A experimental investigation of turbulent flow and heat transfer in elliptical duets

Fully developed turbulent flow and heat transfer to air and water in ducts of elliptical cross section have been investigated experimentally. For the ducts of aspect ratio 2.5 1 and larger, a reduction in the overall heat transfer rate was found in the lower turbulent Reynold's number range (Re<25,000). Similar effects have been noted by investigators of narrow triangular cross sections where flow measurements indicated the possible co-existence of laminar and turbulent flow resulting in localised increases in thermal resistance. It was found that the analogy between momentum and heat transfer could not be applied directly to the larger aspect ratio ducts where significant circumferential variations of wall temperature occurred.

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Zusammenfassung Voll entwickelte turbulente Strömung und Wärmeübertragung an Luft und Wasser in elliptischen Kanälen wurden experimentell untersucht. Für Kanäle mit Achsenverhältnissen von 2,5 1 und größer fand man eine Verringerung des Wärmedurchgangs im Bereich geringer Reynolds-Zahlen (Re < 25 000). Ähnliche Effekte waren von anderen Autoren in engen Dreieckskanälen gefunden worden, wobei man aus Strömungsmessungen das gleichzeitige Auftreten von laminarer und turbulenter Strömung mit örtlicher Zunahme des thermischen Widerstandes folgern konnte. Die Analogie zwischen Impuls- und Wärmeübertragung konnte nicht unmittelbar auf Kanäle mit großem Achsenverhältnis, bei denen die Umfangstemperatur beträchtlich variierte, angewendet werden.

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Nomenclature A cross-sectional area - b duct wall thickness - C_p specific heat at constant pressure - d_e equivalent diameter of noncircular cross-section (=4A/p) - f Fanning friction coefficient - h local heat transfer coefficient (=q_w/(T_w-T_b)) - ¯h average circumferential heat transfer coefficient - k thermal conductivity of fluid - k_w thermal conductivity of wall material - K^* wall conductivity parameter (= k_wb/kd_e) - p wetted perimeter - q_w wall heat flux - T_b bulk fluid temperature - T_w local wall temperature - absolute viscosity - kinematic viscosity (=/) - mass density - Nu Nusselt number (= h d_e/k) - Nu average circumferential Nusselt number (= ¯h d_e/k) - Pr Prandtl number (= C_p/k) - Re Reynolds number (= d_e/) - St Stanton number (= Nu/Re · Pr)     

The effects of temperature-dependent fluid properties on free convective flow along a semi-infinite vertical plate by the presence of radiation

The effects of temperature-dependent density, viscosity and thermal conductivity on the free convective steady laminar boundary layer flow by the presence of radiation for large temperature differences, are studied. The fluid density and the thermal conductivity are assumed to vary linearly with temperature. The fluid viscosity is assumed to vary as a reciprocal of a linear function of temperature. The usual Boussinesq approximation is neglected due to the large temperature difference between the plate and the fluid. The nonlinear boundary layer equations, governing the problem under consideration, are solved numerically by applying an efficient numerical technique based on the shooting method. The effects of the density/temperature parameter n, the thermal conductivity parameter , the viscosity/temperature parameter _r and the radiation parameter F are examined on the velocity and temperature fields as well as the coefficient of heat flux and the shearing stress at the plate.     

Approximate integration op nonstationary equations of a diffusion or thermal boundary layer

An approximate method is proposed for integrating the nonstationary equations of a diffusion or thermal boundary layer using the known steady solution in the planar or axisymmetric case. It is shown that the proposed method is exact in problems involving mass or heat transfer of reacting drops and bubbles in a laminar flow of a viscous incompressible fluid and also particles moving in an ideal fluid. An integral equation is obtained for the local diffusion or heat flux in the case of abrupt activation of a reaction on the surface of a particle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 87–92, September–October, 1982.     

Effects of magnetic field and variable viscosity on forced non‐darcy flow about a flat plate with variable wall temperature in porous media in the presence of suction and blowing

This paper presents a study of the effect of a magnetic field and variable viscosity on steady twodimensional laminar nonDarcy forced convection flow over a flat plate with variable wall temperature in a porous medium in the presence of blowing (suction). The fluid viscosity is assumed to vary as an inverse linear function of temperature. The derived fundamental equations on the assumption of small magnetic Reynolds number are solved numerically by using the finite difference method. The effects of variable viscosity, magnetic and suction (or injection) parameters on the velocity and temperature profiles as well as on the skinfriction and heattransfer coefficients were studied. It is shown that the magnetic field increases the wall skin friction while the heattransfer rate decreases.     

Analysing flow and heat transfer of a viscoelastic fluid over a semi-infinite horizontal moving flat plate

This paper presents a numerical study of the flow and heat transfer of an incompressible homogeneous second grade type fluid above a flat plate moving with constant velocity U. Such a viscoelastic fluid is at rest and the motion is created by the sheet. The effects of the non-Newtonian nature of the fluid are governed by the local Deborah number K (the ratio between the relaxation time of the fluid and the characteristic time of the flow). When , a new analytical solution for this flow is presented and the effects of fluid's elasticity on flow characteristics, dimensionless stream function and its derivatives are analysed in a wide domain of K. A novel result of the analysis is that a change in the flow solution's behaviour occurs when the dimensionless stream function at the edge of the boundary layer, f_∞, equals 1.0. It is found that velocity at a point decreases with increase in the elasticity of the fluid and, as expected, the amount of fluid entrained diminishes when the effects of fluid's elasticity are augmented. In our heat transfer analyses we assume that the surface temperature has a power-law variation. Two cases are studied, namely, (i) the sheet with prescribed surface temperature (PST case) and (ii) the sheet with prescribed heat flux (PHF case). Local similarity heat-transfer solutions are given for PST case when s=2 (the wall temperature parameter) whereas when a similarity solution takes place in the case of prescribed wall heat flux. The numerical results obtained are fairly in good agreement with the aforementioned analytical ones.     

Heat transfer in laminar pulsating flows of fluids with temperature dependent viscosities

The effect of time-dependent pressure pulsations on heat transfer in a pipe flow with constant temperature boundaries is analysed numerically when the viscosity of the pulsating fluid is an inverse linear function of the temperature. The coupled differential equations are solved using Crank-Nicholson semi-implicit finite difference formulation with some modifications.The results indicate local variations in heat transfer due to pulsations. They are useful in the design of heat exchangers working under pulsating flow conditions. The analytical results are presented for both heating and cooling. The conditions under which pulsating flows can augment the heat transfer are discussed. The results are applicable for heat exchangers with fluids of high Prandtl number.     

Thermal free convection from a solid sphere

Summary Thermal free convection from a sphere has been studied by melting solid benzene spheres in excess liquid benzene (Pr=8,3; 10⁸<Gr<10⁹). Overall heat transfer as well as local heat transfer were investigated. For the effect of cold liquid produced by the melting a correction has been applied. Results are compared with those obtained by other workers who used alternative experimental methods.Nomenclature coefficient of heat transfer - d characteristic length, here diameter of sphere - thermal conductivity - g acceleration of free fall - cubic expansion coefficient - T temperature difference between wall and fluid at infinity - kinematic viscosity - density - c specific heat capacity - a thermal diffusivity (=/c) - D diffusion coefficient - Nu dimensionless Nusselt number (=d/) - Nu* the analogous number for mass transfer (=kd/D) - mean value of Nusselt number - Gr dimensionless Grashof number (=gd ³T/ ²) - Gr* the analogous number for mass transfer (=gd ³x/ ²) - Pr dimensionless Prandtl number (=/a) - Sc dimensionless Schmidt number (=/D)     

Numerical study on a vertical plate with variable viscosity and thermal conductivity

Free convection over an isothermal vertical plate immersed in a fluid with variable viscosity and thermal conductivity is studied in this paper. We consider the two-dimensional, laminar and unsteady boundary layer equations. Using the appropriate variables, the basic governing equations are transformed to non-dimensional governing equations. These equations are then solved numerically using a very efficient implicit finite difference scheme known as Crank–Nicolson scheme. The fluid considered in this study is of viscous incompressible fluid of temperature dependent viscosity and thermal conductivity. The effect of varying viscosity and thermal conductivity on velocity, temperature, shear stress and heat transfer rate are discussed. The velocity and temperature profiles are compared with previously published works and are found to be in good agreement.     

The effect of velocity distribution on forced convection laminar flow heat transfer in a pipe at constant wall temperature

In this paper, a theoretical study of heat transfer to a fluid of vanishing viscosity in laminar flow in a pipe is made. The constant wall temperature boundary condition is considered in order to facilitate comparison with other classical solutions. Using velocity profiles of simple geometrical shape, the dependence of the heat transfer on velocity distribution is illustrated. Because of the nature of the idealised flow and heat transfer models, the theoretical results are applicable to all axisymmetric flows. Accordingly, some account of the possible effects of swirl on heat transfer in real flows is given.

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Zusammenfassung Es handelt sich um eine theoretische Untersuchung des Wärmeübergangs in laminarer Rohrströmung bei verschwindender Viskosität. Zum Vergleich mit anderen klassischen Lösungen wurde konstante Wandtemperatur als Randbedingung vorgegeben. Unter Benutzung von Geschwindigkeitsprofilen einfacher Geometrie wurde deren Einfluß auf den Wärmeübergang ermittelt. Diese Ergebnisse sind wegen der gewählten Strömungs- und Wärmeübergangsmodelle auf alle achsensymmetrischen Strömungen anwendbar. Die mögliche Wirkung einer Wirbelströmung auf den Wärmeübergang wird diskutiert.

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Nomenclature =(k/c) Thermal diffusivity - C, C ₁, C₂, C₃, C_n Constants - c Specific heat at constant pressure - D=(2r_w) Diameter - k Thermal conductivity - M _n Root of Bessel Equation,J ₀(M_n)=0 - r Radius - T Temperature - u, Velocity, average velocity - x Axial distance - X, R Function ofx, (r) alone - _n (= 2M _n/r _w ² ) Eigen value - Dynamic viscosity - (=/) Kinematic viscosity - Density - (=(T-T _w)/(T₁-T_w)) Dimensionless temperature - (=(T–T _w)/(T ₁–T _w)) Nusselt number - Pe (=Re·Pr) Péclet number - Pr (= c/k) Prandtl number - Re(=2r_w·v) Reynolds number Suffixes b Bulk - 1 Inlet - w wall     

Free- and forced-convection film condensation from a horizontal elliptic tube with a vertical plate and horizontal tube as special cases

A simple mathematical model is developed for the study of the mixed-convection film condensation with downward flowing vapors onto a horizontal elliptic tube. Analytical analysis for both the local condensate film thickness and heat transfer characteristics under simultaneous effects of interfacial vapor shear and pressure gradient has been performed by adopting a unified geometry parameter, eccentricity e. The present results for two limit cases, e = 0 (circular tube) and e = 1.0 (vertical plate) are in an excellent agreement with the earlier works. For very slow vapor flow, the present result for dimensionless mean heat transfer coefficient reduces to the same form as in the earlier works, , whose value is 0.728 for e = 0 and 0.943 for e = 1.0. As for very fast vapor flow, the dimensionless mean heat transfer coefficient, increase with increasing eccentricity under the effects of pressure gradient caused by potential flow and surface tension.     

A new kinetic model for rapid granular flow

The fluid motion of an assemblage of cohesionless granules is governed by the balance laws of mass, momentum and energy and is special because due to the variability of the mean free path of the particles the density is not preserved and the particle fluctuation energy (= granular temperature) is determined together with the mean fields. Constitutive equations are postulated for the dispersive pressure, viscosity, thermal diffusivity and energy annihilation rate in terms of the mean free path length, the time of encounter between two collisions and geometric and dynamic quantities via dimensional arguments. The model defines the time of encounter to consist of the free flying time between collisions plus the contact duration. The inclusion of the latter brings in the elasticity of the particles and introduces a relaxation mechanism that explains the typical behaviour of shear and fluidized layers better than has been done by previous models.     

Use of turbulent energy balance equation in the theory of turbulent flows along walls

The turbulent flow of an incompressible fluid is considered in a plane channel, a circular tube, and the boundary layer on a flat plate. The system of equations describing the motion of the fluid consists of the Reynolds equations and the mean kinetic energy balance equation for turbulent fluctuations. On the basis of an analysis of experimental data, hypotheses are formulated with respect to the eddy kinematic viscosity and lengthl entering into the expression for specific dissipation of turbulent energy into heat. It is assumed that in the central (outer) region of the flow in a channel, andl are constants, and expressions are taken for them which are used for a free boundary layer; near the walll varies linearly and almost linearly. Results of calculations of the turbulent energy distribution, the mean velocity, and the drag coefficient are in good agreement with the existing experimental data. The values of two empirical coefficients, which enter into the system of equations as the result of the hypotheses, are close to those obtained for a free boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 25–33, May–June, 1973.     

The effect of real viscosity on the heat transfer of water based Al2O3 nanofluids in a two-sided lid-driven differentially heated rectangular cavity

The heat transfer and fluid flow behavior of water based Al₂O₃ nanofluids are numerically investigated inside a two-sided lid-driven differentially heated rectangular cavity. Physical properties which have major effects on the heat transfer of nanofluids such as viscosity and thermal conductivity are experimentally investigated and correlated and subsequently used as input data in the numerical simulation. Transport equations are numerically solved with finite volume approach using SIMPLEC algorithm. It was found that not only the thermal conductivity but also the viscosity of nanofluids has a key role in the heat transfer of nanofluids. The results show that at low Reynolds number, increasing the volume fraction of nanoparticles increases the viscosity and has a deteriorating effect on the heat transfer of nanofluids. At high Reynolds number, the increase in the viscosity is compensated by force convection and the increase in the volume fraction of nanoparticles which results in an increase in heat transfer is in coincidence with experimental results.     

Effects of inclination angle on natural convection in enclosures filled with Cu–water nanofluid

Effects of inclination angle on natural convection heat transfer and fluid flow in a two-dimensional enclosure filled with Cu-nanofluid has been analyzed numerically. The performance of nanofluids is tested inside an enclosure by taking into account the solid particle dispersion. The angle of inclination is used as a control parameter for flow and heat transfer. It was varied from  = 0° to  = 120°. The governing equations are solved with finite-volume technique for the range of Rayleigh numbers as 10³  Ra  10⁵. It is found that the effect of nanoparticles concentration on Nusselt number is more pronounced at low volume fraction than at high volume fraction. Inclination angle can be a control parameter for nanofluid filled enclosure. Percentage of heat transfer enhancement using nanoparticles decreases for higher Rayleigh numbers.     

Numerical calculation of the coefficient of capture of aerosol particles in toroidal channels of circular section

The flow of a liquid (or gas) with aerosol particles suspended in it in channels of different configurations is of great interest in the solution of many practical problems. The aim of the present paper is to develop a method for calculating the hydrodynamics and the heat and concentration transfer of aerosol particles for steady flow of an incompressible fluid in toroidal channels of circular section. The paper uses an implicit difference scheme with different approximations of the convective terms on a nonuniform grid (directed differences, central differences, and the monotonic approximation of Samarskii), which makes it possible to reduce the solution of the system of the original nonlinear partial differential equations to the successive solution of one-dimensional systems [1]. The method proposed by Polezhaev and Gryaznov [2] is used to calculate the boundary conditions for the vorticity. The hydrodynamic equations are solved by means of the difference scheme developed by Khristov [3], and the heat and concentration transfer equations are solved by the difference scheme proposed by Val'tsiferov and Polezhaev [4]. The obtained results make possible a detailed analysis of the dependence of the basic integrated (particle capture coefficient) and local characteristics on the values of the relevant dimensionless numbers, namely, the Dini, Prandtl, and Schmidt numbers, the parameter R/R_k, which characterizes the curvature of the channel, and the dimensionless parameter W_f = fR_G(T_O–T_W)/(pM), which characterizes the rate of thermophoresis.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 159–164, March–April, 1984.     

Development of mixed convection flow over a vertical plate due to an impulsive motion

The development of the mixed convection flow of an incompressible laminar viscous fluid over a semi-infinite vertical plate has been investigated when the fluid in the external stream is set into motion impulsively, and at the same the surface temperature is suddenly raised from its ambient temperature. The problem is formulated in such a way that at time t = 0, it reduces to Rayleigh type of equation and as time t , it tends to Blasius type of equation. The scale of time has been selected such that the traditional infinite region of integration becomes finite which significantly reduces the computational time. The nonlinear coupled singular parabolic partial differential equations governing the unsteady mixed convection flow have been solved numerically by using an implicit finite-difference scheme. The surface shear stress and the heat transfer increase or decrease with time when the buoyancy parameter is greater or less than a certain valve. There is a smooth transition from the initial steady state to the final steady state. The skin friction and heat transfer for the constant heat flux case are more than those of the constant wall temperature case. Also they increase with the buoyancy force.     

Turbulent flow field and heat transfer in a heated circular channel under a reciprocating motion

This study is to simulate the turbulent flow field and heat transfer when cold fluid flows through a finite-length circular channel, which meantime undergoes a reciprocating motion using a numerical method. The mass, momentum and energy conservation equations and turbulent k- equations are derived for a turbulent flow field in a reciprocating coordinate by a coordinate transformation from the stationary coordinate system. The SIMPLE-C scheme is used to investigate heat transfer in a cooling channel undergoing a reciprocating motion by changing parameters and cooling mediums. The parameters are Reynolds number (7170 to 20000) and Pulsating number (1.55 to 4.65). The results show that the averaged Nusselt number increase with increasing both Reynolds number and Pulsating number, and the averaged Nusselt number of cooling oil is lower than that of water at the same incoming flow rate for both the stationary and reciprocating channel.     

Unsteady mixed convection on the stagnation-point flow adjacent to a vertical plate with a magnetic field

An analysis is performed to study the unsteady combined forced and free convection flow (mixed convection flow) of a viscous incompressible electrically conducting fluid in the vicinity of an axisymmetric stagnation point adjacent to a heated vertical surface. The unsteadiness in the flow and temperature fields is due to the free stream velocity, which varies arbitrarily with time. Both constant wall temperature and constant heat flux conditions are considered in this analysis. By using suitable transformations, the Navier–Stokes and energy equations with four independent variables (x, y, z, t) are reduced to a system of partial differential equations with two independent variables (, ). These transformations also uncouple the momentum and energy equations resulting in a primary axisymmetric flow, in an energy equation dependent on the primary flow and in a buoyancy-induced secondary flow dependent on both primary flow and energy. The resulting system of partial differential equations has been solved numerically by using both implicit finite-difference scheme and differential-difference method. An interesting result is that for a decelerating free stream velocity, flow reversal occurs in the primary flow after certain instant of time and the magnetic field delays or prevents the flow reversal. The surface heat transfer and the surface shear stress in the primary flow increase with the magnetic field, but the surface shear stress in the buoyancy-induced secondary flow decreases. Further the heat transfer increases with the Prandtl number, but the surface shear stress in the secondary flow decreases.     

Thermally induced velocity gradients in electroosmotic microchannel flows: the cooling influence of optical infrastructure

An axially non-uniform temperature distribution is shown to induce a disturbance to the electroosmotic flow field in microchannels, causing a significant deviation from the ideal plug-like velocity profile. Such axial temperature gradients are shown to be induced passively by the increased dissipation of Joule heat through the optical infrastructure of a viewing window. A combination of caged-dye-based molecular tagging velocimetry (to determine the cross-stream velocity profiles), fluorescence-based thermometry (to determine the in-channel fluid temperatures), and electrical current measurements are employed. The temperature visualization experiments demonstrate that the fluid is locally cooled in the viewed region, resulting in a local increase in the electric field strength. When large fields are applied, measurements indicate that the fluids temperature in the viewed region can be as much as 30°C less than in the remainder of the capillary. Despite an increase in viscosity, this local cooling results in a locally increased electroosmotic wall velocity which induces a concave velocity profile in the viewed portion and a convex velocity profile elsewhere. Experimentally determined profiles exhibit a variation in velocity across the channel of up to 5%. The cause of this velocity profile curvature is confirmed by comparing the velocity profiles obtained at a range of fields to an analytical solution that includes the effects of temperature on the liquid conductivity and viscosity.     

Rotating disk heat transfer in a fluid swirling as a forced vortex

The paper represents results of an exact solution of a laminar heat transfer problem for a rotating disk in a fluid co-rotating with the disk as a solid body. The angular speed of the fluid is less than the angular speed of the disk. Disks surface temperature varies radially accordingly to a power law. Results for the laminar regime are compared with computations for turbulent heat transfer obtained using an integral method developed earlier. On the basis of the exact solution for laminar flow and basic ideas of the integral methods solution for turbulent flow, an integral method for laminar regime is designed and an approximate analytical solution of the considered problem is derived. Inaccuracies of the laminar approximate solution over the main range of variation of the influencing parameters and Prandtl numbers from 0.71 to 1 do not exceed 2.5%. It is shown that the dependence of the Nusselt number on the ratio of the angular speeds of disk and fluid varying from 0 to 0.3 is weak and has a point of maximum within this region for laminar flow. The obtained results are important in predictions of fluid flow and heat transfer in different types of rotating machinery.