Unsteady heat transfer from a rotating disk to fluids with low Prandtl numbers

Summary An analysis is presented concerning unsteady heat transfer from a rotating disk to a low Prandtl number fluid under the condition of a step change in surface temperature with time. Entire time history results for the surface heat flux are given for Prandtl numbers up to 0.04, and these results are obtained by means of a first-order perturbation about the solution for zero Prandtl number. Steady-state heat transfer predicted by this method agrees almost precisely with exact values.     

Heat and mass transfer of a water film falling down a tilted plate with radiant heating and evaporation

This paper has concerned the heat and mass transfer of a water film falling down a tilted plate with radiant heating and water evaporation. A cluster of physical models was developed for evaluating the properties of heat and mass transfer. A fully implicit control-volume finite-difference procedure was used to solve the coupling equations. The effects of various parameters on heat and mass transfer were investigated. The results showed that the mass fraction of water vapor in ambient atmosphere and the flow turbulence played key roles in the heat and mass transfer. The ambient atmospheric temperature dramatically affected the sensible heat flux. However its effect on the latent heat flux is negligibly small. The magnitude of solar incident flux had an intense influence on the water film temperature. Received on 29 January 1998     

Experimental and theoretical investigations of falling film evaporation

In this study, a mathematical model was developed for falling film evaporation in vacuum using heat transfer relations. An experimental device was designed. experimental set-up which was used was equipped with a triangular weir distribution device and it had the ability to record data up to 3?m. Experiments were performed in a single-effect process with sucrose–water solution varying from 3 to 20% concentration rate of sucrose and we used a vertical tube evaporator with the dimensions of laboratory scale. The model that was developed considers convection, shear stress, viscosity and conjugate heat transfer while most of the previous works ignored these factors. The main factors influencing the heat transfer mechanism performance of the unit were investigated and analyzed. We concluded that the experimental studies are verified by the developed model. Furthermore, it was also concluded that, the heat transfer is affected by the mass flow rate, sucrose concentration rate in solution, film thickness and pressure.     

Heat transfer in non-Newtonian falling liquid film on a horizontal circular cylinder

 This study aims to investigate numerically the laminar flow and heat transfer in a pseudoplastic non-Newtonian falling liquid film on a horizontal cylinder for the constant heat flux and isothermal boundary conditions. The inertia terms are taken into account. An implicit finite difference method is carried out to solve the governing boundary layer equations. The effects of operational parameters on the hydrodynamic and heat transfer characteristics are examined and discussed in detail. The results presented show that the local and average Nusselt numbers varies significantly as a function of the concentration of aqueous carboxymethylcellulose (CMC) solutions and the cylinder diameter. Higher concentration of aqueous CMC solutions generate larger heat transfer coefficients. Finally, a comparison with the experimental and numerical results available in the literature for Newtonian fluids shows clearly that the present analysis is reasonably accurate. Received on 29 March 2001 / Published online: 29 November 2001     

Energy separation of gases with low and high Prandtl numbers

The process of energy separation of a gas with the Prandtl number which is not equal to unity is investigated. The gas flows through a heat exchanger consisting of two coaxial axisymmetric pipes with sub- and supersonic velocities. Heat exchange between the gas streams takes place as a result of the fact that the recovery factor is not equal to unity. The flow is described by one-dimensional gas dynamic equations for averaged quantities.     

Heat transfer characteristics of internally heated liquid pools at high Rayleigh numbers

Detailed numerical analysis is presented for buoyancy driven flow of a Newtonian fluid contained in a square enclosure for high Rayleigh (Ra) numbers. Natural convection is due to internal heating sources, which are assumed to be uniformly distributed within the enclosure. All walls of the cavity are maintained at constant temperature. Flow and heat transfer characteristics are investigated for a Ra number range of 10⁷ to 10¹² while Prandtl (Pr) number is taken to be 7.0. Governing equations (in primitive variables) are discretised using control volume technique based on staggered grid formulation. These equations are solved using SIMPLER algorithm of Patankar. Flow and heat transfer characteristics, streamlines, isotherms and average wall Nusselt (Nu) number, are presented for whole range of Ra number considered. Finally, present results for average wall Nu numbers are compared with experimental observations obtained from open literature. It is concluded that both results are in very good agreement, which confirmed the accuracy of the scaling used for present investigation. Received on 15 November 1999     

Experimental analysis of the local heat transfer coefficient of falling film evaporation with and without co-current air flow velocity

This paper presents the experimental results of the local heat transfer for falling film evaporation of water sheet by solving the inverse heat conduction problem. It is shown that the local heat transfer coefficients increase by increasing the air flow velocity, the film liquid flow rate or decreasing the inlet bulk film temperature. Correlations for the mean heat transfer coefficients in the absence of superimposed flow for the stagnation region, the thermally developed region and the bottom of the heated cylinder are proposed.     

A second-order approximation to natural convection for large Rayleigh numbers and small Prandtl numbers

The problem under investigation is that of fluid flow within an enclosed rectangular cavity. It is assumed that one wall is maintained at a constant temperature T₁ (hot wall) and the other wall is maintained at a constant temperature T₀ (cold wall). At the remaining walls, two separate cases are studied. In the first, an adiabatic boundary condition is assumed. That is, the normal derivative of the temperature function is assumed to be 0. In the second, it is assumed the temperature varies linearly from T₀ to T₁. The purpose of this paper is the application of a second order numerical technique to the problem of fluid flow within a heated closed cavity. The method is a modification of a method developed by Shay¹ and applied to the driven cavity problem. In order to test the viability of this technique, it was decided to extend the technique to the problem of natural convection in a square. Jones² proposed that this problem is suitable for testing techniques that may be applied to a wide range of practical problems such as reactor insulation, cooling of radioactive waste containers, solar energy collection and others.³ The technique makes use of second-order finite difference approximations to all derivatives in the governing equations. Furthermore, second-order approximations are also used to determine boundary vorticities and, when the adiabatic boundary condition is used, for the boundary temperatures as well. In some works, where second-order approximations are used at interior points, second-order boundary approximations have been sacrificed in favour of a more stable, but first-order boundary approximation. The current approximations are generated by writing the unknown value of a function at a given interior node as a linear combination of unknown function values at all of the neighbouring nodes. Then the function values at these neighbouring nodes are expanded in a Taylor series about the given node. Through appropriate regrouping of terms and the use of the equations to the solved, constraints are imposed on the coefficients of the linear combination to yield a second-order approximation. As it turns out, there are more unknowns than constraints and, as a result, we are left with some freedom in choosing coefficients. In this work this freedom was used to choose coefficients in such a way as to maximize stability of the resulting system of equations. In other words, the approximations to the governing partial differential equation are individually determined at each point dependent on the direction of flow in order to generate the best possible stability. This idea is analogous to that used in the derivation of the upwind method. However, the current method is second-order accurate where the upwind method is only first-order accurate. Thus, what is generated is an easily implemented second-order method that yields a system of equations that has proved easy to solve. The system of equations is solved via the method of successive overrelaxation. The stability of the method is shown in the convergence for a wide range of Rayleigh numbers, Prandtl numbers and mesh sizes. Level curves of the stream, vorticity and temperature functions are provided for Rayleigh numbers (Ra) as large as 100,000, Prandtl numbers (Pr) as small as 0.0001, and mesh sizes as small as 0.0125. Values of the Nusselt number have also been calculated through the use of Simpson's rule, and a second order approximation to the normal derivative of the temperature along the cold wall. Comparisons are made with other current works to aid in the verification of this methods' accuracy and also with the first-order upwind method to demonstrate superiority over the first-order method.     

Heat and mass transfer characteristics of a horizontal tube falling film absorber with small diameter tubes

This paper presents experimental results of the heat and mass transfer characteristics of a water–LiBr horizontal tube absorber made of small diameter tubes. The experimental set up includes a tube absorber, a generator, solution distribution system and cooling water system. Three different tube diameters of 15.88, 12.70 and 9.52 mm have been installed inside the absorber to investigate the effect of the tube diameter on the absorber performance. The experimental results show that the heat and mass transfer performance of the absorber increases as the tube diameter decreases. A comparison of the heat and mass transfer coefficients of the present study agree reasonable well with that of the previous studies.     

A new approach to analyze entrance region mass transfer within a falling film

A boundary layer model is developed to analyze diffusion through a laminar falling film for incomplete penetration of the dissolved gas. Regarding the rather intractable nature of the problem, Kantorovich Integral Method is chosen. Accordingly, a mass transfer boundary layer is assumed to keep growing perpendicularly to the falling film until it hits the wall. Such an approach is superior to its preceding artworks based on Higbie’s penetration theory in terms of implementation of more realistic conditions/modeling assumptions. Furthermore, unlike penetration model, this approach gives a criterion whether the diffusion is complete. Comparing the two models, boundary layer model estimates up to 2.3 % larger mass transfer coefficients. Moreover, a sensitivity analysis of liquid velocity distribution upon Sherwood number is conducted. It is found that the local velocity at gas–liquid interface is of highest dominance. Experimental data of SO₂ absorption in water reported in literature is exploited to validate the model. It is shown that boundary layer model better fits the experimental data.     

Heat transfer to evaporating falling films of water and sugar/water solution

A series of heat transfer experiments is performed with pure water and sugar/water solution films flowing down on outside surface of a smooth or fluted vertical stainless steel tube near atmospheric pressure. Data on length-averaged heat transfer coefficient are presented in a range of Prandtl number 2<Pr<20 with wall heat fluxes up to 30 kW/m². The measured heat transfer coefficient for smooth tube does not agree very well with the well-known correlated of Chun & Seban. A new corrected correlation is presented. The fluted tube shows an increased heat transfer performance as compared to the corresponding smooth tube. The increase in the overall heat transfer coefficient turned out to be slightly higher than the increase in the corresponding heat transfer area. Film thickness and spectrum of wave frequencies were measured for smooth tube with a special probe assembly.

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Wärmeübergang bei Verdampfung von Wasser- und Zuckerwasser-Rieselfilmen

Zusammenfassung Es werden Ergebnisse von Verdampfungs-experimenten mitgeteilt, die an Rieselfilmen von Wasser und Zuckerwasserlösungen gewonnen wurden, welche an der Außenseite von glatten oder geriefelten Edelstahlrohren unter Atmosphärendruck abliefen. Sie beziehen sich auf den gemittelten Wärmeübergangskoeffizienten bei Wärmestromdichten von bis zu 30 kW/m² im Prandtl-Zahlenbereich 2<Pr<20. Letztere stimmen bei Glattrohren nicht zufriedenstellend mit den aus der wohlbekannten Korrelation von Chun und Seban folgenden Werten überein, weshalb eine verbesserte Beziehung vorgeschlagen wird. Im Vergleich mit Glattrohren liefern Rohre mit Riefen ein besseres Wärmeübertragungsverhalten, auch wenn die durch Riefelung bewirkte Vergrößerung der Oberfläche berücksichtigt wird. Filmdichte und Spektrum der Wellenfrequenzen wurden an Glattrohren mit einem speziellen Meßgerät ermittelt.

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Heat transfer enhancement through liquid film evaporation into countercurrent moist air flow in a vertical plate channel

A numerical analysis has been carried out to study the heat removal process from hot channel plate through liquid film evaporation into a countercurrent air flow. The influences of the wall heat flux, the inlet Reynolds number of liquid film and the inlet Reynolds number of moist air on the transfer characteristics are investigated detailedly. The Results show that the interface latent heat transfer associated with the film vaporization causes a temperature drop of the heated plate in the entry region of air flow, which is more significant for a system with higherq _w , lowerRe _l,in or largerRe _{c, in} . The overall temperature rise of the heated wall is rather small, as compared with the case without interface latent heat transfer. In addition, the difference in results obtained by the one-dimensional and two-dimensional methods is substantial.Die numerische Untersuchung bezieht sich auf den Wärmetransportprozeß von heißen, plattenförmigen Kanalwänden durch Flüssigfilmverdampfung in gegenströmende Luft. Die Einflüsse des Wärmeflusses, der Reynolds-Zahlen, des Flüssigkeitfilms und der Feuchtluft (jeweils am Eintritt) auf das Wärmeübertragungsverhalten werden eingehend untersucht. Die Ergebnisse zeigen, daß die bei der Filmverdampfung eingespeicherte latente Wärme eine Temperaturabnahme der Heizplatte am Eintritt der Luft bewirkt, die mit den Wärmefluß und steigender Reynolds-Zahl für Feuchtluft zunimmt. Die gesamte Temperaturerhöhung der beheizten Wand ist sehr gering im Vergleich mit dem Fall ohne Latentwärmeaustausch. Darüber hinaus resultieren erhebliche Unterschiede in den Ergebnissen, je nachdem, ob eindimensionale oder zweidimensionale Methoden angewandt werden.The financial support of this study by the engineering division of the National Science Council, Taiwan, R.O.C., through the contract NSC 82-0401-E-150-049 is greatly appreciated.     

Heat transfer to non-Newtonian fluids in laminar flow through rectangular ducts

Heat transfer to non-newtonian fluids flowing laminarly through rectangular ducts is examined. The conservation equations of mass, momentum, and energy are solved numerically with the aid of a finite volume technique. The viscoelastic behavior of the fluid is represented by the Criminale-Ericksen-Filbey (CEF) constitutive equation. Secondary flows occur due to the elastic behavior of the fluid, and, consequently, heat transfer is strongly enhanced. It is observed that shear thinning yields negligible heat transfer enhancement effect, when compared with the secondary flow effect. Maximum heat transfer is shown to occur for some combinations of parameters. Thus, there are optimal combinations of aspect ratio and Reynolds numbers, which depend on the fluid's mechanical behavior. This result can be usefully explored in thermal designs of certain industrial processes.     

Convective heat and mass transfer from a falling film to a laminar gas stream

A numerical study has been made of convective heat and mass transfer from a falling film to a laminar gas stream between vertical parallel plates. The effects of gas-liquid phase coupling, variable thermophysical properties, and film vaporization have been considered. Simultaneous mass, momentum and heat transfer between liquid film and gas stream is numerically studied by solving the respective governing equations for the liquid film and gas stream together. The influences of the inlet liquid temperature and liquid flowrate on the cooling of liquid film are examined for air-water and air-ethanol systems. Results show that the heat transfer from the gas-liquid interface to the gas stream is predominantly determined by the latent heat transfer connected with film evaporation. Additionally, better liquid film cooling is noticed for the system having a higher inlet liquid temperature or a lower liquid flowrate.     

Instabilities of thermal gravitational convection and heat transfer in the Czochralski model at different Prandtl numbers

The results of the calculations of critical Grash of numbers, at which flowfield and temperature fluctuations originate in the axisymmetric and three-dimensional models of crystal growth by pulling from a melt, are presented. The salient features of the convection and heat transfer structure in the zones of stabilization and changeover of dangerous modes are studied over a wide Prandtl number range under different boundary conditions on the melt surface and compared with the experimental data.