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We present a new Nusselt number correlation for spray cooling at large Reynolds numbers and high surface temperatures for water sprays impinging perpendicularly onto a flat plate. A large set of experimental data on spray cooling of hot surfaces with water has been analyzed, including the water temperature effects. For large-scale cooling, such as in industrial processes, large number of injection parameters such as number, type, pressure, and angle of the spray injection has led to a multitude of correlations that are difficult for general and practical applications. However, by synthesizing a set of experimental data where all of the above parameters have been varied, we find that the Nusselt number and therefore the heat transfer coefficient can be cast accurately as a function of the Reynolds number. Water is widely used as the coolant during spray cooling, and has a specific phase change characteristic. At large Reynolds number (Re > 100,000) and surface temperature (Ts > 600°C) ranges, which are of interest in large-scale spray cooling, the effect of water temperature is quite significant as it affects the film boiling close to the surface. This effect also has been parameterized using experimental data.

Enhanced heat transfer is widely used in various industrial processes and devices, including gas-turbine combustor cooling, steel production and metal processing. Heat removal through cooling and evaporation of liquid (most commonly, water) sprays is an attractive method, due to large volumetric flow rate and heat capacity. The thermal performance during spray injection of liquid onto surfaces is normally represented by the associated heat transfer coefficient (HTC) or the Nusselt number, which is a measure of the cooling rate due to forced convection. Characterization of the heat transfer coefficient is important for determining the required cooling capacity and corresponding internal properties of the product. For this reason, a vast amount of work has been devoted to parameterize or model the heat transfer coefficient during spray cooling in different flow regimes [

In large majority of applications as well as in this work, the interest is in large volumetric-flow-rate spray cooling or large Reynolds numbers, and for hot surfaces above the so-called Leidenfrost temperature (T_{Lei}). Here, the Reynolds number is defined as the Re = ρUL/μ, where U is the mean injection velocity of water at the injector exit, L the length scale of the surface to be cooled, and ρ and μ the water density and viscosity, respectively. The injection velocity has been obtained by measuring the volumetric flow rate of water and dividing by the injector area. At these conditions, the flow is turbulent and the cooling occurs through film-boiling in which a stable vapor film exists at the hot surface. The turbulent two-phase flow and heat transfer in these regimes involve complex transport processes, with several more factors to consider than in single-phase heat transfer. The Leidenfrost temperature, for example, has been found to be dependent on the Weber number [_{h}) linearly increasing with the surface temperature, T_{s} [

complete or misleading. For example, by considering T_{s} varying from 200˚C to 1100˚C, Wendelstorf [_{s} increased. Here, the logic is that the primary cooling effect comes from the flow rate, or the Reynolds number effect. Then, other effects such as boiling, void fraction and two-phase flow structure near the surface can be accounted for using fluid property changes and thermodynamic considerations.

An example of the application of spray cooling is the continuous casting, also called strand casting, which is currently the most efficient method for steel production. During continuous casting as shown in

The major function for the secondary cooling is to speed-up the strand solidification to produce steel with required microstructures but without generating undesirable defects [

A good amount of literature exists for impingement cooling at relatively low Reynolds numbers, for applications in gas-turbine cooling or for fundamental understanding of the two-phase heat transfer [_{w} [

C_{1} is a constant that is used for different type of spray nozzle, while m_{1} is used to take into account the spray cooling patterns (spray overlap, injector spacing, etc.). By matching the experimental measurements of the surface temperatures with those predicted by numerical simulation, de Toledo et al. [_{1} =1.23 and C_{1} varies from 0.836 to 0.951, with ^{2}∙K] and Q_{w} in [liter/min]. By fitting the modeling predictions with the temperature measurements, Jacobi et al. [_{1} was smaller, 1.15 to 1.19. In a more recent investigation, Santos et al. [_{1} was even smaller and reported that m_{1} and C_{1} were 0.556 and 366, respectively, with Q_{w} in [L/s]. The form of this kind of correlation is set up for convenient practical applications, but due to the large number of possible spray configurations and injector types, the applicability is limited to a particular set of spray systems and also can be confusing for general use.

Although the twin-fluid (air/water sprays) cooling have been used for many industrial processes such as continuous casting for more than three decades, many investigators still use the correlations original developed for the single-fluid cooling. For example, Zhang et al. [_{1}, modified to C_{1} = 6 for liquid water spraying and C_{1} = 5 for air-mist spraying. In order to include air flow effects more directly, sophisticated HTC models have been developed for twin-fluid nozzle cooling. Based on the measurement by embedded thermocouples, Jacobi et al. [_{h}) is highly dependent on the surface temperature (T_{s}) for various mixing ratio of water and air flows. They observed that the maximum or critical heat flux took place at T_{s} = 200˚C, at which the complete (or maximum) wetting occurred. By re-examining the heat rate data presented by Jacobi et al. [_{w}).

where D_{1} = 14.6, with h_{s}_{,max} and Q_{w} in [kW/m^{2}∙k] and [L/m^{2}∙s], respectively. Indeed, the data showed that not only the HTC increased linearly with the water flow rate but also that it is independent on the air flow rate (Q_{a}), justifying the use of Equation (1). A correlation by Ueta et al. (1990) is of interest due to its explicit dependence on the surface temperature (applicable from T_{s} = 600˚C ~ 1000˚C).

where Q_{a} is the airflow rate, and D_{2}, n_{1}, n_{2} and n_{3} are correlation constants. Notably, the exponent n_{3} for the surface temperature is −0.136, meaning that the heat rate decreases with increasing surface temperature.

Most of the existing correlations, as noted above, include parameters such as the volumetric flow rates and surface temperature; however, they are cast in practical terms, i.e. heat transfer coefficient as a function of volumetric flow rates and empirical constants. In particular, the empirical constants vary over a wide range and are quite different for various spray configurations. The purpose of this work is to use an extensive experimental data set for heat transfer coefficients during spray cooling, and put the data in a standard form so that a universal application of the results is possible. We use the extensive data set obtained by Raudensky and co-workers [

The experiments involved using an array of injectors to spray water onto horizontal flat plates. The experimental details are fully described in Raudensky et al. [

Lechler 660.604, 660.674, and 660.804 nozzles [

Water Pressure [bar] | Flow rate per one nozzle [l/min per nozzle] | ||
---|---|---|---|

660.604 | 660.674 | 660.804 | |

1 | 2.22 | 3.35 | 7.07 |

2 | 3.15 | 4.75 | 10 |

3 | 3.85 | 5.81 | 12.25 |

5 | 4.98 | 7.51 | 15.81 |

The control parameters for industrial applications of spray cooling are injector type/bore diameter, injection pressure, injector lateral and longitudinal spacing. The optimum configuration should lead to uniform and sufficiently rapid cooling of the hot surface. The possible permutations of the above parameters can be overwhelming large, and one needs a universal correlation for the heat transfer rate based on the transport physics during spray cooling. It is evident that heat transfer during spray cooling depends primarily on the Reynolds number, or the volume of the flow in contact with the surface. Thus, many of the injector configuration parameters can be “compressed” into the Reynolds number of the flow. First, the injector lateral spacing only concentrates more or less of the fluid onto the surface, as does the injection pressure which increases the volumetric flow rate. We look at the geometry of the sprays as in

Based on this logic, we can find a generalized Nu-Re correlation, using the data set in which many of the injection parameters were varied. The length scale, L, in the Reynolds number is the lateral dimension of the substrate. The results are plotted in

It is also interesting that the Nusselt number correlation shown in ^{−2} and m = 0.8, but of course with a factor of Pr^{1/3} multiplying the Reynolds number dependence (Nu = CRe^{m}Pr^{1/3}). If we

take the Prandtl number of the water to be 3.42, then the coefficient for the correlation shown in ^{1/3} = 1.9849 × 10^{−2}, which is smaller than that for heat transfer over a flat plate, but still of similar order of magnitude. The explanation is that even though the spray is perpendicular to the plate in the current data set, the underlying heat transfer mechanism is the cooling by the liquid flow spreading over the hot surface at high Reynolds numbers. Due to the geometry of the flow, where a number of jet flows spread over the plate, the heat transfer is apparently less effective, resulting in a smaller Nusselt number as a function of the Reynolds number.

During spray cooling, complex flow structure develops near the hot surface due to the heat flux can easily cause phase change, creating vapor film near the surface. The vapor has much lower thermal conductivity than liquid, and therefore formation of vapor bubbles or films leads to a significant reduction in the heat transfer coefficient, called the Leidenfrost effect. In the current experimental data set, this complex transition can be observed as the temperature is continuously measured during multiple passes of the injector array. The data presented in

The effect of water temperature can also be incorporated through the Nusselt number.

We consider the physics of the heat transfer to be the same in two-phase flow over the surface, and the primary mechanism of the change in the heat transfer coefficient is due to the reduced thermal conductivity of the vapor phase. The definition of the Nusselt number, as in Equation (5), shows that if the thermal conductivity is decreased, then the heat transfer coefficient will be reduced.

Thus, in order to determine the heat transfer rate during two-phase flows, we need to have an estimate of the quality in relation to the water temperature. The physics of film boiling involves water mass flux over a hot surface, where the heat transferred to the water overcomes the heat of vaporization to convert liquid into water vapor. The exact details of this two-phase flow is quite complex. However, since we are interested in the net average heat transfer effect, we can use an energy balance model to find the functional relationship between the water and surface temperatures and the quality.

If we consider the energy balance for flow over a hot surface, undergoing vaporization, we can write the following equation.

Here, Δm is the vaporization rate, and h_{fg} the heat of vaporization. We can see that the enthalpy difference and the convection heat transfer constitute the excess heat available for vaporization. We can also solve for the excess heat.

Thus, the quality x depends on the excess heat represented on the RHS of Equation (8). The exact dependence is the steady-state solution to the energy equation. However, we can deduce its dependence since x = 0 when there is zero excess heat, and x à 1 if the excess heat is very large. This leads to the following form for the x dependence on the excess heat.

If T_{f} or T_{s} is very large, then the exponential term tends to zero in Equation (9) and x à 1. Conversely, if T_{s} is less than T_{sat} then the argument in the exponential term can become a large negative number, and x à 0. Since the precise effects of the mass flow rate on the two temperature difference terms are not known, these two factors are the correlation constants to be used with the experimental data.

We can examine the effects of water temperature, T_{f}, on the heat transfer coefficient using Equations ((5), (6) and (10)). We consider the physics of heat transfer to be dependent only on the Reynolds and Prandtl number, as in Equation (4). This gives the heat transfer coefficient change as a function of the effective thermal conductivity, which changes with the fluid temperature according to Equations ((6) and (10)). The comparison is shown in _{1} and A_{2} have been found through comparison with experimental data and are listed in

Re | A_{1} | A_{2} |
---|---|---|

2.10 × 10^{5} | 0.026 | 0.100 |

2.44 × 10^{5} | 0.038 | 0.050 |

3.21 × 10^{5} | 0.065 | 0.039 |

Experimental data also show that the surface temperature has an effect on the heat transfer coefficient. This is also apparent in Equation (9), where the higher surface temperature provides more excess heat for vaporization of the fluid layer close to the surface. The heat transfer model is able to predict the surface temperature effect reasonably well at low water temperatures of 20˚C and 40˚C. However, at 60˚C the predicted curve is qualitatively different from data, an inflection occurs at surface temperature of about 500˚C. Thus, there may be some additional physics that need to be considered at high fluid temperature, which are currently being investigated in this laboratory. For example, the constants A_{1} and A_{2} in _{1} increases with the Reynolds number. However, the magnitude is small in comparison to the product of the mass flow rate and the specific heat in Equation (9). Thus, perhaps this term should be computed relative to the heat of vaporization. The constant A_{2}, on the other hand, increases with the Reynolds number, reflecting the fact that the Reynolds number enhances the heat transfer coefficient.

A large set of experimental data on spray cooling of hot surfaces with water has been analyzed, including the water temperature effects. For large-scale cooling, such as in industrial processes, large number of injection parameters such as number, type, pressure, and angle of the spray injection has led to a multitude of correlations that is included at times some of these injection parameters. However, by synthesizing a set of experimental data where all of the above parameters have been varied, we find that the Nusselt number and therefore the heat transfer coefficient can be cast as a function only of the Reynolds number. Also, the temperature effects can be accounted for by making use of the energy balance.

This experimental part of the current work was financially supported by the MEYS, under program NPU I, Project LO1202.

Milan Hnizdil,Martin Chabicovsky,Miroslav Raudensky,Tae-Woo Lee, (2016) Heat Transfer during Spray Cooling of Flat Surfaces with Water at Large Reynolds Numbers. Journal of Flow Control, Measurement & Visualization,04,104-113. doi: 10.4236/jfcmv.2016.43010

A = surface area

A_{1}, A_{2} = constants in Equation (10)

C = a constant in Equation (4)

C_{1} = a constant in Equation (1)

c_{p} = specific heat

D_{1} = a constant in Equation (2)

D_{2} = a constant in Equation (3)

h = heat transfer coefficient

h_{fg} = heat of vaporization

h_{s}_{,max} = maximum heat transfer coefficient

k = thermal conductivity

k_{eff} = effective thermal conductivity of the thermal boundary layer

k_{f} = thermal conductivity of the liquid phase

k_{g} = thermal conductivity of the vapor phase

L = length scalem = constants in Equation (1) and Equation (4).

m= exponent for the Reynolds number in Equation (4).

m_{1} = a constant in Equation (1)

n_{1}, n_{2}, n_{3} = correlation constants in Equation (3)

Nu = Nusselt number = hL/k

Pr = Prandtl number

Q_{a} = air flow rate

Q_{w} = water flow rate

q_{excess} = “excess” heat rate in Equation (8)

Re = Reynolds number

T_{f} = water temperature

T_{s} = surface temperature

T_{sat} = saturation temperature

x = quality

μ = viscosity

ρ = density