Experimental and numerical study of developing turbulent flow and heat transfer in convergent/divergent square ducts

 The work reported in this paper is a systematic experimental and numerical study of friction and heat transfer characteristics of divergent/convergent square ducts with an inclination angle of 1^∘ in the two direction at cross section. The ratio of duct length to average hydraulic diameter is 10. For the comparison purpose, measurement and simulation are also conducted for a square duct with constant cross section area, which equals to the average cross section area of the convergent/divergent duct. In the numerical simulation the flow is modeled as being three-dimensional and fully elliptic by using the body-fitted finite volume method and the kɛ turbulence model. The uniform heat flux boundary condition is specified to simulate the electrical heating used in the experiments. The heat transfer performance of the divergent/convergent ducts is compared with the duct with uniform cross section under three constraints (identical mass flow rate, pumping power and pressure drop). The agreement of the experimental and numerical results is quite good except at the duct inlet. Results show that for the three ducts studied there is a weak secondary flow at the cross section, and the circumference distribution of the local heat transfer coefficient is not uniform, with an appreciable reduction in the four corner regions. In addition, the acceleration/deceleration caused by the cross section variation has a profound effect on the turbulent heat transfer: compared with the duct of constant cross section area, the divergent duct generally shows enhanced heat transfer behavior, while the convergent duct has an appreciable reduction in heat transfer performance. Received on 18 September 2000 / Published online: 29 November 2001     

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.     

Low Reynolds number flow inside straight micro channels with irregular cross sections

This paper presents an analysis of the compound effect of finite temperature differences and fluid friction on the existence of an optimum laminar flow regime in singly connected micro channels with complex free flow area cross sections. A widespread conviction has been established that the two competing irreversibility sources in a channel flow with heat transfer lead to the existence of an optimum flow regime. The results presented in this paper clearly shows the opposite. When an objective function is represented by the entropy generation rate per unit heat capacity rate of the fluid stream, the thermodynamic optimum flow regime represents a rather rare occurrence in the laminar region of irregularly shaped ducts. The presence of an extremum is more probable for very small diameters, the ones of an order of magnitude of O(≤10⁻³ m). The analysis is performed for selected ranges of relevant geometric, flow, and thermal parameters of a set of straight micro channels with irregular free flow area cross-sections. The following geometries of the free flow area cross section were investigated: (i) sine duct, (ii) circular duct, (iii) elliptical duct, (iv) moon-shaped ducts, and (v) four-cuspped duct. The range of Reynolds numbers has been established between O(10²) and O(10⁴). The existence of the objective function minimum is confirmed for ducts with an irregular cross section only for very small hydraulic diameters. These minima are relatively weak, and as a general rule, the sets of optimum parameters are close to the onset of turbulence or possibly even in the transitional or turbulent regions. Received on 10 November 1998     

Zur Kondensation an der Wand dampfdurchströmter Kanäle mit veränderlichem Querschnitt

The flow of pure vapor in channels with variable cross section and cooled walls is considered. The balances for mass, energy and momentum and the transport of heat across the film of condensate are dealt by a finite difference method. Selected examples show that convergent channels improve condensation compared with channels of constant cross section. Excessive temperature drop by acceleration, however, must be avoided. Adversed pressure gradient may cause separation of the condensate film. The influence of the following parameters is discussed: gravity, increased friction, entrainment of condensat, cocurrent and countercurrent flow of the cooling medium and finite resistance of heat of channel wall. The shape of velocity profiles of the condensate is shown as a function of the pressure gradient.     

Transient forced convection heat transfer in a channel

A numerical analysis is made of incompressible transient turbulent flow heat transfer between two parallel plates when there is a step jump in space along the channel in wall heat flux or wall temperature. The variation of the fluid velocity and effective diffusivity over the channel cross section are accounted for. The fluid is assumed to have a fully-developed turbulent velocity profile throughout the length of the channel. The thermal responses of the system are obtained by solving energy equation for air by a digital computer. The results are presented in graphical forms. The stability of the finite difference solution is studied and condition for the stability of the difference solution is derived. A method is given to obtain velocity distributions from the distribution of turbulent eddy diffusivity of momentum. Variations of Nusselt numbers are obtained as a function of time and space. Steady-state values are also given and compared with the published results.     

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     

Laminar flow heat transfer for simultaneously developing velocity and temperature profiles in ducts of rectangular cross section

Summary A numerical method is used to solve the heat transfer equations for laminar flow in ducts of rectangular cross section with simultaneously developing temperature and velocity profiles, both for constant wall temperature and for constant heat input per unit length of the duct. Like the solutions for a fully developed velocity profile, the Nusselt number for each aspect ratio is found to increase from a limiting value at large distances from the entry plane to a maximum at the entry plane. The results also show a strong effect of the Prandtl number on the heat transfer coefficients with uniform and fully developed velocity profiles representing the upper and lower limits respectively. Comparisons are made with analytical solutions for circular ducts and parallel plates and with experimental data.     

Experimental study of augmented heat transfer and friction in solar air heater with different orientations of W-Rib roughness

Artificial roughness in the form of ribs is a convenient method for enhancing thermal performance of solar air heaters. This paper presents the experimental investigation of heat transfer and friction factor characteristics of a rectangular duct roughened with W-shaped ribs arranged at an inclination with respect to the flow direction on its underside on one broad wall. W ribs have been tested both pointing in downstream W-down and upstream W-up to the flow. The range of parameters for this study has been decided on the basis of practical considerations of the system and operating conditions. The duct has a width to height ratio (W/H) of 8.0, relative roughness pitch (p/e) of 10, relative roughness height (e/D_h) of 0.03375 and angle of attack of flow (α) of 30-75°. The air flow rate corresponds to Reynolds number between 2300-14,000. The heat transfer and friction factor results have been compared with those for smooth duct under similar flow and thermal boundary condition and thermo-hydraulic performance has been investigated. Thermo-hydraulic performance comparison for different angle of attack of flow shows that W-down arrangement with angle of attack of flow as 60° gives best thermo-hydraulic performance. In addition heat transfer and friction factor correlations have been developed.     

Der Druckverlust bei nichtisothermer,laminarer Rohrströmung von Potenzgesetz-Flüssigkeiten mit verschiedenartigen thermischen Randbedingungen

This work is concerned with numerical computations of pressure-drops in non-isothermal, laminar flow of power law fluids in circular tubes with various thermal wall conditions (constant and variable wall temperature as well as constant and variable wall heat flux). In the paper both temperature dependent rheological fluid properties and viscous dissipation are included. The results of numerical computations are plotted in 7 figures and selected numerical data are presented tabularly. As a result an extended friction factor formula is derived similar to that one introduced by Metzner and Reed. In addition, an approximative friction factor formula is proposed which enables the calculation of local friction factor if only one value of the so called “film temperature” has been previously measured. This formula is valid for all the thermal wall conditions considered in this work. The results of the present paper are intended to be very helpful for design engineers while calculating the pressure losses in some heat exchangers and pipes for transporting polymer melts.     

Similarity solutions for MHD thermosolutal Marangoni convection over a flat surface in the presence of heat generation or absorption effects

The problem of steady, laminar, thermosolutal Marangoni convection flow of an electrically-conducting fluid along a vertical permeable surface in the presence of a magnetic field, heat generation or absorption and a first-order chemical reaction effects is studied numerically. The general governing partial differential equations are converted into a set of self-similar equations using unique similarity transformations. Numerical solution of the similarity equations is performed using an implicit, iterative, tri-diagonal finite-difference method. Comparisons with previously published work is performed and the results are found to be in excellent agreement. Approximate analytical results for the temperature and concentration profiles as well as the local Nusselt and sherwood numbers are obtained for the conditions of small and large Prandtl and Schmidt numbers are obtained and favorably compared with the numerical solutions. The effects of Hartmann number, heat generation or absorption coefficient, the suction or injection parameter, the thermo-solutal surface tension ratio and the chemical reaction coefficient on the velocity, temperature and concentration profiles as well as quantitites related to the wall velocity, boundary-layer mass flow rate and the Nusselt and Sherwood numbers are presented in graphical and tabular form and discussed. It is found that a first-order chemical reaction increases all of the wall velocity, Nusselt and Sherwood numbers while it decreases the mass flow rate in the boundary layer. Also, as the thermo-solutal surface tension ratio is increased, all of the wall velocity, boundary-layer mass flow rate and the Nusselt and Sherwood numbers are predicted to increase. However, the exact opposite behavior is predicted as the magnetic field strength is increased.     

Effects of cross ribs on heat/mass transfer in a two-pass rotating duct

Two-pass internal cooling passage with rib turbulators has been investigated for convective heat/mass transfer under rotating conditions. The flow and heat transfer characteristics in the cooling passage are very complicated so that it is required the detail analysis to design more efficient gas turbine blades. A naphthalene sublimation technique is employed to determine detailed local heat transfer coefficients using the heat and mass transfer analogy. The local heat/mass transfer and flow pattern in the cooling passage are changed significantly according to rib configurations, duct turning geometries and duct rotation speeds. Four different rib configurations are investigated to obtain the combined effects of the angled rib, duct turning and rotation. The results show that the duct rotation generates the heat/mass transfer discrepancy between the leading and trailing walls due to the secondary flows induced by the Coriolis force. The angled ribs generate a single rotating secondary flow with the cross-rib arrangement and the duct turning makes a strong Dean-type vortex. These vortices affect significantly the heat/mass transfer on the duct wall. The overall heat transfer pattern on the leading and trailing surfaces for the first and second passes are dependent on the duct rotation, but the local heat transfer trend is affected mainly by the rib arrangements. In addition, the present study observes the rotating effect in the two-pass smooth duct to obtain the baseline data in comparison with the ribbed duct for various rib arrangements.     

An analytical treatment for combined heat transfer by radiation,convection and conduction within a heat insulating wall structure

An analytical procedure has been proposed to attack a highly conjugate thermal problem associated with radiation, convection and conduction within a heat insulating wall structure. Firstly, an analytical solution is derived for fully-developed mixed convective flow through parallel plates. Secondly, the resulting expressions for convection are coupled with radiation and conduction equations to form a set of heat balance equations for each heat transfer surface. Illustrative calculations are conducted to estimate heat transfer rates through a heat insulating roof structure with an aluminum partition with high reflectivity, subject to solar radiation in summer. The effect of an aluminum partition on the heat insulation is elucidated.     

Flow and heat/mass transfer in a wavy duct with various corrugation angles in two dimensional flow regimes

In this study, two dimensional heat/mass transfer characteristics and flow features were investigated in a rectangular wavy duct with various corrugation angles. The test duct had a width of 7.3 mm and a large aspect ratio of 7.3 to simulate two dimensional characteristics. The corrugation angles used were 100°, 115°, 130°, and 145°. Numerical analysis using the commercial code FLUENT, was used to analyze the flow features. In addition, the oil-lamp black method was used for flow visualization. Local heat/mass transfer coefficients on the corrugated walls were measured using a naphthalene sublimation technique. The Reynolds number, based on the duct hydraulic diameter, was varied from 700 to 5,000. The experimental results and numerical analysis showed interesting and detailed features in the wavy duct. Main flow impinged on upstream of a pressure wall, and the flow greatly enhanced heat/mass transfer. On a suction wall, however, flow separation and reattachment dominantly affected the heat/mass transfer characteristics on the wall. As the corrugation angle decreased (it means the duct has more sharp turn), the region of flow stagnation at the front part of the pressure wall became wider. Also, the position of flow reattachment on the suction wall moved upstream as the corrugation angle decreased. A high heat transfer rate appeared at the front part of the pressure wall due to main-flow impingement, and at the front part of the suction wall due to flow reattachment. The high heat/mass transfer region by the main-flow impingement and the circulation flow induced at a valley between the pressure and suction walls changed with the corrugation angle and the Reynolds number. As the corrugation angle decreased, the flow in the wavy duct changed to transition to turbulent flow earlier.     

Buoyancy-Opposed Darcy’s Flow in a Vertical Circular Duct with Uniform Wall Heat Flux: A Stability Analysis

An analytical and numerical study is presented to show that buoyancy-opposed mixed convection in a vertical porous duct with circular cross-section is unstable. The duct wall is assumed to be impermeable and subject to a uniform heat flux. A stationary and parallel Darcy’s flow with a non-uniform radial velocity profile is taken as a basic state. Stability to small-amplitude perturbations is investigated by adopting the method of normal modes. It is proved that buoyancy-opposed mixed convection is linearly unstable, for every value of the Darcy–Rayleigh number, associated with the wall heat flux, and for every mass flow rate parametrised by the Péclet number. Axially invariant perturbation modes and general three-dimensional modes are investigated. The stability analysis of the former modes is carried out analytically, while general three-dimensional modes are studied numerically. An asymptotic analytical solution is found, suitable for three-dimensional modes with sufficiently small wave number and/or Péclet number. The general conclusion is that the onset of instability selects the axially invariant modes. Among them, the radially invariant and azimuthally invariant mode turns out to be the most unstable for all possible buoyancy-opposed flows.     

Solution of the laminar duct flow problem by a boundary integral equation method

Summary A boundary integral equation method is proposed for approximate numerical and exact analytical solutions to fully developed incompressible laminar flow in straight ducts of multiply or simply connected cross-section. It is based on a direct reduction of the problem to the solution of a singular integral equation for the vorticity field in the cross section of the duct. For the numerical solution of the singular integral equation, a simple discretization of it along the cross-section boundary is used. It leads to satisfactory rapid convergency and to accurate results. The concept of hydrodynamic moment of inertia is introduced in order to easily calculate the flow rate, the main velocity, and the fRe-factor. As an example, the exact analytical and, comparatively, the approximate numerical solution of the problem of a circular pipe with two circular rods are presented. In the literature, this is the first non-trivial exact analytical solution of the problem for triply connected cross section domains. The solution to the Saint-Venant torsion problem, as a special case of the laminar duct-flow problem, is herein entirely incorporated.     

Chemically reactive hydromagnetic flow of a second-grade fluid in a semi-porous channel

An analysis of a second-grade fluid in a semi-porous channel in the presence of a chemical reaction is carried out to study the effects of mass transfer and magnetohydrodynamics. The upper wall of the channel is porous, while the lower wall is impermeable. The basic governing flow equations are transformed into a set of nonlinear ordinary differential equations by means of a similarity transformation. An approximate analytical solution of nonlinear differential equations is constructed by using the homotopy analysis method. The features of the flow and concentration fields are analyzed for various problem parameters. Numerical values of the skin friction coefficient and the rate of mass transfer at the wall are found.     

Comparative study on performance of a zigzag printed circuit heat exchanger with various channel shapes and configurations

Comparative study has been performed with various channel cross-sectional shapes and channel configurations of a zigzag printed circuit heat exchanger (PCHE), which has been considered as a heat exchanging device for the gas turbine based generation systems. Three-dimensional Reynolds-averaged Navier–Stokes equations and heat transfer equations are solved to analyze conjugate heat transfer in the zigzag channels. The shear stress transport model with a low Reynolds number wall treatment is used as a turbulence closure. The global Nusselt number, Colburn j-factor, effectiveness, and friction factor are used to estimate the thermal–hydraulic performance of the PCHE. Four different shapes of channel cross section (semicircular, rectangular, trapezoidal, and circular) and four different channel configurations are tested to determine their effects on thermal–hydraulic performance. The rectangular channel shows the best thermal performance but the worst hydraulic performance, while the circular channel shows the worst thermal performance. The Colburn j-factor and friction factor are found to be inversely proportional to the Reynolds number in cold channels, while the effectiveness and global Nusselt number are proportional to the Reynolds number.     

Flow with heat transfer in the presence of phase transitions on the wall of an annular channel

A solution is presented to the problem of the laminar flow of a heat-transfer vapor in an annular channel of constant cross section in the presence of a liquid film on the outer wall and heating through the inner wall of the channel. The results of test calculations for water vapor are given. The obtained results are analyzed, and it is shown that they make it possible to establish a number of characteristic features of the process, in particular, the possible existence of a self-similar solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 143–146, May–June, 1981.     

An exact analytical solution for creeping Dean flow of Bingham plastics through curved rectangular ducts

In this paper, an exact analytical solution for creeping flow of Bingham plastic fluid passing through curved rectangular ducts is presented for the first time. The closed form of axial velocity distribution, flow resistance ratio, and wall shear stress are derived using bounded Fourier transformation. An extensive investigation on mutual effects of Hedstrom number, curvature ratio, and aspect ratio is conducted. The results indicate that a drag reduction is caused in the flow field by increasing the Hedstrom number. It is shown that unlike the Newtonian creeping Dean flow, the critical aspect ratio (an aspect ratio in which the flow resistance ratio is independent from curvature ratio) does not exist at large enough Hedstrom numbers. Analytical solution also indicated that as Hedstrom number is increased, the value of Poiseuille number is enhanced, and unlike the Newtonian flows, the value of Poiseuille number is not zero at edges of cross section.     

Three-dimensional laminar flow and heat transfer in the entrance region of trapezoidal ducts

The laminar convective flow and heat transfer in a duct with a trapezoidal cross-sectional area are studied numerically. The governing equations are solved numerically by a finite volume formulation in complex three-dimensional geometries using co-located variables and Cartesian velocity components. Details of the numerical method are presented. The accuracy of the method was also established by comparing the calculated results with the analytical and numerical results available in the open literature. The Nusselt numbers are obtained for the boundary condition of a uniform wall temperature whereas the friction factors are calculated for no-slip conditions at the walls. The asymptotic values of the Nusselt numbers, friction factors. incremental pressure drops, axial velocity and momentum rate and kinetic energy correction factors approach the available fully developed values. Various geometrical dimensions of the cross-section are considered.