Expressions For Nusselt Number Research Articles

Heat transfer characteristics of oscillating flow regenerator filled with circular tubes or parallel plates

Under assumption of small perturbation, linear thermoacoustic theory was applied to analyze heat transfer characteristics of compressible oscillating flow in two kinds of simple regenerators filled with circular tubes or parallel plates. Based on the cross-sectional oscillating velocity and temperature distributions, the exact expressions of Nusselt number were derived in complex notation. The Nusselt number is the function of Prandtl number, kinetic Reynolds number Re ω and the third dimensionless variable, D. Here, the D is defined as the ratio of heat transfer capability aroused by mean temperature gradient and gas compressibility. Both the gas compressibility and mean temperature gradient effects were discussed and two corresponding Nusselt numbers were given. In particular, simpler expressions for these two Nusselt numbers were deduced for extreme values of Re ω and D. Finally, combined effect of gas compressibility and non-zero mean temperature gradient on heat transfer characteristics were analyzed via D. The analysis shows that the mean temperature gradient gives predominant contribution to the heat transfer performance of oscillating flow regenerator.

Convective heat transfer in unsteady laminar parallel flows

In laminar, fully developed pipe and channel flows that undergo transients from a known initial state, exact analytical solutions for the momentary velocity field as functionals of the flow rate can be determined from the Navier-Stokes equations, for arbitrary flow unsteadiness [Phys. Fluids 12, 518 (2000)]. For laminar, fully developed duct flows with uniform wall heating that undergo large flow transients, the companion thermal energy equation can be approximated in a form that may also be solved analytically, yielding solutions for the instantaneous temperature field for arbitrary time unsteadiness in both the flow and the wall heat flux. Expressions for Nusselt numbers in convective heat transfer in duct flows with arbitrary temporal flow and heat flux unsteadiness can then be found, which illustrate how the flow and heat flux transient histories determine whether the unsteadiness enhances or reduces the overall heat-transfer effectiveness. These expressions are used to show how significant enhancements or reductions in the average Nusselt number can be achieved in duct flow by applying appropriate temporal flow control.

Combined heat and mass transfer by laminar mixed convection flow from a vertical surface with induced magnetic field

The mixed convection flow over a continuously moving porous vertical plate under the combined buoyancy effects of thermal and mass diffusion has been studied under the action of transverse applied magnetic field taking into account the induced magnetic field, when the plate is subjected to constant heat and mass fluxes. Solutions for the velocity field, temperature distribution, concentration distribution, induced magnetic field and current density are obtained using perturbation technique. Expressions for shear stress, Nusselt number and Sherwood number are also obtained. Its apparent that the effect of Grashof number (Gr) on temperature distribution is significantly greater than the magnetic parameter and Schmidt number and the temperature distribution remains more for Gr<0 than that of Gr>0 for both air and water. It is found that the shear stress decreases with increasing Prandtl number (Pr).

Transient Natural Convection Heat Transfer Correlations for Tube Bundles Immersed in a Thermal Storage

A scale analysis of the transient discharge of a fully mixed thermal storage vessel with an immersed single-tube heat exchanger is extended to provide a generalized expression for the transient natural convection Nusselt number for heat exchangers comprising many tubes. The transient Nusselt number is expressed in terms of the Rayleigh number at the initiation of the discharge (or charge) process and easily measured geometric parameters. Nusselt numbers measured for a 240-tube heat exchanger immersed in a fully mixed 126L storage vessel are well correlated in the proposed form. The applicability of the approach to thermally stratified storage fluids is evaluated for both a single-tube and the 240-tube bundle. For heat exchangers of practical size for solar systems, for example the 240-tube bundle, buoyancy driven flow within the storage is sufficient to mix an initially stratified fluid. In this case, Nusselt numbers during the discharge process are predicted accurately by the proposed transient formulation. However, if the storage fluid remains stratified during discharge, as is the case for an initially stratified vessel with a single-tube heat exchanger, the transient formulation is not recommended.

A Unit Cube-Based Model for Heat Transfer and Fluid Flow in Porous Carbon Foam

Abstract A unit-cube geometry model is proposed to characterize the internal structure of porous carbon foam. The unit-cube model is based on interconnected sphere-centered cubes, where the interconnected spheres represent the fluid or void phase. The unit-cube model is used to derive all of the geometric parameters required to calculate the heat transfer and flow through the porous foam. An expression for the effective thermal conductivity is derived based on the unit-cube geometry. Validations show that the conductivity model gives excellent predictions of the effective conductivity as a function of porosity. When combined with existing expressions for the pore-level Nusselt number, the proposed model also yields reasonable predictions of the internal convective heat transfer, but estimates could be improved if a Nusselt number expression for a spherical void phase material were available. Estimates of the fluid pressure drop are shown to be well-described using the Darcy-Forchhiemer law, however, further exploration is required to understand how the permeability and Forchhiemer coefficients vary as a function of porosity and pore diameter.

Thermal convection for a thermo-dependent yield stress fluid in an axisymmetric horizontal duct

This paper deals with the combined forced and free convection heat transfer of a yield stress fluid in a horizontal duct heated uniformly with a constant heat flux density. It is assumed that (i) the rheological behavior of the fluid can be described by the Herschel–Bulkley model and that the consistency K* varies with temperature T*, as K* = a exp(− bT*); (ii) the variation of the fluid density ρ, with temperature, ρ = ρ e ( 1 - β ( T ∗ - T e ∗ ) ) , is considered important only in the buoyancy term and (iii) the Péclet number is sufficiently large so that it is possible to resort to an asymptotic solution. The aim of this study is to quantify the effect of the rheological properties on the magnitude of the secondary flows induced by the thermo-dependency of K* and ρ. Expressions for local Nusselt number and wall shear stress are given. Finally, to be consistent with the variation along the duct of the axial velocity in the central zone around the axis, due to the thermo-dependency of K* or ρ, the pseudo-plug zone and pseudo-yield surface notions are introduced. Some characteristics of the stresses distribution within the pseudo-plug zone are discussed.

Conjugate heat transfer inside a porous channel

Analytical and numerical analyses have been performed for fully developed forced convection in a fluid-saturated porous medium channel bounded by two parallel plates. The channel walls are assumed to be finite in thickness. Conduction heat transfer inside the channel wall is also accounted and the full problem is treated as a conjugate heat transfer problem. The flow in the porous material is described by the Darcy–Brinkman momentum equation. The outer surfaces of the solid walls are treated as isothermal. A temperature dependent volumetric heat generation is considered inside the solid wall only. Analytical expressions for velocity, temperature, and Nusselt number are obtained after simplifying and solving the governing differential equations with reasonable approximations. Subsequent results obtained by numerical calculations show an excellent agreement with the analytical results.

SIMULTANEOUS NATURAL-CONVECTION HEAT TRANSFER ABOVE AND BELOW AN ISOTHERMAL HORIZONTAL THIN PLATE

Numerical studies were conducted to investigate the simultaneous natural-convection heat transfer above and below an isothermal thin plate in an infinite space. The full two-dimensional conservation equations for laminar flows were solved. The numerical approach was based on the finite-volume technique with a nonstaggered grid arrangement. The SIMPLE algorithm was used for handling the pressure-velocity coupling. Plate width and temperature were used to vary the Rayleigh number over the range 1.0 2 10 5 to 1.7 2 10 7 . The expressions for numerical local and average Nusselt number were correlated, and horizontal velocity and temperature profiles were provided. It was found that the upper-surface Nusselt numbers are smaller than the corresponding lower-surface Nusselt numbers. Numerical results for upward- and downward-facing isothermal horizontal plates (single-sided heat transfer) were also shown for comparison. For these cases, Nusselt numbers on the upward-facing surface are larger than those on the downward-facing surface.

A synergistic approach to parameter estimation in multimode heat transfer

We report the efficacy of the Least Square Residual (LSR) method in parameter estimation when more than one mode of heat transfer is encountered. A vertical flat plate, made of aluminium, to ensure that lumped formulation is valid, is allowed to cool in quiescent air. The novelty arises in coating the plate with a paint of known emissivity, in this case black board paint with e = 0.85 and using the experimentally obtained transient temperature history to develop an expression for convective Nusselt number. The same plate is then given a different coating of unknown emissivity, and using the just established relationship for convective Nusselt number, by a regular parameter estimation using LSR, the emissivity is calculated. The synergy arises by using known information on emissivity to estimate convection in the first case and the reverse to back calculate an unknown emissivity. The ubiquitous vertical flat plate, thus serves as a simple, inexpensive, yet a potent emissivity comparator

Heat Transfer to Liquid Metal Flow in a Porous Saturated Circular Tube with Uniform Wall Temperature: An Exact Solution

A theoretical analysis is presented to investigate the thermal development of liquid metal forced-convection flow in a circular tube filled by a saturated porous medium, with uniform wall temperature and with the effects of axial conduction included. The Darcy model is employed. The analysis leads to expressions for the local Nusselt number as a function of the dimensionless axial coordinate and the Peclet number.

Numerical study of simultaneous natural convection heat transfer from both surfaces of a uniformly heated thin plate with arbitrary inclination

Numerical studies were conducted to investigate the natural convection heat transfer around a uniformly heated thin plate with arbitrary inclination in an infinite space. The numerical approach was based on the finite volume technique with a nonstaggered grid arrangement. For handling the pressure–velocity coupling the SIMPLE-algorithm was used. QUICK scheme and first order upwind scheme were employed for discretization of the momentum and energy convective terms respectively. Plate width and heating rate were used to vary the modified Rayleigh number over the range of 4.8×106 to 1.87×108. Local and average heat transfer characteristics were compared with regarding to the inclination angle. The empirical expressions for local and average Nusselt number were correlated. It has been found that for inclination angle less than 10∘, the flow and heat transfer characteristics are complicated and the average Nusselt number can not be correlated by one equation while for inclination angle larger than 10∘, the average Nusselt number can be correlated into an elegant correlation.

Heat transfer by laminar Hartmann's flow in thermal entrance region with uniform wall heat flux: the Graetz problem extended

Thermally developing laminar Hartmann flow through a parallel plate channel, with prescribed transversal uniform magnetic field, including both viscous dissipation, Joule heating and axial heat conduction with uniform heat flux, is studied analytically by using a functional analysis method. The analytical expressions for the developing temperature and local Nusselt number in the entrance region are obtained in the general case. However, the solution obtained could be applied to any developed flow with heat generation. Applications to Hagen-Poiseuille and Hartmann flows are made. The associated eigenvalues problem is solved analytically to obtain explicit forms of the eigenfunctions, which correspond to Mathieu's functions. Results show that the effects of viscous and Joule dissipation can be neglected for liquid metal, even for large values of Hartmann number M, for which the corresponding Brinkman number Br is very small. Therefore, the heat transfer characteristics in the entrance region are only influenced by Peclet and Hartmann numbers. Results show that as long as M increases, the heat transfer increases until a certain saturation is reached, for which the effect of magnetic field on Nusselt number vanishes completely. Finally, a practical empirical expression for local Nusselt number is proposed in terms of axial distance ξ and Hartmann number M.

A universal entrance nusselt number for internal slip flow

A universal finite Nusselt number is found for laminar slip flow heat transfer at the entrance of a conduit. The Nusselt number expression is valid for both isothermal and isoflux thermal boundary conditions and for any conduit geometry. The value of the entrance Nusselt number is found to be dependent on two nondimensional parameters that include effects of rarefaction, fluid properties, and the fluid/wall interaction.

Heat transfer from a porous sphere in a slow viscous flow

Forced heat convection in the context of slow streaming motion past a permeable sphere is studied, postulating the normal velocity on the surface with Darcian law. The analytic expression for mean Nusselt number, obtained upto O( σ 2 Re 2)under the assumption that the flow Reynolds number Re < 1 and the Prandtl number σ at most of O(1) shows the improved effect of permeability on heat transfer in comparison to the existing results in the literature. Enhanced heat transfer is realised with increasing permeability when axial spin incident with free stream is given to the body.

A simple way to determine the two asymptotic Nusselt number expressions for in-tube, laminar forced convective flows employing the method of lines

The principal goal of this educational article is to present an alternative simple way for the determination of the two asymptotic Nusselt number subdistributions for laminar forced convection tube flows subjected to a uniform surface temperature. Under the assumption of developed velocity and developing temperature, the generalized method of lines (MOL) has been adopted here because it is a powerful computational technique that facilitates the determination of the two asymptotic temperature solutions (one for x → 0 and the other for x → ∞). The semianalytic solution method, MOL, is quite different and possesses unique features that distinguish it from other standard solution methods used in the past. The two asymptotic Nusselt number subdistributions, Nux→0 and Nux→∞, blend themselves into an approximate Nusselt number distribution of high quality. The unprecedented ease of the computational procedure, along with its adequate results, has proved to be a valuable instructional technique in undergraduate courses on heat transfer. © 1998 John Wiley & Sons, Inc. Comput Appl Eng Educ 6: 79–87, 1998

Simulation of convective mass transfer in a solar distillation process

In this communication, an attempt has been made to develop a modified Nusselt number for a trapezoidal cavity, which can be used for evaluation of convective mass transfer in a solar distillation process. The modified expression has been obtained by regression analysis using experimental data and from simulations for different environment conditions. It is observed that there is a reasonable agreement between experimental observations and theoretical results calculated by the modified Nusselt number expression for a higher operating temperature range (≥ 60°C).

Study on liquid laminar free convection with consideration of variable thermophysical properties

The analyses focus on liquid laminar free convection along an isothermal vertical flat plate with consideration of variable fluid thermophysical properties. Measurements on velocity fields were made for water laminar free convection with different plate-surface temperature conditions by LDV (laser doppler velocimeter). The measured results agree very well with the calculated velocity profiles. Expressions for local Nusselt number, Nu x.∞, depending on the temperature gradient dθ/dη| η=0 are developed. A method is proposed to predict accurately the heat transfer of liquid laminar free convection by means of numerical solution of the temperature gradient.

Heat transfer from turbulent air flowing through a cooled isothermal tube

A device was constructed to establish turbulent flow of air in a constant wall-temperature copper tube. The flow was fully developed hydrodynamically but developing thermally. Data on flow rate and fluid temperatures were obtained and reduced to an expression for Nusselt number as a function of Reynolds and Prandtl numbers. An existing equation for developing flow was tested in order to determine how well it fitted the data. The derived equation of this study has a correlation coefficient of 0·974 and a standard deviation of 6·03.

Nusselt number of a sphere in creeping flow

Rates of heat or mass transfer from a spherical droplet translating in a field under creeping flow conditions are examined. The existing expressions for Nusselt numbers are evaluated via an independent numerical analysis.

An exact solution of extended Graetz problem with axial heat conduction

The convective heat transfer properties of a hydrodynamically, fully developed viscous flow in a circular tube are analyzed without using any simplified assumption such as low Reynolds numbers or Peclet numbers. The pipe is then subjected to a step-change in wall temperature. A straightforward approach using the Fourier transform technique is utilized to obtain analytical expressions for temperature distribution, heat flux, and Nusselt numbers. The effects of axial heat conduction are included in these expressions in both the upstream and the downstream directions for Peclet numbers ranging from 0 to ∞. By first taking the Fourier transform of the temperature field and expanding the coefficients of the transformed temperature in terms of the Peclet number, the energy equation with discontinuous wall temperature and longitudinal heat conduction is transformed into a set of ordinary differential equations. This resulting set of equations is then solved successively. The representative curves illustrating the variations of bulk temperature, heat flux, and Nusselt numbers with pertinent parameters are plotted. The asymptotic Nusselt numbers for small, as well as large Peclet numbers, is obtained as 3.6565, compared to the exact classical value of 3.6568. The significance of each curve is also discussed.