular methods of cooling with water include the following: cooling with a water jet (JET) [2,3,4], water spray (WS) [5,6,7,8,9,10] and air-assisted water spray (AAS) [11,12,13]. The cooling process with these methods allows for achieving large amounts of heat dissipating from the cooling surface, which, in practice, is unattainable in the field of single-phase convection.

During JET-cooling, the liquid flowing freely from the nozzle hits the cooled surface. Cooling with a JET is used, among others, in the metallurgical industry during quenching [3] and rolling plates on a hot steel mill [14,15]. This type of cooling is also commonly used during the cooling of turbine components [16].

An equally common method of cooling, which uses water as a coolant, is WS. WS is a mixture of water molecules dispersed in a gaseous medium [17]. There are two most common spraying methods: the hydraulic method (WS) and the pneumatic method (AAS) [17]. In the hydraulic method, water is sprayed through a pressure nozzle. Pneumatic atomization (AAS) is caused by the action of a compressed gas, usually air, on a liquid. Spray-cooling, both hydraulic (WS) and pneumatic (AAS), is used in practice, among others, in microelectronics [18,19,20], ventilation systems [21], nuclear industry [22,23], energy industry [24,25,26,27], the metallurgical industry [5,7,12,28] and many others.

JET-cooling and spray-cooling, both pneumatic (AAS) and hydraulic (WS), are competitive methods in applications where there is a need to dissipate large amounts of heat from the cooled surface. The selection of an appropriate cooling method, ensuring an appropriate rate of heat dissipation from the cooled surface, is particularly important in high-temperature metallurgical processes which include quenching, rolling or the continuous casting of steel or other alloys. Providing an appropriate rate of heat dissipation guarantees the formation of the desired microstructure of the cooled product, which affects the properties of the manufactured product. In publicly available literature, there are several studies in which researchers mainly focused on comparisons: two cooling methods [6,12,29,30] and cooling methods for a specific boiling phase [29]. Alam [29] compared heat transfer during pneumatic (AAS) and hydraulic spray-cooling (WS) in the field of film-boiling. These research studies showed that the heat transfer coefficient (HTC) in the film boiling regime was 10 times higher for pneumatic spray-cooling compared to the hydraulic one. Labergue et al. [30] compared JET-cooling with hydraulic spray-cooling (WS). Researchers showed that WS-cooling provided ideal cooling uniformity, higher heat transfer efficiency and a lower cooling medium consumption than JET-cooling. A comparison of heat transfer during cooling by a hydraulic spray nozzle and JET was presented by Ali et al. [6]. JET-cooling allowed for the dissipation of more heat from the cooled surface compared to WS-cooling. However, the authors demonstrated that the spray nozzle provided a more uniform distribution of water over the cooled surface. Puschmann et al. [12] made a comparison between hydraulic and pneumatic spray-cooling. During cooling with a pneumatic spray, higher values of HTC than during cooling with hydraulic spray were obtained. The authors demonstrated that the biggest impact on HTC is the spray impingement density. In the available literature, there is a lack of papers comparing all three methods of water cooling (JET, WS, AAS) under high-temperature conditions, considering both single-phase and two-phase heat transfer.

The main purpose of this paper was to identify and compare the boundary conditions of heat transfer on a cooled surface of a cylindrical sensor made of Inconel 600 during JET-, WS- and AAS-cooling. Experimental tests were carried out using these three cooling methods. All tests were carried out on the same test stand, with the use of the same sensor and under the same process conditions. To ensure a suitable cooling method, only the nozzle type was changed. These activities aimed to obtain as similar cooling conditions as possible for all the tested methods. The temperature measurements obtained from the experimental test were used as input data for an inverse method build in a computer program. The results of the calculation allowed for determining the boundary conditions of heat transfer on the cooled surface of the sensor. In this paper, the results of the calculation are presented in the form of average values of HTC and HF; also, the local values of HTC are shown. Additionally, the amount of energy dissipating from the sensor surface during boiling was compared to the energy dissipating during forced single-phase convection. The conducted analysis can be useful for researchers dealing with the intensification of heat transfer during water-cooling under two-phase heat transfer conditions, and for entrepreneurs which are looking for solutions that can meet their requirements.

2. Materials and Methods

2.1. Experimental Tests

To determine the boundary conditions of heat transfer using the inverse method, the determination of the temperature changes during cooling inside the cooling element was required. Experimental tests were carried out using a cylindrical sensor consisting of a cylindrical probe and a casing [31]. The cylindrical probe was placed in a 1 mm thick casing which had a 30 mm diameter. Both the cylindrical probe and the casing were made of the Inconel 600 alloy. The diameter of the cylindrical probe and its height were 20 mm (Figure 1 and Figure 2). The space between the cylindrical probe and the casing was filled with air.

The measurements of the sensor temperature changes while cooling with a JET, WS and AAS were carried out on a laboratory stand (Figure 3).

During the experimental tests, the sensor was placed in a holder, which was filled with an insulating material made of aluminosilicates [32] to ensure the stability of the sensor during the impact of the coolant. The sensor was heated in a resistance furnace to a temperature of 900 °C. After reaching a pre-set temperature, the heating process was continued for 30 min to equalize temperature within the entire volume of the sensor. Then, the sensor was transferred to the cooling chamber using an automatic feeder. During transport from the furnace to the cooling chamber, the sensor was cooled with air. The cooling time in the air was about 15–20 s. During the experimental tests, for all analyzed cooling systems, the coolant was fed perpendicular to the front surface of the sensor (Figure 4 and Figure 5). The water temperature, which was measured in the tank with a Pt100 resistance sensor, was 16 °C.

Temperature changes inside the sensor were measured at three points using the K-type thermocouples with a diameter of 1 mm. The thermocouples were placed 1.5 mm below the cooled surface (Figure 6) at distances of 2 mm (P1), 5.2 mm (P2) and 9 mm (P3) from the cylinder axis. An additional thermocouple (P4) was placed 9.7 mm below the cooled surface and 2 mm from the side surface of the sensor to verify the correctness of the numerical calculations of the sensor temperature based on the assumed boundary conditions at the sensor surfaces. All thermocouples were placed in holes with a diameter of 1 mm. After completing the experimental tests, the sensor was ground to verify the thermocouple locations (Figure 6).

The resulting accuracy of the thermocouple was ±0.4% of the measured temperature [33]. The maximum temperature recorded during the experimental measurement was 903 °C (

Table 1. Cooling parameters adopted for numerical calculations.

Type of Cooling	Pressure, MPa	Total Cooling Time, s	Initial Temperature of the Sensor, °C
JET	0.05	70	895
	0.1	80	895
	0.2	80	895
WS	0.05	80	893
	0.1	80	896
	0.2	65	891
AAS	0.05	80	903
	0.1	80	898
	0.2	80	900

). Therefore, the maximum error due to the accuracy class of the thermocouple was ±3.6 °C. Temperature measurements were recorded using the MGCplus data acquisition system with an accuracy of 0.2%. The maximum temperature measurement error due to the accuracy class of the data acquisition system was ±1.8 °C [34]. Therefore, the maximum temperature measurement error, which is the sum of the maximum temperature measurement error due to the accuracy class of the data acquisition system and the thermocouple, was ±5.4 °C.

The experimental tests also included measurements of an amount of water falling on the cooled surface. Measurements of the amount of water were carried out at room temperature. To determine the amount of water falling on the cooled surface, an additional system was installed on the measuring stand (Figure 7). A hollow sleeve was placed in the place where the sensor was located during the cooling tests. The diameter of this sleeve was the same as the diameter of the cylindrical probe (20 mm), so the size of the area from which the water was drained corresponded to the area of the cylindrical probe that was used in the cooling tests. Water entering the sleeve during cooling was collected by a rubber hose which was connected to a measuring cylinder (Figure 8). The volume of water hitting the cooled surface during the measurement (V) was read from the graduated scale of the measuring cylinder. The time at which water was drained into the measuring cylinder (τ_w) was also measured. Water volume measurements were repeated six times for each type of cooling system and each set of cooling parameters. The example results of the measurements conducted for the AAS system at a pressure of 0.2 MPa are presented in Figure 9 and Figure 10. Based on the collected data, the average water flux density falling on the cooled surface was determined using Equation (1):

$$g_w={\displaystyle \frac{\sum _{w=1}^{w=6}\frac{1}{\pi R^2}\left(\frac{V}{{\tau }_w}\right)}{6}}$$

(1)

where

• R—cylinder radius, m;
• V—volume of the water collected in the measuring cylinder, m³;
• τ_w—time of collection of the water in the measuring cylinder, s.

Figure 7. Scheme of the system for measuring the amount of water: 1—nozzle, 2—hollow sleeve, 3—rubber hose, 4—feeder, 5—cooling chamber, 6—measuring cylinder.

Figure 7. Scheme of the system for measuring the amount of water: 1—nozzle, 2—hollow sleeve, 3—rubber hose, 4—feeder, 5—cooling chamber, 6—measuring cylinder.

Figure 8. System for measuring the amount of water.

Figure 8. System for measuring the amount of water.

Figure 9. The volume of water collected in a measuring cylinder at times (τ_w) during the six measurements, AAS, p = 0.2 MPa.

Figure 9. The volume of water collected in a measuring cylinder at times (τ_w) during the six measurements, AAS, p = 0.2 MPa.

Figure 10. Time of collection of the volume of water (V) in the measuring cylinder during the six measurements, AAS, p = 0.2 MPa.

Figure 10. Time of collection of the volume of water (V) in the measuring cylinder during the six measurements, AAS, p = 0.2 MPa.

The average values of the water flux density for all three cooling systems are summarized in

Table 2. The accuracy of water flux density measurements.

Type of Cooling	Pressure, MPa	Average Water Flux Density, m3/(m2∙s)	Accuracy of the Water Flux Density Measurement${\Delta \mathit{g}}_{\mathit{w}}$, m3/(m2∙s)	Relative Error of the Water Flux Density Measurement δ, %
JET	0.05	62.27∙10−3	5.43∙10−3	7.95
	0.1	95.86∙10−3	8.67∙10−3	9.04
	0.2	143.31∙10−3	17.05∙10−3	11.90
WS	0.05	35.26∙10−3	1.89∙10−3	5.36
	0.1	43.72∙10−3	2.43∙10−3	5.56
	0.2	44.38∙10−3	2.83∙10−3	6.38
AAS	0.05	21.38∙10−3	1.30∙10−3	6.08
	0.1	28.24∙10−3	1.63∙10−3	5.77
	0.2	22.47∙10−3	1.36∙10−3	6.05

.

The accuracy of the water flux density measurement may have been affected by the occurrence of the measurement errors results from the experimental procedure. The error in measuring the water flux density may have been related to the accuracy of the measurement of the volume of water collected in the measuring cylinder, the accuracy of the surface area of the sleeve used to collect the water, and the accuracy of the time measurements. This error was determined from Formula (2) [35]:

$${\Delta g}_w=\left|{\displaystyle \frac{\partial g_w}{\partial A}}·\Delta A\right|+\left|{\displaystyle \frac{\partial g_w}{\partial V}}·\Delta V\right|+\left|{\displaystyle \frac{\partial g_w}{\partial {\tau }_w}}·\Delta {\tau }_w\right|$$

(2)

• ${\Delta g}_w$—the accuracy of water flux density measurement, m³/(m²·s);
• g_w—average water flux density, m³/(m²·s);
• A—area of the inlet to the sleeve, m²;
• ΔA—the accuracy of the measurement of the sleeve inlet surface area, m²;
• ΔV—the accuracy of the measuring cylinder, m³;
• Δτ_w—time measurement error, s.
• where

$${\displaystyle \frac{\partial g_w}{\partial A}}={\displaystyle \frac{V}{{\tau }_w}}·\left({\displaystyle \frac{1}{A^2}}\right)=-{\displaystyle \frac{V}{A^2·{\tau }_w}}$$

(3)

$${\displaystyle \frac{\partial g_w}{\partial V}}={\displaystyle \frac{1}{A·{\tau }_w}}$$

(4)

$${\displaystyle \frac{\partial g_w}{\partial {\tau }_w}}={\displaystyle \frac{V}{A}}·\left(-{\displaystyle \frac{1}{{{\tau }_w}^2}}\right)=-{\displaystyle \frac{V}{A·{{\tau }_w}^2}}$$

(5)

The error related to the accuracy of the measurement of the sleeve inlet surface area was determined using Equation (6):

$$\Delta A=\left|{\displaystyle \frac{\partial A}{\partial d}}·\Delta d\right|$$

(6)

• d—diameter of the sleeve inlet, m;
• Δd—accuracy of the calliper, m.
• where

$${\displaystyle \frac{\partial A}{\partial d}}={\displaystyle \frac{\pi }{4}}·2d={\displaystyle \frac{\pi d}{2}}$$

(7)

The diameter of the sleeve was measured with a caliper. The accuracy of the caliper was Δd = 0.00002 m.

The error related to the measurement of the volume of the water collected in the measuring cylinder (ΔV) was caused by the accuracy of the measuring cylinder. The accuracy of the measuring cylinder was ΔV = ±0.0000025 m³.

The time measurement error (Δτ_w) constituted the sum of the reaction time of the person performing the measurement and the accuracy of the stopwatch. This error was determined using Equation (8):

$$\Delta {\tau }_w=2{\tau }_r+{\tau }_{st}$$

(8)

where

• τ_r—human reaction time, s;
• ${\tau }_{st}$—accuracy of the stopwatch, s.

Human reaction time is 0.4 s [36]. The accuracy of the stopwatch was 0.01 s. Therefore, the time measurement error (Δτ_w) was ±0.81 s.

The accuracy of the water flux density measurements, which was determined for each considered cooling system, as well as the relative error of the water flux density, is presented in Table 2. The relative error of the water flux density was determined using Equation (9):

$$\delta ={\displaystyle \frac{\Delta g_w}{g_w}}·100\%$$

(9)

2.2. Heat Conduction Model Used to Identify the Boundary Conditions of the Heat Transfer

The identification of the boundary conditions during cooling with the use of three systems requires the calculation of the temperature field of the cooled cylinder using the heat conduction Equation (10) [4]:

$$\rho c_p\left(t\right){\displaystyle \frac{\partial t\left(r,y,\tau \right)}{\partial \tau }}={\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left[\lambda \left(t\right)r{\displaystyle \frac{\partial t\left(r,y,\tau \right)}{\partial r}}\right]+{\displaystyle \frac{\partial }{\partial y}}\left[\lambda \left(t\right){\displaystyle \frac{\partial t\left(r,y,\tau \right)}{\partial y}}\right]$$

(10)

assuming that

$$\left(\tau >0;0<r<R;0<y<h\right)$$

where

• $\rho$—density of the sensor material, kg/m³;
• $c_p$—specific heat, J/(kg·K);
• t—sensor temperature, °C;
• r, y—cylindrical coordinates;
• τ—time, s;
• λ—thermal conductivity, W/(m·K);
• h—sensor height, m.

The solution of Equation (10) was obtained using the finite element method, which transforms Equation (10) into a system of algebraic linear Equation (11) [37]:

$$K_{ij}{T}_j+C_{ij}{\displaystyle \frac{\partial T_j}{\partial \tau }}=B_i$$

(11)

where

• K_ij—heat conduction matrix;
• C_ij—heat capacity matrix;
• B_i—boundary condition vector.

For one element, matrices K_ij and C_ij, and vector B_i take forms ((12)–(14)):

$$K_{ij}=\sum {\int }_{{\mathsf{\Omega }}_e}\lambda \left({\displaystyle \frac{{\partial H}_i}{\partial r}}{\displaystyle \frac{{\partial H}_j}{\partial r}}+{\displaystyle \frac{{\partial H}_i}{\partial z}}{\displaystyle \frac{{\partial H}_j}{\partial z}}\right)rd{\mathsf{\Omega }}_e+\sum {\int }_{S_e}r{H}_i{H}_j\alpha d{S}_e$$

(12)

$$C_{ij}=\sum {\int }_{{\mathsf{\Omega }}_e}\rho c{H}_i{H}_{j}rd{\mathsf{\Omega }}_e$$

(13)

$$B_i=\sum {\int }_{{\mathsf{\Omega }}_e}{q}_{v}r{H}_{i}d{\mathsf{\Omega }}_e+\sum {\int }_{S_e}{H}_i\left(\alpha T_a\right)rd{S}_e$$

(14)

where

• α—heat transfer coefficient, W/(m²·K);
• T_a—water temperature, °C.

The finite element method was used to determine the temperature field from the heat conduction equation. The description of the finite element method used to solve Equation (10) is presented in the paper [38].

The accuracy of the FEM solution was used for modeling the temperature field of the axially symmetrical sensor [5]. The boundary condition at the water-cooled surface was searched in the form of a function depending on time and location (15). Thus, it was possible to determine the local values of the boundary conditions on the cooled surface:

$$\dot{q}=\alpha (r,\tau )(t_s-t_{\mathit{n}})$$

(15)

where

• $t_s$—temperature of the cooled surface, °C;
• $t_{\mathit{n}}$—coolant temperature, °C.

To solve the inverse problem, the objective function was adopted (16):

$$E\left(p_i\right)={\displaystyle \frac{1}{n m}}\sum _{i=1}^n\sum _{j=1}^m\left\{{\left[{\displaystyle \frac{{te}^{ji}-{t\left(p_i\right)}^{ji}}{\sqrt{1+{\left(\frac{{∆te}^{ji}}{∆\tau }\right)}^2}}}\right]}^2+{\left({\displaystyle \frac{\frac{{∆te}^{ji}}{∆\tau }-\frac{{∆t\left(p_i\right)}^{ji}}{∆\tau } }{{Fs}^{j i}}}\right)}^2\right\}$$

(16)

where

• p_i—vector of unknown parameters;
• m—number of temperature measurements performed by one sensor;
• n—number of temperature sensors;
• ${te}^{ji}$—sample temperature measured by sensor j at time τ_i;
• $t^{ji}$—computed sample temperature at the location of sensor j at time τ_i.

The term Fs ^{j i} was specified as follow (17):

$$\begin{array}{c}{Fs}^{j i}={\displaystyle \frac{{∆te}^{ji}}{∆\tau }} \mathrm{for} \left|{\displaystyle \frac{{∆te}^{ji}}{∆\tau }}\right|>1\\ {Fs}^{ji}=1 \mathrm{f}\mathrm{o}\mathrm{r} \left|{\displaystyle \frac{{∆te}^{ji}}{∆\tau }}\right|\le 1\end{array}$$

(17)

The objective function (16) defines the temperature difference between the measured and computed temperatures. The temperature difference was extended with the norm of the temperature gradients. In this paper [39], it iss shown that the objective function (16) ensures acceptable accuracy of the inverse solution for high temperatures as well as for variable temperature gradients in space and time.

The minimization of the objective function made it possible to determine the values of the unknown vector parameters, p_i. This vector was composed of the HTC value in the nodes of the approximation of HTC in time. The set of p_i parameters determined the change in HTC on the cooled surface as a function of time and the sensor radius.

The variable metric method [40], based on the BGFS algorithm [41,42,43,44], was used to minimize the objective function. The derivative of the objective function with respect to the unknown p_i parameters was estimated numerically. The distribution of the HTC along the radius of the sensor was determined using Equation (18):

$$\alpha \left(r,\tau \right)=\left(1-{7\vartheta }^2+{6\vartheta }^3\right)P_1\left(\tau \right)+\left({8\vartheta }^2-{8\vartheta }^3\right)P_2\left(\tau \right)+\left({2\vartheta }^3-{\vartheta }^2\right)P_3\left(\tau \right)$$

(18)

$$\vartheta ={\displaystyle \frac{r}{R_{max}}}$$

(19)

where

• ϑ—dimensionless sensor radius;
• R_max—maximum sensor radius, m.

The P₁(τ) parameter determined the HTC values ON the sensor axis (r = 0), the P₂(τ) parameter determined the values of this coefficient for r = 1/2Rmax and for the P₃(τ) parameter, r = R_max. The variability of parameters P₁(τ), P₂(τ) and P₃(τ) (20)–(22) over time was approximated by third-degree polynomials [39].

$$P_1\left(\tau \right)=\sum _{i=1}^4{H}_i{P}_{1i}\left({\tau }_i\right)$$

(20)

$$P_2\left(\tau \right)=\sum _{i=1}^4{H}_i{P}_{2i}\left({\tau }_i\right)$$

(21)

$$P_3\left(\tau \right)=\sum _{i=1}^4{H}_i{P}_{3i}\left({\tau }_i\right)$$

(22)

The cooling time was divided into periods in which τ ∈ (τ₁, τ₂). The third-degree Hermite polynomial was used to approximate the HTC distribution (23)–(26) [37]:

$$H_1={\displaystyle \frac{\left(1-\eta \right)\left(9{\eta }^2-1\right)}{16}}$$

(23)

$$H_2={\displaystyle \frac{9\left(1-3\eta \right)\left(1-{\eta }^2\right)}{16}}$$

(24)

$$H_3={\displaystyle \frac{9\left(1+3\eta \right)\left(1-{\eta }^2\right)}{16}}$$

(25)

$$H_4={\displaystyle \frac{\left(1+\eta \right)\left(9{\eta }^2-1\right)}{16}}$$

(26)

where

• η—dimensionless time.

Dimensionless time was determined using Formula (27):

$$\eta ={\displaystyle \frac{2\tau -{\tau }_1-{\tau }_2}{{\tau }_2-{\tau }_1}}$$

(27)

On the rest of the surfaces, a second boundary condition was assumed in the form of HF. The amount of heat transferred in the air gap, between the sensor surface and the inside surface of the casing, was determined using Equation (28) [5]:

$${\dot{q}}_{sz}=\sigma · {\displaystyle \frac{{T_1}^4-{T_2}^4}{\frac{1}{{\epsilon }_1}+\frac{1}{{\epsilon }_2}-1}}+{\displaystyle \frac{{\lambda }_p}{L_{CO}}}·0.22·{\left({\displaystyle \frac{Pr}{0.2+Pr}} ·Ra\right)}^{0.28}·{\left({\displaystyle \frac{h}{L_{CO}}}\right)}^{-{\displaystyle \frac{1}{4}}}·\left(T_1-T_2\right)$$

(28)

where

• σ—Stefan–Boltzmann radiation constant, W/(m²·K⁴);
• T₁—temperature of the inner surface of the casing, K;
• T₂—temperature of the sensor side, K;
• ε₁—emissivity of the sensor surface;
• ε₂—emissivity of the casing surface;
• λ_p—thermal conductivity of air, W/(m·K);
• L_co—distance between sensor and casing, m;
• Pr—Prandtl number;
• Ra—Rayleigh number.

The emissivity of both the cylinder surface and its casing was ε₁ = ε₂ = 0.2. The thermophysical properties of air, necessary to determine the value of Rayleigh and Prandtl numbers, were determined for the average water temperature.

The boundary condition on the lower outer surface of the casing considered heat losses due to natural convection and heat radiation to the environment (29) [5]:

$${\dot{q}}_p=\sigma · {\epsilon }_2·\left({T_2}^4-{T_{ot}}^4\right)+{\displaystyle \frac{{\lambda }_p}{L_o}}·0.474·{Ra}^{0.25}·{Pr}^{0.047}·\left(T_2-T_{ot}\right)$$

(29)

where

• T_ot—ambient temperature, K;
• L_o—diameter of the outer casing, m.

On the outer side surface of the casing, due to the presence of gaps in the insulating material surrounding the sensor (Figure 5), which allowed water to flow off the sensor surface, the boundary condition was defined by Equation (30) [45]:

$$\begin{array}{ll}{{\dot{q}}_b=\alpha }_b\left(t_b-t_{ot}\right)& \\ & =(0.117487t_b-0.466701·{10}^{-3}{t_b}^2+6.932216 ·{10}^{-7}{t_b}^3\\ & -3.661325 ·{10}^{-10}{t_b}^4)\left(t_b-t_{ot}\right)\end{array}$$

(30)

where

• α_b—heat transfer coefficient on the side surface of the casing, W/(m²·K);
• t_b—temperature of the outer surface of the side casing, °C.

Equation (30) describes the boundary condition that is suitable for surface-boiling in channels. Based on this equation, the changes in the temperature of the sensor side surface were possible.

2.3. Numerical Calculations

Numerical calculations that allowed for identifying the HTC were performed for the axially symmetric sensor. Inverse calculations were performed based on measurements of the temperature changes inside the sensor obtained from the experimental cooling tests. Calculations were performed assuming that the thermal conductivity and the specific heat of the sensor could change with temperature. The data given in the data sheet for Inconel 600 [46] were approximated by polynomials (31) and (32):

$$\lambda =2·{10}^{-8}{t}^3-3·{10}^{-5}{t}^2+0.0285t+15.124$$

(31)

$$c_p=3·{10}^{-5}{t}^2+0.1818t+442.85$$

(32)

For the calculations, it was assumed that the density of the Inconel did not change with temperature and was ρ = 8470 kg/m³ [46]. A constant sensor emissivity value of 0.2 was assumed.

The solution to the heat conduction equation was accomplished with the finite element method. Thirty-three elements were used in the direction of the sensor height y and 22 elements in the direction of its radius r (Figure 11). The number of intervals of the boundary condition approximation as a function of cooling time length varied from 7 to 10 intervals.

Uncertainty of the Inverse Solution

The inverse solutions to the boundary condition in space and time at the boundary surface depend on the differences between measured and computed temperatures due to a natural consequence of the objective function formulation, and it may show a significant deviation from the real boundary condition. These differences may result, among others, from the temperature measurement errors. For those reasons, the uncertainty test was performed and the boundary condition resulting from Equation (33) was searched:

$$\alpha (t, r)=\alpha \left(t\right)\left[\left(\left(1-{3\vartheta }^2+{2\vartheta }^3\right)F_1\right)+\left(\left({3\vartheta }^2-{2\vartheta }^3\right)F_2\right)\right]$$

(33)

where

• F₁—scaling factor for the heat transfer coefficient in the sensor axis (r = 0);
• F₂—scaling factor for the heat transfer coefficient in the sensor edge.

$$\alpha \left(t\right)=-1.6544+0.1175t-0.0005t^2+6.9·{10}^{-7}{t}^3-3.33·{10}^{-10}{t}^4$$

(34)

The Inconel cylindrical sensor temperature was simulated using the FEM model with the linear shape functions. Thirty-three elements in the sensor high and 22 elements in the direction of its radius were used. The boundary conditions were defined by heat fluxes: ${\dot{q}}_{sz}$, ${\dot{q}}_p$ and ${\dot{q}}_b$ were specified at the sensor surfaces. The sensor temperatures at four points were recorded as the simulated temperature sensors indications. Then, these temperatures were disturbed by the random errors that simulated the maximum error of the measured temperature. These errors were estimated to have a value of ±5.4 K. The simulated temperature measurement errors assigned to the thermocouple placed at point P2 are shown in Figure 12.

For the inverse solution, the FEM model with 33 elements in the direction of the sensor height and 22 elements in the direction of its radius was employed. The inverse solution was compared to the specific boundary condition obtained from Equation (33). For the inverse solution, the time of cooling was divided into 8 periods. It had given 66 coefficients to be determined from the minimum condition of the objective function (16). The inverse solution converged with the average temperature difference of 2.66 K, with the maximum distance between temperature curves 0.09 K. In Figure 13, a comparison between the measured and calculated temperatures at points P1, P2, P3 and P4 are presented. For all considered points, the results of the calculation stay with good agreement with the measured data. The mean square error of the temperature calculation was 0.61 K. In Figure 14, a comparison of the HTC calculated due to the inverse solution (Inv) with the reference HTC (Ref) obtained from the solution of Equation (33) is shown.

3. Results and Discussion

The assessment of the ability of a liquid to cool a surface is particularly important in engineering practice. This information is needed for the development of cooling technologies in processes like metal quenching or nuclear power. Several methods of assessing the ability of liquid to cool can be found in the literature. One of them is knowledge about the HTC and HF, which are the parameters identified in this paper.

The results of the boundary condition identification are presented as a local HTC, and average HF and HTC variations. The local HTC was determined at four points that were at a distance of r = 3.6 mm, r = 5.6 mm, r = 8.2 mm and r = 9.7 mm from the symmetry axis of the probe.

To compare the average HF and HTC, two areas, A₁ and A₂, were selected (Figure 15). The A₁ area defined the sensor area with a radius r = 5.6 mm, including the area within the range of the direct action of the water jet flowing laminarly from the nozzle (stagnation zone). The A₂ area defined the sensor area with a radius r = 9.7 mm. This area corresponded with the area of spray-nozzle operation during the spray-cooling process. The presentation of the average HF and HTC variation on A₁ and A₂ surfaces shows the differences in the amount of thermal energy removed from the cooled surface with the increase in the determination surface. It also allows for demonstrating the heterogeneity of surface-cooling with the use of all tested systems.

3.1. The Accuracy of the Inverse Calculations

To determine the inverse calculation accuracy, the comparison between measured and calculated temperature was carried out. The analysis of the accuracy of the inverse calculation was performed based on the data obtained from the experimental cooling processes conducted at the water pressure p = 0.1 MPa. The comparison of the temperatures measured at points P1, P2, P3 and P4 (tm) with the temperature calculated at these same points (tc) for AAS, WS and JET cooling systems is presented in Figure 16, Figure 17 and Figure 18.

After leaving the electric furnace, the sensor was cooled in air during transport to the cooling chamber. The effective HTC resulting from heat losses by convection and radiation was identified by the inverse solution. The transport time from the furnace to the cooling chamber was 4 s. The initial temperature field of the sensor used in the numerical solution was generated based on the temperature measurements at time τ = 0. The temperature changes calculated by using FEM were almost identical to those measured at the places where the thermocouples were located (Figure 16, Figure 17 and Figure 18). Good agreement with the temperature changes at point P4 was also obtained. The location of the P4 point was intended to test the validity of the adopted boundary conditions on the rest of the surface. The accuracy of the performed numerical calculations is presented in

Table 3. The accuracy of the numerical calculations.

No.	Type of Cooling	Pressure, MPa	Average Error of Temperature Calculations, K
1	JET	0.05	0.1
2		0.1	0.5
3		0.2	0.9
4	WS	0.05	0.3
5		0.1	0.6
6		0.2	0.4
7		0.05	0.9
8	AAS	0.1	0.5
9		0.2	0.4

. It can be concluded that in all analysed cases, a low value of the average error of temperature calculations less than 1 K was obtained. The error of the inverse solution was derived in the form of the average square error calculated perpendicular to the measurement curve.

The temperature variations indicate that for all cooling systems, the point located in the middle of the sensor radius (P2) was cooled the fastest, but the point P3 located at the edge of the sensor area cooled the slowest. The differences in these values are caused by an uneven supply of water to the cooled surface, which results with the occurrence of different cooling intensity zones.

3.2. Comparison of Average Values of HTC and HF on the Cooled Surface

The assessment of the amount of energy dissipating from the A₁ and A₂ surfaces presented in Figure 19, Figure 20, Figure 21 and Figure 22 was made based on the results of HTC and HF variations. On the presented boiling curves, the first phase of boiling, which is the film-boiling regime, is not observed. The reason for the absence of film-boiling is the high velocity of the liquid falling on the cooled surface and the small distance between the nozzle and the cooled surface. The high velocities of the liquid can prevent the formation of a vapor film which insulates the surface from the liquid. On the boiling curves, only two phases of boiling are observed. In the initial stage of the cooling, a transition boiling occurs. In this phase, a rapid increase in HTC and HF is observed, which continues until the maximum values are reached. After reaching the maximum values, HTC and HF decrease, which is caused by the occurrence of disappearing in time process of nucleate boiling. When the surface temperature drops below 100 °C, forced single-phase convection occurs on the cooled surface.

For each considered cooling system, the increase in the pressure of the supplied medium caused higher values of HTC and HF.

The increase in water pressure caused an increase in the cooling water flux density and the cooling intensity. Such a dependence, for the cooling systems analyzed in this paper, was confirmed by many studies that are presented in the literature [47,48]. Schmidt and Boye [49] indicated that the increase in droplet velocity results in a significant increase in the heat transfer coefficient during spray-cooling.

The analysis of the literature showed that the work on comparing different cooling systems by describing the variability in heat transfer was carried out to a limited extent. In the paper of Puschmann and Specht [12], the results of the study of the variation in the heat transfer coefficient for different cooling systems related to a surface of which the temperature was above the Leidenfrost temperature are presented. The highest HTC values were obtained for AAS cooling. However, in the literature, there is a lack of results comparing the magnitude of the energy dissipating in the entire boiling range. In the presented paper, the results of the investigations performed for the three cooling systems in which the hot plate was cooled from the initial temperature of 900 °C to 60 °C are presented. The assumptions of the experimental conditions were identical for all experiments, which include the water pressure and distance between the nozzle and the cooled surface.

The results of the average HTC and HF variation in A₁ surface versus temperature and time are shown in Figure 19 and Figure 20, respectively. The comparison of the results of the calculations indicates that the highest values of HTC were obtained for the WS cooling system for water pressure p = 0.2 MPa. The lowest HTC values were obtained for the JET system for p = 0.05 MPa (Figure 19). Increasing the determination surface to A₂ area did not change these relations (Figure 21). Similar maximum values of HTC on the A₁ surface were obtained during cooling with the JET system for water pressure p = 0.2 MPa, WS system for p = 0.05 MPa and p = 0.1 MPa, and AAS system for p = 0.1 MPa (Figure 20). On the A₂ surface, a similar maximum HTC, differing from each other by 56 W/(m²·K), was observed for the ASS system for p = 0.2 MPa and WS system for pressures p = 0.05 MPa and p = 0.1 MPa (Figure 21). The mentioned systems allow for achieving similar maximum heat transfer rates with the use of different water flux densities. However, the surface temperature at which the maximum HTC occurs was different for each considered cooling system. The maximum HF value changed in the range from 450 °C to 530 °C depending on the cooling system and cooling parameters (Figure 21 and Figure 22). Changing the size of the determination surface did not influence the change of this range. Similar ranges of the maximum heat flux occurrence for the tested cooling systems are reported in the papers [50,51]. The highest HF on both A₁ and A₂ surfaces was obtained for the AAS system for water pressure p = 0.2 MPa. The lowest HFs were observed for the AAS system for p = 0.05 MPa and the JET system for p = 0.05 MPa (Figure 20 and Figure 22).

The AAS cooling system allows for achieving higher HTC values during transition boiling, which results in a faster reduction in both surface temperature and temperature in the entire sample volume (Figure 16).

The ability of the fluid to receive heat from the cooled surface depends on the type of cooling system and the pressure of the fluid. This dependence is very visible in Figure 23, which shows the thermal characteristics of the coolant for all tested cooling systems. The thermal characteristics of the coolant are presented as F function. Function F defines normalized HF on the A₂ surface for a water flux density of 1 m³ /(m² s). The greatest amount of heat energy removed by water was observed for the AAS cooling system and the lowest for the JET system. The influence of water pressure is ambiguous and depends on the cooling system. For the AAS system, the increase in pressure from 0.1 MPa to 0.2 MPa caused an over 47% increase in the ability of water to dissipate heat. For the WS and JET systems, a decrease in the amount of dissipated energy with increasing the fluid pressure was observed. In the case of the coolant pressure equaling 0.05 MPa, a 29.9% and 67.4% decrease in the ability of the liquid to remove thermal energy from the surface, for the WS and JET systems, respectively, was observed in compared to AAS-cooling. For a coolant pressure of 0.1 MPa, the ability of the coolant to remove heat during WS- and JET-cooling is 39.0% and 69.7% lower than for AAS-cooling. The maximum difference in the possibility of removing the thermal energy, for all considered systems, occurred during cooling, at which the liquid was applied to the cooled surface under a pressure of 0.2 MPa. In this case, the ability of water to dissipate heat during WS- and JET-cooling was reduced by 57.8% and 87.7% in relation to AAS-cooling.

3.3. Cooling Uniformity

A very common problem that occurred during cooling was the lack of uniformity in the removal of heat energy from the surface. A high non-uniformity of cooling under industrial conditions can cause thermal stresses in the material, which can result in the appearance of cracks and material degradation. Choosing a cooling system that allows for the most uniform heat removal from the surface is crucial to prevent the cracking of cooled surfaces.

One of the possible analyses that allows for assessing the uniformity of cooling is the determination of the local changes in the dissipated heat energy from a cooled surface. In this paper, the local changes in HTC versus the surface temperature at four points were determined (Figure 24, Figure 25 and Figure 26). The points were placed on the sensor surface along its radius at r = 3.6 mm, r = 5.6 mm, r = 8.2 mm and r = 9.7 mm. The analysis of the curves described the changes of the local values of HTC along the surface of the sensor cooled with the JET, WS and AAS systems proved the heterogeneity of the cooling process (Figure 24, Figure 25 and Figure 26). Based on the local HTC values at the selected points, the local surface temperatures at these points can be calculated. Further, by calculating the temperature difference between the average temperature and the local temperature over time, a non-uniformity of cooling can be determined (Figure 27). Such an approach allows for observing how much the local temperature at a given point on the surface differs from the average temperature on the surface at a given moment.

The uniformity of the sensor cooling not only depends on the type of convective heat transfer (single-phase/two-phase) but also on the type of boiling phase that is occurred on the surface. At the beginning of cooling, during the transition boiling regime, which lasted maximum 3 s depending on the cooling system, the intensity of heat dissipation increased rapidly with the decrease in surface temperature (Figure 24, Figure 25 and Figure 26). Due to the small cooled area of the sensor, the most uniform cooling conditions during this phase of boiling were obtained for the JET cooling system. The highest differences between the average and local temperatures, amounting to 80 °C, during the transition boiling, occurred in case of cooling the sensor by using the AAS system. However, in next stages of cooling, during nucleate-boiling and forced single-phase convection, AAS-cooling indicated the highest uniformity of cooling. The maximum temperature difference during cooling with the AAS system was 28 °C for nucleate-boiling and only 8 °C when heat was dissipated by single-phase convection. The highest non-uniformity of cooling during nucleate-boiling and single-phase convection was observed during cooling with the WS system. Therefore, the maximum temperature differences were 95 °C during nucleate-boiling and 25 °C during single-phase convection. Therefore, the AAS cooling system is a system that not only guarantees the dissipation of significant amounts of heat from the cooled surface with the least amount of water consumption, but it also ensures a high uniformity of cooling. Due to the atomization of the liquid with compressed air, fine droplets were obtained, which increased the uniformity of cooling. The AAS system can be successfully used under industrial conditions to cool the materials that are particularly sensitive to crack formation which are associated with the presence of high thermal stresses, keeping proper control during transition-boiling. The sensor-cooling with the WS system, despite the largest amounts of heat transferred from the surface, was characterized by high cooling heterogeneity during transition-boiling, nucleate-boiling and single-phase convection. For this reason, it is recommended to use pneumatic not hydraulic atomizers wherever it is necessary to maintain the high uniformity of material cooling.

3.4. Comparison of the Amount of Energy Dissipating from the Surface During Boiling and Forced Single-Phase Convection

In industrial processes, the control of temperature field changes in the entire volume of the cooled material is essential. Changes in the temperature field determine the properties and structure of the material during its production process. In the case of thermally thick materials, a too-high temperature inside the deeper layers of the material after the completion of the cooling process may contribute to an increase in temperature in the surface layers due to heat conduction. To prevent this phenomenon, such materials, after the boiling process is finished, are still cooled in technological line under the conditions of single-phase forced convection. In this paper, the determination of the amount of energy dissipating from the sensor surface A₂ during cooling due to boiling and single-phase forced convection is presented. Additionally, a system that ensures the lowest temperature at point P4 when the surface temperature reached 100 °C and 60 °C was selected. The amount of energy dissipating from a cooled surface resulting from the boiling process was determined using Formula (35):

$$E_B={\int }_{\tau ={\tau }_B}^{\tau ={\tau }_S}HF\left(\tau \right)d\tau$$

(35)

where

• τ_B—time to start boiling, s;
• τ_S—time in which the cooled surface to reach the temperature of 100 °C, s.
• The amount of energy dissipating from a cooled surface caused by single-phase cooling was calculated using Formula (36):

$$E_S={\int }_{\tau ={\tau }_S}^{\tau ={\tau }_E}HF\left(\tau \right)d\tau$$

(36)

where

• τ_E—time of single-phase cooling ending, s.

For all considered types of cooling, the periods at which the individual stages of cooling occurred were defined. In Figure 28, the average HF values on the A₂ surface with marked periods of air cooling, boiling and single-phase forced cooling obtained for cooling the sensor with the AAS system at a pressure of 0.1 MPa are presented.

In this case, the boiling process started at time τ_B and finished when the surface temperature reached 100 °C, at point τ_S. Cooling under the conditions of single-phase forced convection was considered when the sensor surface temperature reached 100 °C within the time in which the surface temperature dropped to 60 °C, τ_E. This approach made it possible to compare the amount of energy dissipating from the cooled surface during each stage of cooling for all three cooling systems (Figure 29). The highest amount of energy dissipating during boiling was obtained for the WS system for all three pressures, with the lowest for the AAS system.

In the case of cooling with the WS system, the amount of energy dissipating from the surface during the boiling process was the highest. Despite that, the value of the temperature at control point P4 was quite high (

Table 4. Temperature reached at point P4 at times τ_S and τ_E.

Type of Cooling	Pressure, MPa	τS, s	Temperature at Point P4 Obtained at Time τS, °C	τE, s	Temperature at Point P4 Obtained at Time τE, °C
AAS	0.05	34	422.3	59	237.6
	0.1	30	447.8	56	254.5
	0.2	29	423.0	53	253.9
WS	0.05	36	390.3	65	214.2
	0.1	36	391.9	62	213.8
	0.2	30	430.7	56	242.1
JET	0.05	39	372.4	68	197.9
	0.1	32	417.9	57	245.2
	0.2	30	433.4	54	252.0

). The lowest temperature at point P4 was achieved for the JET system with a water-supplied pressure of 0.05 MPa. During single-phase convection, a similar trend was observed. The ability of water to remove heat from the cooled surface in this system was the lowest (Figure 23). The JET system guarantees obtaining the lowest temperatures in the deeper layers of the material, both in terms of two-phase and single-phase convection. In the case of cooling thermally thin materials, the WS or the AAS system are the best solution for cooling. The AAS system provides six-times-lower water consumption in relation to the JET system. It also ensures the highest uniformity of temperature distribution during nucleate-boiling and single-phase forced convection (Figure 27). The cooling systems considered ensured almost the same value of temperature at point P4, both for convection with boiling and forced convection, when the cooling process was conducted with the highest water pressure.

4. Conclusions

Water-cooling is an important component of steelmaking technology. The purpose of cooling is not to only remove as much amount of heat as possible in the shortest possible time, but also carry out this operation in a controlled manner. This ensures obtaining a product with the searched properties. As part of this work, three cooling systems used for high-temperature processes were assessed: AAS, WS and JET. The assessment was made by identifying the HTC and HF on the surface of the sensor cooled with water. The identification of the boundary condition of heat transfer was performed using the inverse method. The solution of the heat conduction equation was obtained using the finite element method. The obtained results of the calculations indicate that the accuracy of the inverse solution is high, and the average temperature error ranged from 0.1 K to 0.9 K. The accuracy of the heat conduction model was tested based on the reference boundary condition model similar to the experimental conditions. The average error of the solution was 0.17 K.

Based on the results of the analysis, the following conclusions were defined:

• The analyzed systems did not allow for obtaining a homogeneous distribution of the HTC on the surface. The smallest maximum differences in HTC distribution were obtained for the AAS system (11,200 W/m²K).
• The ability of the fluid to dissipate heat from the surface was the highest for the AAS system and it increased with increasing water pressure (the increase in pressure from 0.1 MPa to 0.2 MPa caused an over 47% increase in the ability of water to dissipate heat). For the other considered systems, the relationship was reversed.
• The highest amount of heat dissipating from the surface in the boiling range was obtained for the WS cooling system. A 17.55% increase in the amount of heat dissipating from the surface at a pressure of 0.05 MPa, a 16% increase at a pressure of 0.1 MPa and a 12.28% increase at a pressure of 0.2 MPa were achieved compared to the amount of heat dissipating during AAS-cooling. The obtained result was influenced by the low thermal conductivity of the tested material and material thickness.

Further research is necessary to investigate the effect of material conductivity on the amount of energy dissipating from a cooled surface and the change in material temperature depending on the used cooling system. This approach will allow for verification if the impact of the adopted cooling system is noticeable for materials with different thermal conductivities, and indicates which cooling systems are suitable for cooling materials with specific properties.