MODELING OF DROPWISE CONDENSATION ON FLAT SURFACES

Transcription

1 13th Intenational Confeence on Heat Tansfe, Fluid Mechanics and Themodynamics MODELING OF DROPWISE CONDENSATION ON FLAT SURFACES Riccado Pain, Andea Penazzato, Stefano Botolin, Davide Del Col * *Autho fo coespondence Depatment of Industial Engineeing Univesity of Padova, Via Venezia, Padova, Italy davide.delcol@unipd.it ABSTRACT Thee ae two ways fo a vapo to condense on a suface: filmwise condensation (FWC) and dopwise condensation (DWC). The inteest in DWC is based on the potential incease of the condensation heat tansfe coefficient (HTC) by 6 to 10 times compaed to the values measued duing filmwise condensation. Fo this eason, seveal eseach goups aound the wold have tied to pomote the dopwise condensation and to descibe the undeneath mechanisms. Such models descibe the phenomena that take place duing dopwise condensation: the nucleation of a doplet until its depatue, the heat exchanged by the dop duing its lifetime and the doplets population on the suface. The pesent pape aims at pesenting some of the models developed in the past yeas which can be used to descibe the DWC pocess. In paticula, similaities and diffeences between the models ae highlighted and thei pedictions ae compaed against expeimental data measued at the Two-phase Heat Tansfe Laboatoy of the Univesity of Padova. INTRODUCTION In 1930, fo the fist time, dopwise condensation was descibed in a scientific pape [1] and the heat tansfe potential of this type of condensation, as compaed to the filmwise condensation mode, was highlighted. Dopwise condensation is usually obtained when the suface is hydophobic: the suface induces the beakage of the liquid film that nomally fully wets the suface, eplacing it with a lage amount of andomly distibuted doplets. The study of the DWC aoused a discontinuous inteest fom the heat tansfe community, and the fist semi-empiical models wee developed only stating fom the second half of the 60s [2]. Afte this peiod, to find othe publications about this topic, we have to wait until the end of 90s and paticulaly until the beginning of new millennium [3 5]. Indeed, in ecent yeas, the mateial science has made majo pogess in the development of new sufaces that could favo dopwise condensation. Moeove, eseaches have focused thei attention on mico- and nanostuctued sufaces displaying supehydophobic chaacteistics [6] which may lead to even highe heat tansfe coefficients pomoting the phenomenon of jumping doplets. The pesent wok deals with dopwise condensation on flat sufaces and thus without consideing the pesence of atificial oughness that, fo example, is necessay to ealize the afoementioned supehydophobic substates. Seveal heat tansfe models that can be used duing DWC on flat sufaces have been poposed in the liteatue and, in the pesent pape, thee diffeent studies have been selected: Le Feve and Rose as epoted in Rose [2], Abu-Oabi [3], and the ecent wok by Kim et al. [4]. The esults calculated by the models ae compaed with some expeimental data taken on hydophobic polished aluminum substates by the pesent eseach goup. Heat tansfe coefficients have been measued in a two-phase themosiphon loop duing pue steam dopwise condensation ove plain vetical sufaces. A high-speed camea has been used to study the diffeent paametes affecting the DWC phenomenon, such as the doplet depating adius and the population of lage doplets. NOMENCLATURE A [m 2 ] aea g [m s -2 ] gavity acceleation G [m s -1 ] dop gowth ate h [J kg -1 ] specific enthalpy h i [W m -2 K -1 ] intefacial heat tansfe coefficient HTC [kw m -2 K -1 ] heat tansfe coefficient ṁ [kg s -1 ] mass flow ate n [m -3 ] small dop population N [m -3 ] lage dop population N s [m -2 ] nucleation sites q [kw] heat flow ate q [kw m -2 ] heat flux [m] adius R [J mol -1 K -1 ] gas constant S [m 2 s -1 ] suface enewal t [ C] tempeatue T [K] tempeatue v [m s -1 ] velocity Geeks α [-] accommodation coefficient γ [-] atio of the specific heat capacities δ [m] thickness θ [ ] contact angle λ [W m -1 K -1 ] themal conductivity ρ [kg m -3 ] density σ [N m -1 ] suface tension τ [s] sweeping peiod 91

2 Subscipts a c d e coat cool l max min SAT v WALL advancing cuvatue dop effective coating coolant liquid maximum minimum eceding satuation vapo wall THEORETICAL MODELS The dopwise condensation stats at molecula level, with the fomation of small clustes of molecules (minimum adius min) in a numbe of pefeential nucleation sites (N s), which gow by diect condensation of steam on them. Subsequently, due to the poximity of the nucleation sites, the dops come into contact with each othe, they coalescence (effective adius e) and, when maximum adius max is eached (the extenal foces exceed the adhesion foce which allows them to emain attached to the suface), the dops begin to slip away. While slipping, the doplets continue to gow fo coalescing with othe doplets that they encounte along thei path leaving the suface clean and available fo the fomation of new nuclei [7]. All the thee models consideed in the pesent pape account fo the afoementioned mechanisms. Since duing DWC the whole suface is coveed by a lage amount of doplets andomly distibuted with diffeent sizes, a distibution function of the doplet population has to be defined. Theefoe, knowing the heat exchanged by a single dop and the numbe of doplets pe unit aea, the heat flux can be calculated though the opeation of integation fom the minimum adius to the maximum one. So, naming n() the "small" doplets population ( < e) and N() the population of "lage" doplets ( > e), it follows that: e max q = q d () n()d + q min d () N()d (1) e The heat tansfe coefficient is calculated as: HTC = q T (2) whee T is the tempeatue diffeence between the satuation tempeatue (t SAT) and the suface tempeatue (t WALL). The minimum doplet adius is defined when a cluste of molecules is in equilibium with the suface and it is calculated as [7] min = 2σv l h lv (t SAT t WALL ) 13th Intenational Confeence on Heat Tansfe, Fluid Mechanics and Themodynamics whee σ is the suface tension, v l is the specific volume of the satuated liquid and h lv the specific enthalpy of evapoation. Le Feve and Rose model (1966) The fist model developed fo dopwise condensation was poposed by Le Feve and Rose [2] in The heat flow ate (3) though a single dop is obtained with a semi-empiical elationship: q d () = ΔT 2σT SAT ρ l h lv K 1 λ l +K 2 ( ) T SAT h lv 2 ρl γ+1 γ 1 [RT SAT 2π ]0.5 (4) whee K 1 and K 2 ae constants imposed by the authos, λ l is the liquid conductivity, ρ l is the liquid density, γ is the atio of the specific heat capacities and R is the ideal-gas constant. At the numeato, in addition to the tempeatue diffeence T, the effect of the doplet cuvatue is taken into account. At the denominato, the fist tem epesents the esistance to the heat conduction though the condensate, meanwhile, the second tem epesents the esistance to the mass tanspot at the inteface between vapo and liquid. The doplet population coves only the lage doplets with the expession: N() = 1 ( ) 2 3 3π 2 max max whee the maximum adius is defined as: max = K 3 [ σ ρg ] 1 2 K 3 is a constant and g is the gavity acceleation. Heeafte, Eq. (5) will be used fo all the thee models. As a final step, the heat flux is detemined by Eq. (1). Abu-Oabi model (1998) The model developed in 1998 by Abu-Oabi [3] was the fist that computes the heat tansfe though a single dop by incopoating the vaious themal esistances fom the vapo to the suface and consides both the populations of small and lage doplets as poposed in [8,9]. The themal esistances can be evaluated as the atio between tempeatue dop and heat flow ate q d; in paticula the tempeatue dop due to the intefacial esistance is: ΔT i = (5) (6) q d h i 2π 2 (7) whee h i is the intefacial heat tansfe coefficient, which includes the accommodation coefficient (α). The expeimental data used in this pape ae taken in satuated conditions of steam, thus α is assumed equal to 1 as suggested in [3]. The themal esistance due to heat conduction though the dop can be estimated fom Eq. (8) ΔT d = q d 4π 2 λ l (8) The pomoting laye on the substate adds a tempeatue dop equal to: ΔT coat = q dδ 4π 2 λ coat (9) whee δ is the laye thickness. Finally, the doplet cuvatue gives a tempeatue vaiation ΔT c = 2T SATσ h lv ρ l (10) 92

3 Fom Eqs. (7-10) and consideing Eq. (3), the heat flow ate though a single dop of adius can be calculated as: q d () = 4π2 (1 min )ΔT ( δ ) λ coat λ l h i (11) With egad to the doplet population, the autho extends the Le Feve and Rose analysis (Eq. (5)) in ode to add the "small" doplet population. Fo the sake of bevity, only the main Eqs. ae epoted in the following. Assuming that in a given suface aea A, the numbe of enteing doplets is equal to the numbe of leaving doplets plus the doplets swept and assuming the gowth ate fo a dop [9] equal to G = d dt the doplet balance in that aea will be: (12) An 1 G 1 t = An 2 G 2 t + Sn t (13) whee n is the numbe of dops pe unit aea pe unit dop adius, S is the ate at which the substate suface is enewed due to sweeping, n is the aveage population density in the size ange 1 to 2, = 2-1, and t is an incement of time. Fo Δ 0, the Eq. (13) becomes G dn dg + n + n = 0 (14) d d τ whee τ = A/S is the sweeping peiod. As a bounday condition, the authos impose that the population of small doplets equals that of lage doplets at e: n( e ) = N( e ) (15) whee e is calculated assuming that the nucleation sites fom a squae aay, e = 1 4N s ; the Eq. (14) can be integated with espect to, obtaining: n() = N( e ) ( e min )(A 2 + A 3 ) e ( min )(A 2 e + A 3 ) eb 1+B 2 (16) whee B 1 = A 2 A 1 τ [ e min ( e min ) 2 min ln ( min )] (17) e min B 2 = A 3 [( A 1 τ e ) min ln ( min )] (18) e min τ = 3 e 2 (A 2 + A 3 ) 2 A 1 [8A 3 e 14A 2 e min +11A 2 e 2 11A 3 min ] 13th Intenational Confeence on Heat Tansfe, Fluid Mechanics and Themodynamics (19) The thee paametes (A 1, A 2, and A 3) in Eqs. (16-19) ae defined as: A 1 = 2ΔT ; A ρ l h 2 = 1 ; A lv λ 3 = 2 + δ (20) l h i λ coat which ae deived fom Eq. (11) by means of Eq. (12) and Eq. (21) (heat exchanged though a dop) q d () = ρ l h lv (2π d dt ) (21) Kim et al. model (2011) Both the pevious models assume that the doplets gow duing dopwise condensation with a hemispheical shape, i.e. a contact angle (θ) between solid and liquid equal to 90. Howeve, liquids wet the suface with diffeent contact angles depending on the suface tension balance at the tiple line [6]. The model of Kim et al. [4] intoduces the contact angle as a vaiable in ode to fill this gap. The authos studied how the contact angle influences the dopwise condensation pefomances in the ange fom 90 to 150. Since the oughness is not consideed, using wate as a woking fluid, the analysis should stop at about 120. A simila appoach to the one poposed by Abu-Oabi is consideed: the themal esistances fom the vapo to the suface ae consideed and, in this case, the doplet gowing angle is also accounted fo. In paticula, the esistance due to conduction in the dop can be obtained fom: ΔT d = q dθ 4πλ l sin θ (22) Eq. (22) changes damatically the conduction though a single doplet, giving to the doplets thei natual spheical shape instead of a flat laye as in Le Feve and Rose and Abu-Oabi. The heat flow though the single dop is calculated as: q d () = ΔTπ 2 (1 min ) δ ( λ coat sin θ 2+ θ 4λ l sin θ + 1 2h i (1 cos θ) ) (23) Also this model consides a division between "lage" and "small" doplets. The coefficients fo "small" doplets population (Eq. 16) ae: A 1 = 2ΔT θ(1 cos (θ)) ; A ρ l h 2 = ; A lv 4 λ l sin (θ) 3 = 1 δ(1 cos (θ)) + 2 h i λ coat sin (θ) 2 (24) In the model, the maximum doplet adius eached duing dopwise condensation on a vetical suface is obtained fom the balance between the capillay foce and the gavity foce: max = 6(cos (θ ) cos (θ a)) sin(θ) π(2 3 cos(θ)+cos 2 (θ)) σ ρ l g (25) whee θ a and θ ae the advancing and eceding contact angles, espectively. THEORETICAL MODELS COMPARISON The fist model poposed by Le Feve and Rose needs the popeties of the fluid (calculated at the satuation tempeatue T sat) and the wall tempeatue as input values. Instead, in the model by Abu-Oabi, due to its inceased complexity, in addition to the popeties of the hydophobic laye (thickness and conductivity), an accommodation coefficient must be povided fo the estimation of the themal esistance at the inteface (Eq. 10). Futhemoe, the model by Abu-Oabi distinguishes the doplet population in two categoies, small doplets and lage doplets, which ae sepaated using the effective adius e, thus N s (Eq. (15)). The model by Kim et al., with a futhe step, adds the contact angle (othewise assumed equal to 90 in the othe coelations). In Table 1, all the input of the models ae summaized and the values used fo the compaison with expeimental data measued by the pesent eseach goup ae epoted. 93

4 13th Intenational Confeence on Heat Tansfe, Fluid Mechanics and Themodynamics Table 1. List of input vaiables consideed in the models. Vaiable Value Le Feve Abu- Kim et al. & Rose Oabi tsat [ C] 108 X X X T [ C] 5 X X X δ [μm] 0.2 X X λcoat [W m -1 K -1 ] 0.2 X X α [-] 1 X X NS [m -2 ] X X θ [ ] 90 X θa [ ] 88.6 X θ [ ] 63.4 X In Fig. 1, a compaison between the doplets population calculated using all the thee models is plotted. The doplet population is the numbe of dops (#) pe unit aea [m -2 ] pe unit adius [m -1 ], thus it is expessed in [# m -3 ]. The majo diffeence among the models is, indeed, the intoduction of the small doplet population fo doplet adius anging between min and e. In this zone, the population balance poposed by Le Feve and Rose oveestimates the numbe of small dops as compaed to the othe models, eaching a maximum of about 4 odes of magnitude nea the min egion. Assuming θ = 90, the model by Abu-Oabi and the model by Kim et al. pedict the same fequency distibution. The small deviation in the "lage" dop aea is due to the diffeent value of the maximum dop adius: in the case of Abu-Oabi model, a semi-empiical expession (Eq. (6)) is used, while the model by Kim et al. consides a foce balance (in Eq. (25)). Fig. 1. Doplet population vesus doplet adius calculated by Le Feve & Rose, Abu-Oabi and Kim models. In Fig. 2, the computed heat flow ate exchanged though a single doplet is plotted vesus the doplet adius: the heat flow ate inceases when the doplet becomes bigge. It should be noticed that, even adopting the same appoach and imposing a contact angle equal to 90, thee is still a significant diffeence between the models by Abu-Oabi and Kim et al. In the model by Abu-Oabi, the conduction esistance is calculated consideing a flat laye with thickness equal to the adius of the single dop (Eq. 8). Instead, Kim et al. accounts fo the spheical shape of doplets (Eq. 22). Fig. 2. Heat flow ate in a single dop vesus the doplet adius calculated with the thee models. Consideing a contact angle equal to 90, the heat tansfe esistance due to heat conduction in the doplet obtained fom Eq. (22) (Kim et al. model) is about 50% highe compaed to the value calculated with the model by Abu-Oabi (Eq. 8). This discepancy esults in lowe heat flow ate computed with the Kim et al. model. The Le Feve and Rose model consides the spheical shape of the doplet adding a constant coective tem (K 1), lowe than 1, to account fo the spheical shape of the dop. COMPARISON WITH EXPERIMENTAL MEASUREMENTS Expeimental measuements have been pefomed by the pesent eseach goup on aluminum samples, which wee teated with a hydophobic laye. The detailed pocedue and the chaacteization of the sample is epoted in [10]. The test ig is a themosiphon loop and satuated vapo is used as opeative fluid. The system consists of fou main components (boiling chambe, test section, cooling wate loop and post-condense) and it allows simultaneous visualization of the condensation pocess. The sample is equipped with six themocouples (two at the inlet, two in the middle and two at the outlet along the steam diection). The specimen is placed inside the test section: the teated suface is exposed to the vapo flow wheeas the opposite suface is cooled by wate. A detailed desciption of the expeimental appaatus, the data eduction technique and the uncetainty analysis can be found in [11,12]. Fig. 3 epots an image taken duing a dopwise condensation test on the aluminum sample: the aea of analysis is highlighted. AREA OF ANALYSIS Fig. 3. Image of an aluminum sample duing a DWC test. 94

5 a) b) c) 13th Intenational Confeence on Heat Tansfe, Fluid Mechanics and Themodynamics located at two diffeent positions z 1 and z 2 fom the suface of the sample: T wall = T z1 + (T z1 T z2 ) z 1 z 2 z 1 (27) Fo each opeating condition, the measuements obtained at the aea of analysis (Fig. 3) ae epoted. The main opeative paametes duing the tests ae: t SAT = 108 C, v VAP = 2.6 m s -1, coolant wate tempeatue fom 10 C to 85 C. A compaison between the expeimental data and the heat tansfe coefficients pedicted by the models is epoted in Fig. 4. As shown in Table 1, each model needs diffeent input and the values actually used to un the models ae epoted in each gaph. The expeimental campaign was pefomed with vapo flow and thus the vapo velocity can affect the doplet depatue adius. In fact, in Eqs. (6) and (25) the vapo shea stess is not taken into account. Fo this eason, the maximum doplet adius is fistly measued fom videos and then it is imposed to all the models as a bounday condition. All the models display a good ageement with the expeimental data, leading to a mean deviation of about 10%. It is woth mentioning that the Le Feve and Rose model needs only the satuation tempeatue (and fluid popeties) as input vaiables. In the case of Abu-Oabi and Kim et al. models, the numbe of nucleation sites is an input vaiable which is chosen hee to get a good pediction of the data. The numbe of nucleation sites used in the Abu-Oabi model is diffeent fom the one adopted fo the Kim et al. model: N s used in the model by Kim et al. must be highe because the Kim et al. model pedicts a lowe doplet heat flow ate (Fig. 2). IMAGES ANALYSIS As mentioned befoe, the test section allows the diect visualization of the dopwise condensation (Fig. 3). Fom the analysis of the images, seveal fundamental aspects of the models can be veified. Table 2 shows the maximum adius max obtained fom the models and fom the visualizations. The measuements have been epeated seveal times in ode to minimize the uncetainty. Fig. 4. Compaison between expeimental and calculated heat tansfe coefficients: a) Le Feve and Rose model; b) Abu- Oabi model; c) Kim et al. model. The main equations used fo the detemination of the heat tansfe coefficient ae summaized below. The local heat flux is obtained by applying the Fouie Law: q loc = λ al ΔT Δz (26) The suface tempeatue T wall is obtained fom a linea intepolation of the tempeatues measued by the themocouples Table 2. Compaison between expeimental and theoetical doplet depating adius. Model Max. Radius [mm] Le Feve & Rose [2] 1.00 Kim et al. [4] 1.26 Expeimental 0.93 The adius measued fom the images is lowe than the adius pedicted by the two models (Eqs. (6) and (25)). A possible explanation is that such discepancy is due to the speed of the vapo. In fact, an assumption of the models is that the steam is in a steady quiescent state, thus at zeo velocity. The maximum doplet adius is impotant fo the lage doplet population (Eq. 5) and fo the calculation of the heat flux being the uppe limit of the integal in Eq. (1). Anothe inteesting infomation achievable by the videos is the lage doplet population. Due to the esolution of the 95

6 13th Intenational Confeence on Heat Tansfe, Fluid Mechanics and Themodynamics optical system, only a small pat of the population is appeciable, with a lowe limit of about 0.1 mm. The measuement has been pefomed in the highlighted aea of the sample (Fig. 3) and, in ode to compae the data with Eq. (5), a statistical appoach has been adopted. The expeimental data have been divided in 20 classes and the numbe of dops fo each class has been calculated. In ode to make the compaison with the model, the integal of the distibution of the doplets fo each peviously defined class is computed. The numbe of doplets fo each class is: b n dops pe class = A N()d (28) a whee A is the aea of the aea of analysis, a and b ae the minimum and maximum adius fo the i-th class, espectively. Fig. 5 shows the esult of the pesent analysis. Fig. 5. Numbe of dops: calculated vesus expeimental esults. Even if the potion of measuable doplets size is vey limited as compaed to the enomous vaiety of doplets size (see Fig. 2), the measuements confim Eq. (5). It can be noted that, since the doplets each the maximum adius in the inlet zone and then they fall down, in the studied zone the maximum adius is significantly lowe. CONCLUSIONS Thee dopwise condensation models available in the liteatue and developed espectively by Le Feve and Rose [2], Abu-Oabi [3] and Kim et al. [4] have been analyzed, highlighting analogies and diffeences between them. The models adopted a simila appoach: the heat flux is calculated fom the doplets population and fom the heat flow ate though a single dop. The heat tansfe coefficients calculated using these models ae compaed against expeimental data taken by the pesent eseach goup. Tests wee pefomed duing steam condensation ove hydophobic aluminum sufaces. The calculated heat tansfe coefficients ae in good ageement with measuements. Futhemoe, high speed visualizations togethe with images analysis ae employed to detemine the population of doplets, at last fo those that can be egaded as lage doplets. REFERENCES [1] Schmidt, E., Schuig, W., Sellschopp, W., Vesuche übe die kondensation von wassedampf in film- und topfenfom., Fosch. Im Ingenieuwes. 1 (1930) [2] J.W. Rose, Dopwise condensation theoy and expeiment: a eview, Poc. Inst. Mech. Eng. Pat A J. Powe Enegy. 216 (2002) doi: / [3] M. Abu-Oabi, Modeling of heat tansfe in dopwise condensation, Int. J. Heat Mass Tansf. 41 (1998) doi: /s (97)00094-x. [4] S. Kim, K.J. Kim, Dopwise Condensation Modeling Suitable fo Supehydophobic Sufaces, J. Heat Tansfe. 133 (2011) doi: / [5] N. Miljkovic, R. Enight, E.N. Wang, Modeling and Optimization of Supehydophobic Condensation, J. Heat Tansfe. 135 (2013) doi: / [6] R. Enight, N. Miljkovic, J.L. Alvaado, K. Kim, J.W. Rose, Dopwise Condensation on Mico- and Nanostuctued Sufaces, Nanoscale Micoscale Themophys. Eng. 18 (2014) doi: / [7] S. Khandeka, K. Mualidha, Dopwise Condensation on Inclined Textued Sufaces, Spinge, [8] H. Tanaka, A Theoetical Study of Dopwise Condensation, J. Heat Tansfe. 97 (1975) [9] H.W. Wen, R.M. Je, On the heat tansfe in dopwise condensation, Chem. Eng. J. 12 (1976) doi: / (76) [10] P. Innocenzi, M.O. Abdiashid, M. Guglielmi, Stuctue and Popeties of Sol-Gel Coatings fom Methyltiethoxysilane and Tetaethoxysilane, J. Sol-Gel Sci. Technol. 3 (1994) doi: /bf [11] D. Del Col, R. Pain, A. Bisetto, S. Botolin, A. Matucci, Film condensation of steam flowing on a hydophobic suface, Int. J. Heat Mass Tansf. 107 (2017) doi: /j.ijheatmasstansfe [12] A. Bisetto, S. Botolin, D. Del Col, Expeimental analysis of steam condensation ove conventional and supehydophilic vetical sufaces, Exp. Them. Fluid Sci. 68 (2015) doi: /j.expthemflusci