Breath Figure Spot: a Recovery Concentration manifestation
BBreath Figure Spot: a Recovery Concentrationmanifestation
Ali Alshehri a,b,+ and Pirouz Kavehpour a a Mechanical and Aerospace Engineering Department, Henry Samueli School of Engineeringand Applied Science, University of California, Los Angeles, CA 90095, USA b Mechanical Engineering Department, King Fahd University of Petroleum and Minerals(KFUPM), Dhahran 31261, Saudi Arabia + Corresponding Author: Phone: +1 (310)-849-7913E-mail: [email protected]
January 12, 2021 a r X i v : . [ phy s i c s . f l u - dyn ] J a n bstract Directing a jet of humid air to impinge on a surface that is cooled below the dewpoint results in micro-sized water droplets. Lord Rayleigh discussed the phenomenonby contrasting clean to flame-exposed glass and called such behaviour Breath Fig-ures (BF). Historically, utilizing dew as a water source was investigated by severalscientists dating back to Aristotle. However, due to the degrading effects of air as anon-condensable gas (NCG) such efforts are limited to small scale water productionsystems and exhaled breath condensate (EBC) technology, to name a few. Recently,the concept of BF has been utilized extensively in the generation of micro-scale poly-mer patterns as a self-assembly process. However, the generation of BF on surfaceswhile being impinged by a humid air jet has not been quantified. In this work, weillustrate that a BF spot generated on a cooled surface is a manifestation of a recoveryconcentration. The concept is analogous to the concept of adiabatic-wall temperaturedefined for heat transfer applications. Upon closer examination of the vapor concentra-tion distribution on a cooled impinged surface, we found that the distribution exhibitsdistinct regimes depending on the radial location from the center of the impingementregion. The first regime is confined within the impingement region, whereas the secondregime lies beyond this radial location including the wall jet region. Scaling analysisas well as numerical solution of the former regime shows that the maximum concentra-tion on the surface is equivalent to its counterpart of a free unbounded jet with similargeometrical conditions. Additionally, the scaling analysis of the latter regime revealsthat the jet speed and standoff distance are not important in determining the recoveryconcentration. However, the recovery concentration is found to vary monotonicallywith the radial location. Our conclusions are of great importance in optimizing jetimpingement where condensation phase change is prevalent.
Keywords: Condensation; Phase Change; Mass Transfer, Breath Figure OMENCLATURE a v constant associated with the Gaussian distribution of the jet velocity a ω constant associated with the Gaussian distribution of the jet concentration C − constants that are independent of jet velocity and standoff distance D Tube diameter, m g Gravitational acceleration, m/s H Standoff distance between tube exit and condensation surface, m L Tube length, m D Tube diameter, m M Molecular weight, kg/mol P Pressure, Pa Q Volumetric flow rate, m /s Re Reynolds number RH Relative humidity D Tube diameter, m r Radial direction in polar coordinate system T Temperature , o C v velocity, m/s z Axial direction in polar coordinate system
Greek symbols ν Kinematic viscosity, m /s ρ Density, kg/m ω Species mass fraction defined as the ratio between the density ofa certain species to the total density at a prescribed state δ Thickness of the boundary layer in the wall jet region, m
Superscripts and subscripts a Property related to air BF Related to the diameter of the Breath Figure spot j Property related to the jet at the tube exit condition3 ax Property related to the maximum value in a velocity/concentrationdistribution, usually located at the center (r=0) o Related to the characteristic velocity r Property related to the recovery concentration which is defined as theconcentration on the wall where no condensation is present s Property related to condensation surface condition v Property related to the water vapor ∞ Evaluated at the ambient conditions4
Introduction
Condensation is a prevalent phenomenon in nature and industry, yet not fully explored. Innature, most living species rely evidently on condensed atmospheric vapor. Moreover, someplants and animals get their share of fresh water by evolutionary modified surfaces thatenhance the condensation process, such as the Darkling beetles [1], and Sequoia Semper-virens [2]. Utilizing the phenomenon in numerous applications has been a course of scientificcuriosity for a very long time, dating back to Aristotle (300 BC). In modern era, utilizingcondensation has gone beyond large scale desalination plants to micro- and nano-scale lithog-raphy techniques [3, 4]. In daily experience, people observe that upon breathing against aglass surface, white traces of condensate are generated. Upon a closer look under the micro-scope, such traces are composed of sessile droplets of a micron size [5–9]. External lightingscatters in all directions from dewed surfaces, therefore, they appear cloudy. However, oldobservations by Aitken [10–12] and Lord Rayleigh [13, 14] discussed that flame-exposed glassdoes not show such cloudiness. In their discussion, they termed such behaviour as breathfigures (BF) for obvious reasons. It is with our present understanding of surface energy effectthat we are aware of wettability importance. Today, the phenomenon has been utilized inself assembly processes to produce honeycomb polymer patterns [4, 15–17].The presence of untraceable amounts of Non-condensable gases (NCG), such as air,in condensation processes has shown to dramatically reduce the condenser efficiency [18,19]. The reason of this reduction is the accumulation of NCG on the liquid-vapor interfaceintroducing a layer that is NCG-rich. The condensation rate becomes solely limited by thediffusion of vapor through this layer. Researchers have shown that heat transfer is thuslimited by this layer’s thermal resistance. Even though experimental studies have beensuccessful in reducing NCG effect by means of vacuuming test chambers to environment[20], it is a highly impractical solution in large scale equipment. NCG can break throughequipment via leak points, which is a problem of its own, or as a chemical reaction productof vapor interacting with the equipment material [21].In efforts to mitigate the negative effect of NCG, other active techniques have beenutilized, such as extended surfaces [22–25]; direct contact between gas and cooling medium526–32]; and different NCG carriers [33, 34]. Even though the former two solutions arepromising, the latter seems to address the problem at its core, i.e. the effect of vapordiffusion through the diffusion layer which in result affects the heat and mass transfer.However, improvements from those techniques come with great material cost (former twotechniques) or industrial impracticality (latter technique). Investigating the problem of NCGfurther shows that the solution lies within two possibilities; (1) increasing heat/mass transfercontact area ( A ); (2) increasing heat/mass transfer coefficient ( h ). The optimal solutionshould be obtained by maximizing the design parameter ( hA ) while minimizing the requiredcost. Jet impingement of heat transfer fluids has shown a great potential in increasing theheat transfer coefficient for single phase [35–40] as well as multi-phase applications [41–47].Utilization of jet impingement to improve condensation heat transfer has not beentackled in literature. Therefore, we present in this paper a first look at the problem. Initially,we pondered upon a sentence Lord Rayleigh wrote in 1911 about the generation of BF. ‘[as]the breath [was] led through a tube[, the] first deposit occurs very suddenly.’ [13] Uponperforming a simple experiment of breathing through a paper straw against a mirror, wenoticed the sudden appearance of a condensate spot. The spot had a shape similar to thestraw exit, a circle of defined boundaries. However, to our surprise, the condensate spot wasweakly influenced by the strength (speed) of our breath and the distance between the mirrorand the straw exit. This led us to build a simple experimental setup to control the mentionedvariables. We show here that condensate spots are manifestations of a recovery concentrationconcept. The recovery concentration concept is analogous to the recovery or adiabatic-walltemperature investigated by Hollworth and Wilson [48, 49]. In their work, they showed thatconsistent results were obtained upon basing Nusselt number correlations on the recoverytemperature difference rather than the apparent temperature difference. In this work, weshow that the recovery concentration manifests itself as a condensate spot which we callBreath Figure (BF) spot. This spot defines the effective area over which condensation ofthe jet’s vapor takes place. Hence, we believe that quantifying this parameter is an essentialstep towards understanding condensation improvement by jet impingement.6 Experimental method
In Figure 1, we show the experimental setup which consists of a humidifier, a flow system,and a condensation surface. Dry air was first directed into a humidifier tank through severalspargers to produce a humid air jet with the desired relative humidity. The humidifier tankwas filled with DI water at room temperature, therefore, resulting in a room-temperaturejet of humid air, T ∞ = T j = 22 o C. The flow rate of the air was controlled by a flow-adjustment valve and was measured using a rotameter (Walfront, model no. LZQ-7). Flowrate ranging from 1 LPM to 10 LPM were used in our experiments. The jet of humidifiedair exited a tube of diameter, D = 3 mm, and a length, L = 60 mm, that was locatedat a varying standoff distance, H = 1 cm to 4.5 cm normal to the condensation surface.The jet impinged normally on the surface in an ambient relative humidity of RH ∞ = 20%. The jet exited the tube in a highly humid condition, RH j = 95 %. This was achievedby placing three spargers (manufactured by Ferroday) to generate around 0.5 micron airbubbles in the humidifier tank, only one sparger is shown in Figure 1 for illustration. Thecondensation surface was an aluminum substrate that was placed on the cold side of aPeltier plate with a thermally conductive paste. The Peltier plate was supplied with anenvironmental chamber and a PID temperature controller (KR ¨USS, DSA100). A range ofsubstrate temperature, T s = o C to 5 o C, was tested to observe the BF spot incipience andsize variation. The temperature of the cold side of the Peltier plate was recorded using anRTD element that was supplied with the PID temperature controller (KR ¨USS). An Infra-red(IR) camera (FLIR, A6753sc), and a flush-mounted k-type thermocouple (OMEGA, HH378)were used to observe the condensation substrate temperature as well as the condensatedroplets. The substrate temperatures measured by the three methods were in agreementwithin 0.1 o C. This rules out any possible heat transfer impedance of condensation due tosurface thermal resistance. Systematic experiments were performed by first adjusting theflow to the desired jet Reynolds number Re j = Q / πνD , where ν is the kinematic viscosityof humid air. At the desired standoff-to-diameter ratio (H/D), the jet exiting the tube wasallowed to impinge on the surface without lowering surface temperature initially. The surfacetemperature was then lowered in steps of 0.5 o C from room temperature. At a certain surface7able 1: Colour/shape code of the experimental conditions for a total of 35 combinationsof H / D and Re j . Under each combination point, the temperature of the surface was variedfrom 22 o C to 5 o C and BF spot diameter was observed. H / D = Re j = ○ ● ○ ● ○ ● ● □ (cid:4) □ (cid:4) □ (cid:4) (cid:4) △ ▲ △ ▲ △ ▲ ▲ ◇ ◆ ◇ ◆ ◇ ◆ ◆ temperature, we denote as the BF spot incipient temperature, BF spot starts to appear. Aswe lowered the surface temperature further, the expansion of the BF spot diameter wasobserved and recorded. The experimental parameters are summarized in Table 1 alongwith the colour/shape code of each data point. It is worth noting that a regular camera(Teledyne Photometrics, CoolSnap HQ2) was used to observe the BF spots. The camerawas inclined with a maximum of 10 o from the horizontal to obtain better visualization ofthe process. In Figure 2, we show a typical BF spot observation from a selected experiment.The image on the left shows a macroscopic view of the BF spot while the right image showsa microscopic view (3X). Due to light scattering from the condensate micro-droplets, a whitetrace was observed upon looking at the condensate deposit. Under the microscope, the BFspot boundary becomes very distinct as it separate between a wet inside and a dry outsideregions. Within the BF spot drop-wise condensation is observed as seen in Figure 2 (right)and Video 1. In Video 1 (supplementary material), we show a time lapse of the growth ofsessile droplets near the BF spot boundary. 8 Results and Discussion
Selected pictures at various experimental conditions are shown in Figure 3(a-c). In Figure3(a), for fixed T j − T s = 18 o C and H / D = 10, the effect of the Reynolds number is shown.We observed that at the lowest Reynolds number, Re j = 500, any obliqueness of the tubefrom the normal to the surface is characterized by a ”tailed” BF spot. The tail is directedopposite to the angle of obliqueness. Adjusting the impingement angle to eliminate the tailserved as an indication of a 90 o -angle impingement in our experiments. It should be notedthat the tail is absent for higher Reynolds numbers for small inclination of the tube. It isworth mentioning that for 1000 < Re j < Re j > T j − T s =18 o C and Re j = 3130 is depicted in Figure 3(b). We observed that BF spot size is invariantwith H / D at least for the tested range of 3.33 to 15. In Figure 3(c), we present the effectof jet-surface temperature difference for Re j = 3130 and H / D = 8.33. As the temperatureof the surface falls below the dew point of the jet center, the BF spot appears. The pointof BF spot inception occurs at lower surface temperature as H / D increases. Also, furtherdecrease in the surface temperature corresponds to an increase in the BF spot diameter.The BF spot diameter keeps increasing with decreasing the surface temperature to the pointat which atmospheric vapor start condensing. Below the atmospheric dew point, BF spotbecomes indistinguishable from sessile droplets that appear on the entire surface.To quantify that behaviour, vapor concentration distribution is inferred from the tem-perature measurements and BF spot size. At any experimental run, the vapor mass fractionat the boundaries of the BF spot was obtained as [50]1 ω s = + M a M v ( P ∞ P v − ) (1)where M a , M v , P ∞ , and P v , are molecular weight of air, molecular weight of water, am-bient pressure, and water vapor pressure at the surface temperature, respectively. In Figure4, we show the distribution of dimensionless vapor mass fraction ( ω j − ω ∞ )/( ω max − ω ∞ ) as a function of normalized BF spot diameter D BF / D at different experimental conditions.9he maximum mass fraction is obtained at the inception of BF spot. It should be notedthat only results of turbulent jets ( Re j > H / D , the mass frac-tion distribution is weakly influenced by Reynolds number. The lowest H / D value shows asteeper drop of vapor mass fraction while increasing H / D has a flattening effect. Further,for H / D >
5, we observe that even standoff distance has a weak influence on the distribu-tion of vapor mass fraction. It is worth noting that for H / D <
5, the free jet is still in thedeveloping region [51]. Therefore, the behaviour becomes similar to a confined jet [52]. InFigure 5, we plot the maximum vapor mass fraction as a function of both H / D and Re j . Thevapor mass fraction is normalized with the jet excess mass fraction ( ω j − ω ∞ ). We recognizethat the laminar jet has a constant maximum vapor concentration for H / D < .
67, whichsuggests that laminar jets lose less vapor content into the ambience compared to their tur-bulent counterparts. This is probably due to the improved mixing of the latter which helpsin dissipating vapor to ambience. However, for turbulent jets, the maximum mass fractionsseem to decrease monotonically with H / D value.The BF spot formation can be described using the following simple thought experiment.Consider a surface that is in thermal equilibrium with the jet and the ambience with the jetcontaining a higher vapor concentration than the ambience. After the humid jet exits thetube, vapor diffuses into the ambience before impinging on the surface. The concentrationprofile of the jet is therefore changed from being uniform to having a Gaussian distributionas shown in several analytical solutions [53, 54]. The value of the maximum concentrationand width of the diffusing jet depends on the distance travelled by the jet as well as ambientthermal and flow conditions. Upon jet impingement, a significant mixing occurs that allowsdissipation of the high vapor concentration near the surface. Without reducing the surfacetemperature, the vapor concentration at the wall has a decaying distribution from a maxi-mum value at the center of the impingement area to a minimum value equivalent to that ofthe ambience further away. Due to the variation of the vapor mass fraction at the surface,there is an equivalent saturation temperature (dew point) variation with radial direction.For a constant surface temperature ( T s ), When T s falls below the dew point at a given radiallocation, condensation will take place from the center up to that radial circumference, hence,10 BF spot forms.In Supplementary material, we present a numerical model that employs our understand-ing of recovery concentration. The jet impingement on a wall is reduced to a two-dimensionalaxisymmetric problem. The jet issuing from the tube has fully developed velocity profileand uniform temperature, and concentration profiles. The continuity, momentum, energyand species equations are solved over the jet vicinity region. Because of the importanceof accounting for turbulence in jet dynamics, standard k − ω formulation is usually pre-ferred [55, 56]. A finite volume solver was utilized to obtain the solution of the governingequations in the desired domain. Solutions of the Re j and H / D combinations were obtainedboth for free jet and wall obstructed cases. Further details and insights could be found inSupplementary material.We first model theoretically the vapor concentration in the impingement region ( D BF ≤ D ). Most importantly, we focus on the maximum concentration value which seems to beinfluenced significantly by the standoff distance rather than Reynolds number according toFigure 5. To obtain a scaling analysis of such behaviour, we resort to the visual observationsobtained from our numerical simulations in Figure S.3 and Figure S.4. The maximum con-centration of a jet impinging on wall corresponds to its counterpart in a free unbounded jetat the same standoff-to-diameter ratio. In other words, the maximum vapor concentration isnot affected by the impingement action. Therefore, we derive the theoretical curve in Figure5 by using a free unbounded jet solution. Using a control volume at the tube exit to anarbitrary axial location in a free unbounded jet, momentum conservation can be written as ρv j πD = πρ ∫ ∞ v rdr (2)where ρ is the overall mixture density and v j is the mean velocity of the jet. Byassuming that the overall mixture density does not vary greatly, which is valid for such lowvapor concentrations. According to previous studies, in the developed region ( H / D > v = v max exp [ − a v ( rH ) ] (3)11here a v is an empirical constant that depends on the tube exit geometry. SubstitutingEq. (3) into Eq. (2), we get the maximum velocity as v max / v j = √ a v / ( D / H ) . If we applyvapor species conservation over the same control volume, we have ρv j ( ω j − ω ∞ ) πD = πρ ∫ ∞ v ( ω − ω ∞ ) rdr (4)where ω i is the vapor mass fraction evaluated at the surface temperature and saturatedconditions, and ω ∞ is the vapor mass fraction evaluated at the ambient temperature andrelative humidity. In general, the concentration profile of the jet has a Gaussian distributionas well. Therefore, one can write the concentration profile as ( ω − ω ∞ ) = ( ω max − ω ∞ ) exp [ − a ω ( rH ) ] (5)where a ω is an empirical constant different from that associated with the velocity profile.Substituting Eq. (3) and Eq. (5) into Eq. (4) and combining the constants yield ω max − ω ∞ ω j − ω ∞ = a v + a ω √ a v DH (6)where the leading constant ( a v + a ω )/√ a v is an empirical value that depends on thetube-exit type and experimental conditions. In table S.1, we present the experimental valuesof the leading constant for the different cases studied. Data of over 105 experiments show tobe well represented by ( a v + a ω )/√ a v = . ±
2. Eq. (6) is depicted in Figure 5 along withthe numerical model result. The theoretical model seem to capture maximum concentrationbehaviour within the experimental uncertainty. On the other hand, a small deviation isobserved for the numerical simulation. Even though, the overall behaviour is capturedby both methods, we believe that both methods have their limitations. The theoreticalmodel assumes velocity and concentration to possess Gaussian distributions, however, severalother profiles, such as a polynomial [59] could be used. The numerical simulation utilizedthe standard k − ω model which is highly sensitive to the inlet and boundary conditions.However, given the simplistic approach of predicting the general behaviour, both methodsoffer excellent predictive tools.Next, we use an analytical approach for ( D BF / D >
5) to analyse the BF spot boundaryin the wall-jet region. Because of the sudden deposition, we can assume that BF spots are12nalogous to the concept of adiabatic wall or recovery temperature. Recovery temperaturehas been discussed in the context of heat transfer of impinging jets [48, 49, 60] as well as highMach number flows [61, 62]. The importance of such parameter emerged from the mismatchbetween surface, jet and ambient temperatures which necessitates entrainment . By the sametoken, we think BF spots are manifestations of a recovery concentration concept that hasnot been discussed in literature as to the author’s knowledge. Here we present a theoreticalmodel of the recovery concentration.In Figure 6, we present a schematic of an imaginary conduit starting from the tubeexit and covering the impinged surface at an arbitrary radial location ( r ). At the boundingsurfaces of the conduit, there is negligible mass transfer or negligible vapor mass concentra-tion gradient. Applying a species mass conservation between the tube exit and the radiallocation on the surface gives ρ πD v ( ω j − ω ∞ ) = ρ ∫ δ v ( ω − ω ∞ )( πr ) dz (7)where δ is the total thickness of the boundary layer. It has been recognized by severalresearchers that upon normalizing the velocity profile with its local maximum value, allvelocity profiles in the wall jet region simplifies to v / v o ∼ f ( z / δ ) [48]. Whereas normalizingthe excess local vapor mass fraction with the excess recovery concentration should result in aself-similar solution. Here we assume that, in the wall jet region, the non-dimensional vapormass fraction is ∼ f ( z / δ ) . Upon performing the normalization, we obtain rδ ( ω r − ω ∞ ) v o ∫ ( vv o )( ω − ω ∞ ω r − ω ∞ ) dzδ = D v j ( ω j − ω ∞ ) (8)For self-similar velocity and concentration profiles, the entire integral is assume to be aconstant ( C ). Simplifying the previous relation gives ω r − ω ∞ ω j − ω ∞ = C ( v j v o )( Dr )( Dδ ) (9)According to the several studies of turbulent jets [48,63,64], the normalization thicknessand velocity in the wall jet region can be correlated as v o v j = C ( HD ) . ( rD ) − . (10)13 D = C ( rD ) (11)Substituting Eq. (10) and Eq. (11) into Eq. (9) and combining the constants result inthe following conclusion ω r − ω ∞ ω j − ω ∞ = C ( HD ) − . ( rD ) − . (12)where C = /( C C C ) is a constant that depends on the tube-exit type and experi-mental conditions. Eq. 12 shows a weak effect of standoff-to-diameter ratio with a power lawof − .
1. Acceptable results within the experimental uncertainty could also be obtained ifthe effect of H / D is absorbed into the leading constant. Figure 7 shows all the experimentaldata along with the theoretical curve given by Eq. (12). The recovery concentration is inde-pendent of the jet Reynolds number at any given standoff-to-diameter ratio. Furthermore,there is no clear effect of standoff-to-diameter ratio in the wall-jet region. This is clear asall data points collapse on a universal curve in that region. The effect of standoff distanceis noticed from Eq. (12) to be very minimal which is in accord to our observation in Figure3 and Figure 4. Table S.1 presents the curve fitting constant obtained for the experimentaldata points. Data of over 1890 experiments show to be well represented by a leading con-stant ( C = . ± .
14) in Eq. (12). We also showed mathematically that the jet velocityhas no effect on the value of maximum vapor concentration. Eq. (6) is depicted in Figure7 where the effect of H / D is pronounced at the center of the impingement region. The BFspot dimension between the center of the impingement region to the wall jet region variessmoothly in a transition region. In conclusion, measurements were made of an isothermal humid air jet exiting a tube intoa stagnant room-condition air. Because the humid air jet has higher vapor content thanthe environment, vapor diffuses from the former to the latter. It has been shown thathumid laminar jets lose less vapor content as they travel into the environment comparedto the turbulent jets because of the improved mixing mechanism of the latter. This wasclear by observing the maximum concentration of the jet as it travel into an ambient air.14n the other hand, for turbulent jets, the vapor content diffusion into the environment isindependent of the jet’s velocity magnitude. The maximum vapor concentration becomesa function of standoff distance only beyond the developing free jet region. We also showedfor the first time that visible BF spots are manifestations of a new concept of a recoveryconcentration. We drew our analogy from the recovery temperature concept in heat transferapplications. The newly found concept is very important in studying species mass transferdue to jet impingement in general. Our findings show that BF spot is the area over whicheffective condensation takes place. Quantification of BF spot size is essential in optimizingthe surface area of condensers as well as their temperatures to obtain effective condensationrates. We also predicted theoretically the concentration distribution on a surface exposedto humid air jet impingement. Eq. (6) and Eq. (12) present important conclusions withwhich concentration distributions on an impinged wall are found. We believe that this studyis of great importance to optimize jet impingement heat and mass transfer rates. Severalapplications could utilize this work’s findings, such as in textile drying, dehumidificationtechnologies or exhaled breath condensate (EBC) technology [65, 66].
Acknowledgment
A. Alshehri would like to express his sincere gratitude to King Fahd University of Petroleumand Minerals (KFUPM), Dhahran, Saudi Arabia.
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Bubble Humidifier
Dry air 𝑔 Flowmeter H D L Figure 1: Schematics of a table-top set-up for observing the BF spots from a jet of humidair under varying parameters namely jet-surface temperature difference ( T j − T s ), jet exitReynolds number ( Re j = v j D / ν ), and standoff-to-diameter ratio ( H / D ).22 F spot boundary Outside BF spot:
Dry region
Within BF spot:
Drop-wise condensation
Figure 2: Typical BF spot formation taken by a regular camera (left image) and a lowmagnification microscope (right image). 23 𝑇 𝑗 − 𝑇 𝑠 = 9 𝑇 𝑗 − 𝑇 𝑠 = 10 𝑇 𝑗 − 𝑇 𝑠 = 13 𝑇 𝑗 − 𝑇 𝑠 = 12𝑇 𝑗 − 𝑇 𝑠 = 11 𝐻/𝐷 = 8.33
𝐻/𝐷 = 10 𝐻/𝐷 = 11.67𝐻/𝐷 = 6.67𝐻/𝐷 = 5 𝑅𝑒 𝑗 = 2230 𝑅𝑒 𝑗 = 3130 𝑅𝑒 𝑗 = 4130 𝑅𝑒 𝑗 = 500 𝑅𝑒 𝑗 = 1340 bc Figure 3: Selected pictures of the BF spots at various conditions. a. BF spots at varyingjet Reynolds number. The selected pictures are for the case of T j − T s = 18 o C and H / D =10. b. BF spots at different standoff-to-diameter ratios. The selected pictures are for thecase of T j − T s = 18 o C and Re j = 3130. c. BF spots at different jet-surface temperaturedifferences. The selected pictures are for the case of H / D = 8.33 and Re j = 3130.24 ( 𝜔 𝑠 − 𝜔 ∞ )( 𝜔 𝑚 𝑎 𝑥 − 𝜔 ∞ ) 𝜔 𝑠 𝜔 𝑚𝑎𝑥 𝐷 𝐵𝐹 Figure 4: Dimensionless concentration distribution on the surface as a function of dimension-less radial distance (or BF spot diameter to tube diameter ratio) ( D BF / D ). Colour/shapecode correspond to Table I. 25 ( 𝜔 𝑚 𝑎 𝑥 − 𝜔 ∞ )( 𝜔 𝑗 − 𝜔 ∞ ) Figure 5: Dimensionless maximum concentration as a function of H / D and Re j . The maxi-mum concentration is obtained at the inception of BF spot point as depicted in Figure 4 andEq. (1). The red-shaded region correspond to the experimental uncertainty in measurements.26 𝑜 (𝑟)𝛿 𝑜 (𝑟)𝐷 𝑟 𝜔 ∞ 𝑇 ∞ = 𝑇 𝑗 𝜔(𝑟, 𝑧)𝑧𝜔(𝑟, 𝑧) 𝑣(𝑟, 𝑧) Imaginary conduitBF spot boundary
Impingement region boundary (
Τ2𝑟 𝐷 < 5 ) 𝐻 Figure 6: Control volume approach for analysing humid air jet impingement. Schematic ofthe imaginary conduit over which vapor mass is conserved. Derivations of Eq. (6) and Eq.(12) depend on the understanding of this schematic. As the humid air exits the tube, vaporstarts to diffuse into the ambience. However, the imaginary conduit boundary is locatedat a radial location where the gradient of vapor concentration is nearly zero, i.e. negligiblediffusion is present. As the stream of vapor-air impinges on the surface, flow changes directionfrom y-direction to r-direction. The velocity and vapor concentration profiles at an arbitraryradial location away from the impingement region are depicted.27 ( 𝜔 𝑠 − 𝜔 ∞ )( 𝜔 𝑗 − 𝜔 ∞ ) Impingement
Region
Wall jet Region Τ ( 𝑇 𝑠 − 𝑇 𝑑 , ∞ )( 𝑇 𝑑 , 𝑗 − 𝑇 𝑑 , ∞ ) Τ𝐻 𝐷 = 15Τ𝐻 𝐷 = 11.67Τ𝐻 𝐷 = 8.33
Eq. (12)Eq. (6)
Figure 7: Recovery concentration distribution. A plot of nondimensional vapor mass fractionand nondimensional surface dew temperature with respect to the extent of BF spot circle.The plot is split into two regions; impingement region ( D BF / D < D BF / D > reath Figure Spot: a Recovery ConcentrationManifestation (Supplementary Material) Ali Alshehri a,b,+ and Pirouz Kavehpour a a Mechanical and Aerospace Engineering Department, Henry Samueli School of Engineeringand Applied Science, University of California, Los Angeles, CA 90095, USA b Mechanical Engineering Department, King Fahd University of Petroleum and Minerals(KFUPM), Dhahran 31261, Saudi Arabia + Corresponding Author: Phone: +1 (310)-849-7913E-mail: [email protected]
January 12, 2021 a r X i v : . [ phy s i c s . f l u - dyn ] J a n able S1: Leading constant results from curve fitting of BF spot diameter, see equation (6)and equation (12). H/D Re j ( a v + a ω ) / √ a v C H/D Re j ( a v + a ω ) / √ a v C Numerical model development
In this section, we utilize the finite volume method (FVM) to obtain the recovery concen-tration at various conditions. As was concluded in the paper, BF spots are manifestationsof the recovery concentration concept. Therefore, impingement of humid air jet on an adi-abatic surface is simulated. The geometrical domain as well as the boundary conditionsare depicted in Fig. S.2a. The problem is reduced to an axisymmetric problem aroundan axis, at which no gradient in state variables is present in the radial direction. The jetoriginates from an inlet section of uniform velocity, temperature, and concentration profiles.The humid air flows through a tube of a length greater than the entry region to ensure fullydeveloped conditions at the tube exit. The jet exits the tube into an ambient conditionof given temperature, pressure and concentration preset to the outlet surfaces depicted inFigure S.2a. The impingement surface as well as the tube surface are characterized by zero2 b Τ ( 𝜔 𝑠 − 𝜔 ∞ )( 𝜔 𝑚 𝑎 𝑥 − 𝜔 ∞ ) Τ ( 𝜔 𝑠 − 𝜔 ∞ )( 𝜔 𝑗 − 𝜔 ∞ ) Eq. (12) Τ ( 𝑇 𝑠 − 𝑇 𝑑 , ∞ )( 𝑇 𝑑 , 𝑗 − 𝑇 𝑑 , ∞ ) Figure S1: Laminar jet experimental results. a. Dimensionless concentration distributionon the surface as a function of dimensionless radial distance ( D BF /D ). Colour and shapecoding correspond to Figure 1(b). b. plot of nondimensional vapor mass fraction and surfacedew-point temperature with respect to the extent of BF spot circle.3eat and mass fluxes. The no-slip condition is applied to both surfaces as well. The flowof the humid air jet is assisted by gravitational force which acts normal to the impingementsurface. The governing equations in the solution domain are given as ∇ · ( ρ −→ v ) = 0 (1) ∇ · ( ρ −→ v −→ v ) = −∇ P + ∇ · τ + ρ −→ g (2) ∇ · ( −→ v ( ρE + P )) = ∇ · ( k ∇ T − X j h j −→ J j ) (3) ∇ · ( ρ j −→ v ) = −∇ · −→ J j (4)where E ≈ h neglecting pressure work and kinetic energy. The total enthalpy is a massweighted average of each species enthalpy. The species enthalpy is given by equation (5). h j = Z TT ref c p,j dT (5)The term −→ J j in equation (3) and equation (4) refers to the diffusive mass flux of each specieswhich is given by Fick’s law. −→ J j = − D j,i ∇ ρ j (6)In order to take care of turbulence, standard k − ω model was implemented. Addingperturbed state variables to equations (1-4), yields the extra term of Reynolds stress ( ρu i u j ).The standard k − ω model solves for two additional equations representing the transport ofturbulent kinetic energy ( k ) and specific rate of dissipation ( ω ). Enough documentationcan be found in many references, therefore are not repeated here [ ? , ? ]. The differentialequations are solved using an FVM in which the domain is discretised into smaller cells as inFig. S.2b. Finer meshing was concentrated where the change in state variables is expected tobe greatest. It is worth noting that a separate simulation was performed on a free unboundedjet for comparison purposes.Figure S.3 presents the contour plots of vapor mass fraction at varying standoff-to-diameter ratios while Fig. S.4 presents those of different Reynolds numbers. We observe thatvapor concentration is maximum in the core of the jet. As the jet advances in the ambience,its vapor content diffuses and the uniform concentration tends to transition smoothly near4 b D Axis ( Τ 𝝏 𝝏𝒓 = 𝟎 ) o u t l e t ( a m b i e n t c o nd i t i o n s ) W a ll ( A d i a b a t i c + n o - s li p ) Pipe ( Adiabatic + no-slip ) I n l e t μm μ m 𝑔 Figure S2: Geometrical configuration of the numerical model. a. geometrical model ofthe axissymetric problem under simulation using FVM. b. Refined meshing of the solutiondomain. 5he perimeter of the jet. Figure S.3 shows that for jets with heights that are 8.33 diametersor less, the maximum vapor concentration coincides with that of the inlet. The maximumvapor concentration then starts to drop due to the diffusion effect with ambience. FromFig. S.4, we notice that for Reynolds numbers of 1340 and higher, there is no significantdifference of the vapor concentration profiles. The case of Reynolds of 500 shows slightlyhigher throw of vapor content. This could be attributed to the low mixing characteristicof laminar flows, therefore, maintaining its vapor content for a longer distance. For all thepresented cases, the introduction of a wall normal to the jet flow direction does not seem tochange the flow upstream. Therefore, further insights of the maximum vapor concentrationat the wall could be obtained from a free jet case corresponding to similar geometric andflow conditions. This conclusion was the basis of the derivation of equation (6) in the mainmanuscript.Figure S.5 depicts the vapor mass fraction normalized with the jet excess vapor massfraction. We observe that within a radial location ( D BF /D ≤ Re ≥ 𝐻 𝐷 = 15 11.67 10
Impinging jet0.0164 𝑔 Free jet
Figure S3: Results of different standoff-to-diameter ratios. Contours of vapor mass fractionat
H/D of 3.33, 5, 6.67, 8.33, 10, 11.67, and 15 (from right to left). Results are for a selectedReynolds number of 4130. At each standoff-to-diameter ratio two cases are presented; (topcontour plot) represents the case were a jet impinges on a wall corresponding to a given
H/D ; (bottom contour plot) represents the case of a free unbounded jet at a similar flowand geometric conditions. 7 .01640.01380.0112 𝑅𝑒 𝑗 = 5001340 Figure S4: Results of different Reynolds numbers. Contours of vapor mass fraction at Re j of500, 1340, 2230, 3120, and 4130. Results are for a selected standoff-to-diameter ratio of 6.67.At each value of Re j , two cases are presented; (top contour plot) represents the case were ajet impinges on a wall corresponding to the given Re j ; (bottom contour plot) represents thecase of a free unbounded jet at a similar flow and geometric conditions.8 ( 𝜔 𝑠 − 𝜔 ∞ )( 𝜔 𝑗 − 𝜔 ∞ ))