- Email: [email protected]
Numerical study of the flow and heat transfer in a turbulent bubbly jet impingement
International Journal of Heat and Mass Transfer 92 (2016) 689–699
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Numerical study of the flow and heat transfer in a turbulent bubbly jet impingement M.A. Pakhomov ⇑, V.I. Terekhov Laboratory of Thermal and Gas Dynamics, Kutateladze Institute of Thermophysics, Siberian Branch of Russian Academy of Sciences, Acad. Lavrent’ev Avenue 1, 630090 Novosibirsk, Russia
a r t i c l e
i n f o
Article history: Received 14 July 2015 Received in revised form 4 September 2015 Accepted 7 September 2015 Available online 26 September 2015 Keywords: Bubbly impinging jet Numerical modeling Heat transfer enhancement Turbulence modification
a b s t r a c t This contribution presents the results of modeling of the flow structure and heat transfer enhancement obtained by adding air bubbles into a turbulent liquid (water) impinging jet. A set of axisymmetrical steady-state RANS equations for the two-phase flow is utilized. The dispersed phase (bubbles) is modeled by the Eulerian approach. Liquid-phase turbulence is computed with the Reynolds stress model, taking into account the effect of bubbles on the carrier phase. The effect of changes in the gas volumetric flow rate ratio and bubble size on the flow structure, wall friction, and heat transfer in a gas–liquid impinging jet is numerically studied. The predictions demonstrate the significant anisotropy of the turbulent fluctuations in the axial and radial directions for the bubbly impinging jet. The addition of a gas phase into the turbulent liquid increases the wall friction (by up to 40% in comparison with the single-phase liquid jet) and heat transfer (by up to 50%). Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Jet impingement is a very effective method for enhancing heat transfer rate between a solid impinging surface and a jet flow in various applications (see comprehensive reviews [1–5]). Potential applications where high heat fluxes should be dissipated while the surface is kept at a relatively low temperature include highpower equipment, glass tempering, cooling of surfaces and preforms, spray painting, rotor blades, and nuclear reactors during emergency conditions [1–5]. The main parameters that affect the heat and mass transfer processes include the distance between the nozzle orifice (or pipe outlet) and the impingement (target) surface, the temperatures of the jet and wall surface, the velocity magnitude of the jet, and the turbulence intensity. Many attempts to modify flow characteristics and increase heat transport in impinging jets have been made in the last decades. Impinging two-phase jets with droplets or bubbles are employed to enhance heat transfer between the flow and the solid surface. Gas-droplet jet impingement on a hot surface removes large amounts of heat because of the latent heat of evaporation [6,7]. One of the ways of increasing the heat transfer rate is by adding a gas (or vapor) phase to an impinging liquid jet [8–13]. The enhancement in heat transfer is attributed to the high turbulence ⇑ Corresponding author. E-mail addresses: [email protected] (M.A. Pakhomov), [email protected] (V.I. Terekhov). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.09.010 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
level in the near-wall region accompanying bubble-induced agitation in a bubbly impinging jet [8]. Serizawa et al. [8] experimentally studied the heat transfer in an impinging air–liquid bubbly jet. The heat transfer coefficient was increased by a factor of 2–4 at a gas volumetric flow rate ratio of b 6 0.53. The enhancement was attributed to the high turbulence level caused by bubble-induced agitation and to acceleration of the liquid phase by the more rapidly moving air. Measurements in a slot impinging bubbly jet by boiling on the target surface were carried out in [9]. The gas volumetric flow rate ratio was b = 0–0.86 and the Reynolds numbers varied Re = 3700–21,000. Significant enhancement of the heat transfer (up to 2.2-fold in comparison with single-phase water impinging jet) was obtained. The experimental investigation of single and multiple confined two-phase jets using Freon R-113 in the presence of boiling was carried out in [10,11]. The regimes of single-phase forced convection and boiling heat transfer were studied in [10,11]. Heat transfer of the liquid–vapor impinging jets was augmented by a factor of 1.2 than that of the single-phase jet impingement [10]. The experimental investigation of heat transfer and pressure drop an in the air bubbles-water impinging jet was performed in [12,13]. The heat transfer enhancement (up to 2 times) in bubbly impinging jet at b = 20–30% was shown. The following increase of the gas volumetric flow rate ratio led to a decrease in the heat transfer enhancement. The turbulent structure in a round bubbly impinging jet without interphase heat transfer was studied experimentally under
690
M.A. Pakhomov, V.I. Terekhov / International Journal of Heat and Mass Transfer 92 (2016) 689–699
Nomenclature CD CP, CPb
drag coefficient of bubbles specific heat capacity of carrier liquid (water) and dispersed (gas bubbles) phases respectively (J kg1 K1) D diffusion coefficient (m2 s1) d bubble diameter (m) 2 Eob Eo ¼ gðq qb ÞdH =r modified Eotvos number hc convective heat transfer coefficient between liquid and bubble (W K1 m2) J, Jb superficial velocity of liquid (water) and bubbles respectively (m s1) k k = huiuii/2 turbulent kinetic energy (m2 s2) Nu Nu ¼ h2R=k Nusselt number P pressure (Pa) Pr Nu = lCp/k Prandtl number R pipe radius (m) R specific gas constant (J kg1 K1) Re 2RUm1/m Reynolds number Reb ¼ jU R jd=m Reynolds number of bubbles, based on the Reb sleep velocity r radial coordinate (m) T temperature (K) jU R j jU R j ¼ jU U L j sleep (interphase) velocity (m s1) Ui components of mean velocity (m s1) Um1 bulk mean velocity of the liquid flow at the pipe edge (m s1) friction velocity (m s1) U⁄ hu0 i; hv 0 i intensity of velocity fluctuations in axial and radial directions respectively (m s1) turbulent heat flux (m K s1) hu0j ti 0 0 hu v i turbulent Reynolds stress (m2 s2) We We ¼ qjU R j2 d=r Weber number
the superposition of external periodic excitation in [14]. The shear stress was measured on the target surface at a change of b = 0–12%. The effect of suppression of the large-scale structures was observed for high gas contents. It was found out that an addition of the gas phase at b < 12% leads to a significant increase in the wall shear stress (up to 40%) [13]. Recently, an experimental study of isothermal bubbly impinging turbulent jet by a combination of optical planar fluorescence for bubble imaging (PFBI) and particle image velocimetry (PIV) and particle tracking velocimetry (PTV) methods was carried out in [14]. The bubbly jet was investigated at different gas volume flow rate ratios of b = 0–4.2% at a fixed Reynolds number, Re = 12,000. Distributions of local void fraction a and mean and fluctuating velocities of both phases were measured. Bubbles increase the liquid phase turbulence at the nozzle edge when there is an additional supply of bubbles into the liquid jet. Then, they suppress the turbulence fluctuations compared to the single-phase flow in the downstream direction at x/(2R) > 0.3. The bubbles significantly intensify turbulence fluctuations due to an increase of the slip velocity between the phases close to the impingement surface [15]. The above mentioned studies were focused on the influence of gas volumetric flow rate ratio on heat transfer and turbulent structure in impinging jet. The effect of the size of dispersed phase on these characteristics was poorly studied. The authors found no studies on numerical study of flow structure and heat transfer in the impinging jets. A numerical study of an isothermal bubbly flow under plunging free round jet conditions using mono- and polydispersed approaches was performed in [16]. The turbulence is modeled by a k–e model with taking into account the bubble-induced turbulence. The bubble break-up and coalescence processes were
x y
axial coordinate (m) distance normal from the wall (m)
Greek letters volume fraction of the dispersed phase a local void fraction of bubbles b b = Jb/(Jb + J) gas volumetric flow rate ratio e dissipation of the turbulent kinetic energy (m2 s3) k heat conductivity (W K1 m1) l dynamic viscosity (kg m1 s1) m kinematic viscosity (m2 s1) q density (kg m3) r surface tension (N m1) 2 s s ¼ 3l4qReb db C D dynamic relaxation time of bubbles (s)
U
sW sH
wall friction (Pa) sH ¼ C Pb qb d2 =ð12kb YÞ thermal relaxation time of bubbles (s)
Subscripts 0 single-phase flow 1 parameter under initial conditions L liquid T turbulent parameter W parameter on the wall condition b bubble m mean-mass parameter Acronym CV control volume RANS Reynolds averaged Navier–Stokes equations SMC second moment closure
considered in the polydispersed air–water flow by using the inhomogeneous MUlti SIze Group (MUSIG) model [17]. Several studies of heat transfer in bubble flow in pipes have been presented in the literature [18–20]. The complexity of modeling bubbly turbulent flows with interphase heat transfer is associated with the large number of different physical multiscale phenomena involved, such as heat transfer, bubble coalescence and break-up processes. The aim of this work is to carry out a numerical simulation of the effect of bubbles size and gas volumetric flow rate ratio on flow structure and heat transfer modification in the turbulent bubbly jet impingement. This study may be interesting for scientists and engineers dealing with the problem of heat transfer enhancement in power equipment. The paper is organized as follows. In Section 2, the governing equations employed in this study for the simulation of a two-phase turbulent bubbly impinging jet are introduced. The numerical methods are described in Section 3. In Section 4, a numerical study of bubbly jet impingement with heat transfer is presented. Section 5 summarizes the main findings and conclusions. 2. Problem statement and governing equations 2.1. Physical model The modeling of dispersed phase is accomplished by the Eulerian two-fluid approach that treats the particulate phase as a continuous medium with properties analogous to those of a liquid [21]. In the two-fluid approach, both phases are considered as interacting continua. This technique involves the solution of a second set of Navier–Stokes-like equations in addition to those of the carrier (liquid) phase. Furthermore, the significance of liquid–par-
691
M.A. Pakhomov, V.I. Terekhov / International Journal of Heat and Mass Transfer 92 (2016) 689–699
ticle interactions is reflected in the momentum and kinetic energy coupling terms via the mean and the fluctuating drag force, respectively. The motion of the incompressible bubbly turbulent flow in the impinging jet is considered. The bubbles’ behavior in turbulent liquid and their back action on the flow is determined by drag, gravity lift, virtual mass, wall lubrication forces, turbulent transport, and turbulent diffusion. The effect of break-up and coalescence of bubbles is not taken into account. In order to account for the interaction between phases, that is, momentum exchange and heat and mass transfer, the conservation equations have to be extended by appropriate source/sink terms. The Eulerian approach is based on kinetic equations for a one-point probability density function of particle coordinates, velocity, and temperature in the turbulent Gaussian fluid flow fields [22–26]. Properties such as the mass of particles per unit volume are considered as a continuous property and the particle velocity is the averaged velocity over an average control volume. The interfacial transfer of mass, momentum, and energy requires averaging over the computational control volume too. Effects of pressure drop of the liquid and heat up of the gas phase on the bubble size are taken into account. The system of RANS equations for the carrier phase, the turbulence model and the equations for the dispersed phase are written in a form appropriate to axisymmetric flow, but for reason of brevity, are given below in general Cartesian tensor form. 2.2. The set of basic equations for the liquid phase The mean liquid flow is treated as a steady-state, incompressible and axisymmetrical flow and it is described by continuity, two-momentum and energy equations with taking into account the effect of bubbles presence. The system of RANS equations for the two-phase is given as @½ð1UÞqU i ¼0 @xi @½ð1UÞqU i U j UÞP ¼ @ð1 þ @x@ ð1 @xi @xi j @½ð1UÞqC P T ¼ @x@ i ð1 @xi
i UÞ l @U qhu0i u0j i þ qg þ ML @xj
6UðTT b Þ @T 0 @U UÞ k @x q C C Pb qb g t < u0j t > @x P huj ti d i j ð1Þ
Here M L ¼ Mb is the interphase interaction; g is the gravity force and gt is the coefficient of involvement of bubbles into the thermal fluctuational motion of liquid phase [22,24]. The set of Eq. (1) is written with taking into account the effect of bubbles on the transport process in the carrier (liquid) phase. The values of turbulent Prandtl number is equal to PrT = 0.9. The turbulent heat flux in the carrier phase is determined according to the Boussinesq hypothesis:
mT @T
PrT @xj
@½ð1 UÞU j hu0i u0j i @xj
:
¼ ð1 UÞðdij þ Pij þ Aij eij Þ þ S1
" 2 # @½ð1 UÞU j e e @U i ¼ ð1 UÞ C e1 Pk þ mT @xj k @xj e2 @ k @e e C e hu0l u0k i þ C e 3 S1 : C e2 k @xk e @xl k
ð2Þ
ð3Þ
Last terms in Eqs. (2) and (3) include an additional summand for description of interfacial interaction [30]. The right hand side of Eq. (2) describes turbulent diffusion, production of turbulent stresses from the shear averaged motion, redistribution of pulsation energy between different components of velocity fluctuations caused by correlation of pressure and deformation rate pulsations, and viscous dissipation, including interfacial interaction S1. The diffusion term (turbulent mixing at interaction of fluctuation velocity components) takes form
dij ¼ C S
@ k 0 0 @ D 0 0E hul uk i ui uj : @xk e @xl
Production of turbulent stresses from the averaged motion (intensity of energy transfer from averaged to fluctuation motion)
Pij ¼
q ¼ ð1 UÞP=ðRTÞ
< u0j t >¼
cost than the LES model. The second-moment closure of the turbulent flow is based on the solution to the system of transport equations for the components of Reynolds stresses. The SMC model predicts the turbulent Reynolds stresses directly from differential equations and allows us to compute the anisotropic flow. In the present study, the low-Reynolds number second-moment closure of [28,29] is employed. It is modified for the presence of bubbles [30]:
u0i u0k
@U j @xk
þ hu0j u0k i
@U i : @xk
The pressure-strain term considering energy transfer between different components of pulsations due to pressure-deformation rate correlations takes form [28]
Aij ¼ C 1
2 1 hu0i u0j i dij k C 2 Pij Pkk dij þ AW ij ; 3 3 k
e
ð4Þ
where the near-wall effects in Eq. (4) are written as [28]. The isotropic concept is used for dissipative term: eij 23 dij e. Last summands in the right part of Eqs. (2) and (3) take into account the back effect of dispersed phase on transport processes. The last term in the right side of the Eq. (2) S1 determines the additional turbulence production in liquid in the wake of the bubbles [30]
S1 ¼
3C i C D UjU R j3 : 4d
The constants of the turbulence model are presented in [28]: 2.3. The turbulence model of the carrier phase (liquid) The two-equation turbulence models have some well-known shortcomings. These models do not describe turbulent stress anisotropy and it leads to considerable errors in modeling strongly non-equilibrium flows with high velocity gradients and curvatures. The measurements reported in [27] demonstrated a significant anisotropy of the turbulent pulsations of axial and radial coordinates for a single-phase axisymmetrical impinging jet. One way to treat this anisotropy is to apply the model of Reynolds stress transport or second-moment closure (SMC). They are more complex from the point of view of computations than ke models. SMC is more detailed than the two-equation models, has a lower computational
CS = 0.18,
C1 = 1.8,
C2 = 0.6,
CW1 = 0.5,
CW2 = 0.08,
C 1w2 ¼ 0:1;
C 2w2 ¼ 0:4, Cl = 2.5 Ce1 = 1.44, Ce2 = 1.92, Ce = 0.18. The constants of the model of [26] are Ci = 0.1, Ce3 = 1.44. 2.4. The system of basic equations for the dispersed phase (bubbles) The dispersed phase (bubbles) is modeled by the Eulerian approach, which treats the particulate phase as a continuous medium with properties analogous to those of a liquid. This technique involves the solution of a second set of Navier–Stokes-like equations in addition to those of the carrier (liquid) phase. Furthermore, the significance of liquid–gas interactions is reflected in the momentum and kinetic energy coupling terms via the mean and
692
M.A. Pakhomov, V.I. Terekhov / International Journal of Heat and Mass Transfer 92 (2016) 689–699
fluctuating forces, respectively. The system of axisymmetric mean transport equations for the dispersed phase has the form
@ðUqb U bi Þ ¼0 @xi @ðUqb U bi U bj Þ @ðqUhubi ubj iÞ 1 @ðqb Db UÞ @ðUPÞ þ ¼ Mb @xi s @xj @xj @xj
ð5Þ
@ðUqb C Pb T b Þ q 1 @ðDbH UÞ ¼ hc UðT T b Þ b ; @xj sH sH @xj
ð6Þ
0
0
where Db, DbH are the turbulent diffusivity and bubbles turbulent heat transport tensors respectively [24] and hc is the convective heat transfer coefficient between liquid and bubble. The set of kinetic stresses hu0bi u0bj i equations in the dispersed phase are given in the form of [24]. The first term on the left hand side of Eq. (5) considers convective transport of the dispersed phase. The first, second, and third terms on the right side of Eq. (5) characterize the transport of bubbles due to the turbulent diffusion, the pressure drop, and the interphase interaction respectively.
the direction of the lift force changes its sign for gas–liquid flows if a substantial deformation of the bubbles occurs. The expression for prediction of lift force (IV) has the form [33] and CL is the lift coefficient by [36]
8 > < min½0:288tanhð0:121Reb Þ; f ðEoÞ; Eo 6 4 C L ¼ f ðEoÞ; 4 6 Eo 6 10 : > : 0:27; Eo > 10 Here f ðEoÞ ¼ 0:00105Eo3 0:0159Eo2 0:0204Eo þ 0:474 is the correction function [36] taking into account the bubble deforma2
Eo ¼ gðq qb ÞdH =r is the modified Eotvos number, p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 dH ¼ d 1 þ 0:163Eo0:757 is the maximum horizontal dimension of the bubble [37]. The coefficient CL changes its sign at a bubble diameter of d = 5.8 mm for our conditions. The coefficient CT in the term taking the turbulent diffusion (V) is CT = 0.1 [38]. Near the wall surface of the velocity difference between the bubble and the internal flow is less than the one between the bubble and the external flow. It causes the pressure gradient, and therefore bubble does not slip onto the wall. In order to handle this behavior of bubbles near the wall an additional wall lubrication force (VI) is introduced by [39]. tion,
2.5. Interface forces The summand for prediction of interphase interaction in the monodispersed flow has the form [20,31]
3.1. Numerical procedures
M L ¼ M b
3qC D @U @U ¼ UU R jU R j þ C m qU U j i U bj bi þ qb Ug 4d @xj @xj @U i @U j @U C T qkjU R j C L UqjU R j @xj @xi @xi d 2aqjU R j2 C W1 C W2 : 2yb d
ð7Þ
The interphase interaction in Eq. (7) M liq i is considered by taking into account the effect of following forces: drag force, added mass, gravity and Archimedes forces, lift force, turbulent diffusion, wall lubrication force and turbulent migration (turbophoresis) force [20,31]. In the transverse direction the lift and wall lubrication forces, turbulent diffusion and turbophoresis act, determining together the radial distribution of the void fraction in the vertical tube cross-section. However, the major influence on the radial distribution of bubbles is given by the lift and wall lubrication forces. The drag coefficient CD of the deformable bubbles in the drag force (I) is determined by the formula of [32]
C D ¼ C We!0 þ DC D ðC We!1 C We!0 Þ; where C We!0 is the drag coefficient of non-deformable spherical bubbles at Weber numbers We ? 0, C We!1 ¼ 8=3 þ 24=Reb is the drag coefficient at large Weber numbers We ? 1, and h i 1:6 is the interpolation function DC D ¼ tanh 0:0038ðWeRe0:2 b Þ describing transition from zero to a large Weber number. In accordance to [32] the following relation can be written:
C We!0 ¼
8 < :
3. Numerical procedures, boundary conditions, and validation of the model
24 Reb
3 1 þ 16 Re0:687 ; b
0:44;
Reb 6 103
The developed model employs ‘‘in-house” code to simulate a turbulent bubbly impinging jet. The mean transport equations and the SMC model are solved with a control volume method on a staggered grid. The QUICK scheme is used to approximate the convective terms, and the second-order accurate central difference scheme is adopted for the diffusion terms. The velocity correction is used to satisfy continuity through the SIMPLEC algorithm, which couples velocity and pressure. The computational domain is a cylinder of 30R in the radial direction and height H. The computational grid, which is nonuniform in both the axial and the radial direction, is applied with a high resolution near the solid boundary (the impinging surface) and in the axis region. The logarithmic function is used to produce the non-uniformity in both the axial and the radial direction. The first cell from the solid wall is located at a distance y+ = yU⁄/ m = 0.3–0.5 from the wall, where U⁄ is the friction velocity obtained for the flow in the pipe outlet. At least 10 control volumes have been generated to resolve the mean velocity field and turbulence quantities in the viscosity-affected near-wall region (y+ < 10). Grid sensitivity studies are carried out to determine the optimum grid resolution that gives the mesh-independent solution. For all numerical investigations performed in the study, a basic grid with 200 256 control volumes (CVs) along the axial and radial directions is used. Grid convergence is verified for three grid sizes: m1: 100 128 = 1.28 104, m2: 200 256 = 5.12 104, and m3: 400 400 = 1.6 105 CVs. The profiles of Nusselt numbers along the radial coordinate are shown in Fig. 1 at H/(2R) = 2. 3.2. Boundary conditions
3
Reb > 10 :
The coefficient for prediction of the effect of added mass (II) is Cm = 0.5 [33]. The lift force considers the interaction of the bubble with the shear field of the liquid per unit volume. The well-known lift force formulation for two-phase flows, which has a positive coefficient CL, acts in the direction of decreasing liquid velocity. Experimental [34] and numerical [35] studies have shown that
The results of preliminary simulations for the single-phase water flow in a pipe with a length of 150R are used for the determination of velocity components and turbulence at the pipe outlet. These conditions are sufficient to achieve fully developed turbulent flow. The symmetry conditions are set on the stagnation line. Noslip conditions are set on the target surface. The conditions at the outlet edges of the computational domain are set @/=@r ¼ 0 for
M.A. Pakhomov, V.I. Terekhov / International Journal of Heat and Mass Transfer 92 (2016) 689–699
693
Fig. 1. Grid convergence test. Re = 2.3 104, TW = 313 K, T1 = Tb1 = 293 K, H/(2R) = 2, b = 5%, d1 = 1 mm. 1 – 100 128 control volumes, 2 – 200 256, 3 – 400 400.
all variables. The components of Reynolds stresses are determined at the same points on the control volume surface as the corresponding components of mean velocities by using the method of [28,29]. The water and bubbles inlet temperatures in the impinging jet and temperature in ambient quiescent medium are T1 = Tb1 = T/ = 293 K. 3.3. Validation analysis for the single-phase impinging turbulent jet In the first step, a comparison is carried out with measurements of turbulent quantities and heat transfer characteristics [27] in a single-phase air impinging round jet. These results were published in [7] for air and gas-droplet turbulent impinging jets. Our predictions agree well with the data of other studies for a wide range of distances between the pipe outlet and the impinging surface and Reynolds numbers. A good agreement is obtained with the correlations for the liquid impinging jets presented in [4] (the maximal difference in the Nusselt number is up to 6%). This is the basis for the study of the more complicated case of impinging bubbly jet. 4. Numerical results and discussion All simulations have been carried out for monodispersed air bubbles and water mixture at atmospheric pressure for the downward flow. The effect of break-up and coalescence of bubbles is not taken into account. The pipe’s inner diameter is 2R = 20 mm. The mean velocity of the gas flow at the pipe edge varies within the range Um1 = 0.25–1 m/s. The magnitude of bubble velocity at the inlet cross-section Ub1 = 0.8Um1. The Reynolds number of the jet varies within the range Re = 2RUm1/m = (0.8–6) 104. The wall temperature is TW = 313 K and is kept constant throughout the simulations. The initial temperature of gas and water at the pipe edge is T1 = Tb1 = 293 K. The distance to the obstacle varies within the range H/(2R) = 1–10. The bubble diameter varies in the range d = 0–5 mm and the gas flow rate ratio varies in the range b = 0–10%. 4.1. Turbulent flow structure The mean total velocity distributions of single-phase water (1), liquid (water) in two-phase flow (2), and bubbles (3) for various
distances from the stagnation point are shown in Fig. 2. The addition of bubbles affects the magnitude of liquid velocity. The profiles of liquid velocity in the two-phase impinging jet are generally similar to those in the single-phase flow. The liquid velocity in the presence of gas bubbles (2) is higher than the corresponding value in the single-phase flow (1). This effect increases with a rise of the bubble size. The liquid velocity is higher than the velocity of the gas phase (3). Flow acceleration occurs initially when moving away from the stagnation point. The velocity profiles have a pronounced jet profile with a maximum located in the wall region at x/(2R) = 1. The flow velocity decreases on moving away from the stagnation point due to jet expansion when it mixes with the surrounding quiescent medium and the jet shape of the profile becomes less obvious. Distributions of axial (normal to wall) and radial (parallel to wall) fluctuations and Reynolds stresses of liquid phase are shown at H/(2R) = 2 in Fig. 3 for several stations downstream of the stagnation point. Here, line 1 corresponds to the single-phase liquid jet and lines 2 and 3 are those for the water in bubbly impinging jet for various bubble diameters. The predictions demonstrate the significant anisotropy of the turbulent fluctuations in both the axial and radial directions for the bubbly impinging jet. The increase in the intensity of liquid velocity fluctuations in the two-phase flow (up to 25%) is shown in comparison with the single-phase flow (1). The growth of the size of the dispersed phase causes an increase in liquid turbulence. Additional production of liquid phase turbulence is explained by vortex formation (flow separation) on streamlining of the large gas bubbles by the liquid. The maximal values of axial and radial fluctuations of the two-phase jet velocity are observed at a distance of r/(2R) = 2 (see Fig. 3a and b). Turbulence production decreases with development of the near-wall jet along the impinging surface due to near-wall jet mixing with the ambient medium and uplifting of air bubbles; the value of radial fluctuations in the gas–liquid flow is close to the corresponding value for the single-phase water flow. The profiles of Reynolds shear stress along the transverse coordinate are shown in Fig. 3c. The maximal value of Reynolds stress is observed at a distance of r/(2R) 3 from the stagnation point. The local maximum of hu0 m0 i is situated in the wall zone. These conclusions correspond qualitatively to the data of measurements [27] for the single-phase air impinging jet and our previous predictions for the gas-droplet jet impingement [7].
694
M.A. Pakhomov, V.I. Terekhov / International Journal of Heat and Mass Transfer 92 (2016) 689–699
a
b
Fig. 2. The profiles of total mean liquid velocities in the two-phase flow (a) and for various gas bubbles size (b). Re = 2.3 104, Um1 = 0.5 m/s, 2R = 20 mm, H/(2R) = 2, TW = 313 K, T1 = Tb1 = 293 K. (a) d = 1 mm. 1 – single-phase jet (b = 0), 2 – liquid at b = 5%, 3 – gas bubbles at b = 5%. (b) b = 5%. 1 – Single-phase jet (d = 0), 2 – d = 1 mm, 3 – 3 mm.
Transverse distributions of the local void fraction a along the length of the impinging surface are presented in Fig. 4. For small distances from the stagnation point, it is characteristic for the location of the local maximum of air bubble concentration to be near the wall, r/(2R) 6 2; then, due to the action of buoyancy forces and extension of the wall jet, the local maximum of a is located at some distance from the target wall. The thickness of the layer of ‘‘pure” liquid depends on b. With the increase of the volumetric gas flow rate ratio, the thickness of the zone that is free of bubbles increases.
an increase in the wall friction (up to 40%). Moreover, this effect increases with a rise of the bubble diameter (see Fig. 5a). It is explained by an increase in turbulence generation at the separated flow around large air bubbles and the increasing gradient of liquid velocity near the wall. The effect of the diameter of air bubbles on heat transfer is shown in Fig. 5b. The local Nusselt number at a constant wall temperature was determined by the difference between the wall temperature and the temperature of liquid at the pipe outlet:
Nu ¼ 4.2. Wall friction and heat transfer The effect of bubble size on shear stress distributions on the wall surface and heat transfer is shown in Fig. 5. The wall friction is calculated by relationship: sW ¼ lð@V=@yÞW . It is shown numerically that the addition of gas phase to the turbulent liquid causes
ð@T=@yÞW 2R ; TW T1
where ð@T=@yÞW is the gradient of liquid phase temperature on the wall. The increase in the bubble size intensifies heat transfer between the two-phase jet and target, and this is consistent with our previous data for the vertical downflows in the pipe [18]. The addition of gas bubbles leads to a 1.5-fold increase in heat transfer
M.A. Pakhomov, V.I. Terekhov / International Journal of Heat and Mass Transfer 92 (2016) 689–699
695
a
b
c
Fig. 3. The effect of air bubbles diameter on distributions of axial (normal to the wall) (a), radial fluctuations (parallel to the wall) (b) and Reynolds shear stress (b). Dashed lines are the single-phase water impinging jet (1), solid and dotted lines are two-phase air–water (2, 3) jet. Re = 2.3 104, Um1 = 0.5 m/s, 2R = 20 mm, H/(2R) = 2, TW = 313 K, T1 = Tb1 = 293 K, b = 5%. 1 – Single-phase impinging jet (b = 0), 2 – d = 1 mm, 3 – 3 mm.
696
M.A. Pakhomov, V.I. Terekhov / International Journal of Heat and Mass Transfer 92 (2016) 689–699
Fig. 4. The local void fraction profiles downstream of the stagnation point. Re = 2.3 104, H/(2R) = 2, TW = 313 K, T1 = Tb1 = 293 K, d = 1 mm. 1 – b = 2%, 2 – 5%, 3 – 10%.
a
Fig. 6. Heat transfer at the stagnation point for various distance between pipe outlet and target surface. d = 1 mm, Re = 2.3 104, Um1 = 0.5 m/s. 1 – H/(2R) = 2, 2 – 4, 3 – 6, 4 – 10.
b
Fig. 5. The effect gas bubbles size on distributions of wall friction (a) and heat transfer rate (b) along the radial coordinate. Re = 2.3 104, Um1 = 0.5 m/s, 2R = 20 mm, H/(2R) = 2, TW = 313 K, T1 = Tb1 = 293 K, b = 5%. 1 – d = 0 (single-phase flow at b = 0%), 2 – 0.5 mm, 3 – 1, 4 – 3.
Fig. 7. The effect of Reynolds numbers on the Nusselt numbers at the stagnation point at H/(2R) = 2. d = 1 mm, Re = 2.3 104, Um1 = 0.5 m/s, 2R = 20 mm, TW = 313 K, T1 = Tb1 = 293 K. 1 – semi-empirical formula Nu ¼ 0:93Re0:5 Pr0:4 of [4], 2 – b = 0 (single-phase water flow), 3 – 5%, 4 – 10%.
M.A. Pakhomov, V.I. Terekhov / International Journal of Heat and Mass Transfer 92 (2016) 689–699
intensity in the range of r/(2R) < 2 (the zone of flow stagnation and the gradient zone). Then, down the flow at r/(2R) 5 (the region of development of the near-wall jet), the value of heat transfer in the two-phase jet with small bubbles (d = 1 mm) almost corresponds to this value for the single-phase impinging water jet. This occurs due to a decrease in the dispersed phase concentration in the near-wall part of the jet at its expansion and uplifting of gas bubbles. Two maxima in the distribution of the local heat transfer coefficient are explained by the small distance between the cooled surface and the pipe outlet and it agrees with data for the single-phase impinging jets [1–5]. The effect of distance between the pipe outlet and target surface on the heat transfer rate at the stagnation point in the bubbly impinging jet is presented in Fig. 6. The maximum value of heat transfer in the two-phase impinging jet is obtained at H/(2R) = 6 and the minimum magnitude of heat transfer is observed at H/ (2R) = 4. The same trend is revealed for the single-phase jets [1– 5]. The reason for this is that the length of the jet potential core is approximately six pipe diameters and the fully developed impinging jet reaches the obstacle in this case [1–5]. This correlates with our previous simulations of the gas-droplets jet impingement [7]. Then, the heat transfer decreases because the distance from the pipe outlet to the impingement surface increases due to attenuation of the axial gas velocity. The Reynolds number characterizes the effect of the jet velocity on the heat transfer rate between the heated impingement surface and the jet in single-phase impinging jets [1–5]. The effect of one of the key parameters of the impinging jet, the Reynolds number (Re), on the heat transfer rate at the stagnation point is shown in Fig. 7. Here, line 1 corresponds to the single-phase water jet in the form of Nu0 Re0.5 [1–5]. The magnitude of heat transfer at the stagnation point is maximal for all investigated distances H/(2R) (see Fig. 8). The Re number is high enough to assure a turbulent flow regime. It is noted that the heat transfer at the stagnation point is 1.4–2.2 times higher than that in the laminar impinging jet [1–5]. Heat transfer in the bubbly impinging jet is up to 2.2fold higher than in the single-phase water jet impingement. An increase in the Reynolds number leads to augmentation of the heat transfer for all values of gas flow rate ratios. A higher amount of liquid flow with gas bubbles reaches the heated surface, causing
697
enhancement of the heat transfer rate between the target surface and the impinging jet.
5. Comparison with experimental results Transverse profiles of liquid radial mean velocities in bubbly impinging jet at several stations from the stagnation point are shown in Fig. 8. The points are the experimental results of [15], and the lines are the predictions of the present study. The jet flow was formed by using the Vitoshinsky axisymmetric nozzle with an outlet diameter 2R = 15 mm. The target surface was placed normal to the flow direction at the nozzle-to-plate distance H/(2R) = 3. The working liquid was a mixture of water and 10% volume fraction of ethanol. The liquid and bubble temperatures were kept constant at T1 = Tb = 303 K. The Reynolds number defined from the liquid bulk velocity at the nozzle outlet Um1 = 0.93 m/s and the nozzle diameter was Re = 2RUm1/m = 1.2 104. The bubbles’ size was d = 0.85 mm and their gas flow rate ratio was b = 1.2%. The flow direction was upward. The distributions of the radial mean velocities of the liquid are similar to those in the wall single-phase jet. There is quantitative agreement with the measurements of [14] (the difference is up to 5%) except in the region very close to the wall (y/H < 0.03). The effect of the gas volumetric flow rate ratio of air bubbles on the modification of wall friction is shown in Fig. 9. Points indicate the experiments carried out in [14] and lines are the simulations conducted in the current study. The measurements were carried out by using the electrodiffusion method. The nozzle diameter was 2R = 10 mm. The profile of the averaged axial velocity on the nozzle edge was close to the uniform one. The momentum thickness at x/(2R) = 0.15 was 0.1 mm. The degree of turbulence on the nozzle edge was u0 /Um1 = 0.005–0.008 at its axis and 0.05– 0.06 in the mixing layer. The Reynolds number, based on the mean liquid velocity on the nozzle edge, was Re = 2RUm1/m = 4.04 104, the nozzle-to-target distance was H/(2R) = 2, the average bubble diameter equaled d = 200 lm, and the bubble size scattering was low so the diameter of gas bubbles in experiments was assumed to be constant. The volumetric gas flow rate ratio changed within the range of b = 0–12.1%. The impingement surface was vertical
Fig. 8. Transverse profiles of liquid radial mean velocities in two-phase impinging jet at several stations from the stagnation point. Points are the measurements of [15], lines are author’s simulations. H/(2R) = 3, d = 0.85 mm, b = 1.2%, Re = 1.2 104, Um1 = 0.93 m/s, 2R = 15 mm, T1 = Tb = 303 K.
698
M.A. Pakhomov, V.I. Terekhov / International Journal of Heat and Mass Transfer 92 (2016) 689–699
Fig. 9. The distributions of wall friction in the impinging jet for various gas flow rate ratio. Points are the measurements of [14], lines are author’s simulations. Re = 4.04 104, H/(2R) = 2, 2R = 10 mm, d = 200 lm. 1 – b = 0 (single-phase liquid flow), 2 – b = 3.4%, 3 – 5.9%, 4 – 12.1%.
and the nozzle was mounted horizontally. The results of numerical predictions of wall friction agree well with the measurement data of [14]: the maximal difference is not more than 10–15%. The addition of gas bubbles increases the wall friction (by up to 35%) in comparison with the single-phase water jet and the effect increases with a rise of b. This is consistent with the data in Fig. 5 and is characteristic for our simulations and the measurements of [14]. The second obvious local maximum in the distribution of wall friction is observed at r/(2R) 2 as for the heat transfer. The reasons of this phenomenon are the wall jet accelerates while moving along the flat plate and laminar–turbulent transition [4,5]. The behavior of wall friction for small values of b is similar to that one for the single-phase jet. The second local maximum becomes smoother only for the maximal value of gas volumetric flow rate ratio (b = 12.1%). The changes in the Nusselt number along the radial coordinate for the round impinging two-phase jet are presented in Fig. 10. Points indicate measurements by [8] and lines are the authors’ predicted results. The profiles of liquid and gas velocities at the pipe outlet were obtained by preliminary predictions of the water flow with air bubbles in the pipe with the length of 90R. The gas velocity
Fig. 10. The effect of air bubbles gas flow rate ratio on heat transfer distributions along the radial coordinate. Points are the measurements of [8], curves are author’s predictions. Re = 2.81 104, H/(2R) = 3, 2R = 11 mm, d = 1.4 mm, J = 2 m/s. 1 – b = 0 (single-phase liquid flow); 2 – Jb = 0.5 m/s, b = 20%; 3 – 1.2%, 37.5%.
at the pipe outlet was not measured in [8], and therefore, its value was obtained by preliminary calculation of the two-phase flow in the pipe. The addition of bubbles into impinging jet intensifies heat transfer significantly (more than in 1.5 times as compared to the single-phase impinging jet) due to turbulization of the near-wall zone of the impinging surface by the bubbles and it agrees with results of [8]. It should be noted that for the experiments of [8] and our numerical simulations, the main heat transfer intensification occurs in the stagnation zone, and this proves the data of our computations (see Figs. 5 and 6).
6. Conclusions The turbulent flow structure and heat transfer of a bubbly turbulent impinging jet are numerically investigated. The set of axisymmetrical steady-state RANS equations for the two-phase flow is utilized. The dispersed phase is modeled by the Eulerian approach. Liquid phase turbulence is computed with the Reynolds stress model for two-phase flow. The interaction between phases (two-way coupling) is considered by the addition of extra terms in the equations for the mean and the fluctuating motion. The interfacial model considers interfacial momentum transfer terms arising from drag, lift, wall lubrication, and turbulent dispersion force for the bubbles. The effect of a change in gas volumetric flow rate ratio and bubble size on the flow structure and heat transfer in the gas–liquid impinging jet is studied. The profiles of the mean liquid velocity in the two-phase impinging jet for the small values of b are in general similar to those for the single-phase flow. The liquid velocity in the presence of gas bubbles is higher than the corresponding value in the single-phase flow. The predictions demonstrated the significant anisotropy of the turbulent fluctuations in both the axial and radial directions for the bubbly impinging jet. An increase in the intensity of liquid velocity fluctuations in the two-phase flow (up to 25% as compared to the single-phase liquid jet) is shown in comparison with the single-phase flow. The increasing size of the dispersed phase increases the liquid turbulence. It is shown numerically that addition of the gas phase to the turbulent liquid increases the wall friction (by up to 40%). An increase in the bubble diameter intensifies heat transfer in the two-phase impinging jet (by up to 50%). The increase in wall friction and heat transfer is
M.A. Pakhomov, V.I. Terekhov / International Journal of Heat and Mass Transfer 92 (2016) 689–699
limited by the stagnation and gradient areas at r/(2R) 6 2. Two distinctive maxima of the Nusselt number distributions are observed for small distances between the pipe outlet and heated target surface H/(2R) = 2 in the bubbly impinging jets. Comparison of the modeling results with experimental data for the cases of isothermal [14,15] and non-isothermal [8] impinging jets at different orientations (upflows [15], downflows [8], and horizontal [14] impinging jets) has shown that the developed approach allows simulation of bubbly turbulent impinging jets. Acknowledgements This turbulence model was developed by the supporting of the Russian Scientific Foundation (Project No. 14-19-00402) and numerical results were obtained by financial support of the Russian Foundation for Basic Research (Project No. 15–08–03909). The authors are thankful to Mr. K.S. Pervunin (IT SB RAS, Novosibirsk) for providing access to the experimental database of upward bubbly impinging jets in electronic form. References [1] H. Martin, Heat and mass transfer between impinging gas jets and solid surface, Adv. Heat Transfer 13 (1977) 1–60. [2] E.P. Dyban, A.I. Mazur, Convective Heat Transfer in Jet Flows around Bodies, Naukova Dumkova, Kiev, 1982 (in Russian). [3] K. Jambunathan, E. Lai, M.A. Moss, B.L. Button, A review of heat transfer data for single circular jet impingement, Int. J. Heat Fluid Flow 13 (1992) 106–115. [4] B.W. Webb, C.F. Ma, Single-phase liquid jet impingement heat transfer, Adv. Heat Transfer 26 (1995) 105–217. [5] N. Zuckerman, N. Lior, Jet impingement heat transfer: physics, correlations, and numerical modeling, Adv. Heat Transfer 39 (2006) 565–631. [6] M. Garbero, M. Vanni, U. Fritsching, Gas/surface heat transfer in spray deposition processes, Int. J. Heat Fluid Flow 27 (2006) 105–122. [7] M.A. Pakhomov, V.I. Terekhov, Second moment closure simulation of flow and heat transfer in a gas-droplets turbulent impinging jet, Int. J. Therm. Sci. 60 (2012) 1–12. [8] A. Serizawa, O. Takahashi, Z. Kawara, T. Komeyama, I. Michiyoshi, Heat transfer augmentation by two-phase bubbly flow impinging jet with a confining wall, in: G. Hetsroni (Ed.), Proc. of the 9th Int. Heat Transfer Conference, vol. 4, Jerusalem, Israel, Paper 10-EH-16, 1990, pp. 93–98. [9] D.A. Zumbrunnen, M. Balasubramanian, Convective heat transfer enhancement due to gas injection into an impinging liquid jet, ASME J. Heat Transfer 117 (1995) 1011–1017. [10] C.T. Chang, G. Kojasoy, F. Landis, S. Downing, Confined single- and multiple-jet impingement heat transfer – II. Turbulent two-phase flow, Int. J. Heat Mass Transfer 38 (1995) 843–851. [11] D.E. Hall, F.P. Incropera, R. Viskanta, Jet impingement boiling from a circular free-surface jet during quenching: Part 2 two-phase jet, ASME, J. Heat Transfer 123 (2001) 911–917. [12] K. Choo, S.J. Kim, Heat transfer and fluid flow characteristics of two-phase impinging jets, Int. J. Heat Mass Transfer 53 (2010) 5692–5699. [13] D. Trainer, J. Kim, S.J. Kim, Heat transfer and flow characteristics of air-assisted impinging water jets, Int. J. Heat Mass Transfer 64 (2013) 501–513. [14] S.V. Alekseenko, D.M. Markovich, V.I. Semenov, Turbulent structure of a gas– liquid impinging jet, Fluid Dyn. 37 (2002) 684–694.
699
[15] S.V. Alekseenko, V.M. Dulin, D.M. Markovich, K.S. Pervunin, Experimental investigation of turbulence modification in bubbly axisymmetric jets, J. Eng. Thermophys. 24 (2015) 101–112. [16] F.Z. Kendil, E. Krepper, A.B. Salah, D. Lucas, A. Mataoui, Numerical study of a bubble plume generated by bubble entrainment from an impinging jet, Nucl. Eng. Des. 241 (2011) 4111–4121. [17] E. Krepper, Th. Frank, D. Lucas, H.-M. Prasser, P.J. Zwart, The inhomogeneous MUSIG model for the simulation of polydispersed flows, Nucl. Eng. Des. 238 (2008) 1690–1702. [18] R.T. Lahey Jr., D.A. Drew, The analysis of two-phase flow and heat transfer using a multidimensional, four-field, two-fluid model, Nucl. Eng. Des. 204 (2001) 29–44. [19] G.H. Yeoh, J.Y. Tu, Population balance modelling for bubbly flows with heat and mass transfer, Chem. Eng. Sci. 59 (2004) 3125–3139. [20] M.A. Pakhomov, V.I. Terekhov, Simulation of the turbulent structure of a flow and heat transfer in an ascending polydisperse bubble flow, Technol. Phys. 60 (2015) 1268–1276. [21] D.A. Drew, Mathematical modeling of two-phase flow, Ann. Rev. Fluid Mech. 15 (1983) 261–291. [22] I.V. Derevich, L.I. Zaichik, Particle deposition from a turbulent flow, Fluid Dyn. 23 (1988) 722729. [23] M.W. Reeks, On a kinetic equation for the transport of particles in turbulent flows, Phys. Fluids A 3 (1991) 446–456. [24] L.I. Zaichik, A statistical model of particle transport and heat transfer in turbulent shear flows, Phys. Fluids 11 (1999) 1521–1534. [25] J. Pozorski, J.-P. Minier, Probability density function modeling of dispersed two-phase turbulent flows, Phys. Rev. E 59 (1999) 855–863. [26] L.I. Zaichik, A.P. Skibin, S.L. Solov’ev, Simulation of the distribution of bubbles in a turbulent liquid using a diffusion-inertia model, High Temp. 42 (2004) 101–107. [27] D. Cooper, D.C. Jackson, B.E. Launder, G.X. Liao, Impinging jet studies for turbulence model assessment – I. Flow field experiments, Int. J. Heat Mass Transfer 36 (1993) 2675–2684. [28] T.J. Craft, B.E. Launder, New wall-reflection model applied to the turbulent impinging jet, AIAA J. 30 (1992) 2970–2972. [29] T.J. Craft, L.J.W. Graham, B.E. Launder, Impinging jet studies for turbulence model assessment II. An examination of the performance of four turbulence models, Int. J. Heat Mass Transfer 36 (1993) 26852697. [30] M. Lopez de Bertodano, S.J. Lee, R.T. Lahey Jr., D.A. Drew, The prediction of twophase turbulence and phase distribution using a Reynolds stress model, ASME J. Fluids Eng. 112 (1990) 107–113. [31] O.N. Kashinsky, P.D. Lobanov, M.A. Pakhomov, V.V. Randin, V.I. Terekhov, Experimental and numerical study of downward bubbly flow in a pipe, Int. J. Heat Mass Transfer 49 (2006) 3717–3727. [32] E. Loth, Quasi-steady shape and drag of deformable bubbles and drops, Int. J. Multiphase Flow 34 (2008) 523–546. [33] D.A. Drew, R.T. Lahey Jr., The virtual mass and lift force on a sphere in rotating and straining in viscid flow, Int. J. Multiphase Flow 13 (1987) 113–121. [34] A. Tomiyama, I. Sou, I. Zun, N. Kanami, T. Sakaguchi, Effects of Eotvoes number and dimensionless liquid volumetric flux on lateral motion of a bubble in a laminar duct flow, Adv. Multiphase Flow (1995) 3–15. [35] E.A. Ervin, G. Tryggvason, The rise of bubbles in a vertical shear flow, ASME J. Fluids Eng. 119 (1997) 443–449. [36] A. Tomiyama, H. Tamai, I. Zun, S. Hosokawa, Transverse migration of single bubbles in simple shear flows, Chem. Eng. Sci. 57 (2002) 1849–1958. [37] R.M. Wellek, A.K. Agrawal, A.H.P. Skelland, Shapes of liquid drops moving in liquid media, AIChE J. 12 (1966) 854–860. [38] M. Politano, P. Carrica, J. Converti, A model for turbulent polydisperse twophase flow in vertical channel, Int. J. Multiphase Flow 29 (2003) 1153–1182. [39] S.P. Antal, R.T. Lahey Jr., J.F. Flaherty, Analysis of phase distribution in fully developed laminar bubbly of two-phase flow, Int. J. Multiphase Flow 17 (1991) 635–652.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Numerical study of the flow and heat transfer in a turbulent bubbly jet impingement M.A. Pakhomov ⇑, V.I. Terekhov Laboratory of Thermal and Gas Dynamics, Kutateladze Institute of Thermophysics, Siberian Branch of Russian Academy of Sciences, Acad. Lavrent’ev Avenue 1, 630090 Novosibirsk, Russia
a r t i c l e
i n f o
Article history: Received 14 July 2015 Received in revised form 4 September 2015 Accepted 7 September 2015 Available online 26 September 2015 Keywords: Bubbly impinging jet Numerical modeling Heat transfer enhancement Turbulence modification
a b s t r a c t This contribution presents the results of modeling of the flow structure and heat transfer enhancement obtained by adding air bubbles into a turbulent liquid (water) impinging jet. A set of axisymmetrical steady-state RANS equations for the two-phase flow is utilized. The dispersed phase (bubbles) is modeled by the Eulerian approach. Liquid-phase turbulence is computed with the Reynolds stress model, taking into account the effect of bubbles on the carrier phase. The effect of changes in the gas volumetric flow rate ratio and bubble size on the flow structure, wall friction, and heat transfer in a gas–liquid impinging jet is numerically studied. The predictions demonstrate the significant anisotropy of the turbulent fluctuations in the axial and radial directions for the bubbly impinging jet. The addition of a gas phase into the turbulent liquid increases the wall friction (by up to 40% in comparison with the single-phase liquid jet) and heat transfer (by up to 50%). Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Jet impingement is a very effective method for enhancing heat transfer rate between a solid impinging surface and a jet flow in various applications (see comprehensive reviews [1–5]). Potential applications where high heat fluxes should be dissipated while the surface is kept at a relatively low temperature include highpower equipment, glass tempering, cooling of surfaces and preforms, spray painting, rotor blades, and nuclear reactors during emergency conditions [1–5]. The main parameters that affect the heat and mass transfer processes include the distance between the nozzle orifice (or pipe outlet) and the impingement (target) surface, the temperatures of the jet and wall surface, the velocity magnitude of the jet, and the turbulence intensity. Many attempts to modify flow characteristics and increase heat transport in impinging jets have been made in the last decades. Impinging two-phase jets with droplets or bubbles are employed to enhance heat transfer between the flow and the solid surface. Gas-droplet jet impingement on a hot surface removes large amounts of heat because of the latent heat of evaporation [6,7]. One of the ways of increasing the heat transfer rate is by adding a gas (or vapor) phase to an impinging liquid jet [8–13]. The enhancement in heat transfer is attributed to the high turbulence ⇑ Corresponding author. E-mail addresses: [email protected] (M.A. Pakhomov), [email protected] (V.I. Terekhov). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.09.010 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
level in the near-wall region accompanying bubble-induced agitation in a bubbly impinging jet [8]. Serizawa et al. [8] experimentally studied the heat transfer in an impinging air–liquid bubbly jet. The heat transfer coefficient was increased by a factor of 2–4 at a gas volumetric flow rate ratio of b 6 0.53. The enhancement was attributed to the high turbulence level caused by bubble-induced agitation and to acceleration of the liquid phase by the more rapidly moving air. Measurements in a slot impinging bubbly jet by boiling on the target surface were carried out in [9]. The gas volumetric flow rate ratio was b = 0–0.86 and the Reynolds numbers varied Re = 3700–21,000. Significant enhancement of the heat transfer (up to 2.2-fold in comparison with single-phase water impinging jet) was obtained. The experimental investigation of single and multiple confined two-phase jets using Freon R-113 in the presence of boiling was carried out in [10,11]. The regimes of single-phase forced convection and boiling heat transfer were studied in [10,11]. Heat transfer of the liquid–vapor impinging jets was augmented by a factor of 1.2 than that of the single-phase jet impingement [10]. The experimental investigation of heat transfer and pressure drop an in the air bubbles-water impinging jet was performed in [12,13]. The heat transfer enhancement (up to 2 times) in bubbly impinging jet at b = 20–30% was shown. The following increase of the gas volumetric flow rate ratio led to a decrease in the heat transfer enhancement. The turbulent structure in a round bubbly impinging jet without interphase heat transfer was studied experimentally under
690
M.A. Pakhomov, V.I. Terekhov / International Journal of Heat and Mass Transfer 92 (2016) 689–699
Nomenclature CD CP, CPb
drag coefficient of bubbles specific heat capacity of carrier liquid (water) and dispersed (gas bubbles) phases respectively (J kg1 K1) D diffusion coefficient (m2 s1) d bubble diameter (m) 2 Eob Eo ¼ gðq qb ÞdH =r modified Eotvos number hc convective heat transfer coefficient between liquid and bubble (W K1 m2) J, Jb superficial velocity of liquid (water) and bubbles respectively (m s1) k k = huiuii/2 turbulent kinetic energy (m2 s2) Nu Nu ¼ h2R=k Nusselt number P pressure (Pa) Pr Nu = lCp/k Prandtl number R pipe radius (m) R specific gas constant (J kg1 K1) Re 2RUm1/m Reynolds number Reb ¼ jU R jd=m Reynolds number of bubbles, based on the Reb sleep velocity r radial coordinate (m) T temperature (K) jU R j jU R j ¼ jU U L j sleep (interphase) velocity (m s1) Ui components of mean velocity (m s1) Um1 bulk mean velocity of the liquid flow at the pipe edge (m s1) friction velocity (m s1) U⁄ hu0 i; hv 0 i intensity of velocity fluctuations in axial and radial directions respectively (m s1) turbulent heat flux (m K s1) hu0j ti 0 0 hu v i turbulent Reynolds stress (m2 s2) We We ¼ qjU R j2 d=r Weber number
the superposition of external periodic excitation in [14]. The shear stress was measured on the target surface at a change of b = 0–12%. The effect of suppression of the large-scale structures was observed for high gas contents. It was found out that an addition of the gas phase at b < 12% leads to a significant increase in the wall shear stress (up to 40%) [13]. Recently, an experimental study of isothermal bubbly impinging turbulent jet by a combination of optical planar fluorescence for bubble imaging (PFBI) and particle image velocimetry (PIV) and particle tracking velocimetry (PTV) methods was carried out in [14]. The bubbly jet was investigated at different gas volume flow rate ratios of b = 0–4.2% at a fixed Reynolds number, Re = 12,000. Distributions of local void fraction a and mean and fluctuating velocities of both phases were measured. Bubbles increase the liquid phase turbulence at the nozzle edge when there is an additional supply of bubbles into the liquid jet. Then, they suppress the turbulence fluctuations compared to the single-phase flow in the downstream direction at x/(2R) > 0.3. The bubbles significantly intensify turbulence fluctuations due to an increase of the slip velocity between the phases close to the impingement surface [15]. The above mentioned studies were focused on the influence of gas volumetric flow rate ratio on heat transfer and turbulent structure in impinging jet. The effect of the size of dispersed phase on these characteristics was poorly studied. The authors found no studies on numerical study of flow structure and heat transfer in the impinging jets. A numerical study of an isothermal bubbly flow under plunging free round jet conditions using mono- and polydispersed approaches was performed in [16]. The turbulence is modeled by a k–e model with taking into account the bubble-induced turbulence. The bubble break-up and coalescence processes were
x y
axial coordinate (m) distance normal from the wall (m)
Greek letters volume fraction of the dispersed phase a local void fraction of bubbles b b = Jb/(Jb + J) gas volumetric flow rate ratio e dissipation of the turbulent kinetic energy (m2 s3) k heat conductivity (W K1 m1) l dynamic viscosity (kg m1 s1) m kinematic viscosity (m2 s1) q density (kg m3) r surface tension (N m1) 2 s s ¼ 3l4qReb db C D dynamic relaxation time of bubbles (s)
U
sW sH
wall friction (Pa) sH ¼ C Pb qb d2 =ð12kb YÞ thermal relaxation time of bubbles (s)
Subscripts 0 single-phase flow 1 parameter under initial conditions L liquid T turbulent parameter W parameter on the wall condition b bubble m mean-mass parameter Acronym CV control volume RANS Reynolds averaged Navier–Stokes equations SMC second moment closure
considered in the polydispersed air–water flow by using the inhomogeneous MUlti SIze Group (MUSIG) model [17]. Several studies of heat transfer in bubble flow in pipes have been presented in the literature [18–20]. The complexity of modeling bubbly turbulent flows with interphase heat transfer is associated with the large number of different physical multiscale phenomena involved, such as heat transfer, bubble coalescence and break-up processes. The aim of this work is to carry out a numerical simulation of the effect of bubbles size and gas volumetric flow rate ratio on flow structure and heat transfer modification in the turbulent bubbly jet impingement. This study may be interesting for scientists and engineers dealing with the problem of heat transfer enhancement in power equipment. The paper is organized as follows. In Section 2, the governing equations employed in this study for the simulation of a two-phase turbulent bubbly impinging jet are introduced. The numerical methods are described in Section 3. In Section 4, a numerical study of bubbly jet impingement with heat transfer is presented. Section 5 summarizes the main findings and conclusions. 2. Problem statement and governing equations 2.1. Physical model The modeling of dispersed phase is accomplished by the Eulerian two-fluid approach that treats the particulate phase as a continuous medium with properties analogous to those of a liquid [21]. In the two-fluid approach, both phases are considered as interacting continua. This technique involves the solution of a second set of Navier–Stokes-like equations in addition to those of the carrier (liquid) phase. Furthermore, the significance of liquid–par-
691
M.A. Pakhomov, V.I. Terekhov / International Journal of Heat and Mass Transfer 92 (2016) 689–699
ticle interactions is reflected in the momentum and kinetic energy coupling terms via the mean and the fluctuating drag force, respectively. The motion of the incompressible bubbly turbulent flow in the impinging jet is considered. The bubbles’ behavior in turbulent liquid and their back action on the flow is determined by drag, gravity lift, virtual mass, wall lubrication forces, turbulent transport, and turbulent diffusion. The effect of break-up and coalescence of bubbles is not taken into account. In order to account for the interaction between phases, that is, momentum exchange and heat and mass transfer, the conservation equations have to be extended by appropriate source/sink terms. The Eulerian approach is based on kinetic equations for a one-point probability density function of particle coordinates, velocity, and temperature in the turbulent Gaussian fluid flow fields [22–26]. Properties such as the mass of particles per unit volume are considered as a continuous property and the particle velocity is the averaged velocity over an average control volume. The interfacial transfer of mass, momentum, and energy requires averaging over the computational control volume too. Effects of pressure drop of the liquid and heat up of the gas phase on the bubble size are taken into account. The system of RANS equations for the carrier phase, the turbulence model and the equations for the dispersed phase are written in a form appropriate to axisymmetric flow, but for reason of brevity, are given below in general Cartesian tensor form. 2.2. The set of basic equations for the liquid phase The mean liquid flow is treated as a steady-state, incompressible and axisymmetrical flow and it is described by continuity, two-momentum and energy equations with taking into account the effect of bubbles presence. The system of RANS equations for the two-phase is given as @½ð1UÞqU i ¼0 @xi @½ð1UÞqU i U j UÞP ¼ @ð1 þ @x@ ð1 @xi @xi j @½ð1UÞqC P T ¼ @x@ i ð1 @xi
i UÞ l @U qhu0i u0j i þ qg þ ML @xj
6UðTT b Þ @T 0 @U UÞ k @x q C C Pb qb g t < u0j t > @x P huj ti d i j ð1Þ
Here M L ¼ Mb is the interphase interaction; g is the gravity force and gt is the coefficient of involvement of bubbles into the thermal fluctuational motion of liquid phase [22,24]. The set of Eq. (1) is written with taking into account the effect of bubbles on the transport process in the carrier (liquid) phase. The values of turbulent Prandtl number is equal to PrT = 0.9. The turbulent heat flux in the carrier phase is determined according to the Boussinesq hypothesis:
mT @T
PrT @xj
@½ð1 UÞU j hu0i u0j i @xj
:
¼ ð1 UÞðdij þ Pij þ Aij eij Þ þ S1
" 2 # @½ð1 UÞU j e e @U i ¼ ð1 UÞ C e1 Pk þ mT @xj k @xj e2 @ k @e e C e hu0l u0k i þ C e 3 S1 : C e2 k @xk e @xl k
ð2Þ
ð3Þ
Last terms in Eqs. (2) and (3) include an additional summand for description of interfacial interaction [30]. The right hand side of Eq. (2) describes turbulent diffusion, production of turbulent stresses from the shear averaged motion, redistribution of pulsation energy between different components of velocity fluctuations caused by correlation of pressure and deformation rate pulsations, and viscous dissipation, including interfacial interaction S1. The diffusion term (turbulent mixing at interaction of fluctuation velocity components) takes form
dij ¼ C S
@ k 0 0 @ D 0 0E hul uk i ui uj : @xk e @xl
Production of turbulent stresses from the averaged motion (intensity of energy transfer from averaged to fluctuation motion)
Pij ¼
q ¼ ð1 UÞP=ðRTÞ
< u0j t >¼
cost than the LES model. The second-moment closure of the turbulent flow is based on the solution to the system of transport equations for the components of Reynolds stresses. The SMC model predicts the turbulent Reynolds stresses directly from differential equations and allows us to compute the anisotropic flow. In the present study, the low-Reynolds number second-moment closure of [28,29] is employed. It is modified for the presence of bubbles [30]:
u0i u0k
@U j @xk
þ hu0j u0k i
@U i : @xk
The pressure-strain term considering energy transfer between different components of pulsations due to pressure-deformation rate correlations takes form [28]
Aij ¼ C 1
2 1 hu0i u0j i dij k C 2 Pij Pkk dij þ AW ij ; 3 3 k
e
ð4Þ
where the near-wall effects in Eq. (4) are written as [28]. The isotropic concept is used for dissipative term: eij 23 dij e. Last summands in the right part of Eqs. (2) and (3) take into account the back effect of dispersed phase on transport processes. The last term in the right side of the Eq. (2) S1 determines the additional turbulence production in liquid in the wake of the bubbles [30]
S1 ¼
3C i C D UjU R j3 : 4d
The constants of the turbulence model are presented in [28]: 2.3. The turbulence model of the carrier phase (liquid) The two-equation turbulence models have some well-known shortcomings. These models do not describe turbulent stress anisotropy and it leads to considerable errors in modeling strongly non-equilibrium flows with high velocity gradients and curvatures. The measurements reported in [27] demonstrated a significant anisotropy of the turbulent pulsations of axial and radial coordinates for a single-phase axisymmetrical impinging jet. One way to treat this anisotropy is to apply the model of Reynolds stress transport or second-moment closure (SMC). They are more complex from the point of view of computations than ke models. SMC is more detailed than the two-equation models, has a lower computational
CS = 0.18,
C1 = 1.8,
C2 = 0.6,
CW1 = 0.5,
CW2 = 0.08,
C 1w2 ¼ 0:1;
C 2w2 ¼ 0:4, Cl = 2.5 Ce1 = 1.44, Ce2 = 1.92, Ce = 0.18. The constants of the model of [26] are Ci = 0.1, Ce3 = 1.44. 2.4. The system of basic equations for the dispersed phase (bubbles) The dispersed phase (bubbles) is modeled by the Eulerian approach, which treats the particulate phase as a continuous medium with properties analogous to those of a liquid. This technique involves the solution of a second set of Navier–Stokes-like equations in addition to those of the carrier (liquid) phase. Furthermore, the significance of liquid–gas interactions is reflected in the momentum and kinetic energy coupling terms via the mean and
692
M.A. Pakhomov, V.I. Terekhov / International Journal of Heat and Mass Transfer 92 (2016) 689–699
fluctuating forces, respectively. The system of axisymmetric mean transport equations for the dispersed phase has the form
@ðUqb U bi Þ ¼0 @xi @ðUqb U bi U bj Þ @ðqUhubi ubj iÞ 1 @ðqb Db UÞ @ðUPÞ þ ¼ Mb @xi s @xj @xj @xj
ð5Þ
@ðUqb C Pb T b Þ q 1 @ðDbH UÞ ¼ hc UðT T b Þ b ; @xj sH sH @xj
ð6Þ
0
0
where Db, DbH are the turbulent diffusivity and bubbles turbulent heat transport tensors respectively [24] and hc is the convective heat transfer coefficient between liquid and bubble. The set of kinetic stresses hu0bi u0bj i equations in the dispersed phase are given in the form of [24]. The first term on the left hand side of Eq. (5) considers convective transport of the dispersed phase. The first, second, and third terms on the right side of Eq. (5) characterize the transport of bubbles due to the turbulent diffusion, the pressure drop, and the interphase interaction respectively.
the direction of the lift force changes its sign for gas–liquid flows if a substantial deformation of the bubbles occurs. The expression for prediction of lift force (IV) has the form [33] and CL is the lift coefficient by [36]
8 > < min½0:288tanhð0:121Reb Þ; f ðEoÞ; Eo 6 4 C L ¼ f ðEoÞ; 4 6 Eo 6 10 : > : 0:27; Eo > 10 Here f ðEoÞ ¼ 0:00105Eo3 0:0159Eo2 0:0204Eo þ 0:474 is the correction function [36] taking into account the bubble deforma2
Eo ¼ gðq qb ÞdH =r is the modified Eotvos number, p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 dH ¼ d 1 þ 0:163Eo0:757 is the maximum horizontal dimension of the bubble [37]. The coefficient CL changes its sign at a bubble diameter of d = 5.8 mm for our conditions. The coefficient CT in the term taking the turbulent diffusion (V) is CT = 0.1 [38]. Near the wall surface of the velocity difference between the bubble and the internal flow is less than the one between the bubble and the external flow. It causes the pressure gradient, and therefore bubble does not slip onto the wall. In order to handle this behavior of bubbles near the wall an additional wall lubrication force (VI) is introduced by [39]. tion,
2.5. Interface forces The summand for prediction of interphase interaction in the monodispersed flow has the form [20,31]
3.1. Numerical procedures
M L ¼ M b
3qC D @U @U ¼ UU R jU R j þ C m qU U j i U bj bi þ qb Ug 4d @xj @xj @U i @U j @U C T qkjU R j C L UqjU R j @xj @xi @xi d 2aqjU R j2 C W1 C W2 : 2yb d
ð7Þ
The interphase interaction in Eq. (7) M liq i is considered by taking into account the effect of following forces: drag force, added mass, gravity and Archimedes forces, lift force, turbulent diffusion, wall lubrication force and turbulent migration (turbophoresis) force [20,31]. In the transverse direction the lift and wall lubrication forces, turbulent diffusion and turbophoresis act, determining together the radial distribution of the void fraction in the vertical tube cross-section. However, the major influence on the radial distribution of bubbles is given by the lift and wall lubrication forces. The drag coefficient CD of the deformable bubbles in the drag force (I) is determined by the formula of [32]
C D ¼ C We!0 þ DC D ðC We!1 C We!0 Þ; where C We!0 is the drag coefficient of non-deformable spherical bubbles at Weber numbers We ? 0, C We!1 ¼ 8=3 þ 24=Reb is the drag coefficient at large Weber numbers We ? 1, and h i 1:6 is the interpolation function DC D ¼ tanh 0:0038ðWeRe0:2 b Þ describing transition from zero to a large Weber number. In accordance to [32] the following relation can be written:
C We!0 ¼
8 < :
3. Numerical procedures, boundary conditions, and validation of the model
24 Reb
3 1 þ 16 Re0:687 ; b
0:44;
Reb 6 103
The developed model employs ‘‘in-house” code to simulate a turbulent bubbly impinging jet. The mean transport equations and the SMC model are solved with a control volume method on a staggered grid. The QUICK scheme is used to approximate the convective terms, and the second-order accurate central difference scheme is adopted for the diffusion terms. The velocity correction is used to satisfy continuity through the SIMPLEC algorithm, which couples velocity and pressure. The computational domain is a cylinder of 30R in the radial direction and height H. The computational grid, which is nonuniform in both the axial and the radial direction, is applied with a high resolution near the solid boundary (the impinging surface) and in the axis region. The logarithmic function is used to produce the non-uniformity in both the axial and the radial direction. The first cell from the solid wall is located at a distance y+ = yU⁄/ m = 0.3–0.5 from the wall, where U⁄ is the friction velocity obtained for the flow in the pipe outlet. At least 10 control volumes have been generated to resolve the mean velocity field and turbulence quantities in the viscosity-affected near-wall region (y+ < 10). Grid sensitivity studies are carried out to determine the optimum grid resolution that gives the mesh-independent solution. For all numerical investigations performed in the study, a basic grid with 200 256 control volumes (CVs) along the axial and radial directions is used. Grid convergence is verified for three grid sizes: m1: 100 128 = 1.28 104, m2: 200 256 = 5.12 104, and m3: 400 400 = 1.6 105 CVs. The profiles of Nusselt numbers along the radial coordinate are shown in Fig. 1 at H/(2R) = 2. 3.2. Boundary conditions
3
Reb > 10 :
The coefficient for prediction of the effect of added mass (II) is Cm = 0.5 [33]. The lift force considers the interaction of the bubble with the shear field of the liquid per unit volume. The well-known lift force formulation for two-phase flows, which has a positive coefficient CL, acts in the direction of decreasing liquid velocity. Experimental [34] and numerical [35] studies have shown that
The results of preliminary simulations for the single-phase water flow in a pipe with a length of 150R are used for the determination of velocity components and turbulence at the pipe outlet. These conditions are sufficient to achieve fully developed turbulent flow. The symmetry conditions are set on the stagnation line. Noslip conditions are set on the target surface. The conditions at the outlet edges of the computational domain are set @/=@r ¼ 0 for
M.A. Pakhomov, V.I. Terekhov / International Journal of Heat and Mass Transfer 92 (2016) 689–699
693
Fig. 1. Grid convergence test. Re = 2.3 104, TW = 313 K, T1 = Tb1 = 293 K, H/(2R) = 2, b = 5%, d1 = 1 mm. 1 – 100 128 control volumes, 2 – 200 256, 3 – 400 400.
all variables. The components of Reynolds stresses are determined at the same points on the control volume surface as the corresponding components of mean velocities by using the method of [28,29]. The water and bubbles inlet temperatures in the impinging jet and temperature in ambient quiescent medium are T1 = Tb1 = T/ = 293 K. 3.3. Validation analysis for the single-phase impinging turbulent jet In the first step, a comparison is carried out with measurements of turbulent quantities and heat transfer characteristics [27] in a single-phase air impinging round jet. These results were published in [7] for air and gas-droplet turbulent impinging jets. Our predictions agree well with the data of other studies for a wide range of distances between the pipe outlet and the impinging surface and Reynolds numbers. A good agreement is obtained with the correlations for the liquid impinging jets presented in [4] (the maximal difference in the Nusselt number is up to 6%). This is the basis for the study of the more complicated case of impinging bubbly jet. 4. Numerical results and discussion All simulations have been carried out for monodispersed air bubbles and water mixture at atmospheric pressure for the downward flow. The effect of break-up and coalescence of bubbles is not taken into account. The pipe’s inner diameter is 2R = 20 mm. The mean velocity of the gas flow at the pipe edge varies within the range Um1 = 0.25–1 m/s. The magnitude of bubble velocity at the inlet cross-section Ub1 = 0.8Um1. The Reynolds number of the jet varies within the range Re = 2RUm1/m = (0.8–6) 104. The wall temperature is TW = 313 K and is kept constant throughout the simulations. The initial temperature of gas and water at the pipe edge is T1 = Tb1 = 293 K. The distance to the obstacle varies within the range H/(2R) = 1–10. The bubble diameter varies in the range d = 0–5 mm and the gas flow rate ratio varies in the range b = 0–10%. 4.1. Turbulent flow structure The mean total velocity distributions of single-phase water (1), liquid (water) in two-phase flow (2), and bubbles (3) for various
distances from the stagnation point are shown in Fig. 2. The addition of bubbles affects the magnitude of liquid velocity. The profiles of liquid velocity in the two-phase impinging jet are generally similar to those in the single-phase flow. The liquid velocity in the presence of gas bubbles (2) is higher than the corresponding value in the single-phase flow (1). This effect increases with a rise of the bubble size. The liquid velocity is higher than the velocity of the gas phase (3). Flow acceleration occurs initially when moving away from the stagnation point. The velocity profiles have a pronounced jet profile with a maximum located in the wall region at x/(2R) = 1. The flow velocity decreases on moving away from the stagnation point due to jet expansion when it mixes with the surrounding quiescent medium and the jet shape of the profile becomes less obvious. Distributions of axial (normal to wall) and radial (parallel to wall) fluctuations and Reynolds stresses of liquid phase are shown at H/(2R) = 2 in Fig. 3 for several stations downstream of the stagnation point. Here, line 1 corresponds to the single-phase liquid jet and lines 2 and 3 are those for the water in bubbly impinging jet for various bubble diameters. The predictions demonstrate the significant anisotropy of the turbulent fluctuations in both the axial and radial directions for the bubbly impinging jet. The increase in the intensity of liquid velocity fluctuations in the two-phase flow (up to 25%) is shown in comparison with the single-phase flow (1). The growth of the size of the dispersed phase causes an increase in liquid turbulence. Additional production of liquid phase turbulence is explained by vortex formation (flow separation) on streamlining of the large gas bubbles by the liquid. The maximal values of axial and radial fluctuations of the two-phase jet velocity are observed at a distance of r/(2R) = 2 (see Fig. 3a and b). Turbulence production decreases with development of the near-wall jet along the impinging surface due to near-wall jet mixing with the ambient medium and uplifting of air bubbles; the value of radial fluctuations in the gas–liquid flow is close to the corresponding value for the single-phase water flow. The profiles of Reynolds shear stress along the transverse coordinate are shown in Fig. 3c. The maximal value of Reynolds stress is observed at a distance of r/(2R) 3 from the stagnation point. The local maximum of hu0 m0 i is situated in the wall zone. These conclusions correspond qualitatively to the data of measurements [27] for the single-phase air impinging jet and our previous predictions for the gas-droplet jet impingement [7].
694
M.A. Pakhomov, V.I. Terekhov / International Journal of Heat and Mass Transfer 92 (2016) 689–699
a
b
Fig. 2. The profiles of total mean liquid velocities in the two-phase flow (a) and for various gas bubbles size (b). Re = 2.3 104, Um1 = 0.5 m/s, 2R = 20 mm, H/(2R) = 2, TW = 313 K, T1 = Tb1 = 293 K. (a) d = 1 mm. 1 – single-phase jet (b = 0), 2 – liquid at b = 5%, 3 – gas bubbles at b = 5%. (b) b = 5%. 1 – Single-phase jet (d = 0), 2 – d = 1 mm, 3 – 3 mm.
Transverse distributions of the local void fraction a along the length of the impinging surface are presented in Fig. 4. For small distances from the stagnation point, it is characteristic for the location of the local maximum of air bubble concentration to be near the wall, r/(2R) 6 2; then, due to the action of buoyancy forces and extension of the wall jet, the local maximum of a is located at some distance from the target wall. The thickness of the layer of ‘‘pure” liquid depends on b. With the increase of the volumetric gas flow rate ratio, the thickness of the zone that is free of bubbles increases.
an increase in the wall friction (up to 40%). Moreover, this effect increases with a rise of the bubble diameter (see Fig. 5a). It is explained by an increase in turbulence generation at the separated flow around large air bubbles and the increasing gradient of liquid velocity near the wall. The effect of the diameter of air bubbles on heat transfer is shown in Fig. 5b. The local Nusselt number at a constant wall temperature was determined by the difference between the wall temperature and the temperature of liquid at the pipe outlet:
Nu ¼ 4.2. Wall friction and heat transfer The effect of bubble size on shear stress distributions on the wall surface and heat transfer is shown in Fig. 5. The wall friction is calculated by relationship: sW ¼ lð@V=@yÞW . It is shown numerically that the addition of gas phase to the turbulent liquid causes
ð@T=@yÞW 2R ; TW T1
where ð@T=@yÞW is the gradient of liquid phase temperature on the wall. The increase in the bubble size intensifies heat transfer between the two-phase jet and target, and this is consistent with our previous data for the vertical downflows in the pipe [18]. The addition of gas bubbles leads to a 1.5-fold increase in heat transfer
M.A. Pakhomov, V.I. Terekhov / International Journal of Heat and Mass Transfer 92 (2016) 689–699
695
a
b
c
Fig. 3. The effect of air bubbles diameter on distributions of axial (normal to the wall) (a), radial fluctuations (parallel to the wall) (b) and Reynolds shear stress (b). Dashed lines are the single-phase water impinging jet (1), solid and dotted lines are two-phase air–water (2, 3) jet. Re = 2.3 104, Um1 = 0.5 m/s, 2R = 20 mm, H/(2R) = 2, TW = 313 K, T1 = Tb1 = 293 K, b = 5%. 1 – Single-phase impinging jet (b = 0), 2 – d = 1 mm, 3 – 3 mm.
696
M.A. Pakhomov, V.I. Terekhov / International Journal of Heat and Mass Transfer 92 (2016) 689–699
Fig. 4. The local void fraction profiles downstream of the stagnation point. Re = 2.3 104, H/(2R) = 2, TW = 313 K, T1 = Tb1 = 293 K, d = 1 mm. 1 – b = 2%, 2 – 5%, 3 – 10%.
a
Fig. 6. Heat transfer at the stagnation point for various distance between pipe outlet and target surface. d = 1 mm, Re = 2.3 104, Um1 = 0.5 m/s. 1 – H/(2R) = 2, 2 – 4, 3 – 6, 4 – 10.
b
Fig. 5. The effect gas bubbles size on distributions of wall friction (a) and heat transfer rate (b) along the radial coordinate. Re = 2.3 104, Um1 = 0.5 m/s, 2R = 20 mm, H/(2R) = 2, TW = 313 K, T1 = Tb1 = 293 K, b = 5%. 1 – d = 0 (single-phase flow at b = 0%), 2 – 0.5 mm, 3 – 1, 4 – 3.
Fig. 7. The effect of Reynolds numbers on the Nusselt numbers at the stagnation point at H/(2R) = 2. d = 1 mm, Re = 2.3 104, Um1 = 0.5 m/s, 2R = 20 mm, TW = 313 K, T1 = Tb1 = 293 K. 1 – semi-empirical formula Nu ¼ 0:93Re0:5 Pr0:4 of [4], 2 – b = 0 (single-phase water flow), 3 – 5%, 4 – 10%.
M.A. Pakhomov, V.I. Terekhov / International Journal of Heat and Mass Transfer 92 (2016) 689–699
intensity in the range of r/(2R) < 2 (the zone of flow stagnation and the gradient zone). Then, down the flow at r/(2R) 5 (the region of development of the near-wall jet), the value of heat transfer in the two-phase jet with small bubbles (d = 1 mm) almost corresponds to this value for the single-phase impinging water jet. This occurs due to a decrease in the dispersed phase concentration in the near-wall part of the jet at its expansion and uplifting of gas bubbles. Two maxima in the distribution of the local heat transfer coefficient are explained by the small distance between the cooled surface and the pipe outlet and it agrees with data for the single-phase impinging jets [1–5]. The effect of distance between the pipe outlet and target surface on the heat transfer rate at the stagnation point in the bubbly impinging jet is presented in Fig. 6. The maximum value of heat transfer in the two-phase impinging jet is obtained at H/(2R) = 6 and the minimum magnitude of heat transfer is observed at H/ (2R) = 4. The same trend is revealed for the single-phase jets [1– 5]. The reason for this is that the length of the jet potential core is approximately six pipe diameters and the fully developed impinging jet reaches the obstacle in this case [1–5]. This correlates with our previous simulations of the gas-droplets jet impingement [7]. Then, the heat transfer decreases because the distance from the pipe outlet to the impingement surface increases due to attenuation of the axial gas velocity. The Reynolds number characterizes the effect of the jet velocity on the heat transfer rate between the heated impingement surface and the jet in single-phase impinging jets [1–5]. The effect of one of the key parameters of the impinging jet, the Reynolds number (Re), on the heat transfer rate at the stagnation point is shown in Fig. 7. Here, line 1 corresponds to the single-phase water jet in the form of Nu0 Re0.5 [1–5]. The magnitude of heat transfer at the stagnation point is maximal for all investigated distances H/(2R) (see Fig. 8). The Re number is high enough to assure a turbulent flow regime. It is noted that the heat transfer at the stagnation point is 1.4–2.2 times higher than that in the laminar impinging jet [1–5]. Heat transfer in the bubbly impinging jet is up to 2.2fold higher than in the single-phase water jet impingement. An increase in the Reynolds number leads to augmentation of the heat transfer for all values of gas flow rate ratios. A higher amount of liquid flow with gas bubbles reaches the heated surface, causing
697
enhancement of the heat transfer rate between the target surface and the impinging jet.
5. Comparison with experimental results Transverse profiles of liquid radial mean velocities in bubbly impinging jet at several stations from the stagnation point are shown in Fig. 8. The points are the experimental results of [15], and the lines are the predictions of the present study. The jet flow was formed by using the Vitoshinsky axisymmetric nozzle with an outlet diameter 2R = 15 mm. The target surface was placed normal to the flow direction at the nozzle-to-plate distance H/(2R) = 3. The working liquid was a mixture of water and 10% volume fraction of ethanol. The liquid and bubble temperatures were kept constant at T1 = Tb = 303 K. The Reynolds number defined from the liquid bulk velocity at the nozzle outlet Um1 = 0.93 m/s and the nozzle diameter was Re = 2RUm1/m = 1.2 104. The bubbles’ size was d = 0.85 mm and their gas flow rate ratio was b = 1.2%. The flow direction was upward. The distributions of the radial mean velocities of the liquid are similar to those in the wall single-phase jet. There is quantitative agreement with the measurements of [14] (the difference is up to 5%) except in the region very close to the wall (y/H < 0.03). The effect of the gas volumetric flow rate ratio of air bubbles on the modification of wall friction is shown in Fig. 9. Points indicate the experiments carried out in [14] and lines are the simulations conducted in the current study. The measurements were carried out by using the electrodiffusion method. The nozzle diameter was 2R = 10 mm. The profile of the averaged axial velocity on the nozzle edge was close to the uniform one. The momentum thickness at x/(2R) = 0.15 was 0.1 mm. The degree of turbulence on the nozzle edge was u0 /Um1 = 0.005–0.008 at its axis and 0.05– 0.06 in the mixing layer. The Reynolds number, based on the mean liquid velocity on the nozzle edge, was Re = 2RUm1/m = 4.04 104, the nozzle-to-target distance was H/(2R) = 2, the average bubble diameter equaled d = 200 lm, and the bubble size scattering was low so the diameter of gas bubbles in experiments was assumed to be constant. The volumetric gas flow rate ratio changed within the range of b = 0–12.1%. The impingement surface was vertical
Fig. 8. Transverse profiles of liquid radial mean velocities in two-phase impinging jet at several stations from the stagnation point. Points are the measurements of [15], lines are author’s simulations. H/(2R) = 3, d = 0.85 mm, b = 1.2%, Re = 1.2 104, Um1 = 0.93 m/s, 2R = 15 mm, T1 = Tb = 303 K.
698
M.A. Pakhomov, V.I. Terekhov / International Journal of Heat and Mass Transfer 92 (2016) 689–699
Fig. 9. The distributions of wall friction in the impinging jet for various gas flow rate ratio. Points are the measurements of [14], lines are author’s simulations. Re = 4.04 104, H/(2R) = 2, 2R = 10 mm, d = 200 lm. 1 – b = 0 (single-phase liquid flow), 2 – b = 3.4%, 3 – 5.9%, 4 – 12.1%.
and the nozzle was mounted horizontally. The results of numerical predictions of wall friction agree well with the measurement data of [14]: the maximal difference is not more than 10–15%. The addition of gas bubbles increases the wall friction (by up to 35%) in comparison with the single-phase water jet and the effect increases with a rise of b. This is consistent with the data in Fig. 5 and is characteristic for our simulations and the measurements of [14]. The second obvious local maximum in the distribution of wall friction is observed at r/(2R) 2 as for the heat transfer. The reasons of this phenomenon are the wall jet accelerates while moving along the flat plate and laminar–turbulent transition [4,5]. The behavior of wall friction for small values of b is similar to that one for the single-phase jet. The second local maximum becomes smoother only for the maximal value of gas volumetric flow rate ratio (b = 12.1%). The changes in the Nusselt number along the radial coordinate for the round impinging two-phase jet are presented in Fig. 10. Points indicate measurements by [8] and lines are the authors’ predicted results. The profiles of liquid and gas velocities at the pipe outlet were obtained by preliminary predictions of the water flow with air bubbles in the pipe with the length of 90R. The gas velocity
Fig. 10. The effect of air bubbles gas flow rate ratio on heat transfer distributions along the radial coordinate. Points are the measurements of [8], curves are author’s predictions. Re = 2.81 104, H/(2R) = 3, 2R = 11 mm, d = 1.4 mm, J = 2 m/s. 1 – b = 0 (single-phase liquid flow); 2 – Jb = 0.5 m/s, b = 20%; 3 – 1.2%, 37.5%.
at the pipe outlet was not measured in [8], and therefore, its value was obtained by preliminary calculation of the two-phase flow in the pipe. The addition of bubbles into impinging jet intensifies heat transfer significantly (more than in 1.5 times as compared to the single-phase impinging jet) due to turbulization of the near-wall zone of the impinging surface by the bubbles and it agrees with results of [8]. It should be noted that for the experiments of [8] and our numerical simulations, the main heat transfer intensification occurs in the stagnation zone, and this proves the data of our computations (see Figs. 5 and 6).
6. Conclusions The turbulent flow structure and heat transfer of a bubbly turbulent impinging jet are numerically investigated. The set of axisymmetrical steady-state RANS equations for the two-phase flow is utilized. The dispersed phase is modeled by the Eulerian approach. Liquid phase turbulence is computed with the Reynolds stress model for two-phase flow. The interaction between phases (two-way coupling) is considered by the addition of extra terms in the equations for the mean and the fluctuating motion. The interfacial model considers interfacial momentum transfer terms arising from drag, lift, wall lubrication, and turbulent dispersion force for the bubbles. The effect of a change in gas volumetric flow rate ratio and bubble size on the flow structure and heat transfer in the gas–liquid impinging jet is studied. The profiles of the mean liquid velocity in the two-phase impinging jet for the small values of b are in general similar to those for the single-phase flow. The liquid velocity in the presence of gas bubbles is higher than the corresponding value in the single-phase flow. The predictions demonstrated the significant anisotropy of the turbulent fluctuations in both the axial and radial directions for the bubbly impinging jet. An increase in the intensity of liquid velocity fluctuations in the two-phase flow (up to 25% as compared to the single-phase liquid jet) is shown in comparison with the single-phase flow. The increasing size of the dispersed phase increases the liquid turbulence. It is shown numerically that addition of the gas phase to the turbulent liquid increases the wall friction (by up to 40%). An increase in the bubble diameter intensifies heat transfer in the two-phase impinging jet (by up to 50%). The increase in wall friction and heat transfer is
M.A. Pakhomov, V.I. Terekhov / International Journal of Heat and Mass Transfer 92 (2016) 689–699
limited by the stagnation and gradient areas at r/(2R) 6 2. Two distinctive maxima of the Nusselt number distributions are observed for small distances between the pipe outlet and heated target surface H/(2R) = 2 in the bubbly impinging jets. Comparison of the modeling results with experimental data for the cases of isothermal [14,15] and non-isothermal [8] impinging jets at different orientations (upflows [15], downflows [8], and horizontal [14] impinging jets) has shown that the developed approach allows simulation of bubbly turbulent impinging jets. Acknowledgements This turbulence model was developed by the supporting of the Russian Scientific Foundation (Project No. 14-19-00402) and numerical results were obtained by financial support of the Russian Foundation for Basic Research (Project No. 15–08–03909). The authors are thankful to Mr. K.S. Pervunin (IT SB RAS, Novosibirsk) for providing access to the experimental database of upward bubbly impinging jets in electronic form. References [1] H. Martin, Heat and mass transfer between impinging gas jets and solid surface, Adv. Heat Transfer 13 (1977) 1–60. [2] E.P. Dyban, A.I. Mazur, Convective Heat Transfer in Jet Flows around Bodies, Naukova Dumkova, Kiev, 1982 (in Russian). [3] K. Jambunathan, E. Lai, M.A. Moss, B.L. Button, A review of heat transfer data for single circular jet impingement, Int. J. Heat Fluid Flow 13 (1992) 106–115. [4] B.W. Webb, C.F. Ma, Single-phase liquid jet impingement heat transfer, Adv. Heat Transfer 26 (1995) 105–217. [5] N. Zuckerman, N. Lior, Jet impingement heat transfer: physics, correlations, and numerical modeling, Adv. Heat Transfer 39 (2006) 565–631. [6] M. Garbero, M. Vanni, U. Fritsching, Gas/surface heat transfer in spray deposition processes, Int. J. Heat Fluid Flow 27 (2006) 105–122. [7] M.A. Pakhomov, V.I. Terekhov, Second moment closure simulation of flow and heat transfer in a gas-droplets turbulent impinging jet, Int. J. Therm. Sci. 60 (2012) 1–12. [8] A. Serizawa, O. Takahashi, Z. Kawara, T. Komeyama, I. Michiyoshi, Heat transfer augmentation by two-phase bubbly flow impinging jet with a confining wall, in: G. Hetsroni (Ed.), Proc. of the 9th Int. Heat Transfer Conference, vol. 4, Jerusalem, Israel, Paper 10-EH-16, 1990, pp. 93–98. [9] D.A. Zumbrunnen, M. Balasubramanian, Convective heat transfer enhancement due to gas injection into an impinging liquid jet, ASME J. Heat Transfer 117 (1995) 1011–1017. [10] C.T. Chang, G. Kojasoy, F. Landis, S. Downing, Confined single- and multiple-jet impingement heat transfer – II. Turbulent two-phase flow, Int. J. Heat Mass Transfer 38 (1995) 843–851. [11] D.E. Hall, F.P. Incropera, R. Viskanta, Jet impingement boiling from a circular free-surface jet during quenching: Part 2 two-phase jet, ASME, J. Heat Transfer 123 (2001) 911–917. [12] K. Choo, S.J. Kim, Heat transfer and fluid flow characteristics of two-phase impinging jets, Int. J. Heat Mass Transfer 53 (2010) 5692–5699. [13] D. Trainer, J. Kim, S.J. Kim, Heat transfer and flow characteristics of air-assisted impinging water jets, Int. J. Heat Mass Transfer 64 (2013) 501–513. [14] S.V. Alekseenko, D.M. Markovich, V.I. Semenov, Turbulent structure of a gas– liquid impinging jet, Fluid Dyn. 37 (2002) 684–694.
699
[15] S.V. Alekseenko, V.M. Dulin, D.M. Markovich, K.S. Pervunin, Experimental investigation of turbulence modification in bubbly axisymmetric jets, J. Eng. Thermophys. 24 (2015) 101–112. [16] F.Z. Kendil, E. Krepper, A.B. Salah, D. Lucas, A. Mataoui, Numerical study of a bubble plume generated by bubble entrainment from an impinging jet, Nucl. Eng. Des. 241 (2011) 4111–4121. [17] E. Krepper, Th. Frank, D. Lucas, H.-M. Prasser, P.J. Zwart, The inhomogeneous MUSIG model for the simulation of polydispersed flows, Nucl. Eng. Des. 238 (2008) 1690–1702. [18] R.T. Lahey Jr., D.A. Drew, The analysis of two-phase flow and heat transfer using a multidimensional, four-field, two-fluid model, Nucl. Eng. Des. 204 (2001) 29–44. [19] G.H. Yeoh, J.Y. Tu, Population balance modelling for bubbly flows with heat and mass transfer, Chem. Eng. Sci. 59 (2004) 3125–3139. [20] M.A. Pakhomov, V.I. Terekhov, Simulation of the turbulent structure of a flow and heat transfer in an ascending polydisperse bubble flow, Technol. Phys. 60 (2015) 1268–1276. [21] D.A. Drew, Mathematical modeling of two-phase flow, Ann. Rev. Fluid Mech. 15 (1983) 261–291. [22] I.V. Derevich, L.I. Zaichik, Particle deposition from a turbulent flow, Fluid Dyn. 23 (1988) 722729. [23] M.W. Reeks, On a kinetic equation for the transport of particles in turbulent flows, Phys. Fluids A 3 (1991) 446–456. [24] L.I. Zaichik, A statistical model of particle transport and heat transfer in turbulent shear flows, Phys. Fluids 11 (1999) 1521–1534. [25] J. Pozorski, J.-P. Minier, Probability density function modeling of dispersed two-phase turbulent flows, Phys. Rev. E 59 (1999) 855–863. [26] L.I. Zaichik, A.P. Skibin, S.L. Solov’ev, Simulation of the distribution of bubbles in a turbulent liquid using a diffusion-inertia model, High Temp. 42 (2004) 101–107. [27] D. Cooper, D.C. Jackson, B.E. Launder, G.X. Liao, Impinging jet studies for turbulence model assessment – I. Flow field experiments, Int. J. Heat Mass Transfer 36 (1993) 2675–2684. [28] T.J. Craft, B.E. Launder, New wall-reflection model applied to the turbulent impinging jet, AIAA J. 30 (1992) 2970–2972. [29] T.J. Craft, L.J.W. Graham, B.E. Launder, Impinging jet studies for turbulence model assessment II. An examination of the performance of four turbulence models, Int. J. Heat Mass Transfer 36 (1993) 26852697. [30] M. Lopez de Bertodano, S.J. Lee, R.T. Lahey Jr., D.A. Drew, The prediction of twophase turbulence and phase distribution using a Reynolds stress model, ASME J. Fluids Eng. 112 (1990) 107–113. [31] O.N. Kashinsky, P.D. Lobanov, M.A. Pakhomov, V.V. Randin, V.I. Terekhov, Experimental and numerical study of downward bubbly flow in a pipe, Int. J. Heat Mass Transfer 49 (2006) 3717–3727. [32] E. Loth, Quasi-steady shape and drag of deformable bubbles and drops, Int. J. Multiphase Flow 34 (2008) 523–546. [33] D.A. Drew, R.T. Lahey Jr., The virtual mass and lift force on a sphere in rotating and straining in viscid flow, Int. J. Multiphase Flow 13 (1987) 113–121. [34] A. Tomiyama, I. Sou, I. Zun, N. Kanami, T. Sakaguchi, Effects of Eotvoes number and dimensionless liquid volumetric flux on lateral motion of a bubble in a laminar duct flow, Adv. Multiphase Flow (1995) 3–15. [35] E.A. Ervin, G. Tryggvason, The rise of bubbles in a vertical shear flow, ASME J. Fluids Eng. 119 (1997) 443–449. [36] A. Tomiyama, H. Tamai, I. Zun, S. Hosokawa, Transverse migration of single bubbles in simple shear flows, Chem. Eng. Sci. 57 (2002) 1849–1958. [37] R.M. Wellek, A.K. Agrawal, A.H.P. Skelland, Shapes of liquid drops moving in liquid media, AIChE J. 12 (1966) 854–860. [38] M. Politano, P. Carrica, J. Converti, A model for turbulent polydisperse twophase flow in vertical channel, Int. J. Multiphase Flow 29 (2003) 1153–1182. [39] S.P. Antal, R.T. Lahey Jr., J.F. Flaherty, Analysis of phase distribution in fully developed laminar bubbly of two-phase flow, Int. J. Multiphase Flow 17 (1991) 635–652.