A two-phase model for laminar film condensation from steam-air mixtures in vertical parallel-plate channels

Detailed results are presented for laminar film condensation from steam-air mixtures flowing downward in vertical flat-plate channels. The mixture flow is laminar and saturation conditions prevail at the inlet. A fully coupled implicit numerical approach is used that achieves excellent convergence behavior, even for high inlet gas mass fractions. The detailed results include velocity, temperature, and gas mass fraction profiles, as well as axial variations of film thickness, pressure gradient and Nusselt number. The effects of a wide range of changes in the four independent variables (the inlet-to-wall temperature difference and the inlet values of gas concentration, Reynolds number, and pressure) on the film thickness, axial pressure gradient, and the local and average Nusselt numbers are carefully examined. It was found that increases in inlet concentration of noncondensable gas caused significant decreases in the film thickness, local Nusselt number, and axial pressure gradient. An analytical solution for the film thickness and velocity field at the end of condensation path was developed and shown to be the asymptotic value of the numerical results for large distances along the channel.     

An analytical investigation into filmwise condensation on a horizontal tube in a porous medium with suction at the tube surface

An analytical investigation is performed into the problem of steady filmwise condensation flow over the outside surface of a horizontal tube embedded in a porous medium with suction at the tube surface. As in classical film condensation problems, an assumption is made that the condensate and vapor layers meet at a common boundary rather than being separated by an intermediary two-phase zone. Furthermore, it is assumed that the condensate film has constant properties and conforms to Darcy’s law within the porous medium. By introducing an effective suction function to represent the effect of the wall suction on the thickness of the liquid film, both the local condensate film thickness and the local Nusselt number are derived using a simple numerical shooting method. The analytical results indicate that the mean Nusselt number depends on the Darcy number, the Jakob number, the Rayleigh number and the suction parameter. Furthermore, it is found that the local Nusselt number has a maximum value at the upper surface of the horizontal tube and reduces toward zero at the lower surface as a result of the finite thickness of the condensate layer.     

Numerical study of condensing a small concentration of vapour inside a vertical tube

The purpose of this study is to analyse the combined heat and mass transfer of liquid film condensation from a small steam–air mixtures flowing downward along a vertical tube. Both liquid and gas stream are approached by two coupled laminar boundary layer. An implicit finite difference method is employed to solve the coupled governing equations for liquid film and gas flow together with the interfacial matching conditions. The effects of a wide range of changes of three independent variables (inlet pressure, inlet Reynolds number and wall temperature) on the concentration at exit tube, local Nusselt and Sherwood numbers, film thickness, accumulated condensate rate and temperature are carefully examined. The numerical results indicate that in the case of condensing a small concentration of vapours from a mixture, the resistance to heat and mass transfer by non-condensable gas becomes very intense. The comparisons of average Nusselt number and local condensate heat transfer coefficient with the literature results are in good agreement.     

Natural convection on one side of a vertical wall embedded in a Brinkman-porous medium coupled with film condensation on the other side

A theoretical study of conjugate natural convection and film condensation in porous media is reported. The natural convection phenomenon takes place along one side of a vertical impermeable wall and the condensation phenomenon along the other side. This wall constitutes the interface between two spaces filled with fluid-saturated porous media. The flow in both porous spaces is modelled on the basis of the Brinkman-modified Darcy momentum equation which satisfies the condition of zero velocity on a solid boundary. The temperature and flow fields in the porous medium are completely determined in the natural convection side as well as in the condensation side of the wall. In addition, the dependence of the wall heat flux and temperature distributions on height and on a number of dimensionless groups relevant to the problem is thoroughly documented. Important results pertinent to the impact of the problem parameters on the overall heat leak from the condensation space to the natural convection space are also reported. These results are presented with the help of the Nusselt number. Finally, the effect of the wall thermal resistance on the heat and fluid flow characteristics of the system is determined.     

R410A condensation inside a commercial brazed plate heat exchanger

This paper presents the heat transfer coefficients and the pressure drop measured during HFC-410A condensation inside a commercial brazed plate heat exchanger: the effects of saturation temperature, refrigerant mass flux and vapour super-heating are investigated. The heat transfer coefficients show weak sensitivity to saturation temperature and great sensitivity to refrigerant mass flux and vapour super-heating. At low refrigerant mass flux (<20 kg/m² s) the saturated vapour condensation heat transfer coefficients are not dependent on mass flux and are well predicted by Nusselt [W. Nusselt, Die oberflachenkondensation des wasserdampfes, Energy 60 (1916) 541–546, 569–575] analysis for vertical surface: the condensation process is gravity controlled. For higher refrigerant mass flux (>20 kg/m²s) the saturated vapour condensation heat transfer coefficients depend on mass flux and are well predicted by Akers et al. [W.W. Akers, H.A. Deans, O.K. Crosser, Condensing heat transfer within horizontal tubes, Chem. Eng. Prog. Symp. Series 55 (1959) 171–176] equation: forced convection condensation occurs. In the forced convection condensation region the heat transfer coefficients show a 30% increase for a doubling of the refrigerant mass flux. The condensation heat transfer coefficients of super-heated vapour are 8–10% higher than those of saturated vapour and are well predicted by Webb [R.L. Webb, Convective condensation of superheated vapor, ASME J. Heat Transfer 120 (1998) 418–421] model. A simple linear equation based on the kinetic energy per unit volume of the refrigerant flow is proposed for the computation of the frictional pressure drop.     

Experimental study on condensation heat transfer in vertical minichannels for new refrigerant R1234ze(E) versus R134a and R236fa

Experimental condensation heat transfer data for the new refrigerant R1234ze(E), trans-1,3,3,3-tetrafluoropropene, are presented and compared with refrigerants R134a and R236fa for a vertically aligned, aluminum multi-port tube. Local condensation heat transfer measurements with such a multi-microchannel test section are very challenging due to the large uncertainties related to the heat flux estimation. Presently, a new experimental test facility was designed with a test section to directly measure the wall temperature along a vertically aligned aluminum multi-port tube with rectangular channels of 1.45 mm hydraulic diameter. Then, a new data reduction process was developed to compute the local condensation heat transfer coefficients accounting for the non-uniform distribution of the local heat flux along the channels. The condensation heat transfer coefficients showed the expected decrease as the vapor quality decreased (1.0-0.0) during the condensation process, as the mass velocity decreased (260-50 kg m⁻² s⁻¹) and as the saturation temperature increased (25-70 °C). However, the heat transfer coefficients were not affected by the condensing heat flux (1-62 kW m⁻²) or by the entrance conditions within the tested range. It was found that the heat transfer performance of R1234ze(E) was about 15-25% lower than for R134a but relatively similar to R236fa. The experimental data were then compared with leading prediction methods from the literature for horizontal channels. In general, the agreement was poor, over-predicting the high Nusselt number data and under-predicting the low Nusselt number data, but capturing the mid-range quite well. A modified correlation was developed and yielded a good agreement with the current database for all three fluids over a wide range of operating conditions.     

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.     

Effect of vapor condensation on forced convection heat transfer of moistened gas

The forced convection heat transfer with water vapor condensation is studied both theoretically and experimentally when wet flue gas passes downwards through a bank of horizontal tubes. Extraordinarily, discussions are concentrated on the effect of water vapor condensation on forced convection heat transfer. In the experiments, the air–steam mixture is used to simulate the flue gas of a natural gas fired boiler, and the vapor mass fraction ranges from 3.2 to 12.8%. By theoretical analysis, a new dimensionless number defined as augmentation factor is derived to account for the effect of condensation of relatively small amount of water vapor on convection heat transfer, and a consequent correlation is proposed based on the experimental data to describe the combined convection–condensation heat transfer. Good agreement can be found between the values of the Nusselt number obtained from the experiments and calculated by the correlation. The maximum deviation is within ±6%. The experimental results also shows that the convection–condensation heat transfer coefficient increases with Reynolds number and bulk vapor mass fraction, and is 1∼3.5 times that of the forced convection without condensation.     

Entropy generation in non-Newtonian fluids due to heat and mass transfer in the entrance region of ducts

A new solution for the Graetz problem (hydrodynamically developed forced convection in isothermal ducts) extended to power-law fluids and mass transfer with phase change at the walls is presented. The temperature and concentration spatial distributions in the corresponding entrance regions are obtained for two geometries (parallel-plates duct and circular pipe) in terms of appropriate dimensionless parameters. They are used to illustrate the effects of the fluid nature on the velocity, temperature and concentration distributions, on the axial evolution of the sensible and latent Nusselt numbers as well as on the local entropy generation rate due to velocity, temperature and concentration gradients.     

Mass transfer enhancement by capillary waves at a liquid–vapour interface

We present experimental results showing that large amplitude capillary waves at a liquid–vapour interface substantially enhance the interfacial heat and mass transfer. The experiments have been conducted in a circular cylinder that is partially filled with a wetting liquid of low boiling point temperature and pressurized by its vapour. The interfacial capillary waves are sub-harmonically excited by oscillating the circular cylinder at 50 Hz with forcing amplitude A in the direction normal to the liquid surface. The upper part of the test cell containing the vapour is heated to a temperature slightly below the boiling point temperature at the operating pressure. When the interface is at rest, the pressure decrease due to condensation is small. However, in the presence of interfacial capillary waves the rate of pressure decrease is substantial. The results show that the vapour condensation rate with respect to the diffusive vapour flux at an undisturbed interface, which is a Nusselt number, increases with the square of the wave amplitude that is proportional to the forcing amplitude. A model is developed that expresses the pressure variation in terms of Jacob number, the temperature gradient in the liquid at the interface and the capillary wave motion. This model allows extrapolation of the results to other fluids and configurations.     

Fluid flow and heat transfer characteristics for a square prism (blockage ratio = 0.1) placed inside a wind tunnel

Experimental investigations in fluid flow and heat transfer have been carried out to study the effect of wall proximity due to flow separation around a square prism at Reynolds number 2.6 × 10⁴, blockage ratio 0.1, different height-ratios and various angles of attack. The static pressure distribution has been measured on all faces of the square prism. The results have been presented in the form of pressure coefficient, drag coefficient for various height-ratios. The pressure distribution shows positive values on the front face whereas on the rear face negative values of the pressure coefficient have been observed. The positive pressure coefficient for different height-ratios does not vary too much but the negative values of pressure coefficient are higher for all points on the surface as the bluff body approaches towards the upper wall of the wind tunnel. The drag coefficient decreases with the increase in angle of attack as the height-ratio decreases. The maximum value of drag coefficient has been observed at an angle of attack nearly 50° for the square prism at all height-ratios. The heat transfer experiments have been carried out under constant heat flux condition. Heat transfer coefficient are determined from the measured wall temperature and ambient temperature and presented in the form of Nusselt number. Both local and average Nusselt numbers have been presented for various height-ratios. The variation of local Nusselt number has been shown with non-dimensional distance for different angles of attack. The variation of average Nusselt number has also been shown with different angles of attack. The local as well as average Nusselt number decreases as the height-ratio decreases for all non-dimensional distance and angle of attack, respectively, for the square prism. The average Nusselt number for the square prism varies with the angle of attack. But there is no definite angle of attack at which the value of average Nusselt number is either maximum or minimum.     

Radiation effect on mixed convection over an isothermal cone in porous media

The radiation effect on the mixed convection flow of an optically dense viscous fluid adjacent to an isothermal cone embedded in a saturated porous medium with Rosseland diffusion approximation is numerically investigated. The entire regime of the mixed convection is included, as the mixed convection parameter of χ varies from 0 (pure free convection) to 1 (pure forced convection). The transformed nonlinear system of equations is solved by using an implicit finite difference method. Numerical results are given for the dimensionless temperature profiles and the local Nusselt number for various values of the mixed convection parameter χ, the cone angle parameter m, the radiation-conduction parameter R _d and the surface temperature parameter H. The local Nusselt number decreases initially, reaches a minimum in the intermediate value of χ and then increases gradually. It is apparent that increasing the cone angle parameter m enhances the local Nusselt number. The local Nusselt number is significantly increased for the large values of the radiation-conduction parameter R _d and the surface temperature parameter H, i.e., radiation effect becomes pronounced. Received on 25 October 1999     

Heat transfer characteristics of an annular turbulent impinging jet with a confined wall measured by thermosensitive liquid crystal

The present paper describes the heat transfer characteristics of an annular turbulent impinging jet with a confined wall. The local temperature distribution on the impingement surface was measured using a thermosensitive liquid crystal sheet and an image processor. The net heat flux was evaluated by considering the heat conduction in the heated substrate and the thermal radiation between an upper confining insulated wall and an impingement surface. Distributions of the temperature and Nusselt number on the impingement surface were captured in two-dimensional maps. Effects of the diameter ratio of the annular nozzle, the space between nozzle and impingement surface and the Reynolds number on radial distributions of the local Nusselt number were examined. Experimental formulas of the local Nusselt number were obtained in power-law expressions of r/r_p for the major and minor flow regions.     

Effect of an Inserted Porous Layer Located at a Wall of a Parallel Plate Channel on Forced Convection Heat Transfer

A theoretical study is performed on heat and fluid flow in partially porous medium filled parallel plate channel. A uniform symmetrical heat flux is imposed onto the boundaries of the channel partially filled with porous medium. The dimensional forms of the governing equations are solved numerically for different permeability and effective thermal conductivity ratios. Then, the governing equations are made dimensionless and solved analytically. The results of two approaches are compared and an excellent agreement is observed, indicating correctness of the both solutions. An overall Nusselt number is defined based on overall thermal conductivity and difference between the average temperature of walls and mean temperature to compare heat transfer in different channels with different porous layer thickness, Darcy number, and thermal conductivity ratio. Moreover, individual Nusselt numbers for upper and lower walls are also defined and obtained. The obtained results show that the maximum overall Nusselt number is achieved for thermal conductivity ratio of 1. At specific values of Darcy number and thermal conductivity ratio, individual Nusselt numbers approach to infinity since the value of wall temperatures approaches to mean temperature.     

Forced convection with laminar pulsating flow in a tube

The heat transfer model of laminar pulsating flow in a tube in rolling motion is established. The correlations of velocity, temperature and Nusselt number are obtained. The effects of several parameters on Nusselt number are investigated. The theoretical results are consistent with experimental data. Then the results are evaluated with Nield and Kuznetsov’s results. It is found that Nield and Kuznetsov’s results are not applicable for the laminar pulsating flow in nuclear power systems in ocean environments.     

Numerical investigation of natural convection inside an inclined parallel‐walled channel

Steady two‐dimensional natural convection in an inclined parallel‐walled channel was investigated numerically. The full elliptic forms of conservation equations were solved together and the velocity vectors, temperature contours and local and average Nusselt number distribution were obtained. The comparisons of local and average Nusselt number with published experimental and numerical results indicate very good agreement. Results are presented for a single aspect ratio, L/b=24, over the range of Rayleigh number of 3–1000 and angle of inclination 0–90°. The results indicate that the overall channel average Nusselt number is reduced as the inclination angle is increased. Significant reductions in the overall Nusselt number are exhibited at high angle of channel inclination. Copyright © 2005 John Wiley & Sons, Ltd.     

Hydrodynamic and heat transfer characteristics of laminar flow past a parabolic cylinder with constant heat flux

 Steady, two-dimensional, symmetric, laminar and incompressible flow past parabolic bodies in a uniform stream with constant heat flux is investigated numerically. The full Navier–Stokes and energy equations in parabolic coordinates with stream function, vorticity and temperature as dependent variables were solved. These equations were solved using a second order accurate finite difference scheme on a non-uniform grid. The leading edge region was part of the solution domain. Wide range of Reynolds number (based on the nose radius of curvature) was covered for different values of Prandtl number. The flow past a semi-infinite flat plate was obtained when Reynolds number is set equal to zero. Results are presented for pressure and temperature distributions. Also local and average skin friction and Nusselt number distributions are presented. The effect of both Reynolds number and Prandtl number on the local and average Nusselt number is also presented. Received on 5 July 2000     

Numerical analysis of a symmetrically heated vertical channel with obstruction

 In this paper, a numerical investigation of laminar natural convection flows in a vertical channel with obstructions is carried out. The main purpose was to analyze the effects of the locations of symmetric obstructions. The computations were performed in a two-dimensional domain and a symmetric uniform wall temperature has been taken as thermal boundary condition. The governing equations were solved using a control volume method and the SIMPLER algorithm for the velocity–pressure coupling was employed. The profiles of the local Nusselt number were given for three different locations of the obstructions. The variation of the average Nusselt number and inlet flow rate versus the modified Rayleigh number were investigated. The results demonstrated that the average Nusselt number decreases as the distance of the obstructions from the inlet increases. Received on 17 January 2000     

The effect of thermal dispersion on free convection film condensation on a vertical plate with a thin porous layer

An analytical solution to the problem of condensation by natural convection over a thin porous substrate attached to a cooled impermeable surface has been conducted to determine the velocity and temperature profiles within the porous layer, the dimensionless thickness film and the local Nusselt number. In the porous region, the Darcy–Brinkman–Forchheimer (DBF) model describes the flow and the thermal dispersion is taken into account in the energy equation. The classical boundary layer equations without inertia and enthalpyterms are used in the condensate region. It is found that due to the thermal dispersion effect, the increasing of heat transfer is significant. The comparison of the DBF model and the Darcy–Brinkman (DB) one is carried out.     

Transient heat transfer from a solid sphere translating at low reynolds number

In this paper the heat transfer from a solid spherical particle translating at low Reynolds number is analytically examined for the limit of vanishing Peclet number. The temperature distribution within the solid sphere is treated as fully transient while the fluid phase is considered to be quasisteady. The temperature field in the dispersed phase is obtained by a singular perturbation expansion of the Acrivos-Taylor type. The time-dependence in the solid phase is handled by means of the Laplace transform. This approach allows the temperature and heat flux continuity conditions at the solid-liquid interface to be exactly satisfied. The solution in the time domain appears in the form of infinite series which have associated with them a set of eigenvalues for every order of the perturbation expansion. The Nusselt number to orderPe, however, depends only on the leading order eigenvalues. An analytical limit of the Nusselt number for large values of time is also obtained.