Transient laminar forced convection with arbitrary variation in the wall heat flux

 The aim of the work is to present a detailed numerical study of the transient forced laminar convection flow over a flat plate, when thermal conditions are due to arbitrary wall heat flux variations in space. The energy governing equation is modelled using the Karman–Pohlhausen integral approach in the wide range of Prandtl numbers. The influence of both the thermal problem nature (transient heating and/or cooling processes) and the wall flux function on the resulting mathematical expressions is evidenced and the thermal boundary layer thickness behaviour is discussed. In addition, a particular attention has been focused on both the change in sign of the flux and the duration of the transient heating and cooling. Detailed thermal responses and convective heat coefficient evolutions due to the change of wall conditions are presented. Received on 14 April 2000 / Published online: 29 November 2001     

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     

Laminar forced convection with sinusoidal wall heat flux distribution: axially periodic regime

Stationary and laminar forced convection in a circular tube with a sinusoidal axial distribution of wall heat flux is studied under the hypothesis that both axial heat conduction and viscous dissipation in the fluid are negligible. Two cases are considered: a sinusoidal wall heat flux distribution with a vanishing mean value; a sinusoidal wall heat flux distribution which does not change its sign. In both cases, the temperature field and the local Nusselt number are evaluated analytically in the fully developed region, i.e. where the local Nusselt number depends periodically on the axial coordinate. It is shown that, in the first case, the fully developed region presents an infinite sequence of axial positions where the local Nusselt number is singular. In these positions, the wall heat flux has a non-vanishing value even if the wall temperature equals the bulk temperature.     

Nonstationary laminar heat convection in a closed domain for a given heat flux

Characteristic modes of the time development of nonstationary heat convection in a closed planar domain upon a sudden supply of heat from the lateral surface are considered for Rayleigh numbers 10³–10⁷. Estimates of the boundaries of the beginning of the influence of convection on the temperature field and the buildup of a quasistationary convection mode in the range of Rayleigh and Fourier numbers are given. Characteristics of the circulation flow, the singularities of the temperature-field configuration and of the heat transfer from the wall to the fluid, are investigated. The mechanism for the origination and disappearance of vertical temperature differences, caused by convection, and the dependence of the vertical temperature differences on the Rayleigh and Fourier numbers, on the thermal mode of the boundary, and the domain geometry, are considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 109–117, July–August, 1970.The author is grateful to T. D. Pirumov and T. V. Volokitin for assistance in performing the computations.     

On the laminar forced convection with axial conduction in a circular tube with exponential wall heat flux

The values of the fully developed Nusselt number for laminar forced convection in a circular tube with axial conduction in the fluid and exponential wall heat flux are determined analytically. Moreover, the distinction between the concepts of bulk temperature and mixing-cup temperature, at low values of the Peclet number, is pointed out. Finally it is shown that, if the Nusselt number is defined with respect to the mixing-cup temperature, then the boundary condition of exponentially varying wall heat flux includes as particular cases the boundary conditions of uniform wall temperature and of convection with an external fluid.

---

Über laminare Zwangskonvektion mit Längswärmeleitung in einem Kreisrohr mit exponentiell veränderlichem Wandwärmefluß

Zusammenfassung Es werden die Endwerte der Nusselt-Zahlen für vollausgebildete laminare Zwangskonvektion in einem Kreisrohr mit Längswärmeleitung und exponentiell veränderlichem Wandwärmefluß analytisch ermittelt. Besondere Betonung liegt auf dem Unterschied zwischen den Konzepten für die Mittel- und die Mischtemperatur bei niedrigen Peclet-Zahlen. Schließlich wird gezeigt, daß bei Definition der Nusselt-Zahl bezüglich der Mischtemperatur die Randbedingung exponentiell veränderlichen Randwärmeflusses die Spezialfälle konstanter Wandtemperatur und konvektiven Wärmeaustausches mit einem umgebenden Fluid einschließt.

---

Nomenclature A _n dimensionless coefficients employed in the Appendix - Bi Biot numberBi=h _e r ₀/ - c _n dimensionless coefficients defined in Eq. (17) - c _p specific heat at constant pressure of the fluid within the tube, [J kg^–1 K^–1] - f solution of Eq. (15) - h ₁,h ₂ specific enthalpies employed in Eqs. (2) and (4), [J kg^–1] - h _e convection coefficient with a fluid outside the tube, [W m^–2 K^–1] - rate of mass flow, [kg s^–1] - Nu bulk Nusselt number,2r ₀ q _w /[(T _w –T _b )] - Nu _H fully developed value of the bulk Nusselt number for the boundary condition of uniform wall heat flux - Nu _T fully developed value of the bulk Nusselt number for the boundary condition of uniform wall temperature - Nu ^* mixing Nusselt number,2r ₀ q _w /[(T _w –T _m )] - Nu _C ^* fully developed value of the mixing Nusselt number for the boundary condition of convection with an external fluid - Nu _H ^* fully developed value of the mixing Nusselt number for the boundary condition of uniform wall heat flux - Nu _T ^* fully developed value of the mixing Nusselt number for the boundary condition of uniform wall temperature - Pe Peclet number, 2r ₀/ - q ₀ wall heat flux atx=0, [W m^–2] - q _w wall heat flux, [W m^–2] - r radial coordinate, [m] - r ₀ radius of the tube, [m] - s dimensionless radius,s=r/r ₀ - T temperature, [K] - T ₀ temperature constant employed in Eq. (14), [K] - T reference temperature of the fluid external to the tube, [K] - T _b bulk temperature, [K] - T _m mixing or mixing-cup temperature, [K] - T _w wall temperature, [K] - u velocity component in the axial direction, [m s^–1] - mean value ofu, [m s^–1] - x axial coordinate, [m] Greek symbols thermal diffusivity of the fluid within the tube, [m² s^–1] - exponent in wall heat flux variation, [m^–1] - dimensionless parameter - dimensionless temperature =(T _w –T)/(T _w –T _b ) - ^* dimensionless temperature ^*=(T _w –T)/(T _w –T _m ) - thermal conductivity of the fluid within the tube, [W m^–1 K^–1] - density of the fluid within the tube, [kg m^–3]     

Combined natural and forced convection in a laminar wall jet along a vertical plate with uniform surface heat flux

The steady state flow and heat transfer characteristics of the combined natural and forced convection in a two dimensional, laminar, incompressible wall jet over a vertical wall are obtained for constant wall heat flux boundary condition. The velocity and temperature distribution are assumed to be power series, where the zeroth term corresponds to that for a plane wall jet in the absence of buoyancy effects. Numerical results for the momentum and thermal series functions are presented for a Prandtl number of 0.73. Wall values of the momentum and thermal series functions are presented for Prandtl numbers ranging from 0.01 to 1000.Nomenclature Gr* modified Grashof number - k thermal conductivity - Nu Nusselt number - Pr Prandtl number - q _w heat flux at the wall - Re Reynolds number - T temperature - u velocity component in x-direction - v velocity component in y-direction - x co-ordinate along the plane wall - y co-ordinate normal to the wall - () gamma function - non-dimensional co-ordinate defined in (6) - non-dimensional temperature - dynamic viscosity - kinematic viscosity - non-dimensional co-ordinate defined in (6) - density - w values at the wall - values at large distances away from the wall     

Developing laminar forced convection in eccentric annuli

The paper presents a boundary-layer model describing the laminar forced convection heat transfer in the entry region of eccentric annuli. A finite-difference numerical algorithm is developed for solving this model. Numerical results are presented for the developing velocity profiles and the pressure drop in annuli of radius ratio 0.5 and 0.9 over a dimensionless eccentricity ranged from 0.1 to 0.8. Heat transfer parameters are presented for a fluid of Pr=0.7 under the conditions of an isothermally heated inner wall while the outer wall is kept at the inlet fluid temperature. Received on 18 March 1997     

Numerical prediction for laminar forced convection heat transfer in parallel-plate channels with streamwise-periodic rod disturbances

A numerical study has been performed for the periodically fully-developed flow in two-dimensional channels with streamwise-periodic round disturbances on its two walls. To accurately describe the round disturbance boundary condition, a body fitted grid was used. The flow and heat transfer have been studied in the range of Reynolds number, Re=50–700, and Prandtl number Pr=0.71. The influences of disturbance parameters and Reynolds number on heat transfer and friction have been investigated in detail. Some of the solutions have been examined using both steady and unsteady finite difference schemes; and the same results have been obtained. The results show that different flow patterns can occur with different deployments of the disturbances. With appropriate configuration of the disturbances, the Nusselt number can reach a value four times greater than in a smooth channel at the same condition, with the penalty of a much greater pressure drop. On the other hand, if the disturbances are not deployed properly, augmentation of heat transfer cannot be acquired. © 1998 John Wiley & Sons, Ltd.     

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.

---

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.

---

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     

Analysis of laminar flow forced convection heat transfer in the entrance region of a circular pipe

A numerical solution, for incompressible, steady-state, laminar flow heat transfer in the combined entrance region of a circular tube is presented for the case of constant wall heat flux and constant wall temperature. The development of velocity profile is obtained from Sparrow's entrance region solution. This velocity distribution is used in solving the energy equation numerically to obtain temperature profiles. Variation of the heat transfer coefficient for these two different boundary conditions for the early stages of boundary layer formation on the pipe wall is obtained. Local Nusselt numbers are calculated and the results are compared with those given byUlrichson andSchmitz. The effect of the thermal boundary conditions is studied by comparing the uniform wall heat flux results with uniform wall temperature.     

Combined forced and natural convection in laminar film flow

A mathematical model for the flow and heat transfer in a gravity-driven liquid film is presented, in which the strict Boussinesq approximation is adopted to account for buoyancy. A similarity transformation reduces the governing equations to a coupled set of ordinary differential equations. The resulting two-parameter problem is solved numerically for Prandtl numbers ranging from 1 to 1000. Favourable buoyancy arises when the temperatureT _w of the isothermal surface is lower than the temperatureT ₀ of the incoming fluid, and the principal effects of the aiding buoyancy are to increase the wall shear and heat transfer rate. For unfavourable buoyancy (T _w>T ₀), the buoyancy force and gravity act in opposite directions and the flow in the film boundary layer decelerates, whereas the friction and heat transfer are reduced. The observed effects of buoyancy diminish appreciably for higher Prandtl numbers.     

Numerical study of transient laminar natural convection heat transfer over a sphere subjected to a constant heat flux

Transient laminar natural convection over a sphere which is subjected to a constant heat flux has been studied numerically for high Grashof numbers (10⁵ ≤ Gr ≤ 10⁹) and a wide range of Prandtl numbers (Pr = 0.02, 0.7, 7, and 100). A plume with a mushroom-shaped cap forms above the sphere and drifts upward continuously with time. The size and the level of temperature of the transient cap and plume stem decrease with increasing Gr and Pr. Flow separation and an associated vortex may appear in the wake of the sphere depending on the magnitude of Gr and Pr. A recirculation vortex which appears and grows until “steady state” is attained was found only for the very high Grashof numbers (10⁵ ≤ Gr ≤ 10⁹) and the lowest Prandtl number considered (Pr = 0.02). The appearance and subsequent disappearance of a vortex was observed for Gr = 10⁹ and Pr = 0.7. Over the lower hemisphere, the thickness of both the hydrodynamic (δ_H) and the thermal (δ_T) boundary layers remain nearly constant and the sphere surface is nearly isothermal. The surface temperature presents a local maximum in the wake of the sphere whenever a vortex is established in the wake of the sphere. The surface pressure recovery in the wake of the sphere increases with decreasing Pr and with increasing Gr. For very small Pr, unlike forced convection, the ratio δ_T/δ_H remains close to unity. The results are in good agreement with experimental data and in excellent agreement with numerical results available in the literature. A correlation has also been presented for the overall Nusselt number as a function of Gr and Pr.     

The analogy between fluid friction and heat transfer of laminar forced convection on a flat plate

This paper has examined the validity of the analogies between heat transfer and fluid friction when they are applied to laminar forced convection on a flat plate. For the case of uniform wall temperature, all the analogies are valid for Prandtl number do not differ greatly from 1, and the Colburn analogy is consistent with numerical data forPr1. However, the previous analogies are not valid for the case of uniform wall heat flux. For fluids ofPr1 under uniform heating, the Colburn analogy factor is 0.72. In addition, these analogies are not applicable to the fluids of small Prandtl number. For the sake of completeness, this work has developed the analogies between heat and momentum transfer over the range of 0.001 Pr 0.1 for both the cases of constant wall temperature and heat flux.Die Gültigkeit der Analogien zwischen Wärmeübergang und Flüssigkeitsreibung wird für den Fall der erzwungenen Konvektion an einer ebenen Platte überprüft. Unter der Randbedingung gleichförmiger Wandtemperatur sind alle Analogien für Prandtl-Zahlen in der Nähe von 1 gültig und die Colburn-Analogie stimmt mit numerisch gewonnenen Ergebnissen fürPr1 überein. Die vorgenannten Analogien sind allerdings nicht gültig, wenn als Randbedingung gleichförmiger Wandwärmefluß unterstellt wird. Für Fluide mitPr1 und gleichförmigem Wärmefluß ist der Analogiefaktor nach Colburn gleich 0,72. Auch für Flüssigkeiten mit kleinen Prandtl-Zahlen gelten die Analogien nicht. Der Vollständigkeit halber wurden in dieser Arbeit die Analogien zwischen Wärme- und Impulsaustausch für den Bereich 0,001 Pr 0,1 entwickelt, und zwar sowohl für gleiche Temperatur wie gleichen Wärmefluß an der Wand.     

Combined heat and mass transfer by laminar natural convection from a vertical plate with uniform heat flux and concentration

This paper studies combined heat and mass transfer by laminar natural convection from a vertical plate maintained with uniform surface heat flux and species concentration. Very accurate finite-difference solutions of a set of nonsimilarity equations have been obtained for most practical gaseous solutions (Pr?=?0.7, 0.21 ≤ Sc ≤ 2.1) and aqueous solutions (Pr?=?7, 140 ≤ Sc ≤ 1400). Variations of heat and mass transfer rates with buoyancy ratio and Lewis number are presented. Precise correlations have been developed for predicting heat and mass transfer rates of natural convection arising from single (solutal or thermal) buoyancy force and dual buoyancy forces.     

Turbulent forced convection heat transfer of non-Newtonian nanofluids

Forced convection heat transfer of non-Newtonian nanofluids in a circular tube with constant wall temperature under turbulent flow conditions was investigated experimentally. Three types of nanofluids were prepared by dispersing homogeneously γ-Al₂O₃, TiO₂ and CuO nanoparticles into the base fluid. An aqueous solution of carboxymethyl cellulose (CMC) was used as the base fluid. Nanofluids as well as the base fluid show shear-thinning (pseudoplastic) rheological behavior. Results indicate that the convective heat transfer coefficient of nanofluids is higher than that of the base fluid. The enhancement of the convective heat transfer coefficient increases with an increase in the Peclet number and the nanoparticle concentration. The increase in the convective heat transfer coefficient of nanofluids is greater than the increase that would be observed considering strictly the increase in the effective thermal conductivity of nanofluids. Experimental data were compared to heat transfer coefficients predicted using available correlations for purely viscous non-Newtonian fluids. Results show poor agreement between experimental and predicted values. New correlation was proposed to predict successfully Nusselt numbers of non-Newtonian nanofluids as a function of Reynolds and Prandtl numbers.