Momentum Transfer Equations Research Articles

Time-dependent shell model for spreading of impurities locally injected into hot plasmas

The spreading of impurities locally injected into hot magnetized plasmas is modelled on the basis of a fluid description for electrons, impurity neutrals and ions in low charge states in the vicinity of the injection position. The elaborated model takes into account that impurity particles of different charges are localized predominantly inside the nested shells whose extensions along and across the magnetic field increase with the charge. By integrating fluid equations for particle, parallel momentum and heat transfer, ordinary differential equations for the time evolution of characteristic shell dimensions, densities and temperatures of impurity ion species are deduced. The proposed approach is applicable under conditions where the local impurity density can noticeably exceed the density of the background plasma ions, but the effects of plasma cooling due to the presence of impurity are negligible. The results of calculations for plasma conditions typical in experiments with impurity seeding in tokamaks are presented.

Numerical simulation of double-diffusive natural convective flow in an inclined rectangular enclosure in the presence of magnetic field and heat source

Double-diffusive natural convective flow in an inclined rectangular enclosure with the shortest sides being insulated and impermeable is investigated numerically. Constant temperatures and concentration are imposed along the longest sides of the enclosure. In addition, a uniform magnetic field is applied perpendicular to the longest sides. Laminar regime is considered under steady-state condition. The transport equations for continuity, momentum, energy and species transfer are solved using the finite volume technique. The validity of the numerical code used is ascertained and good agreement was found with published results. The numerical results are reported for the effect of thermal Rayleigh number on the contours of streamline, temperature, and concentration. In addition, results for the average Nusselt and Sherwood numbers are presented and discussed for various parametric conditions. This study is done for constant Prandtl number, Pr = 0.7; aspect ratio, A = 2 and Lewis number, Le = 2. Computations are carried out for thermal Rayleigh number ranging from 10 3 to 5 × 10 5, inclination angle range of 0° ≤ γ ≤ 180°, dimensionless heat generation and absorption coefficients range of −40 ≤ Φ ≤ 40, buoyancy ratio range of −5 ≤ N ≤ 5 and the Hartmann number range of 0 ≤ Ha ≤ 70.

Numerical simulation of double-diffusive natural convective flow in an inclined rectangular enclosure in the presence of magnetic field and heat source, part A: Effect of Rayleigh number and inclination angle

Double-diffusive convective flow in an inclined rectangular enclosure with the shortest sides being insulated and impermeable is investigated numerically. Constant temperatures and concentration are imposed along the longest sides of the enclosure. In addition, a uniform magnetic field is applied in a horizontal direction. Laminar regime is considered under steady state condition. The transport equations for continuity, momentum, energy and species transfer are solved using the finite volume technique. The validity of the numerical code used is ascertained and good agreement was found with published results. The numerical results are reported for the effect of thermal Rayleigh number on the contours of streamline, temperature, and concentration. In addition, results for the average Nusselt and Sherwood numbers are presented and discussed for various parametric conditions. This study was done for constant Prandtl number, Pr=0.7, aspect ratio, A=2, Lewis number, Le=2, the buoyancy ratio, N=1, Hartmann number, Ha=10 and the dimensionless heat generation, Φ=1. Computations are carried out for RaT ranging from 103 to 5*105 and inclination angle range of 0°⩽γ⩽180°.

Mechanism of stability of the shear flow of a two-layer system of viscous liquids

171 The evolution of the morphology of multicompo� nent liquid media in the course of mechanical treat� ment is related, first of all, to the development of hydrodynamic instability at the interface. In the initial stage of the shear flow, this is caused by an exponential increase in the amplitude A(t) = A0e kΔct of capillary waves over a wide range of wavenumbers k. At positive values of the rate of change in the perturbation ampli� tude , the flow is unstable, while at negative values of this rate, it is stable. It was shown (1) that the Cou� ette and Poiseuille shear flows can be destabilized by applying weak longwave perturbations to the inter� face of a twolayer system of incompressible Newto� nian liquids at certain values of the viscosity ratio m and the thickness ratio n of the layers and at arbitrarily low Reynolds numbers. This qualitatively distin� guishes the instability of this type from the transition to turbulent flow. It was demonstrated (2, 3) that the change in the amplitude of short and long waves is caused by a phase shift of the velocity perturbation wave induced by inertial liquid transfer. At the same time, the mechanism of instability of shear flow at arbitrary wavelengths remained unclear. Solving this question requires involving numerical methods of analysis of the equations of momentum transfer of media with nonuniform viscosity distributions. In this work, for the first time, we performed a direct numerical modeling of the simple shear flow instability development of a twolayer system of New� tonian liquids over a wide range of wavelengths. We determined the conditions for the transition between the asymptotic solutions corresponding to longand Δ c shortwave perturbations. We also showed that the inertial shift of vortices of flow rate perturbations is proportional to the growth rate of the interface pertur� bation amplitude.

Successive linearisation analysis of unsteady heat and mass transfer from a stretching surface embedded in a porous medium with suction/injection and thermal radiation effects

AbstractIn this study, we investigate the application of the new successive linearisation method (SLM) to the problem of unsteady heat and mass transfer from a stretching surface embedded in a porous medium with suction/injection and thermal radiation effects. The governing nonlinear momentum, energy and mass transfer equations are successfully solved numerically using the SLM approach coupled with the spectral collocation method for iteratively solving the governing linearised equations. Comparison of the SLM results for various flow parameters against numerical results and other published results, obtained using the homotopy analysis method (HAM) and Runge–Kutta methods, for related problems indicates that the SLM is a very powerful tool which is much more accurate and efficient than other methods. The SLM converges much faster than the traditional methods like the HAM and is very easy to implement. © 2011 Canadian Society for Chemical Engineering

Turbulent Mixed Convection Inside a Vertical Tube With Wetted Wall

The authors present a numerical investigation of turbulent mixed convection inside a wetted—wall tube. The mixture air and water vapor flows downwards the vertical tube. Turbulent momentum, heat and mass transfer equations combined with a low Reynolds number model are discretized by using the finite volume method. The algebraic system equations deduced from this discretization are solved with the Thomas and Gauss algorithms. Results show that the evaporative cooling of the wetted wall and the air flow through the tube can be produced by low heat flux applied to the tube wall.

Numerical studies on flow and heat transfer in membrane helical-coil heat exchanger and membrane serpentine-tube heat exchanger

The flow and heat transfer characteristics of synthesis gas (syngas) in membrane helical-coil heat exchanger and membrane serpentine-tube heat exchanger under different operating pressures, inlet velocities and pitches are investigated numerically. The three-dimensional governing equations for mass, momentum and heat transfer are solved using a control volume finite difference method. The realizable k- ε model is adopted to simulate the turbulent flow and heat transfer in heat exchangers. There flows syngas in the channels consisting of the membrane helical coils or membrane serpentine tubes, where the operating pressure varies from 0.5 to 3.0 MPa. The numerically obtained heat transfer coefficients for heat exchangers are in good agreement with experimental values. The results show that the syngas tangential flow in the channel consisting of membrane helical coils is significant to the heat transfer enhancement to lead to the higher average heat transfer coefficient of membrane helical-coil heat exchanger compared to membrane serpentine-tube heat exchanger. The syngas tangential velocity in the membrane helical-coil heat exchanger increases along the axial direction, and it is independent of the gas pressure, increasing with the axial velocity and axial pitch rise and decreasing with the radial pitch rise.

1D Computational model of a two-phase R744 ejector for expansion work recovery

A one-dimensional mathematical model of the R744 two-phase ejector for expansion work recovery is presented in this paper. Governing equations were formulated for all passages of the ejector based on the differential equations for mass, momentum, and energy balance as well as a differential representation for the equation of state. For two-flow sections (mixer and diffuser) closing equations for mass, momentum and energy transfer between the primary and secondary flow were introduced. This model utilises the Delayed Equilibrium Model along with the Homogeneous Nucleation Theory for the purpose of the metastable state analysis for a transcritical flow with delayed flashing over the motive nozzle. The thermal properties model was based on a real fluid approach, where the REFPROP 8.0 database was used. Based on the results of experimental tests performed at SINTEF Energi Laboratory, the model was validated for a typical range of operating conditions. The range of available simulation output allowed for the creation of 1D profiles of local values for the flow variables and the computation of the overall indicators, such as pressure lift and expansion work recovery efficiency.

Prediction of Interfacial Mass Transfer in Liquid-liquid Systems with Moving Interfaces

A numerical method is developed that allows mass transfer at moving interfaces to be rigorously captured. The advantage of this method is its ability to consider arbitraryform boundary conditions at the phase interface. The method is applied to the test system comprising the movement of a rising toluene droplet in a quiescence aqueous phase, with acetone being the transferring component. The level set method is employed for the description of the droplet movement, whereas the velocity and concentration fields are determined from the continuity, momentum and mass transfer equations. The concentration contours obtained are found to be in qualitative agreement with the available data from the literature.

Convective heat transfer in a channel partially filled with a porous medium

In this work we solve effective-medium equations for modeling momentum and heat transfer in a parallel-plate channel partially filled with a porous insert. In order to avoid specifying the boundary conditions at the fluid–porous boundary, we solve equations involving position-dependent coefficients (i.e., a one-domain approach). The solution of the momentum-transport problem is carried out using implicit integral equation formulations based on Green’s functions, whereas the heat transfer problem was solved numerically using the finite element method. The simulations were performed in terms of several values of the porosity, the Péclet number, the size of porous insert and the thermal conductivities ratio. In agreement with previous works, it was found that the thermal performance is improved by either increasing the size of the porous insert or by favoring mixing inside the channel. A drawback of this approach is the high computational demand associated to modeling transport in the vicinity of the porous medium and the fluid. In this way, the extents and limitations about the use of a one-domain formulation are exposed in a practical application. The results from this work should serve as motivation for more experimental and theoretical research; in particular, the derivation and application of jump boundary conditions.

Hierarchy of the models of classical mechanics of inhomogeneous fluids

The methods of perturbation theory and integral representations are used to analyze the general properties of a system of equations of the mechanics of inhomogeneous fluids including the equations of momentum, mass, and temperature transfer. We also consider various submodels of this system, including the reduced systems in which some kinetic coefficients are equal to zero and degenerate systems in which the variations of density or some other variables are neglected. We analyze both regularly perturbed and singularly perturbed solutions of the system. In the case of reduction or degeneration of solutions, the order of the system decreases. In this case, regularly perturbed solutions are preserved (with certain modifications) but the number of singularly perturbed components participating in the formation of the boundary layers on contact surfaces and their analogs in the bulk of the fluid, i.e., the elongated high-gradient interlayers, decreases. The interaction between all components of the currents is nonlinear, despite the fact that their characteristic scales are different.

Numerical study of the vertical solidification process

A parametric study of chemical segregation during the solidification of concentrated binary alloy under uniform magnetic field effect was performed. The purpose of this work is to present and discuss a numerical solution of phase change problem for a vertical Bridgman system with varying temperature gradient at the boundary. During growth, three regions appear; liquid, mushy and solid zones. The mushy zone is assimilated to porous medium. The transport equations for continuity, momentum, energy and spices transfer are solved. Numerical results for flow in the bath, solute distribution and radial segregation have been obtained for different values of the governing parameters. The effects of various physical parameters like Rayleigh number and Hartmann number on dimensionless heat and mass transfer during phase change under magnetic field effect characteristics are discussed in detail. In particular, it has been found that increase in Rayleigh number increase convection in the melt and complicates the solute distribution in the mushy zone, while increase in the strength of the magnetic field leads to decrease the radial segregation and stratified the solute distribution during growth.

Numerical solution of the momentum and heat transfer equations for a hydromagnetic flow due to a stretching sheet of a non-uniform property micropolar liquid

A study of the hydromagnetic flow due to a stretching sheet and heat transfer in an incompressible micropolar liquid is made. Temperature-dependent thermal conductivity and a non-uniform heat source/sink render the problem analytically intractable and hence a numerical study is made using the shooting method based on Runge–Kutta and Newton–Raphson methods. The two problems of horizontal and vertical stretching are considered to implement the numerical method. The former problem involves one-way coupling between linear momentum and heat transport equations and the latter involves two-way coupling. Further, both the problems involve two-way coupling between the non-linear equations of conservation of linear and angular momentums. A similarity transformation arrived at for the problem using the Lie group method facilitates the reduction of coupled, non-linear partial differential equations into coupled, non-linear ordinary differential equations. The algorithm for solving the resulting coupled, two-point, non-linear boundary value problem is presented in great detail in the paper. Extensive computation on velocity and temperature profiles is presented for a wide range of values of the parameters, for prescribed surface temperature (PST) and prescribed heat flux (PHF) boundary conditions.

Displacements of ridges and subduction zones in the models of mantle convection with lithospheric plates

In the existing kinematic theory of the tectonics of lithospheric plates, the position and the parameters of plates are assigned a priori in the initial and boundary conditions. However, in the self-consistent dynamic theory, the properties of oceanic plates (as well as the structure of the mantle’s convection) should appear automatically as the solution of the differential equations of energy, mass, and momentum transfer for a viscous fluid. In this case, the viscosity of the mantle’s substance as a function of temperature, pressure, shear stress, and chemical composition must be taken from the data of laboratory measurements. In the present work, the results of the numerical solution of the equations of convection are presented in the problem formulation mentioned above on a simple model of heated viscous fluid with properties that correspond to the mantle’s substance.

Mixed convection magnetohydrodynamic heat and mass transfer past a stretching surface in a micropolar fluid-saturated porous medium under the influence of Ohmic heating, Soret and Dufour effects

A numerical model is developed to examine the combined effects of Soret and Dufour on mixed convection magnetohydrodynamic heat and mass transfer in micropolar fluid-saturated Darcian porous medium in the presence of thermal radiation, non-uniform heat source/sink and Ohmic dissipation. The governing boundary layer equations for momentum, angular momentum (microrotation), energy and species transfer are transformed to a set of non-linear ordinary differential equations by using similarity solutions which are then solved numerically based on shooting algorithm with Runge–Kutta–Fehlberg integration scheme over the entire range of physical parameters with appropriate boundary conditions. The influence of Darcy number, Prandtl number, Schmidt number, Soret number and Dufour number, magnetic parameter, local thermal Grashof number and local solutal Grashof number on velocity, temperature and concentration fields are studied graphically. Finally, the effects of related physical parameters on local Skin-friction, local Nusselt number and local Sherwood number are also studied. Results showed that the fields were influenced appreciably by the Soret and Dufour effects, thermal radiation and magnetic field, etc.

Numerical simulation of double-diffusive mixed convective flow in rectangular enclosure with insulated moving lid

The present study is concerned with the mixed convection in a rectangular lid-driven cavity under the combined buoyancy effects of thermal and mass diffusion. Double-diffusive convective flow in a rectangular enclosure with moving upper surface is studied numerically. Both upper and lower surfaces are being insulated and impermeable. Constant different temperatures and concentration are imposed along the vertical walls of the enclosure, steady state laminar regime is considered. The transport equations for continuity, momentum, energy and spices transfer are solved. The numerical results are reported for the effect of Richardson number, Lewis number, and buoyancy ratio on the iso-contours of stream line, temperature, and concentration. In addition, the predicted results for both local and average Nusselt and Sherwood numbers are presented and discussed for various parametric conditions. This study was done for 0.1 ≤ Le ≤ 50 and Prandtl number Pr = 0.7. Through out the study the Grashof number and aspect ratio are kept constant at 10 4 and 2 respectively and −10 ≤ N ≤ 10, while Richardson number has been varied from 0.01 to 10 to simulate forced convection dominated flow, mixed convection and natural convection dominated flow.

Nonlinear electron oscillations in a viscous and resistive plasma

Nonlinear, spatially periodic, long-wavelength electrostatic modes of an electron fluid oscillating against a motionless ion fluid (Langmuir waves) are given, with viscous and resistive effects included. The cold plasma approximation is adopted, which requires the wavelength to be sufficiently large. The pertinent requirement valid for large amplitude waves is determined. The general nonlinear solution of the continuity and momentum transfer equations for the electron fluid along with Poisson's equation is obtained in simple parametric form. It is shown that in all typical hydrogen plasmas, the influence of plasma resistivity on the modes in question is negligible. Within the limitations of the solution found, the nonlinear time evolution of any (periodic) initial electron number density profile ne(x,t=0) can be determined (examples). For the modes in question, an idealized model of a strictly cold and collisionless plasma is shown to be applicable to any real plasma, provided that the wavelength λ>>λmin(n(0),Te) , where n(0)=const and Te are the equilibrium values of the electron number density and electron temperature. Within this idealized model, the minimum of the initial electron density n(e)(xmin,t=0) must be larger than half its equilibrium value, n(0)/2 . Otherwise, the corresponding maximum n(e)(xmax,t=τ(p)/2) , obtained after half a period of the plasma oscillation blows up. Relaxation of this restriction on n(e)(x,t=0) as one decreases λ , due to the increase of the electron viscosity effects, is examined in detail. Strong plasma viscosity is shown to change considerably the density profile during the time evolution, e.g., by splitting the largest maximum in two.

An autonomous system for thermal convection of viscoelastic fluids in a porous layer using a thermal nonequilibrium model

Thermal convection of viscoelastic fluids saturating a horizontal porous layer heated from below is analyzed using a thermal nonequilibrium model to take account of the interphase heat transfer between the fluid and the solid. The viscoelastic character of the flow is considered by a modified Darcy’s law. An autonomous system with five differential equations is deduced by applying the truncated Galerkin expansion to the momentum and heat transfer equations. The effects of interphase heat transfer H on the thermal convection of viscoelastic fluids in a porous medium are analyzed and discussed. The results show that the weak interphase heat transfer tends to stabilize the steady convection.

Forced and thermocapillary convection in silicon Czochralski crystal growth in semispherical crucible

In order to understand the influence of a semispherical crucible geometry combined with different convection modes as a thermocapillary convection, natural convection and forced convection, induced by crystal rotation, on melt flow pattern in silicon Czochralski crystal growth process, a set of numerical simulations are conducted using Fluent Software. We solve the system of equations of heat and momentum transfer in classical geometry as cylindrical and modified crystal growth process geometry as cylindro-spherical. In addition, we adopt hypothesis adapted to boundary conditions near the interface and calculations are executed to determine temperature, pressure and velocity fields versus Grashof and Reynolds numbers. The analysis of the obtained results led to conclude that there is advantage to modify geometry in comparison with the traditional one. The absence of the stagnation regions of fluid in the hemispherical crucible corner and the possibility to control the melt flow using the crystal rotation enhances the quality of the process comparatively to the cylindrical one. The pressure field is strongly related to the swirl velocity.

Simulation studies to optimize the process of plasma spray deposition of yttrium oxide

Simulation studies on the thermal behaviour of yttrium oxide particles in a thermal plasma jet was carried out with the objective of controlling and optimization of the plasma spray process. The 'in-flight' behaviour of yttrium oxide particles in the plasma jet was studied by solving the heat transfer and momentum transfer equations using the velocity and temperature distribution in the plasma jet obtained from a two-dimensional model. In particular, the effect of particle size, thermal power of the torch and the torch operating parameters like gas flow rates were considered to calculate the heat transfer and momentum transfer to the particle. Results of simulation studies agree quite well with the experimental results on variation of deposition efficiency with power and particle size. The complete description of the model with the results obtained for the typical operating parameters of our plasma spray torch is presented in the paper.