An analytical solution to non-axisymmetric heat transfer with viscous dissipation for non-Newtonian fluids in laminar forced flow

Forced convection heat transfer in a non-Newtonian fluid flow inside a pipe whose external surface is subjected to non-axisymmetric heat loads is investigated analytically. Fully developed laminar velocity distributions obtained by a power-law fluid rheology model are used, and viscous dissipation is taken into account. The effect of axial heat conduction is considered negligible. The physical properties are assumed to be constant. We consider that the smooth change in the velocity distribution inside the pipe is piecewise constant. The theoretical analysis of the heat transfer is performed by using an integral transform technique – Vodicka’s method. An important feature of this approach is that it permits an arbitrary distribution of the surrounding medium temperature and an arbitrary velocity distribution of the fluid. This technique is verified by a comparison with the existing results. The effects of the Brinkman number and rheological properties on the distribution of the local Nusselt number are shown.     

The effects of viscous dissipation on laminar heat transfer to power law Ruids in tubes

This work is concerned with laminar heat transfer in circular tubes to power law fluids with temperature-dependent rheological properties. Basic equations governing the problem under consideration are derived including effects of viscous dissipation but neglecting internal heat generation. The derived equations are solved numerically by means of finite-difference scheme and for the boundary conditions of constant wall temperature. The results are presented in form of graphs and compared to the computations carried out by other authors. A very good agreement is stated. The present work is intended to be an extension of the paper which has been recently published by Krishnan and Sastri [5].

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Der Einfluß viskoser Dissipation auf die Wärmeübertragung bei laminarer Rohrströmung von Ostwald-de Waele Flüssigkeiten

Zusammenfassung In dem vorliegenden Beitrag wird die Wärmeübertragung bei laminarer Rohrströmung von denjenigen nicht-Newtonschen Flüssigkeiten untersucht, die dem Potenzansatz nach Ostwald-de Waele gehorchen. Es werden Grundgleichungen für den zu behandelnden Vorgang hergeleitet, unter besonderer Berücksichtigung viskoser Dissipation sowie der Temperaturabhängigkeit rheologischer Parameter des strömenden Mediums. Der Einfluß innerer Wärmequellen wird hingegen vernachlässigt. Zur numerischen Lösung der hergeleiteten Differentialgleichungen wird die Methode der finiten Differenzen eingesetzt, unter der Voraussetzung, daß die Temperatur der Rohrwand unverändert bleibt (Randbedingungen erster Art). Die gewonnenen Ergebnisse wurden graphisch dargestellt sowie mit den von anderen Autoren erarbeiteten Resultaten verglichen. Es wurde dabei eine recht gute Übereinstimmung festgestellt. Der hier vorliegende Beitrag soll als eine Verallgemeinerung des Aufsatzes [5] gehalten werden, der neulich in dieser Zeitschrift von Krishnan und Sastri veröffentlicht wurde.

     

Effects of temperature-dependence of viscosity and viscous dissipation on laminar flow heat transfer in circular tubes

Heat transfer effects of variable viscosity and viscous dissipation for heated developing laminar flows in circular tubes have been investigated. Three studies are reported covering a comprehensive range of input data for the case of constant wall heat flux. Initially the program was used to predict the effect on heat transfer of temperature-dependent viscosity via a general temperature power relation. In addition, predictions were made for nine particular fluids covering a range of Prandtl numbers from 0.025 to 12 500, and a range of Brinkman numbers from 1.8 × 10^?10 to 6.8 × 10³. A more detailed study was made for two particular oils covering a range of practical interest. For the liquids considered their viscosity temperature-dependence resulted in enhancement of heat transfer, whereas for fluids with a Prandtl number <200 the effect of viscous dissipation was negligible, and for fluids of a Brinkman number > × 10^?2 the outcome was a reduction of heat transfer. A numerical instability problem occurred for situations of very high viscous dissipation which limited the length of duct that could be examined.     

Combined convection and radiation heat transfer to thermally developing laminar droplet flow in concentric annuli

This paper presents numerical results for combined convection and radiation heat transfer to a laminar mist flow in the thermal entrance region of a concentric annulus with a heated core at constant wall temperature and an insulated outer wall. The saturated droplets in the mist flow are considered as equivalent heat sinks distributed in the superheated vapor stream. Numerical calculations are performed for the variations of droplet size, mean vapor velocity, and the local Nusselt number in the streamwise direction until the single-phase fully-developed condition is reached. The important role of the saturated droplets on combined convection and radiation heat transfer to mist flow is clearly demonstrated.

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Kombinierte Wärmeübertragung durch Konvektion und Strahlung im thermischen Einlauf einer laminaren Tröpfchenströmung in einem konzentrischen Ringspalt

Zusammenfassung Dieser Artikel stellt numerische Ergebnisse für kombinierte Wärmeübertragung durch Konvektion und Strahlung im thermischen Einlauf einer laminaren Tröpfchenströmung in einem konzentrischen Ringraum mit beheiztem Kern bei konstanter Wandtemperatur und isolierter Außenwand dar. Die gesättigten Tröpfchen wirken als verteilte Wärmesenken im überhitzten Dampfstrom. Numerische Berechnungen werden unter Variation des Tröpfchendurchmessers, der durchschnittlichen Dampfgeschwindigkeit und der Nusselt-Zahl durchgeführt, bis eine einphasige vollausgebildete Strömung erreicht ist. Der wichtige Einfluß der gesättigten Tröpfchen auf die kombinierte Wärmeübertragung durch Konvektion und Strahlung wird klar gezeigt.

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Nomenclature A liquid loading parameter, defined in Eq. (3) - A _d heat transfer area of droplets per unit volume of vapor - A _w heat transfer area of heated wall per unit volume of vapor - C wall superheat parameter, defined in Eq. (5) - C _p specific heat of vapor - D dimensionless droplet diameter,d/d ₀ - D _h hydraulic diameter, 2(r ₀–r _i) - d droplet diameter - d ₀ droplet diameter at thermal entrance (x=0) - E dimensionless parameter, defined in Eq. (6) - H dimensionless parameter, defined in Eq. (7) - F _w–d geometric view factor - h _d heat transfer coefficient for evaporating droplets - h _p0 heat transfer coefficient of non-evaporating droplet or solid sphere with diameter ofd ₀ - k thermal conductivity of vapor - n droplet number density (number of droplets per unit volume of vapor) - n ₀ droplet number density at thermal entrance (x=0) - Nu _x local Nusselt number, defined by Eq. (17) - Pr Prandtl number of vapor,C _p/k - Q _r radiative heat transfer to droplets (per unit volume of vapor) - q _w heat flux at the inner wall - R dimensionless radial position,r/r _i - Re Reynolds number of vapor, 2 _v V₀ r _i/ - r radial position - r _i radius of inner tube - r _o radius of outer tube - S heat sink parameter, defined in Eq. (4) - T temperature of vapor - T _m bulk mean temperature of vapor - T _s saturated temperature - T _w inner wall temperature - V mean vapor velocity - V fully-developed vapor velocity, given in Eq. (12) - V ₀ mean vapor velocity atx=0 - x axial position in thermal entrance region - X dimensionless axial position, (x/r _i)/(Re·Pr) - z ₀ flow quality atx=0 Greek symbols ₀ vapor void fraction atx=0 - ratio of radius,r _i/r₀ - _d emissivity of droplets - _w emissivity of inner heated wall - dimensionless vapor temperature, defined in Eq. (9) - _m dimensionless vapor mean temperature, given by Eq. (14) - _wi dimensionless inner wall temperature - _wo dimensionless outer wall temperature - dynamic viscosity of vapor - _l liquid density - _v vapor density - Stefan-Boltzmann constant     

Slip-flow heat transfer in a microchannel with viscous dissipation

Forced convection flow in a microchannel with constant wall temperature is studied, including viscous dissipation effect. The slip-flow regime is considered by incorporating both the velocity-slip and the temperature-jump conditions at the surface. The energy equation is solved for the developing temperature field using finite integral transform. To increase β_v Kn is to increase the slip velocity at the wall surface, and hence to decrease the friction factor. Effects of the parameters β_v Kn, β, and Br on the heat transfer results are illustrated and discussed in detail. For a fixed Br, the Nusselt number may be either higher or lower than those of the continuum regime, depending on the competition between the effects of β_v Kn and β. At a given β_v Kn the variation of local Nusselt number becomes more even when β becomes larger, accompanied by a shorter thermal entrance length. The fully developed Nusselt number decreases with increasing β irrelevant to β_v Kn. The increase in Nusselt number due to viscous heating is found to be more pronounced at small β_v Kn.     

The developing heat transfer and fluid flow in micro-channel heat sink with viscous heating effect

The numerical modeling of the conjugate heat transfer and fluid flow through the micro-heat sink was presented in the paper, considering the viscous dissipation effect. Three different fluids with temperature dependent fluid viscosity are considered: water, dielectric fluid HFE-7600 and isopropanol. The square shape of the cross-section is considered with D _h  = 50 μm with a channel length L = 50 mm. As most of the reported researches dealt with fully developed fluid flow and constant fluid properties in this paper the thermal and hydro-dynamic developing laminar fluid flow is analyzed. Two different heat transfer conditions are considered: heating and cooling at various Br. The influence of the viscous heating on local Nu and Po is analyzed. It was shown that for a given geometry the local Po and Nu numbers are strongly affected by the viscous heating. Moreover the Po number attains the fully developed value as the external heating is equal with the internal viscous heating.     

Analysis of viscous dissipation effect on thermal entrance heat transfer in laminar pipe flows with convective boundary conditions

Consideration is given to the influence of viscous dissipation on the thermal entrance region laminar pipe flow heat transfer with convective boundary condition. The Eigenfunction series expansion technique is employed to solve the governing energy equation. The results for axial distributions of dimensionless bulk and wall temperatures, local Nusselt number as well as modified local Nusselt number are presented graphically forNu ₀ =0.1, 2, and 100. The complicated variations of conventional local Nusselt number is due to the inappropriate definition of conventional heat transfer coefficient in this problem. A modified local heat transfer coefficient, based on the difference of bulk fluid temperature and wall temperature, is introduced. Its value can clearly indicate the extent and the direction of heat exchange between the fluid in the pipe and the ambient. The effects of outside Nusselt number are also investigated. Significant viscous dissipation effects have been observed for large Br.     

Heat transfer in flow of nonlinear fluids with viscous dissipation

The rotational flow of viscoplastic fluids between concentric cylinders is examined while dissipation due to viscous effects through the energy balance. The viscosity of fluid is simultaneously dependent on shear rate and temperature. Exponential dependence of viscosity on temperature is modeled through Nahme law, and the shear dependency is modeled according to the Carreau equation. Hydrodynamically, stick boundary conditions are applied, and thermally, both constant temperature and constant heat flux on the exterior of cylinders are considered. The governing motion and energy balance equations are coupled adding complexity to the already highly correlated set of differential equations. Introduction of Nahme number has resulted in a nonlinear base flow between the cylinders. As well, the condition of constant heat flux has moved the point of maximum temperature toward the inner cylinder. Taking viscous heating into account, the effects of parameters such as Nahme and Brinkman numbers, material time and pseudoplasticity constant on the stability of the flow are investigated. Moreover, the study shows that the total entropy generation number decreases as the fluid elasticity increases. It, however, increases with increasing Nahme and Brinkman numbers.     

Fully-developed heat transfer in annuli for viscoelastic fluids with viscous dissipation

Analytical solutions are obtained for heat transfer in concentric annular flows of viscoelastic fluids modeled by the simplified Phan-Thien–Tanner constitutive equation. Solutions for thermal and dynamic fully developed flow are presented for both imposed constant wall heat fluxes and imposed constant wall temperatures, always taking into account viscous dissipation.Equations are presented for the normalized temperature profile, the bulk temperature, the inner and outer wall temperatures and, through their definitions for the inner and outer Nusselt numbers as a function of all relevant non-dimensional parameters. Some special results are discussed in detail. Given the complexity of the derived equations, for ease of use compact exact expressions are presented for the Nusselt numbers and programmes to calculate all quantities are made accessible on the internet. Generally speaking, fluid elasticity is found to increase the heat transfer for imposed heating at the wall, especially in combination with internal heat generation by viscous dissipation, whereas for imposed wall temperatures it reduces heat transfer when viscous dissipation is weak.     

Effects of viscous dissipation on heat transfer in an oscillating flow past a flat plate

Theoretical investigation has been carried out of laminar thermal boundary layer response to harmonic oscillations in velocity associated with a progressive wave imposed on a steady free stream velocity and convected in the free stream direction. Series solutions are derived both to velocity and temperature field and the resulting equations are solved numerically. The functions affecting the temperature field are shown graphically for different values of Prandtl number. It is observed that there is more reduction in the rate of heat transfer for P _r<1 and a rise in the rate of heat transfer for P _r>1 due to the presence of oscillatory free-stream.Nomenclature u, v velocity components in the x and y direction - x, y Cartesian coordinate axes - t time - U, U ₀ instantaneous value of and mean free stream velocity - density of fluid - kinematic viscosity - T, T _w, T temperature of the fluid, wall and free stream fluid - c _p specific heat at constant pressure - thermal diffusivity - amplitude of free stream velocity - frequency - _p non-dimensional temperature (T–T /T _w–T ) - P _r Prandtl number (c _p/K) - E _c Eckert number (U ₀ ² /c _p(T _w–T )) - a parameter ( ) - ₀ boundary layer thickness of the oscillation of a harmonic oscillation of frequency ( ) - ordinary boundary layer thickness ( ) - time-averaged, time-independent external velocity - A, B, C, D, E, K, L, M, N, P functions used in expansion for u and - Nu Nusselt number (hx/k) - T _w–% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8V4rqqrFfpeea0Jc9yq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepGe9fr-xfr-x% frpeWZqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcadaGcaa% qaaiaadAhacaWG4bGaai4laiqadwfagaqeaaWcbeaakiaacMcaaaa!3CA6!\[(\sqrt {vx/\bar U} )\] - k thermal conductivity     

Effects of viscous dissipation on laminar forced convection with axially periodic wall heat flux

The laminar forced convection in a circular duct is investigated in the case of a sinusoidal axial variation of the wall heat flux. The axial heat conduction in the fluid is neglected, while the effect of viscous dissipation is taken into account. The heat transfer in the thermally developed region, where the temperature is the sum of a linear function and a periodic function of the axial coordinate, is analysed. Both the temperature field and the local Nusselt number are evaluated analytically. Comparisons with the solution in the absence of viscous heating are performed. It is shown that the effect of viscous dissipation on the temperature field may be relevant especially in the case of a sinusoidal wall heat flux distribution with a vanishing mean value. Received on 24 July 1998     

Finite element solution of heat and mass transfer flow with Hall current, heat source, and viscous dissipation

The aim of the paper is to investigate the effect of heat and mass transfer on the unsteady magnetohydrodynamic free convective flow with Hall current, heat source, and viscous dissipation. The problem is governed by the system of coupled non-linear partial differential equations whose exact solution is difficult to obtain. Therefore, the problem is solved by using the Galerkin finite element method. The effects of the various parameters like Hall current, Eckert number, heat source parameter, Prandtl number, and Schmidt number on the velocity components, the temperature, and the concentration are also examined through graphs.     

Viscous dissipation effects on heat transfer in flow over an inclined plate

The effects of viscous dissipation are considered for natural convection flow past a semi-infinite inclined plate with variable surface temperature. Velocity and temperature profiles, skin friction, and rate of heat transfer are obtained. The effects of Grashof and Prandtl numbers, inclination angle, exponent in the wall temperature variation law, and viscous dissipation parameter on the flow are discussed. It is shown that the time required to reach steady states increases with increasing Prandtl number of the fluid. In addition, an increase in the plate temperature due to viscous dissipation was found to lead to a rise in the average skin friction and a decrease in the average Nusselt number.     

The effects of viscous dissipation on heat transfer to power law fluids in arbitrary cross-sectional ducts

In this paper, the analytical study of forced convection heat transfer to power-law fluids in arbitrary cross-sectional ducts with finite viscous dissipation is undertaken. Both the flow and heat transfer develop simultaneously from the entrance of the duct the walls of which are maintained at a constant temperature different from the entering fluid temperature. The governing conservation equations written in curvilinear coordinates are solved using the Line-Successive-Over relaxation (LSOR) method. Numerical results of dimensionless heat transfer coefficients and temperature profiles are presented for the trapezoidal, triangular, circular and square ducts. For cooling, viscous dissipation generally augments heat transfer. At low values of Brinkman number (Br0.1), the cooling effect dominates over viscous heating in the entrance region. AsBr is increased, the location where viscous dissipation becomes important shifts closer to the entrance until a value is reached for which the effect of viscous dissipation is always predominant irrespective of the axial location. When the walls are heated, for a non-zero Brinkman number, theNu _X* distribution exhibits a singularity from the negative side of theNu _X* axis. As the power-law index increases, the position of this singularity shifts closer to the entrance of the duct. Far downstream of the duct, for a fixedn, Nu _X* attains an asymptotic value which is independent ofBr and is at least thrice that for forced convection without viscous dissipation.Es wird die analytische Studie des Wärmeübergangs (freie Konvektion, begrenzte viskose Dissipation) bei mit Potenzansatz beschriebenen Fluiden in Rohrleitungen mit verschiedenen Querschnitten durchgeführt. Die Strömung und der Wärmeübergang entwickeln sich gleichzeitig ab dem Eingang der Leitungen. Die Wände der Rohrleitungen werden auf konstanter Temperatur gehalten, welche ungleich der Temperatur der einströmenden Flüssigkeiten ist. Die in Gaußschen Koordinaten geschriebenen Erhaltungsgleichungen werden mit der LSOR-(Line-Successive-Over-Relaxation-) Methode gelöst. Die numerischen Ergebnisse der dimensionslosen Wärmeübergangskoeffizienten und der Temperaturprofile werden für trapezförmige, dreieckige und runde Querschnitte vorgelegt. Beim Kühlen erhöht die viskose Dissipation in der Regel den Wärmeübergang. Bei kleinen Brinkman-Zahlen (Br0,1) dominieren die Kühlungseffekte über das viskose Aufheizen im Einlaufbereich. Wenn die Br-Zahlen erhöht werden, verschiebt sich die Gegend, in der die viskosen Effekte überwiegen, in Richtung Einlauf solange, bis eine Br-Zahl erreicht wird, bei der die Dissipation, unabhängig von der axialen Entfernung, immer dominant ist.Wenn die Wände bei Br-Zahlen ungleich Null beheizt werden, weist dieNu _X*-Verteilung eine Singularität auf der negativen Seite derNu _X*-Achse auf. Wenn der Index des Potenzansatzes steigt, nähert sich die Singularität dem Einlauf der Rohrleitung. Weit strömungsabwärts, für ein festesn, nimmtNu _X*, unabhängig von der Br-Zahl, asymptotische Werte an. Diese Werte erreichen das Dreifache derjenigen für erzwungene Konvektion ohne viskose Dissipation.     

The effect of flexible tube vibration on pressure drop and heat transfer in heat exchangers considering viscous dissipation effects

The pressure drop and heat transfer coefficient in tube bundle of shell and tube heat exchangers are investigated considering viscous dissipation effects. The governing equations are solved numerically. Because of temperature-dependent viscosity the equations should be solved simultaneously. The flexible tubes vibration is modeled in a quasi-static method by taking the first tube of the row to be in 20 asymmetric positions with respect to the rest of the tubes which are assumed to be fixed and time averaging the steady state solutions corresponding to each one of these positions .The results show that the eccentricity of the first tube increases pressure drop and heat transfer coefficients significantly comparing to the case of rigid tube bundles, symmetrically placed. In addition, these vibrations not only compensate the effect of viscous dissipations on heat transfer coefficient but also increase heat transfer coefficient. The constant viscosity results obtained from our numerical method have a good agreement with the available experimental data of constant viscosity for flexible tube heat exchangers.     

Effect of viscous dissipation and heat source on flow and heat transfer of dusty fluid over unsteady stretching sheet

This paper investigates the problem of hydrodynamic boundary layer flow and heat transfer of a dusty fluid over an unsteady stretching surface.The study considers the effects of frictional heating(viscous dissipation) and internal heat generation or absorption.The basic equations governing the flow and heat transfer are reduced to a set of non-linear ordinary differential equations by applying suitable similarity transformations.The transformed equations are numerically solved by the Runge-Kutta-Fehlberg-45 order method.An analysis is carried out for two different cases of heating processes,namely,variable wall temperature(VWT) and variable heat flux(VHF).The effects of various physical parameters such as the magnetic parameter,the fluid-particle interaction parameter,the unsteady parameter,the Prandtl number,the Eckert number,the number density of dust particles,and the heat source/sink parameter on velocity and temperature profiles are shown in several plots.The effects of the wall temperature gradient function and the wall temperature function are tabulated and discussed.     

Self-similar channel flow of a viscous gas with heat transfer

We examine self-similar flows of a viscous gas in long, smooth channels with a special heat transfer law at the wall, corresponding to the same Mach number profile at all cross sections.     

Coupled heat transfer and viscous flow,and magnetic effects in weld pool analysis

A finite element formulation and analysis is developed to study coupled heat transfer and viscous flow in a weld pool. The thermal effects generate not only buoyancy forces but also a variation in the surface tension which acts to drive the viscous flow in the molten weld pool. A moving phase boundary separates molten and solid material. Numerical experiments reveal the nature of the highly convective flow in the weld pool and the associated thermal profiles. The relative importance of buoyancy, surface tension, phase change, convection, etc. are examined. We also consider the sensitivity of the solution to the finite element mesh and related non-linear numerical instabilities. Of particular interest is the coupling of the thermal and viscous flow fields for the case when radial flow is inward or outward.     

Effect of viscous dissipation on mixed convection heat transfer in a vertical tube with uniform wall heat flux

 The laminar and parallel flow of a Newtonian fluid in a vertical cylindrical duct with circular cross section has been analysed. Both the viscous dissipation effect and the buoyancy effect have been taken into account. The momentum balance equation and the energy balance equation have been solved by means of a perturbation method, in the case of a uniform heat flux prescribed at the wall of the duct. The velocity distribution, the temperature distribution, the Nusselt number and the Fanning friction factor have been evaluated analytically. Moreover, the velocity and temperature of the fluid have been compared with those obtained in two special cases: forced convection with viscous dissipation (i.e. negligible buoyancy effect); mixed convection with negligible effects of viscous dissipation. Received on 26 June 2000     

Effects of viscous dissipation on thermally developing forced convection in a porous saturated circular tube with an isoflux wall

The viscous dissipation effect on forced convection in a porous saturated circular tube with an isoflux wall is investigated on the basis of the Brinkman flow model. For the thermally developing region, a numerical study is reported while a perturbation analysis is presented to find expressions for the temperature profile and the Nusselt number for the fully developed region. The fully developed Nusselt number found by numerical solution for the developing region is compared with that of asymptotic analysis and a good degree of agreement is observed.