Forced convection heat transfer to a power-law fluid in arbitrary cross-section ducts

A numerical method is developed to predict the three-dimensional forced convection laminar incompressible flow of a power law fluid in arbitrary cross-section straight ducts. The continuity equation and boundary layer forms of the energy and momentum equations in rectangular coordinates are transformed into new orthogonal coordinates with boundaries coinciding with the coordinate surfaces. The resulting equations are solved using the finite difference technique. The numerical scheme is capable of handling different hydrodynamic and thermal entry boundary conditions but results are only presented for uniform inlet velocity and temperature profiles and isothermal wall. To demonstrate the wider applicability of the method local heat transfer coefficients and pressure drop in square, trapezoidal and regular pentagonal ducts are computed as functions of pertinent thermal and hydrodynamic parameters.     

Nonlinear thermovibrational convection regimes in a horizontal, exothermic fluid layer

Finite-amplitude convection in a plane, horizontal, exothermic fluid layer in the presence of a static gravitational field and longitudinal high-frequency oscillations is investigated. Nonlinear regimes established after the loss of stability are studied. A numerical finite difference procedure is used. The flow configurations are determined, along with the nonlinear hydrodynamic and heat-transfer characteristics. Maps of the regimes are plotted for the superposition of thermovibrational and thermogravitational excitation mechanisms. It is shown that the intensification of heat transfer by fully developed convection lowers the fluid temperature and raises the threshold of thermal explosion. Deceased. Ural Branch of the Russian Academy of Sciences, Perm. Translated from Fizika Goreniya i Vzryva, Vol. 31, No. 6, pp. 89–98, November–December, 1995.     

Oscillatory convection in a horizontal porous layer saturated with a viscoelastic fluid

When a horizontal porous layer saturated with a viscoelastic liquid is heated from below, the onset conditions of thermal convection are found to be functions of Darcy-Rayleigh number, wave number, and viscoelastic properties. In this study, the linear stability was studied analytically in order to investigate the viscoelastic effects of saturated liquids on the onset conditions in connection with oscillatory instabilities at the threshold of stationary convection. It is suggested that the resulting oscillatory instabilities occur at lower values of Darcy-Rayleigh number than the critical value for the stationary convection. From the occurrence of oscillatory instabilities of viscoelastic liquid, it is expected that the periodic motion should be replaced by stationary modes in a horizontal porous layer.     

Laminar natural convection heat transfer from a slender vertical cone to a power-law fluid

A theoretical analysis of laminar natural convection heat transfer from a slender vertical cone to a power-law fluid has been done by the approximate integral method. Assuming unequal thermal and momentum boundary layer thicknesses, and using the appropriate boundary and compatibility conditions, a “similar” solution of the boundary layer equations has been obtained for high Prandtl numbers. Due to lack of experimental data for power-law fluids, a comparison of the predictions of the present analysis has been done with the experimental data available for Newtonian fluids and good agreement has been found.     

Buoyancy-driven convection in a horizontal fluid layer under uniform volumetric heat sources

Buoyancy effect in an internally heated horizontal fluid layer is considered under the linear stability analysis. The horizontal fluid layer is confined between a rigid adiabatic lower boundary and a rigid isothermal upper boundary. The onset of thermal convection is analyzed by using the propagation theory which transforms partial disturbance equations into ordinary ones similarly under the principle of exchanges of stabilities. The eigenvalue problem is solved by the method of rapidly converging power series. In addition, the connection of stability condition to the fully developed heat transport is investigated. Results show that the critical time to mark cellular convection has increased with a decrease in the Prandtl number. Based on the present stability criteria, a new correlation of the Nusselt number is produced as a function of both the Rayleigh number and the Prandtl number. It is shown that the present correlation on thermal convection compares reasonably with existing experimental data of water.     

Nonisothermal flow of a generalized power-law fluid in converging sections for rubber extrusion

A method for predicting pressure drop and temperature distribution in convergent sections is proposed. The generalized power-law rheological equation of state is used. Only the shear component of the converging flow is considered in order to estimate its contribution to total flow in such sections. A finite-difference scheme was used to simulate the flow. The method was applied to an ethylenepropylene terpolymer compound using different angles of approach for which the pressure drop and the extrudate temperature had been measured experimentally. Comparison with theoretically predicted data showed excellent agreement for small angles, while for large angles the inclusion of extensional effects was found to be necessary. The contribution of viscous dissipation effects was found to be negligible for the converging sections tested.     

Combined forced and free convection flow past a horizontal flat plate

The problem of simultaneous forced and free convection flow of a Newtonian fluid past a hot or cold horizontal flat plate is investigated by means of numerical solutions of the full equations of motion and thermal energy subject only to the Boussinesq approximation. These solutions span the parameter ranges 10 ≤ Re ≤ 100, 0.1 ≤ Pr ≤ 10, and –2.215 ≤ Gr/Re^5/2 ≤ 2.215 where Re, Pr, and Gr are based on the ambient free stream fluid properties and the overall plate length l. When Gr > 0, the boundary flow near the plate surface is accelerated relative to the corresponding forced convection flow, with a resulting increase in both the local skin friction and heat transfer coefficients. When Gr < 0, the boundary flow is decelerated, the local skin friction and heat transfer are decreased, and the flow actually separates for Gr/Re^5/2 < –0.8 when Pr = 0.7. In the latter circumstance, an increasing degree of upstream influence is observed as Gr/Re^5/2 is further decreased.     

Natural convection in a heat generating fluid bounded by two horizontal concentric cylinders

Convective flow and heat transfer of a Boussinesq fluid contained between two horizontal concentric cylinders is investigated under the effects of two driving mechanisms – an externally-imposed temperature gradient across the annulus, and a uniform internal heat generation. Numerical results for flow field and temperature distribution are obtained in terms of four dimensionless parameters, namely the radius ratio, R, the Prandtl number, Pr, the Rayleigh number, Ra*, and the ratio, S, between the characteristic temperature induced by internal heating and the applied temperature difference between the boundaries. Depending on the value of S, the flow pattern is made up of either one or two vortices in each half cavity, and heat is transferred into or out of the cavity through the hot wall. In particular, for a certain value of the applied temperature difference, the hot wall apparently acts as a thermally-insulated boundary, the internal heat is completely lost through the cold wall, and the fluid undergoes a transition from a bicellular to a unicellular flow regime.     

Free convection in power-law fluids from a heated sphere

In this work, the governing field equations describing heat transfer from a heated sphere immersed in quiescent power-law fluids have been solved numerically. In particular, consideration has been given to elucidate the role of Grashof number (Gr), Prandtl number (Pr) and power-law index (n), on the value of the Nusselt number (Nu) for a sphere in the natural convection regime. Further insights are provided by presenting streamline and constant temperature contours. The results presented herein encompass the following ranges of conditions: 10≤Gr≤10⁷; 0.72≤Pr≤100 and 0.4≤n≤1.8 thereby covering both shear-thinning and shear-thickening types of fluid behaviours. Broadly, all else being equal, shear-thinning behaviour can enhance the rate of heat transfer by up to three-fold where as shear-thickening can impede it up to ～30?40% with reference to that in Newtonian fluids. The paper is concluded by presenting detailed comparisons with the scant experimental data and the other approximate treatments of this problem available in the literature.     

Free convection boundary-layer flow over horizontal plates and discs

Free convection flows near horizontal surfaces are discussed for those cases in which a similarity solution can be obtained. Particular cases considered are those of asymptotically large and vanishingly small Prandtl numbers. The solution for the relevant velocity, temperature and pressure functions has to be carried-out numerically. For the particular cases when the Prandtl number is either sufficiently large, or sufficiently small, this integration would yield general functions, no longer dependent on the Prandtl number. Numerical data are included.     

Lattice Boltzmann simulation of power-law fluid flow in the mixing section of a single-screw extruder

The single-screw extruder is commonly used in polymer processing where the performance of the mixing section is significant in determining the quality of the final product. It is therefore of great interest to simulate the flow field in a single-screw extruder. In this paper simulations of non-Newtonian fluids in a single-screw extruder are performed using the lattice Boltzmann model.     

Mixed convection mass transfer studies of opposing and aiding flow in a parallel plate electrochemical flow cell

Studies of combined natural and forced convection in a vertical parallel plate electrochemical cell in laminar conditions in cases of opposing and aiding flow are reported. In an ongoing project it was necessary to identify conditions in which natural convection had no significant influence on mass transfer rates at the cell walls so that data could be validly compared with purely laminar flow computational models. For the different electrode lengths investigated, natural convection dominated at low Reynolds number and there was no Reynolds number dependence. At high Reynolds number the data approached the laminar flow solution. At intermediate Reynolds number, however, there existed a distinct region where free and forced convection were significant. At high electrolyte concentrations data did not merge with laminar flow equations until Re=1000 and low electrolyte concentration data for the large plate could not be compared with numerical predictions below Re of 250. An attempt was made to compare the data with those of other workers on combined forced and natural convection heat and mass transfer.     

Onset of buoyancy-driven convection in the horizontal fluid layer heated from below with time-dependent manner

The onset of buoyancy-driven convection in an initially isothermal, quiescent fluid layer heated from below with time-dependent manner is analyzed by using propagation theory. Here the dimensionless critical time Τ_c to mark the onset of convective instability is presented as a function of the Rayleigh number Ra_Ø and the Prandtl number Pr. The present stability analysis predicts that Τ_c decreases with increasing Pr for a given Ra_Ø. The present predictions compare reasonably well with existing experimental results. It is found that in deep-pool systems the deviation of temperature profiles from conduction state occurs starting from a certain time Τ?(2～4) Τ_c .     

Onset of buoyancy-driven convection in the horizontal fluid layer subjected to ramp heating from below

The onset of buoyancy-driven convection in an initially isothermal, quiescent fluid layer heated from below with a constant heating rate is analyzed by the propagation theory. Here the dimensionless critical time τ_c to mark the onset of convective motion is presented as a function of the Rayleigh number Ra_φ and the Prandtl number Pr. The present stability analysis predicts that for a given large Ra_φ, τ_c decreases with increasing Pr and it is independent of the conditions of the upper boundary. For deep-pool systems, the deviation of the temperature profile from conduction state occurs starting from a certain time τ_o≅4τ_c. The present predictions are compared with other models and existing experimental results in the whole time domain.     

Flow of a temperature dependent power-law model fluid between parallel plates: An approximation for flow in a screw extruder

This paper formulates concisely the problem of fully developed flow of a temperature dependent power-law fluid between parallel plates. An exact solution, in the absence of pressure gradients, is developed. The method has been validated by a comparison with an independent numerical solution. This work is applicable to melt transport in shallow partly filled screw channels, drag flow regimes in the metering zone of a screw extruder and, generally, to flows in hydrodynamically thin layers associated with negligible pressure gradients. A numerical example is included dealing with the extrusion of polypropylene. A general discussion is given concerning the utility of the new theory. Effects of pressure gradients are included in the discussion.     

Two-dimensional unsteady flow of power-law fluids over a cylinder

The unsteady flow of incompressible power-law fluids over an unconfined circular cylinder in cross-flow arrangement has been studied numerically. The two-dimensional (2-D) field equations have been solved using a finite volume method based solver (FLUENT 6.3). In particular, the effects of the power-law index (0.4?n?1.8) and Reynolds number (40?Re?140) on the detailed kinematics of the flow (streamline, surface pressure and vorticity patterns) and on the macroscopic parameters (drag and lift coefficients, Strouhal number) are presented in detail. The periodic vortex shedding and the evolution of detailed kinematics with time are also presented to provide insights into the nature of flow. The two-dimensional flow transits from steady to unsteady behaviour at a critical value of the Reynolds number Re∼(40-50) and the von-Karman vortex street is observed beyond the critical Reynolds number (Re). Obviously, both the lift coefficient and Strouhal number values are zero for the steady flow, but their values increase with the increasing Reynolds number (Re) in the unsteady flow regime. For highly shear-thickening fluids (n=1.8), the flow becomes unsteady at Re=40 and unsteadiness in the flow appears at Re=50 for all values of power-law index (n). As expected, the evolution of the kinematics and vortex shedding show a complex dependence on the flow parameters near the transition in the flow. For a fixed value of the Reynolds number (Re), the drag coefficient increases and lift coefficient decreases with increasing value of the power-law index (n). For a fixed value of the power-law index (n), the drag coefficient gradually increases with the Reynolds number (Re). Similar to the drag coefficient, lift coefficient also shows a complex dependence on the power-law index (n) near the transition zone. The value of the Strouhal number (St) decreases with the increasing value of the power-law index (n) at a fixed value of the Reynolds number (Re).     

Non-Newtonian power-law flow across a confined triangular bluff body in a channel

Wall effects on the flow of incompressible non-Newtonian power-law fluids across an equilateral triangular cylinder confined in a horizontal plane channel have been investigated for the range of conditions: Reynolds number, Re=1–40, power-law index, n=0.4–1.8 (covering shear-thinning, Newtonian and shear-thickening behaviors) and blockage ratio=0.125–0.5. Extensive numerical results on flow pattern, wake/recirculation length, individual and overall drag coefficients, variation of pressure coefficient on the surface of the triangular cylinder and so forth are reported to elucidate the combined effect of power-law index, blockage ratio and Reynolds number. The size of vortices decreases with an increase in the value of the blockage ratio and/or power-law index. For a fixed value of the Reynolds number, individual and overall drags decrease with decrease in power-law index and/or blockage ratio in steady confined flow regime. Simple correlations of wake length and drag are also obtained for the range of settings considered.