Prandtl Number Liquid Research Articles

Thermocapillary convection in shallow liquid layers for moderate Prandtl number liquids using different cylinder configurations

In this paper, the influence of thermocapillary convection in a shallow annular pool using different cylinder configurations is numerically investigated. An axisymmetric, two-dimensional, cylindrical geometry with two concentric cylinders kept at two different temperatures constitutes the model. The main cylinder has a radius of 40 mm and the secondary cylinder has a radius of 20 mm. The effects of Prandtl numbers (Pr = 6.7, 16.1, and 25.2) and layer depths (d = 3.0 mm and 11.0 mm) on the streamlines, isotherms, velocity, and temperature profiles at various locations of the flow are investigated. Depending on the position of the secondary cylinder, three different configurations: 1) an internal cylinder 2) an external cylinder, and a 3) central cylinder is analysed. The flow is characterized by a rapid inward flow along the hot outer wall towards the colder inner wall. In deeper pool at lower Prandtl numbers, flow patterns are characterized by a single rolling cell structure. Conversely, secondary cell formations appear in shallow pool regardless of the Prandtl number. Based on the comparative analysis of various cylinder configurations, it is clear that the external cylinder generates greater velocities and temperatures along its surface. The maximum disparity in surface velocity between an external cylinder and other configurations is 7%, 12%, and 13% for Prandtl numbers of 6.7, 16.1, and 25.2 respectively. Therefore, positioning the cylinder externally promotes swifter convection and more effective mixing of the fluid.

New correlations for bubble departure and liftoff diameters in vertical subcooled boiling flow under low-pressure conditions

The bubble departure and liftoff diameters are crucial parameters affecting wall boiling heat transfer in subcooled boiling flow. This study aimed to develop new correlations for these diameters of a growing bubble on the heated wall in vertical subcooled boiling flow. By analyzing a force balance model, the major non-dimensional parameters were derived to express the bubble diameters. Especially, it revealed that, in addition to the Reynolds number, the Capillary number should be considered to better account for the effect of mass flux on the bubble departure diameter. Finally, the correlations of the two bubble diameters were developed by utilizing the available experimental data. The mean absolute percentage errors of the new correlations were 18.57% and 22.16%, respectively for the bubble departure and liftoff diameters when compared with experimental data. The applicable ranges of correlation of bubble departure diameter cover the superheated Jacob number of 0.0056−0.035, subcooled Jacob number of 0.0081−0.086, liquid Prandtl number of 1.006−1.76, product of Capillary and Reynolds numbers of 181.6−575.23, and density ratio of 0.0006−0.05. For the bubble liftoff diameter, the applicable ranges are 0.0014 – –0.045 for the superheated Jacob number, 0.0024–0.111 for the subcooled Jacob number, 1.29–1.75 for the liquid Prandtl number, 2883.4–99,716.21 for the Reynolds number, and 0.000615–0.00175 for the density ratio.

Condensation Performance of Symmetrical and Parallel Capsule Plate Heat Exchangers

Condensation characteristics of symmetrical and parallel capsule plate heat exchangers were experimentally investigated in this article. Plate heat exchangers with parallel capsule corrugation, symmetrical capsule corrugation, and chevron-type corrugation are manufactured as the test sections, each containing 41 plates. Pure steam and water are the hot and cold fluids, respectively. The experiments covered a range of equivalent Reynolds number from 9,000 to 25,000, mass flux of condensing fluid from to 18 to 49 kg/(m2·s), liquid Prandtl number from 1.37 to 1.71, average vapor quality from 0.5 to 0.7, and condensation pressure from 110 to 240 kPa. The correlations of condensation heat transfer coefficient and friction factor of the symmetrical and parallel capsule plate heat exchangers are developed. Compared with chevron-type plate heat exchangers, capsule plate heat exchangers show low friction pressure drop, and 33% higher energy efficiency index for condensation flow and heat transfer. Specifically, the energy efficiency index of the symmetrical capsule plate heat exchanger is 8.6% higher than that of the parallel capsule plate heat exchanger, indicating the superiority of counter-rotating longitudinal vortices over co-rotating longitudinal vortices in condensation, similar to what is observed in single-phase flow and heat transfer.

A theory of rheological film condensation

The article provides a theoretical investigation of the laminar film-condensation of shear-thinning fluids onto an isothermal, vertical plate in the presence of gravity, interfacial shear, wall friction, and inertia. The liquid motion occurs toward gravity, overcoming the resistances of interfacial shear, wall friction, and inertia. The liquid motion induces momentum to the adjacent stagnant saturated vapor by interfacial shear. The vapor motion, in turn, influences the liquid film’s heat transfer and fluid dynamics. For film condensation, it is well-known that the film, along the length of the plate, can be grouped in two separate heat transfer regimes, viz. conduction-dominated regime and convection-dominated regime. For the first time, we have identified three distinct fluid dynamic regions across the film (viz. R1, R2, and R3) unique to power-law fluids. R1 is bounded between the condensing surface and the locus of Newtonian shear stress. R2 is bounded between the locus of Newtonian shear stress and the locus of zero shear stress. Beyond R2, R3 is stretched up to the liquid-vapor interface. The transverse temperature gradient is subtly different in these three fluid dynamic regions. We examine flow physics on a non-dimensional platform. The critical role of the power-law index in determining wall-friction and condensate’s mass flow rate and the impact of liquid Prandtl number in achieving the degree of subcooling has been nicely illustrated. We expose various limiting conditions for simplifying the robust mathematical theory. An analytical expression for the liquid’s longitudinal velocity has been derived, neglecting inertia and interfacial shear.

Correlation of single-phase convective heat transfer on spray cooled plain surfaces with high Prandtl number liquids

Spray cooling with transmission oils is an innovative cooling concept for electrical machines, whereby the coolant wets the end windings directly. The thermal design process of such machines requires reliable correlations of the heat transfer coefficients obtainable. Thereby, the majority of the literature only focuses on low Prandtl number coolants at ideal nozzle distances. The transferability of the existing heat transfer models to highly viscous coolants above and below the optimal nozzle distance has yet to be investigated. Therefore, the heat transfer coefficients of spray cooling on plain surfaces with aqueous glycerol solutions (model fluid) and a transmission oil of type ATF VI was investigated. The model fluid serves the purpose of covering the entire range of material properties of commercial transmission oils with only one liquid. Prandtl numbers ranging from 80 to 340 were investigated. Various cases of spray coverage were researched by varying the nozzle distance (2mm−100mm), the spray angle (20∘−80∘) and the size of the spray cooled surface (6.35×6.35mm2;12.7×12.7mm2). Regarding the case of the optimal nozzle distance, where the spray impact area just inscribes the cooled surface, the experimental data was validated with models from the literature. Large deviations of up to 450% resulted at nozzle distances above and below the optimum. Therefore, a dimensionless correlation of the heat transfer coefficient is proposed which includes all cases of spray coverage investigated, exceeding the applicability range of the existing models. The model describes the experimental data with a mean absolute percentage error of 15%.

Predictive method for flow condensation heat transfer in plain channels

The flow condensation heat transfer (FCHT) in plain channels has a variety of applications in many industrial sectors, and thus it is important to predict FCHT coefficients accurately. This paper compiled a large compound FCHT database containing 5607 data points and 30 fluids and presented a strategy for developing a new correlation of FCHT coefficients. The parameter range of the database is the hydraulic diameter D = 0.493–20 mm, vapor quality x = 0.003–0.992, mass flux G = 24–1533 kg/m2 s, heat flux q = 2.9–422 kW/m2, reduced pressure pr = 0.04–0.95, liquid Prandtl number Prl = 1.7–8.5, liquid Reynolds number Rel = 11.6–5.3 × 104, and gas Reynolds number Reg = 75.1–9.1 × 105. Based on the database and strategy, a new general correlation with substantially better accuracy was developed, which is applicable to plain channels of various sizes and a broad operational parameter range. It predicts the database with a mean absolute deviation (MAD) of 14.1%, while the best existing ΔT-independent correlation predicts the database with an MAD of 20.2%. The applicability of the new and 38 existing correlations to individual fluids was assessed. The new correlation performs best for 8 of the 14 fluids that have more than 50 data points in the entire database, while the most accurate existing one performs best only for 2 of them. The Fang number Fa plays an important role in the new correlation, implying that it relates to the fundamental mechanisms of both boiling and condensation heat transfer.

Conjugate Heat Transfer Simulations of High Prandtl Number Liquid Jets Impinging on a Flat Plate

Conjugate Heat Transfer Simulations of High Prandtl Number Liquid Jets Impinging on a Flat Plate

Dynamic mode decomposition of oscillatory thermo-capillary flow in curved liquid bridges of high Prandtl number liquids under microgravity

• Influence of free surface heat transfer on the onset of oscillatory instability. • Thermo-capillary flow in curved liquid bridges under microgravity conditions. • Critical value for the onset of oscillatory flow instability. • ‘Dynamic mode decomposition’ technique to reveal the inherent flow dynamics. • Quantification of each characteristic mode. Three-dimensional and oscillatory thermo-capillary flow in a liquid bridge of 5 cSt silicone oil ( Pr = 67) has been numerically investigated by performing dynamic mode decomposition (DMD) analysis applied to the time-resolved thermal data generated by the ‘method of snapshots’. For various heat transfer conditions defined on the free surface of the curved liquid bridge, a series of dynamic modes has been extracted to unfold the underlying physical mechanism that governs the considered dynamical system. Unstable and neutral modes primarily accountable for the transition to the supercritical state established in the liquid bridge have been identified by decomposing the acquired temperature disturbances into spatial and temporal coherent structures. Under microgravity conditions, unsteady and three-dimensional computations are carried out in a liquid bridge of different volume ratios ( VR = 0.9 and 1.1) using the finite volume method (FVM) to capture the dynamical evolution of temperature disturbances at the onset of oscillatory instability. The present numerical investigation is mainly aimed at providing broader insight into the complex fluid dynamic processes associated with the oscillatory thermal convection by determining the growth/decay rate and frequency of each characteristic mode with the help of DMD spectra. For each oscillation pattern, the dominant behavior of oscillatory thermo-capillary flow is revealed by its coherent structures. The characteristic modes having more energy content (dominant modes) are also analyzed by the norm of modes || ϕ || by demonstrating the frequency spectra of DMD analysis.

Experiments on thermocapillary-buoyancy convection of medium Prandtl number liquids in annular pools heated from inner cylinder

A series of terrestrial experiments on thermocapillary-buoyancy convection of medium Prandtl number liquids (Pr = 6.7-25.2) were performed in annular pools with different depths and a heated inner cylinder. The temperature fluctuation pattern on the free surface was recorded by the schlieren method. Experimental results indicate that thermocapillary-buoyancy convection could weaken and the critical Marangoni number of the flow transition increases with the Prandtl number. When the flow destabilizes, the rotating petal-like structure and the spoke pattern appear respectively in the annular pools with the depths of 4-5 mm and 6-8 mm. As the Marangoni number increases, the flow becomes oscillating as a result of the thermal perturbation induced by the thermocapillary effects. Besides, with the increasing liquid pool depth, the strengthened buoyancy force has an impact on the basic flow and the thermal fields. Thus, the temperature fluctuation frequency of the oscillating petal-like structure is more than that of the oscillating spoke pattern. Different from the cylindrical geometry heated from the outer cylinder, the radial waves and the petals appear respectively near the outer cylinder and the inner cylinder. Furthermore, the wavenumber decreases as the liquid pool depth increases.

On three-dimensional flow structures and oscillatory instability due to free surface heat loss in half-floating zones under microgravity

Present work describes a numerical investigation concerning the response of oscillatory thermo-capillary convection to free surface heat loss in a cylindrical half floating zone of high Prandtl number liquid (Pr=67) under microgravity conditions. Unsteady and three-dimensional computations have been carried out in the liquid domain using the finite volume method (FVM) to reveal the oscillatory thermo-fluid dynamic regimes at the onset of instability. The influence of different free surface heat loss conditions on the bifurcation from the basic steady axisymmetric state has been discussed in terms of critical azimuthal wavenumber, m. The dynamic evolution of the temperature disturbances at supercritical conditions are characterized by ‘pulsating’ and ‘rotating’ modes of spatio-temporal convection. The development of counter-propagating hydrothermal waves in azimuthal direction is found to be followed by an even mode of convection characterised by m = 2. The liquid bridge system exhibiting the well-known transition from periodic (m=2) to quasi-periodic behaviour (m=1) with an increase in Biot number, Bi has been analysed from both spatial and temporal points of view. The existence of both symmetric and asymmetric modes of the supercritical state for different Bi are presented. The dominant fluid dynamic information of 3D oscillatory state are extracted from a specific set of disturbance temperature data by employing a data-driven computational technique ‘dynamic mode decomposition’ (DMD). Various neutral, stable and unstable dynamic modes responsible for bifurcation to the 3D supercritical state are depicted with the help of DMD spectrum and the frequency spectrum.

Rayleigh-Bénard convection in a newtonian liquid bounded by rigid isothermal boundaries

A non-linear stability analysis of Rayleigh-Bénard convection is reported in the paper for the case of rigid isothermal boundaries. The Ginzburg-Landau equation with cubic non-linearity is obtained. Out of the three fixed points of the equation, one arises in the pre-onset regime and two during post-onset. The pre-onset one is unstable at post-onset times and hence we have a pitchfork bifurcation point. Comparing heat transports it is found that the Nusselt number of the rigid isothermal boundary condition is less than that of the free isothermal case. Prandtl number influence on the Nusselt number is negligible in very high Prandtl number liquids at all times and at long times this is true for all liquids. Using vital information from the procedure leading to the Ginzburg-Landau equation a minimal Fourier-Galerkin expansion is proposed to derive a generalized Lorenz model for rigid-isothermal boundaries. The Hopf bifurcation point signaling onset of chaos is obtained by transforming the Lorenz model of rigid isothermal boundaries to a standard form resembling the classical one. The possibility of regular, chaotic, periodic and mildly chaotic motions in the Lorenz system is shown through the plots of the maximum Lyapunov exponent and the bifurcation diagram.

Near-Wall Determination of the Turbulent Prandtl Number Based on Experiments, Numerical Simulation and Analytical Models

The Reynolds-averaged computation of turbulent flow with heat transfer most commonly models the turbulent heat flux as directly related to the turbulent flux of momentum through the turbulent Prandtl number. Its significant deviation from a uniform bulk flow value for high molecular Prandtl numbers needs to be adequately described to predict accurately the heat transfer. The present study derives a model for the near-wall variation of this important parameter, used as input into an analytical solution of heated turbulent pipe flow. The basic functional form of the profile of the turbulent Prandtl number is determined from direct numerical simulations (DNS), and experimental data are used for model calibration. The analytically predicted Nusselt numbers agree very well with experimental measurements, proving the reliability of the proposed model for the turbulent Prandtl number also for Reynolds numbers well beyond the scope of DNS. The validation against experiments further highlights the significant effect of the temperature-dependent material properties of the considered high Prandtl number liquids. Numerical simulations often discard this aspect to reduce the computational effort. The present combination of DNS, analytical solution, and experiments appears as a convenient approach for modeling turbulent key quantities such as the turbulent Prandtl number, which is well applicable to other convective flow conditions and Prandtl number regimes, as well.

Thermal transport due to liquid jet impingement on superhydrophobic surfaces with isotropic slip: Isoflux wall

Thermal transport due to a liquid water jet impinging an isoflux superhydrophobic surface with isotropic slip was modeled analytically. An integral analysis of the transport equations resulting in a system of ordinary differential equations was solved numerically. Impingement on superhydrophobic surfaces greatly reduces the heat transfer that occurs relative to a smooth surface due to gas trapped in cavities on the surface. This results in an apparent slip velocity and temperature jump at the surface. Local and average Nusselt numbers are presented as a function of radial position (0 to 45 jet radii), jet Reynolds number (3×103 to 1.5×104), liquid Prandtl number (2 to 11), normalized slip length (0 to 0.2), and normalized temperature jump length (0 to 0.2). All results are compared to classical (no-slip, no temperature jump) behavior on a smooth surface. Although local Nusselt numbers for the isoflux scenario are greater than the corresponding isothermal case, the difference in Nusselt number between these two heating conditions becomes negligible as the temperature jump length increases to quantities realizable on superhydrophobic surfaces. These results may be utilized to explore heat transfer degradation in applications where smooth surfaces are replaced by superhydrophobic surfaces to avoid fouling.

Manipulation of Colloidal Droplets in Space and the Instability of Thermocapillary Convection in Large Prandtl Number Liquid Bridge in Microgravity

In 2016 and 2017, SJ-10 and TG-2 satellites were launched. In this short paper, we report recent progress on the studies of manipulation of colloidal droplets and instability of thermocapillary convection in large Prandtl number liquid bridge that based on the space experiments boarding SJ-10 and TG-2 satellites, separately. It was shown that the colloidal droplets can be successfully formed and manipulated in microgravity through the patterned substrates. In another aspect, the coffee-ring effect was observed at the first time in space. For the studies of the instability of thermocapillary convection in large Prandtl number liquid bridge in microgravity, our experiments in TG-2 broadened the way of such kind of study and abundant experimental results are emerging.

Instabilities of thermocapillary flows between counter-rotating disks under microgravity conditions

Instabilities of thermocapillary flows between counter-rotating disks under microgravity conditions are investigated by linear stability analysis. The basic-state and perturbation equations are solved using the Chebyshev-collocation method. For small Prandtl number liquids (Pr ≤ 0.01), bifurcation of thermocapillary flows between counter-rotating disks is found to be a 3D oscillatory state for the Coriolis number τ ≤ 100, except at certain Coriolis number where the most unstable perturbation is 3D stationary state. The critical capillary Reynolds number is a function of Prandtl number, Coriolis number and aspect ratio.Energy analysis shows that the perturbation energy consists of the viscous dissipation, the work done by surface tension and the interaction between the perturbation flow and the basic flow, respectively. For small Prandtl number liquids (Pr ≤ 0.01), the perturbation energy mainly comes from the interaction between the perturbation and the basic flow, which suggests that the instability mechanism is hydrodynamic. The interaction between the perturbation and the basic flow in the azimuthal direction becomes negative when a moderate rotation is applied on the disks, and the moderate rotation can stabilize the thermocapillary flows for small Prandtl number liquids.

On boundary conditions in liquid sodium convective experiments

Turbulent convection of liquid sodium in a cylindrical cell, heated at one end face and cooled at the other, inclined to the vertical at angle 0 and π/4 is studied experimentally and numerically by solving the Oberbeck-Boussinesq equations with the LES (Large Eddy Simulation) approach for small-scale turbulence. The aspect ratio is one, i.e. cylinder length is equal to diameter L = D = 200 mm. The simulations were done using fixed heat flux thermal boundary conditions for the cylinder faces. To resolve the general problem of boundary condition in convective experiment with low Prandtl number liquids, a special kind of heat exchanger were designed for the experimental setup. Each heat exchanger is a temperature-controlled MHD (magnetohydrodynamic) stirrer, filled with sodium and separated from the convective cell by a thin copper plate. We demonstrate the efficiency of MHD stirring for the temperature control. In convective experiments the Rayleigh number, determined by the cylinder diameter, was in the range from 4.7 · 106 to 1.7 · 107. We show that the structure of the flow and the efficient heat transfer strongly depend on the inclination angle.

Thermocapillary flow instabilities of medium Prandtl number liquid in rotating annular pools

The linear stability of thermocapillary flow for a medium Prandtl number liquid (Pr = 6.7) is investigated in the annular pools rotating along counterclockwise direction. The neutral Reynolds numbers for the incipience of instabilities are determined by linear stability analysis and the underlying mechanisms are analyzed by energy budgets. Four types of oscillatory instabilities are predicted for different aspect ratios over a wide range of Taylor number. The energy budgets reveal that all the instabilities are basically driven by the hydrothermal wave instability mechanism of moderately large-Prandtl-number liquid. The pool rotation influences the basic flow and enables the disturbances to receive energy in different ways. When the system rotation is initiated, the axisymmetric steady thermocapillary flow becomes unstable typically by a hydrothermal wave instability (type I) for all aspect ratios. The slow rotation destabilizes the basic flow whereas the fast rotation stabilizes it when the Taylor number exceeds a certain threshold value. For large Taylor numbers, two types of flow instabilities (types II and III) occur in sequence with increasing aspect ratio. While type IV arises in a range of Taylor number between those of the type I and type II for the layer of A = 0.25. The critical Reynolds numbers for the onset of the later three oscillatory bifurcations all increase with increasing Taylor number.

A numerical study of the transitions of laminar natural flows in a square cavity

This paper deals with the numerical study of heat and mass transfer occurring in a cavity filled with a low Prandtl number liquid. The model includes the momentum, energy and mass balance equations. These equations are discretized by a finite volume technique and solved in the framework of a custom SIMPLER method developed in FORTRAN. The effect of the problem characteristic parameters, namely the Lewis and Prandtl numbers, on the instability of the flow and related solute distribution is studied for positive and negative thermal and solutal buoyancy forces ratio. Nusselt and Sherwood numbers are derived for values of the Prandtl number ranging between 10-2 and 101 with the Lewis number spanning the interval 1-104. The results show unicellular and multi-cellular flows featured by solute and kinematic boundary layers with thickness depending on the Prandtl number.

MyrrhaFoam: A CFD model for the study of the thermal hydraulic behavior of MYRRHA

Numerical analysis of the thermohydraulic behavior of the innovative flexible fast spectrum research reactor, MYRRHA, under design by the Belgian Nuclear Research Center (SCK•CEN) is a very challenging task. The primary coolant of the reactor is Lead Bismuth Eutectic, LBE, which is an opaque heavy liquid metal with low Prandtl number. The simulation tool needs to involve many complex physical phenomena to be able to predict accurately the flow and thermal field in the pool type reactor.In the past few years, within the frame of a collaboration between SCK•CEN and the von Karman Institute, a new platform, MyrrhaFoam, was developed based on the open source simulation environment, OpenFOAM. The current tool can deal with incompressible buoyancy corrected steady/unsteady single phase flows. It takes into account conjugate heat transfer in the solid parts which is mandatory due to the expected high temperature gradients between the different parts of the reactor. The temperature dependent properties of LBE are also considered. MyrrhaFoam is supplemented with the most relevant thermal turbulence models for low Prandtl number liquids up to date.

The influence of surfactants on thermocapillary flow instabilities in low Prandtl melting pools

Flows in low Prandtl number liquid pools are relevant for various technical applications and have so far only been investigated for the case of pure fluids, i.e., with a constant, negative surface tension temperature coefficient ∂γ/∂T. Real-world fluids containing surfactants have a temperature dependent ∂γ/∂T > 0, which may change sign to ∂γ/∂T < 0 at a critical temperature Tc. Where thermocapillary forces are the main driving force, this can have a tremendous effect on the resulting flow patterns and the associated heat transfer. Here we investigate the stability of such flows for five Marangoni numbers in the range of 2.1 × 106 ≤ Ma ≤ 3.4 × 107 using dynamic large eddy simulations, which we validate against a high resolution direct numerical simulation. We find that the five cases span all flow regimes, i.e., stable laminar flow at Ma ≤ 2.1 × 106, transitional flow with rotational instabilities at Ma = 2.8 × 106 and Ma = 4.6 × 106, and turbulent flow at Ma = 1.8 × 107 and Ma = 3.4 × 107.