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Numerical Investigation of the Flow Dynamics and Evaporative Cooling of Water Droplets Impinging onto Heated Surfaces: an Effective Approach to Identify Spray Cooling Mechanisms Jiannan Chen, Zhen Zhang, Ruina Xu, Xiaolong Ouyang, and Pei-Xue Jiang Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b02205 • Publication Date (Web): 16 Aug 2016 Downloaded from http://pubs.acs.org on August 19, 2016
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A Manuscript Entitled
Numerical Investigation of the Flow Dynamics and Evaporative Cooling of Water Droplets Impinging onto Heated Surfaces: an Effective Approach to Identify Spray Cooling Mechanisms By Jian-nan Chen1, Zhen Zhang2, Rui-na Xu1, Xiao-long Ouyang1, Pei-xue Jiang1* 1. Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Beijing Key Laboratory for CO2 Utilization and Reduction Technology, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China; 2. Institute of Nuclear and New Energy Technology, Collaborative Innovation Center of Advanced Nuclear Energy Technology, Key Laboratory of Advanced Reactor Engineering and Safety of Ministry of Education, Tsinghua University, Beijing 100084, China
Submitted to Langmuir
Please address all correspondence to Professor P.X. Jiang Department of Thermal Engineering Tsinghua University Beijing 100084, CHINA Tel: +86 10 62772661 Fax: +86 10 62770209 Email:
[email protected] 1
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Numerical Investigation of the Flow Dynamics and Evaporative Cooling of Water Droplets Impinging onto Heated Surfaces: an Effective Approach to Identify Spray Cooling Mechanisms Jian-nan Chen1, Zhen Zhang2, Rui-na Xu1, Xiao-long Ouyang1, Pei-xue Jiang1* 1. Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Beijing Key Laboratory for CO2 Utilization and Reduction Technology, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China; 2. Institute of Nuclear and New Energy Technology, Collaborative Innovation Center of Advanced Nuclear Energy Technology, Key Laboratory of Advanced Reactor Engineering and Safety of Ministry of Education, Tsinghua University, Beijing 100084, China *
Corresponding author: Phone: +86-10-62772661, Fax: +86-10-62770209 Email:
[email protected] Abstract Numerical investigations of the dynamics and evaporative cooling of water droplets impinging onto heated surfaces can be used to identify spray cooling mechanisms. Droplet impingement dynamics and evaporation are simulated using the presented numerical model. VOF method is used in the model to track the free surface. The contact line dynamics was predicted from a dynamic contact angle model with the evaporation rate predicted by a kinetic theory model. A species transport equation was solved in the gas phase to describe the vapor convection and diffusion. The numerical model was validated by experimental data. The physical effects including the contact angle hysteresis and the thermocapillary effect are analyzed to offer guidance for future numerical models of droplet impingement cooling. The effects of various parameters including surface wettability, surface temperature, droplet velocity, droplet size and droplet temperature were numerical studied from the standpoint of spray cooling. The numerical simulations offer profound analysis and deep insight into the spray cooling heat transfer mechanisms.
Key words droplet impingement cooling; spray cooling; evaporation; dynamic contact angle; VOF 1. Introduction For the past 50 years, tracking Moore’s Law [1] has been the goal of the semiconductor industry in the development of integrated circuits. The chip generates more heat per surface area as it becomes more compact and complex. The IC (integrated circuit) power densities continue to increase as illustrated in Figure 1 [2]. Many other high power electronic devices like IGBT (insulated-gate bipolar transistor) modules and solid state
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lasers generate even more heat when pursuing higher performance and more capability. Such industrial applications are need more effective cooling schemes to ensure reliability. Spray cooling is a heat transfer process where many small droplets impact the heated surface. The heat transfer during this process includes the forced convection heat transfer when the droplets impinge and spread on the surface and the evaporative heat transfer as the droplets being heated. Spray cooling becomes a hot issue in thermal management because of its high cooling efficiency, uniform cooling surface temperature and reliable working performance. It has been reported that spray cooling can cool a surface with more than 1000 W/cm heat flux with low cost [3].
Figure 1 IC power density trends for the projected ITRS integration density and performance
Spray cooling has a wide range of applications and has attracted much interest for many years. Estes and Mudawar [4] experimentally correlated the droplet diameter and critical heat flux for spray cooling. They suggested that all attempts to correlate the present spray cooling CHF data to the drop velocity have been unsuccessful and that the droplet diameter most strongly affects the spray cooling CHF. They believed that smaller drops with their greater surface area to volume ratio than larger drops can more effectively utilize their sensible and latent heat than larger drops. Their research showed that both the single-phase heat transfer coefficient and the CHF increase with decreasing droplet diameter. Chen et al. [5, 6] also studied the effects of droplet size and droplet velocity on the heat transfer coefficient and the CHF but with different conclusion. Their results indicated that droplet velocity had the largest impact on the CHF and that the droplet diameter had negligible effect. The research of both Estes and Mudawar [4] and Chen et al. [5, 6] confirmed that dilute sprays are more effective than dense sprays. Horacek et al. [7] obtained time and space resolved heat transfer distributions using an array of individually controlled microheaters with the liquid–solid contact area and the three-phase contact line length visualized and measured using a Total Internal Reflectance (TIR) technique. Their experiment results indicated that the heat transfer rate is directly related to the liquid-solid contact line length. Kim suggested that these contact line length heat transfer mechanisms should be further verified and clarified. [8]. However, such experiments require sophisticated techniques and are very difficult. Nanostructure has been used to modify the heated surface when phase-change cooling scheme is used to improve the cooling efficiency. Zhang et al. [9] modified the surface with CNT films and their spray cooling tests on such surfaces showed that the heat transfer is enhanced. They suggested that the nanostructures improved the surface wettability so that the heat transfer is enhanced. Droplet impingement cooling heat transfer is also enhanced on nanostructured surfaces reported by Alvarado and Lin [10]. They suggested that the surface wettability is improved by the nanostructures with the improved surface wettability promoting the liquid-vapor phase change. However, insight into the liquid flow inside the droplet and the evaporation process is needed to validate this theory. Spray cooling heat transfer mechanisms are very complex because spray cooling is affected by many factors. The influential factors are nozzle-to-surface distance [11], inclination angle [12] and working fluid [13], and droplet parameters [4~6]. The droplets parameters are the most dominant. However, the inability to
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independently control the droplets parameters make it almost impossible to thoroughly investigate the heat transfer mechanism experimentally [14]. This may explain the contradictions between many studies and leaves questions unanswered such as the mechanism that relates the heat transfer with the contact line dynamics, the partitioning of the energy transfer between the single- and two-phase processes, and the mechanism that triggers CHF [15]. Numerical study on the single droplet evaporative cooling becomes a key issue to further understand spray cooling heat transfer mechanisms because the droplet parameters can be controlled independently in numerical models. Healy et al. [16] indicated that numerical simulations of single droplet impingement cooling is important to predict the spray cooling heat transfer because it can provide many fluid dynamics details. Thus, numerical investigations of the dynamics and evaporative cooling of water droplets impinging onto heated surfaces is an effective approach for investigating the spray cooling mechanisms. Droplet impact dynamics and evaporation heat transfer have been extensively investigated since they are key issues in many applicable techniques such as spray cooling, spray coating, ink-jet printing and pesticide spraying. Early studies proposed simple analytical models [17~19] of the droplet impact dynamics to calculate the droplet deformation while the flow dynamics details could not be offered. Calculations of the heat transfer between a surface and the impinging droplets requires detailed information about the droplet shape and temperature during impact, which can only be obtained by a complete solution of the continuity, momentum, and energy equations [20]. Harlow and Shannon [21] first used the marker and cell (MAC) method to solve the Navier-Stokes equations to study the fluid dynamics during droplet impact on a solid surface. They neglected the viscous and surface tension effects to simplify the equations with satisfactory results only for the initial spreading stage after the droplet impact at which time these forces are negligible compared to inertial effects. The Volume-of-fluid (VOF) method developed by Hirt and Nichols [22] is a very popular methods for simulating free surfaces and has been used to successfully simulate the droplet impact dynamics. Bussman et al. [23, 24] demonstrated the efficiency of the VOF method for studying the flow dynamics of droplets impacting onto complex geometries. The fluid flow at the liquid–solid–gas contact line must be accurately predicted in realistic droplet impact models. Droplet impact models usually specify the contact line boundary condition by assigning a solid-liquid contact angle which determines the liquid-gas interface shape near the contact line. Many studies have used an equilibrium solid-liquid contact angle, which is called a static contact angle in this paper, as the assigning angle [25~28]. However, the apparent contact angle can differ greatly from the static contact angle during droplet impact due to the contact-angle hysteresis. Fukai et al. [29] took the contact-angle hysteresis effect into account by assigning a constant advancing contact angle and a constant receding contact angle. They found that incorporation of different advancing and receding angles improved the results with the substrate wettability found to significantly affect all phases of the spreading process. Pasandideh-Fard et al. [30] went a step further and used a dynamic contact angle obtained from experiment observations and updated after each time step. The droplet diameter during spreading and at equilibrium was more accurately predicted by using observed dynamic contact angles in the model rather than a static contact angle. However, the model overpredicted the droplet diameters during recoil. A similar work by Gunjal et al. [31] used experimentally observed dynamic contact angles in a VOF-based model to study the dynamics of droplet impact on a solid surface. Their research indicated that the dynamic variation of the contact angle had a significant effect for liquid-solid systems with contact angles smaller than 90° with the agreement between the simulations and experimental results improved by using experimentally observed dynamic contact angles instead of a static contact angle. Other studies have used empirical [32] or theoretical [33] models to predict dynamic contact angle in the VOF-based models to simulate the droplet impact flow and heat transfer because experiment observations of dynamic contact angles are not always possible. Zhang [34] suggested that Blake’s molecular kinetics based dynamic contact model [35] was more accurate than experimentally observed dynamic contact angle when used to specify the contact line boundary condition in VOF-based models because the inertial effect is taken into account twice when experimental data is used.
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Droplet evaporation is another key issue in droplet impingement cooling which has been extensively investigated since Maxwell [36] in 1877 and Langmuir [37] in 1918 with most models based on many assumptions and simplifications of the fluid flow [38, 39], droplet shape [40] and liquid evaporation [41] to simplify the numerical solution or for the convenience of applying to engineering simulations. However, fundamentally studies of droplet impingement cooling for spray cooling needs accurate numerical model to provide the details about the droplet impact dynamics, droplet deformation, liquid flow inside the droplet and vapor convection and diffusion which directly affect the droplet evaporation. This paper gives a numerical model that accurately describes the droplet impingement dynamics and the evaporation. VOF method is used to simulate the free surface. The contact line dynamics is predicted using Blake’s molecular kinetics theory based dynamic contact angle model [35] with the evaporation rate predicted using Schrage’s kinetics based evaporation rate model [42]. A species transport equation is solved in the gas phase to describe the vapor convection and diffusion. After validation against experimental data, the numerical model is used to fundamentally study the droplet impingement dynamics and evaporative cooling processes to further understand the spray cooling heat transfer mechanisms. The effects of the dynamic contact angle and thermocapillarity are analyzed to show their importance in numerical simulations. Droplets of different sizes and velocities impinging onto heated surfaces are numerically investigate to study the effects of various droplet parameters. The surface wettability effect is also evaluated to provide evidence that nanostructures enhance dropwise evaporative cooling improving the surface wettability. Droplets impinging onto different temperature surfaces are also numerically studied and shed some light on the mechanism by which CHF is triggered. 2. Experimental In the VOF method, the phase volume fraction, , is defined as,
=
!
(1)
2.1 Conservation equations The droplet shape, fluid flow, heat transfer and vapor distribution are described by the various conservation equations. Complete, accurate conservation equations are needed for accurate simulations. A VOF equation is solved to track phase interfaces, " & [ ' ( ) + ∇ ∙ ' ( ..../) - = 01$ (2) # & $
There are only two phases: liquid phase and gas phase in this study, so n=2. Only the liquid phase volume fraction, 2 , is caculated and the gas phase volume fraction is calculated from the fact that the sum of the volume fractions must equal unity. 01$ is the volume fraction source term produced by the phase-change at the interface due to evaporation. All the variables and properties are shared by both phases and are represented as volume-averaged values. Then the transport properties, such as the density here, are given as volume averaged values, ( = ∑45" ( = 2 (2 + 61 − 2 8(9 (3) The momentum equation is shown as below. & ...../ 6(-/8 + ∇ ∙ 6(-/-/8 = −∇: + ∇ ∙ [;6∇-/ + ∇-/ < 8= + (>/ + ? (4) @ & where -/ is the velocity vector and : is the pressure. .../ ?@ is the body force vector (which is nonzero only at the .../ liquid-gas interface) due to surface tension. ?@ directly determines the droplet deformation and the liquid circulation inside the droplet so the definition of this body force vector is key for accurate numerical .../ simulations of the droplet impingement dynamics and cooling heat transfer. ? @ is given by calculating :4 and : . The capillary pressure, :4 , in the normal direction is given calculated as, :4 = A ∙ B (5) 5
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where A is the surface tension coefficient and B is the interfacial curvature. The pressure in the tangential direction caused is given by, : = ∇ A Referring to the CSF model [43], the body force is calculated as, |∇1D |# .../ ? 6ABIJ + ∇ A8 @ =E F
'#D G#H )
where IJ is the unit surface normal. The unit normal, IJ, is defined as, ./ 4
IJ = |4./|
I./ is the surface normal calculated as the gradient of the liquid phase volume fraction: I./ = ∇2 B = ∇ ∙ IJ
(6) (7)
(8) (9) (10)
Figure 2 Illustration of the composition of the body force
The heat transfer is governed by the energy equation, & 6(K8 + ∇ ∙ '-/6(K + L8) = ∇ ∙ 6M∇N8 + 0O & where the energy, E, is the mass-averaged energy, K=
∑P $QE 1$ #$ O$ ∑P $QE 1$ #$
=
1D #D OD G1H #H OH 1D #D G1H #H
(11) (12)
0O is the source term produced by the heat production or heat loss at the interface due to the phase change. The liquid will evaporate into the surrounding gas. For the cases in this paper, the liquid droplet impingement cooling takes place in the ambient air. The gas phase is then a binary mixture of vapor and air as the liquid evaporates at the interface. The liquid phase changes to the vapor phase at the interface due to evaporation with the generated vapor moving in the air by diffusion and convection. The evaporation rate is strongly related to the vapor mass transfer rate which is dominated by diffusion and convection. Therefore, a species transport equation is solved in the gas phase, &'#$ 1$ R$S )
+ ∇ ∙ '( -/TU ) = ∇ ∙ '( V" ∇TU ) + 0U where T is the mass fraction. The binary diffusion coefficient, V" , was given as[44], &
V" =
"WXY < E.[\ ] E
E E E G _F ^E ^F
E F
`a6∑S bSE 8Y G6∑S bSF 8Y c
(13)
(14)
where T is the temperature, M is the molecular weight, P is the pressure and -U are the special diffusion parameters [44].
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2.2 Liquid-gas interface conditions An important part of the numerical model is to define the evaporation rate at the liquid-gas interface. Many studies [40, 41] have predicted the evaporation rate based on Fick’s law by assuming that the vapor concentration at the interface is saturated to calculate the local vapor concentration gradient. This assumption may be applicable when the liquid is not volatile or the liquid temperature is far below the saturation temperature so the evaporation is relatively slow. However, as mentioned in the introduction, the dropwise evaporation cooling rate is quite large due to the large latent heats with liquid-vapor phase change. The surface temperature is always near or even above the liquid saturation temperature when droplets impinge and cool the hot surfaces and the evaporation is quite strong and the liquid-gas interface is in a non-equilibrium state which makes the assumption inappropriate. Molecular kinetic theory gives the total evaporating mass flux at a liquid-gas interface, d , as [42], d =
@e
f@e
g
h
ij
6
`klm 6
