The extended Graetz problem with Dirichlet wall boundary conditions

The extension of the Graetz problem to include axial conduction has been of interest in view of its application to a number of low Peclet number heat or mass transfer situations. Past efforts in dealing with this problem have been plagued with uncertainties arising from expansion in terms of “eigenfunctions” and “eigenvalues” belonging to a nonselfadjoint operator. The uncertainties spring from a lack of basis for the assumptions that no complex eigenvalues exist and that the calculated eigenvectors originate from a complete set. Other methods have been entirely numerical. The present work produces an entirelyanalytical solution to the Graetz problem for the Dirichlet boundary condition based on a selfadjoint formalism resulting from a decomposition of the convective diffusion equation into a pair of first order partial differential equations. Physically, the decomposition views the convective diffusion process as a pair of stipulations on how the temperature (or concentration) and theaxial energy (or mass) flow through a partial tube cross-section vary with radial and axial distances. The solution obtained is simple, and readily computed. To whom correspondence may be addressed     

Maximum density effects on convective instability of horizontal plane poiseuille flows in the thermal entrance region

A linear stability analysis is used to study the conditions marking the onset of secondary flow in the form of longitudinal vortices for plane Poiseuille flow of water in the thermal entrance region of a horizontal parallel-plate channel by a numerical method. The water temperature range under consideration is 0∼30°C and the maximum density effect at 4°C is of primary interest. The basic flow solution for temperature includes axial heat conduction effect and the entrance temperature is taken to be uniform at far upstream location jackie=−∞ to allow for the upstream heat penetration through thermal entrance jackie=0. Numerical results for critical Rayleigh number are obtained for Peclet numbers 1, 10, 50 and thermal condition parameters (λ ₁, λ ₂) in the range of −2.0≤λ ₁≤−0.5 and −1.0≤λ ₂≤1.4. The analysis is motivated by a desire to determine the free convection effect on freezing or thawing in channel flow of water.     

Forced convection heat transfer of Giesekus viscoelastic fluid in pipes and channels

A theoretical solution is presented for the convective heat transfer of Giesekus viscoelastic fluid in pipes and channels, under fully developed thermal and hydrodynamic flow conditions, for an imposed constant heat flux at the wall. The fluid properties are taken as constant and axial conduction is negligible. The effect of Weissenberg number (We), mobility parameter (α) and Brinkman number (Br) on the temperature profile and Nusselt number are investigated. The results emphasize the significant effect of viscous dissipation and fluid elasticity on the Nusselt number in all circumstances. For wall cooling and the Brinkman number exceeds a critical value (Br ₁), the heat generated by viscous dissipation overcomes the heat removed at the wall and fluid heats up longitudinally. Fluid elasticity shifts this critical Brinkman number to higher values.     

The extended Graetz problem with piecewise constant wall heat flux for laminar and turbulent flows inside concentric annuli

In the present paper, the heat transfer characteristics in the thermal entrance region of concentric annuli have been analysed for laminar and turbulent internal flow. Axial heat conduction effects in the fluid have been taken into account. The present paper shows an exact analytical solution for the problem of a piecewise uniform wall heat flux. The obtained analytical solution for the extended Graetz problem is as simple and efficient to compute as the related solution of the parabolic problem. The obtained results show the effect of axial heat conduction in the fluid for a semi-infinite heated section as well as for a finite length of the heated section. It is shown, that for a finite length of the heated section, axial heat conduction effects might be important even for higher Peclet number.     

Laminar heat transfer in the entrance region of ducts

A finite difference technique is used for the evaluation of the rate of heat transfer in the thermal entrance region of ducts with axial conduction. The velocity profile is fully developed and flow in a tube and between parallel plates is studied. Local and average Nusselt numbers and mixing temperatures are presented as a function of the Péclet number. A criterion is also established which proves useful for predicting the conditions under which axial conduction may be ignored.Nomenclature C transformation constant - c _v specific heat, constant volume - D _h hydraulic diameter - h local convective film coefficient, Eq. (15) - h* local convective film coefficient, Eq. (16) - h _m ^* mean convective film coefficient, Eq. (17) - k thermal conductivity - Nu local Nusselt number, hD _h/k - Nu* local Nusselt number, h*D _h/k - Nu _m ^* mean Nusselt number, hQD _h/k - Pe Péclet number, D _h v _m/ - q rate of heat transfer - r radial coordinate - r _o tube radius - R nondimensional radial coordinate, r/r _o - S transformed axial coordinate, Eq. (10) - T temperature - T _e entrance temperature - T _m mixing temperature, Eq. (18) - T _w wall temperature - v _z axial velocity - v _m mean axial velocity - V nondimensional axial velocity, v _z /v _m - y transverse coordinate in parallel plate flow - y _o half width of parallel plate duct - Y nondimensional transverse coordinate, y/y _o - z axial coordinate - Z nondimensional axial coordinate, z/r _o or z/y _o - Z ⁺ nondimensional axial coordinate divided by Peclet number, Z/Pe - thermal diffusivity - nondimensional temperature, (T–T _w)/(T _e–T _w) - mean nondimensional temperature, - _m nondimensional mixing temperature, Eq. (22) - density - i axial position index - j radial or transverse position index     

Measurement of heat transfer coefficient and axial dispersion coefficient using temperature oscillations

A periodic transient test technique based on the axial dispersion model is proposed for the determination of both heat transfer coefficients and axial dispersion coefficients in heat exchangers. The model uses a parameter called the axial dispersive Peclet number to account for the deviation of the flow pattern from ideal plug flow. It takes both axial dispersion in the fluid and axial heat conduction in the wall into account and is solved analytically by means of a complex Fourier transform. Experiments conducted on dented copper tubes show that axial dispersion has a significant effect on the dynamic temperature response of a heat exchanger.     

Axial heat conduction effects in the entrance region of circular ducts

This paper describes the transport of thermal energy within a small distance after an abrupt wall temperature change in a circular duct. In general, the axial conduction becomes significant when the Peclet number is small. The results indicate that the inclusion of axial conduction in the fluid substantially increases the wall heat flux at near the thermal inlet location. The exact series solution leads to a modified Graetz type problem. This exact solution is augmented by an asymptotic solution describing the wall heat flux near the thermal entrance location.     

Wiener–Hopf solution of rewetting of an infinite cylinder with internal heat generation

The two-dimensional quasi-steady conduction equation governing conduction controlled rewetting of an infinite cylinder with heat generation has been solved by Wiener–Hopf technique. The analytical solution yields the quench front temperature as a function of various model parameters such as Peclet number, Biot number and dimensionless heat generation rate. Also, the dry out heat generation rate is obtained by setting the Peclet number equal to zero, which gives the maximum permissible heat generation so as to prevent the dry out of the coolant.     

Laminar heat transfer characteristics of internally finned tube with sinusoidal wavy fin

Comparative numerical study of laminar heat transfer characteristics of annular tubes with sinusoidal wavy fins has been conducted both experimentally and numerically with Re = 299–1,475. The uniform heat flux is imposed on the tube outside wall surface. Two tube materials (copper and stainless steel) are considered. It is found that the fluid temperature profile is not linear but convex along the flow direction due to the axial heat conduction in tube wall, and the effects of axial heat conduction on the heat transfer decreases with an increase in Reynolds number or decrease in tube wall thermal conductivity. The axial distributions of local Nusselt number could reach periodically fully developed after 3–5 cycles. The convectional data reduction method based on the traditional method should be improved for tube with high thermal conductivity or low Reynolds numbers, Otherwise, the heat transfer performance of internally finned tube may be underestimated.     

Effect of variable thermal conductivity and viscosity on single phase convective heat transfer in slip flow

For a variety of fields in which micro-mechanical systems and electronic components are used, fluid flow and heat transfer at the microscale needs to be understood and modeled with an acceptable reliability. In general, models are prepared by making some extensions to the conventional theories by including the scaling effects that become important for microscale. Some of these effects are; axial conduction, viscous dissipation, and rarefaction. In addition to these effects, temperature variable thermal conductivity and viscosity may become important in microscale gas flows due to the high temperature gradients that may exist in the fluid. For this purpose, simultaneously developing, single phase, laminar and incompressible air flow in a microtube and in the micro gap between parallel plates is numerically analyzed. Navier–Stokes and energy equations are solved and the variation of Nusselt number along the channel is presented in tabular and graphical forms as a function of Knudsen, Peclet, and Brinkman numbers, including temperature variable thermal conductivity and viscosity.     

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.     

Heat dispersion in stationary mixed convection flow about horizontal surfaces in porous media

An analysis has been performed to study the influence of velocity dependent dispersion on transverse heat transfer in mixed convection flow above a horizontal wall of prescribed temperature in a saturated porous medium. The Boussinesq approximation and boundary layer analysis were used to numerically obtain gravity affected temperature and velocity distributions within the frames of Darcy's law and a total thermal diffusivity tensor comprising both of constant coefficient heat conduction and velocity proportional mechanical heat dispersion. Dependending on Pe_∞, the molecular Peclét number basing on the effective thermal diffusivity and the velocity of the oncoming flow, density coupling has distinct influences on heat transfer rates between the wall surface and the porous medium flow region. For small Peclét numbers, when heat conduction is the prevailing mechanism, wall heat fluxes are the higher the larger the density difference between the oncoming and the near wall fluid is. The opposite is true for larger Peclét numbers, when mechanical heat dispersion is the main cause of heat spreading. For Pe_∞ tending to infinity these wall heat fluxes approach finite maximum values in the total heat diffusivity model, they grow beyond any limit if only constant coefficient heat conduction is considered. Thus, the inclusion of mechanical heat dispersion effects yields physically more realistic predictions. Received on 18 September 1996     

Laminar velocity profile development in straight pipes of circular cross section

Velocity profile development has been studied experimentally in Newtonian and some non-Newtonian fluids. The entry length for the development of 99% of the terminal axial velocity from an initially flat profile has been found to be given byZ _e = 1.1–0.112N(Re) for laminar flow Reynolds numbers between 1 and 1500 with Newtonian fluids. There were substantial increases in this length for weakly visco-elastic aqueous solutions of polyethylene oxide and polyacrylamide in the Reynolds number range (between 1 and 10) where these have been studied.     

Rewetting of an infinite tube with a uniform heating

 The two-dimensional quasi-steady conduction equation governing conduction controlled rewetting of an infinite tube, with outer surface flooded and the inside surface subjected to a constant heat flux, has been solved by Wiener–Hopf technique. The solution yields the quench front temperature as a function of various model parameters such as Peclet number, Biot number and dimensionless heat flux. Also, the dryout heat flux is obtained by setting the Peclet number equal to zero, which gives the maximum sustainable heat flux to prevent the dryout of the coolant. Received on 6 September 2000 / Published online: 29 November 2001     

Two-dimensional adiabatic forced convection at low péclet number

This paper deals with heat transfer to two-dimensional laminar Poiseuille flow with arbitrary heat fluxes specified on the walls. The solution is first obtained for fluxes which are symmetric and antisymmetric delta functions and from these the general solution is obtained by integration. The solutions include the effects of axial conduction and also of preheating of the incoming fluid. Detailed solutions are given for the cases Pé=1 and Pé=.5.     

Flow of a multicomponent gas between parallel permeable surfaces

The present paper gives an exact solution of the equations describing the flow of a multicomponent gas between two parallel permeable planes, one of which moves relative to the other with constant velocity (i. e., we study a flow of the Couette type).Notation y coordinate - u, v velocity components - density - c_i mass concentration of i-th component - I_i diffusional flux of i-th component - H enthalpy - T temperature - m molecular weight - viscosity coefficient - heat conduction coefficient - c_p mixture specific heat - D_ij the binary diffusion coefficients - P Prandtl number - S_ij Schmidt number - N total number of components - n number of components in injected gas - l distance between planes Indices i, j component numbers - w applies to quantities for y=0 - ^* applies to quantities for y=l     

Numerical and experimental investigation of thermal convection for a thermodependent Herschel-Bulkley fluid in an annular duct with rotating inner cylinder

Thermal convection for an incompressible Herschel-Bulkley fluid along an annular duct, whose inner cylinder is rotating and outer is at rest, is analyzed numerically and experimentally. The outer cylinder is heated at constant heat flux density and the inner one is assumed adiabatic. The first part of this study deals with the effect of the rheological behavior of the fluid and that of the rotation of the inner cylinder on the flow field and heat transfer coefficient. All the physical properties are assumed constant and the flow is assumed fully developed. The critical Rossby number Ro_c = (R₁Ω/U_d)_c, for which the dimension of the plug flow is reduced to zero is determined with respect to the flow behavior index, the radius ratio and the Herschel-Bulkley number for axial flow. The rotation of the inner cylinder induces a decrease of the axial velocity gradient at the outer cylinder thereby reducing the heat transfer between the heated wall and the fluid. The second part of this study introduces the variation of the consistency K with temperature and analyzes the evolution of the flow pattern and heat transfer coefficient along the heating zone. Two cases are distinguished depending on the Rossby number: (i) Ro < Ro_c, the plug flow dimension increases along the heating zone; (ii) Ro < Ro_c, the decrease of K with temperature leads to the reappearance of the plug flow. For high angular velocities, it is possible to have a plug zone attached to the outer cylinder. Finally, a correlation is proposed for the Nusselt number. It shows clearly that the effect of thermodependency of K on the heat transfer becomes more important with increasing rotational velocity of the inner cylinder.     

Laminar forced convection conjugate heat transfer over a flat plate

Coupled heat transfer between laminar forced convection along and conduction inside a flat plate wall is theoretically studied. The laminar convective boundary layer is analyzed by employing the integral technique. The energy equations for the fluid and the plate wall are combined under the condition of the continuity in the temperature and heat flux at the fluid-solid interface. The analysis results in a simple formal solution. Expressions have been obtained for calculating local Nusselt number, wall heat flux and temperature along the plate, all are functions of the local Brun number, Br _x , which is a measure of the ratio of the thermal resistance of the plate to that of the convective boundary layer. The results indicate that for Br _x ≥0.15, neglecting the plate resistance will results in an error of more than 5% in Nusselt number. Comparison of the present solution with other previous studies has been made. The solution may be of a considerable theoretical and practical interest. Received on 19 August 1998     

Transient heat and mass transfer in a natural circulation loop

Comprehensive work has been performed by theoretical and numerical methods in order to study the steady state, transient and stability characteristics of a double diffusive natural circulation loop. It was found that the behavior of the flow in the system depends on the initial conditions and on the location of the state in the seven-parameter space of the thermal and saline Rayleigh numbers,Ra _T ,Ra _S , the modified Prandtl and Schmidt numbers,Pr, Sc, the dimensionless heat and mass transfer coefficients,H _T ,H _S , and the “aspect ratio” (between the height and width) of the loop, γ. Numerical results are presented here, showing the flow in each of the five regions formed in the stability chart. The steady state solutions include convection (constant velocity flow), conduction (no-flow) and periodic with constant amplitude and frequency. Two main new results were obtained: long term periodic oscillations where the amplitude is not symmetric around the conduction solution, and an overshoot of the velocity in transients before reaching the stable convection solutions. In the monotonic instability region of the conduction solution, convection solutions (constant velocity flow) develop, and in the global stability region the flow decays to the conduction solution (no flow), regardless of the initial conditions.