A model to predict the effect of surface wettability on critical heat flux

International Communications in Heat and Mass Transfer 39 (2012) 1500–1504

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A model to predict the effect of surface wettability on critical heat flux☆ Hai Trieu Phan a, b, Rémi Bertossi a, b, Nadia Caney a, b, Philippe Marty a, b,⁎, Stéphane Colasson b a b

UJF-Grenoble 1/Grenoble-INP/CNRS, LEGI UMR 5519, Grenoble, F-38041, France CEA, LITEN/DTS/LETH, 17 rue des Martyrs, 38054 Grenoble cedex 9, France

a r t i c l e

i n f o

Available online 24 October 2012 Keywords: Pool boiling Critical heat flux Contact angle Wettability

a b s t r a c t Critical heat flux (CHF) in pool boiling experiments corresponds to the heat flux at which a vapor film is formed on the heated surface resulting from the replacement of liquid by vapor adjacent to this surface. Poor thermal conductivity of vapor can severely deteriorate heat transfer. It is important that systems operate below this limit which is a strong limitation to heat transfer due to the huge increase of the thermal resistance near the wall. The concept of macro- and micro-contact angles has been introduced in a previous paper (Phan et al., 2010 [28]) to describe the bubble growth processes. In this paper, an explicit relation between the bubble departure diameter and the contact angle has been presented. Based on these results, we propose a model of critical heat flux, taking into account the effects of the wettability of the fluid, whose property is known to strongly influence boiling heat transfer. A new correlation for CHF, dependent on the contact angle, is proposed. It is found in fair agreement with existing experimental results concerning subcooled boiling to describe the variation of CHF with wettability. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction The critical heat flux (CHF) is the maximum heat flux at which nucleate boiling heat transfer sustains high cooling efficiency. When a surface is submitted to CHF, evaporation of the liquid close to the heated wall occurs. The consequence is the augmentation of the temperature leading to a deterioration of the material. It is then of primary interest to delay the CHF occurrence in order to enhance heat transfer efficiency of the system. Many recent studies proved that critical heat flux can be enhanced by modifying surface wettability [1–6]. Over the past several years, studies on pool boiling have demonstrated that the addition of nanoparticles in a fluid can significantly increase CHF. A number of investigations show that this CHF enhancement can be related to the modification of the heated surface properties due to nanoparticle deposition. A number of nanofluid boiling studies have reported up to 100% enhancements in pool boiling CHF [1,2]. Many models of CHF have been developed. Most of them do not take into account the wettability of the fluid. Bonilla and Perry [7] and Cichelli and Bonilla [8] first presented basic correlations based on their experimental data using organic liquids. Kutateladze [9,10] proposed a correlation (similar to Eq. (1)) for CHF based on a critical velocity of bubbles and gave a correlation dependent to a parameter C (equal to κ−1/2 in Eq. (1)) that can be determined thanks to experimental data. ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author at: CEA, LITEN/DTS/LETH, 17 rue des Martyrs, 38054 Grenoble cedex 9, France. E-mail address: [email protected] (P. Marty). 0735-1933/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.icheatmasstransfer.2012.10.019

Borishanskii [11] developed an analytical expression of the constant C only dependent on fluid physical properties. Zuber [12], who argued that CHF is caused by Taylor and Helmholtz instabilities, proposed a new value for C equal to 0.131. The theory of Chang [13] which considers that CHF is reached when the Weber number We reaches a critical value provided that another value for C is equal to 0.098. In all these models, C is never linked to the contact angle. Rohsenow and Griffith [14] presented another correlation for CHF considering that increased packing of the heating surface with bubbles at higher fluxes inhibited the liquid flow to the heating surface. Haramura and Katto [15] based their analysis on assuming that Kelvin–Rayleigh instabilities can occur in a macrolayer under the bubble, resulting in coalescence of vapor stems; they finally found an equation similar to the one of Kutateladze [9,10]. Guan et al. [16] proposed a new mechanistic model for predicting CHF in horizontal pool boiling systems based on the critical vapor velocity in the bubble. Chung and No [17] also developed a model of CHF, based on the direct observation of the physical boiling phenomena and using a nucleate boiling limitation model which can predict a heat transfer in a nucleate boiling region including CHF. In all these models, the influence of the fluid wettability is never taken into account. Kirishenko and Cherniakov [18] developed a correlation with the contact angle as a parameter. Diesselhorst et al. [19] noticed that this equation gives higher values of CHF for high contact angles. This correlation was found to be inaccurate for water. Ahn et al. [20] proposed a new concept of flow boiling model based on the wetting zone fraction which is given as a function of the wettability. Kim et al. [21] presented an analytical model associating the wettability and the nucleation site density. Wang and Dhir [22] developed a

H.T. Phan et al. / International Communications in Heat and Mass Transfer 39 (2012) 1500–1504

provide an analytical expression for κ. Kim et al. [21] provided an expression for κ using elementary geometry and Lord Rayleigh's formula for the volume of static liquid meniscus:

Nomenclature heat exchange area, m 2 diameter, m maximum diameter, m bubble emission frequency, Hz surface tension force, N horizontal component of Fσ, N vertical component of Fσ, N gravity, m/s 2 latent heat of phase change, J/kg constant, – length, m capillary length, m liquid mass flux, kg/s pressure, Pa heat flux density, W/m 2 radius, m temperature, °C or K

Ah D D* f Fσ Fσ_h Fσ_v g Hlv K L Lc _ m P q R T

κ¼

κ¼

1=2

h
 i1=4 H lv σ ρl −ρg g

   −1 1 þ cosθ −2 2 π þ ð1 þ cosθÞ cosφ 16 π 4

ð3Þ

2. Concept of macro- and micro-contact angles and influence on bubble characteristics

correlation between the wettability and the nucleation site density. Chu and Yu [23] finally proposed an expression of CHF based on the fractal distribution of nucleation sites on boiling surfaces. They emphasize on the fact that wall superheat, contact angle and physical properties of fluid have important effects on CHF. Two models of critical heat flux (CHF) that account for the effect of wettability on CHF are those of Theofanous and Dinh [24] and Kandlikar [25], both based on the assumption that CHF occurs as an irreversible expansion of a hot/dry spot. Theofanous and Dinh [24] (similar to the one of Kutateladze [9,10]) derived the following expression to predict CHF: ρv

ð2Þ

where θ is the contact angle and φ is the heater orientation angle relative to horizontal. Contrary to the model of Theofanous and Dinh [24], the model developed by Kandlikar [25] allows one to directly take into account the surface wettability for both hydrophilic and hydrophobic surfaces. In this paper, another approach is proposed to express and characterize critical heat flux as a function of the fluid wettability. The present theoretical model of CHF is based on the concept of macro- and micro-contact angles — physical notions which are recalled in the following part. This model aims at giving a new physical point of view to interpret the pool-boiling critical heat flux.

Subscripts b bubble c capillary cl contact line d departure l liquid v vapor lv liquid/vapor CHF critical heat flux CL contact line ∞ infinite

−1=2

  sinθ π=2−θ −1=2 1− − : 2 2 cosθ

According to the authors, the above expression only applies for hydrophilic surfaces (θ b 90°). Based on a force balance on the whole bubble, Kandlikar [25] provides the following correlation for κ:

Greek symbols δ thickness, m κ geometric parameter, – φ orientation angle, degree or rad θ macro contact angle, degree or rad θμ micro contact angle, degree or rad ρ density, kg/m 3 σ liquid–vapor surface tension, N/m

qCHF ¼ κ

1501

The contact angle is usually measured at room temperature (25 °C) by depositing a liquid droplet on the sample surface. The surface and the droplet are at the same temperature, thus there is no heat exchange between them. This contact angle is denoted as θ°. However, during the nucleate boiling, the bubble is formed by the liquid evaporation caused by the heat transfer from the wall to the liquid. The liquid is now at a saturated temperature and the contact angle θ is, in general, lower than θ°, when the saturation temperature is higher than the room temperature (cf. Fig. 1a). Besides, under boiling conditions, the balance of the forces: solid–liquidσsl, liquid–vapor σlv, and solid–vapor σsv, becomes unstable due to the non-zero heat flux imposed at the solid–liquid interface. Particularly, for hydrophilic surfaces, this heat flux causes evaporation of the liquid micro-layer underneath the bubble. The thinner this layer, the lower the heat conductive resistance in the liquid and consequently the higher the heat flux passing through. In these conditions, close to the triple contact line (TCL), the heat transfer is extremely high and creates a liquid evaporation with a rate that is much higher than in the surrounding areas. Therefore, the curvature of the liquid–vapor interface changes, leading to the emergence of another contact angle named “microcontact angle” θμ. In contrast, the contact angle θ is relatively at a larger scale. It is consequently named “macro-contact angle”.

Vapor

Liquid

θ

θ°

ð1Þ

where κ is a surface dependent parameter, which is large for a poorly wetting surface but small for a highly wetted surface. Contrary to Kutateladze [9,10], which considered C as a constant, Theofanous and Dinh [24] envision that κ varies with surface properties and notably with wettability. However, Theofanous and Dinh [24] did not

θμ

a)

b)

Fig. 1. Contact angle of a liquid droplet: a) at 25° without any heat transfer and b) at saturation temperature on a heated surface: macro-contact angle θ and micro-contact angle θμ.

The surface tension force Fσ, which is applied to the TCL (cf. Fig. 2), is determined by the micro-contact angle and not by the macro-contact angle. When the nucleation is initiated, close to the TCL, the liquid evaporation may cause a micro-contact angle greater than 90°, as described by Mitrovic [26] (cf. Fig. 1b). Due to the horizontal component of the surface tension force Fσ_h, the liquid in the micro-layer moves away from the bubble axis and the TCL expands from A to B (cf. Fig. 2a). Along with the liquid movement, the micro-contact angle decreases as a result of the restoration of the surface energies balance. At position B, the micro-contact angle is equal to 90° and the surface tension force stops displacing the TCL (cf. Fig. 2b). However, the liquid inertia and the energy minimization of the system will result in a decline of the micro-contact angle to a value close to that of the macro-contact angle (cf. Fig. 2c). The horizontal component of the surface tension force reappears, but this time it moves the liquid toward the bubble axis by reducing the TCL radius. At position C (cf. Fig. 2d), the TCL finally disappears and the bubble detaches from the wall. Then the bubble growth process can restart. The micro-contact angle has an important contribution to nucleate boiling. Firstly, it directly affects the vertical component of the surface tension force Fσ_v, which contributes to maintain the bubble on the wall. In addition, it creates the TCL movement and thus affects the force equilibrium caused by liquid inertia and viscosity. Indeed, when the TCL expands from A to B, the bubble becomes bigger and the inertia of the liquid surrounding the bubble exerts a reaction force to maintain it on the wall. But when the TCL retracts from B to C, the liquid goes forward to the bubble axis, enabling the bubble departure. During the bubble growth, the macro-contact angle changes, describing a hysteresis cycle according to the hysteresis phenomenon: it decreases when the liquid recedes and increases when the liquid advances. 3. Model development As shown in the previous section, during a period of bubble growth, liquid movement occurs in the microlayer zone beneath the bubble. Assuming in a first approach that this layer is similar to a cylinder made by the contact line of radius Rcl and the thickness δ when the bubble reaches its maximum size (cf. Fig. 3), the time-averaged liquid mass flux (cf. Fig. 4) towards the bubble axis can be estimated as: _ l ¼ f ρl π R2cl δ m

ð4Þ

Bubble axis

H.T. Phan et al. / International Communications in Heat and Mass Transfer 39 (2012) 1500–1504

Liquid

1502

Rcl

Microlayer zone

δ

Fig. 3. Schematic view of the microlayer zone at maximum bubble size.

where f is the bubble emission frequency, ρl is the liquid density, and Rcl and δ are the radius of the contact line and the thickness of the microlayer at maximum bubble size, respectively. The bubble emission frequency can be determined from the bubble departure diameter using the correlation of Zuber [27] as: " #1=4 σ g ðρl −ρv Þ 1 f ¼ 0:59  : Db ρ2l

ð5Þ

The approach developed to characterize CHF in this study uses a standard model for CHF in which the model for bubble departure based on micro- and macro-contact angles by Phan et al. [28] is introduced. Using a force balance at the TCL (at position B in Fig. 2 where θμ =90°), including contact pressure forces, surface tension and buoyancy, and mass conservation during liquid evaporation, Phan et al. [28] show that Rcl can be obtained as: Rcl ¼

rffiffiffi  3 ρl −1=2 −1=2 tan θ Lc 2 ρv

ð6Þ

where Lc is the capillary length given by: Lc ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ : g ðρl −ρv Þ

ð7Þ

Bubble axis

Vapour Liquid Fσ ,h Fσ ,v A

a)

θ θ

Fσ B

b)

Detaching

90°

bubble

C

B

d)

c)

C A

B

e

Fig. 2. Movement of the contact line during bubble growth: a) the bubble growth initiates, b) the contact line stops displacing when θμ = 90°, c) the contact line starts moving toward the bubble axis, d) the bubble detaches from the wall and e) hysteresis of θ and θμ.

H.T. Phan et al. / International Communications in Heat and Mass Transfer 39 (2012) 1500–1504

1503

The critical heat flux (CHF) occurs when all the liquid is vaporized, i.e. x = 1. This implies:

Time

Growth time

qCHF ¼

Waiting time

h i 1 _ l C p;l ðT s −T ∞ Þ þ H lv : m Ah

ð15Þ

Eqs. (10) and (15) imply:

θ µ = 90°

qCHF ¼ qCHF;45B  tan θ

Waiting

−1=3

ð16Þ

wherein qCHF,45° is the critical heat flux at θ = 45°, which is determined as:

Grow Fig. 4. Simplified time evolution of the liquid mass flow in the microlayer zone.

qCHF;45 B ¼

i 1 Kf  Ka h C p;l ðT s −T ∞ Þ þ H lv : Ah Kd

ð17Þ

The mean heat exchange area has the following form: The model proposed by Phan et al. [28] permits one to obtain the expression of the diameter of bubble departure Db and of the liquid microlayer thickness at the departure of the bubble δ as:

Db ¼

δ¼

rffiffiffi!1=3    1=3 3 ρl −1=2 ρl −1=6 6 −1 tan θ Lc 2 ρv ρv

rffiffiffi  3 ρl −1=2 1=2 Lc tan θ : 2 ρv

ð8Þ

ð9Þ

Relation (8) has been obtained thanks to mass conservation during liquid evaporation, and geometric considerations using the expression of Rcl given by expression (6). Relation (9) is then obtained arguing the fact that δ/Rcl = tan θ. Thereby, Eq. (4) becomes: _l¼ m

Kf  Ka −1=3 tan θ Kd

ð10Þ

where Kf, Kd and Ka are constants depending on the fluid properties: " #1=4 σ g ðρl −ρv Þ K f ¼ 0:59 ρ2l

Kd ¼

ð11Þ

rffiffiffi!1=3    1=3 3 ρl −1=2 ρl 6 −1 Lc 2 ρv ρv

"rffiffiffi  #3 3 ρl −1=2 K a ¼ π ρl Lc : 2 ρv

ð12Þ

ð13Þ

The inlet temperature of the liquid flow is assumed to be equal to the bulk temperature that corresponds to the temperature of liquid in the boiling vessel far from the sample heater. Hence, the energy balance implies the following expression of the heat flux: q¼

1 _ m Ah l

h

C p;l ðT s −T ∞ Þ þ H lv x

i

ð14Þ

where Ah is the mean heat exchange area between the heated surface and the fluid, Cp,l is the liquid specific-heat capacity, Ts and T∞ are the saturation temperature and the temperature of the bulk liquid respectively, Hlv is the latent heat of vaporization and x is the vapor quality.

Ah ¼

π 2 D 4 x

ð18Þ

where Dx corresponds to the mean diameter of heat exchange surface between the heated surface and the fluid. At a first approximation, it pffiffiffi is taken as Dx ¼ Db =2 2. For a contact angle of 45°, the model of bubble departure diameter gives: K Dx ¼ pdffiffiffi : 2 2

ð19Þ

Eqs. (17), (18) and (19) imply: qCHF;45 B ¼

i 32 K f  K a h C p;l ðT s −T ∞ Þ þ H lv : 3 π Kd

ð20Þ

This critical heat flux depends on the value of the bulk temperature T∞. However, for water, T∞ is relatively close to Ts so that the term Cp,l(Ts − T∞) is very small compared to the latent heat Hlv (about 20 times lower). In our experimental set-up [29], Ts = 100 °C and T∞ = 85 °C. For refrigerants, as FC-72 for example, this assumption is less valid because for Ts = 60 °C, Hlv is only about 5 times greater than Cp,l(Ts − T∞). The analytical Eq. (20) allows determining the CHF; this expression is based on the concept of macro- and micro-contact angles, the present model of bubble departure diameter and the energy conservation. The CHF predicted by the correlation of Kandlikar [25], the correlation of Kim et al. [21] and the present model are compared to the experimental CHF given by recent literature studies on water and nanofluids [21,25,30–32]. In Fig. 5, the present model shows the best agreement with the trend of CHF evolution as a function of contact angle with 80% of data included within ±30%. This model (curve 1) is significantly in better agreement with experiments than the model developed by Kim et al. [21] (curve 3). Besides, contrary to the model developed by Kandlikar [25] (curve 2), the present model seems to show a similar behavior to experimental results obtained by Kim and Kim [30] for low contact angles for which CHF appears to increase. 4. Conclusion The concept of macro- and micro-contact angle was introduced in a previous study [28] to describe the bubble growth process. It represents a new approach to characterize this evolution. Based on these previous results, the present paper enables the development of a new expression for the critical heat flux, taking into account the contact angle effects. This expression is in fair agreement with experimental results of the literature obtained in subcooled boiling. Additional experiments need to be performed, with the same

1504

H.T. Phan et al. / International Communications in Heat and Mass Transfer 39 (2012) 1500–1504

6000

Kim et al. [21] Kim & Kim [30] Liaw & Dhir [31] Coursey & Kim [32]

5000

CHF (kW/m²)

4000

This model

1

3000

Kim et al. [21]

3

2000

2 1000

Kandlikar [25], ϕ = 0°

0 0

30

60

90

Contact angle (°) Fig. 5. Evolution of the CHF with the contact angle for water and water-based nanofluids (models predictions are shown in full lines).

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