Poroz ortamlarda ısı geçişi

Abstract

Bu tez çalışmasında genel olarak poroz ortamdaki ısı geçişi incelenmektedir. Poroz yataklardaki akis, katı cidar direnci, yüksek hızlarda atalet kayıpları, duvar civarı porozite değişimi ve dispersiyon veya gözenek karışımı seklindeki ısı farklılığı gibi Darcy kanununun ihmal ettiği etkileri gerektirir. Bu etkiler önemli ölçüde sınır tabakadaki hız ve sıcaklık prof iller i ni değiştirmektedir. Sınır ve atalet etkileri, sınır tabakadaki momentum geçişini yavaşlatarak ısı geçişini azaltırlar. Homojen olmayan bir ortama sebep olan porozite değişimi duvar civarında geçiş oranını arttıran büyük bir yerel hız değişimine sebep olur. Dispersiyon etkisi ise sıcaklık gradyenini arttırarak ısı geçiş oranını arttırmaktadır. Poroz ortamdaki dispersiyon etkisinden dolayı enerji denklemine ilave edilen terimin içerdiği dispersiyon difüzitesinin, akım yönündeki hız bileşeni ile doğru orantılı olduğu kabul edilmektedir. Dispersiyon katsayısı olarak adlandırılan orantı sabiti ise problemlerin çözümündeki karmaşıklığı gidermek için sabit olarak alınmaktadır. Bununla birlikte poroz yatakların geometrileri gözönüne alındığı takdirde bu kabul geçerli olmamaktadır. Poroz ortamdaki ısı geçişinin hesaplanmasında efektif ısıl iletkenliğinin nasıl alınacağı önemlidir. Kullanılan katı -akışkan sistemi için uygun ampirik ifadeyi seçmek gerekir. Aksi takdirde bulunan sonuç, gerçek değerin çok üstünde olabilir. Bu çalışmada dispersiyon için teorik bir model kurulmuştur. Değişik geometriler için literatürde bulunan ısıl iletkenlik ifadeleri bir araya getirilip karşılaştırılması yapılmaktadır. Ayrıca eşit sıcaklıktaki paralel plakalar ile sınırlı bir poroz kanal boyunca laminer akıştaki ısı geçişi denklemleri verilip sonuçları irdelenmektedir.

Heat transfer in porous media has been attracting the attention of an increasingly large number of investigators in recent years. This increased use of porous media has made it essential to have a better understanding of the associated transport processes. Most analytical studies deal primarily with the mathematical formulation based on Darcy "s law which neglects the effects of a solid boundary, inertial forces and variable porosity on flow through porous media. In many applications, for example, packed-bed catalytic reactors, the porosity is variable; therefore it is important to consider these boundary, inertia and variable porosity effects. The non-Darcian effects which include the no- slip wall and inertia effects, decrease the flow and heat transfer rate, while the nonhomogenei ty effect enhances the heat transfer. For packed spheres, the nonhomogenei t y in permeability due to packing of spheres near the solid wall results in a strong flow-channeling effect that significantly increases the heat transfer. The other effect in porous media heat transfer is dispersion which causes an increase in the heat transfer. The precence of an external solid boundary leads to a momentum boundary layer concept for flow in the porous medium. Models which are established to consider the flow in porous media incorporate these effects introduced above through the use of volume averaging. The volume-averaging process integrates the general equations of motion over a small pore volume which contains both fluid solid phases. These equations are then combined with empirical relations for the permeability, inertial coefficient and dispersion diffusivity. The volume-averaged equations then include both global effects such as confining boundaries and local pore effects such as dispersion. The analysis of flow and heat transfer is usually based on the transport equations resulting VIII from the differantial balance laws. In flow through a porous medium the pressure drop caused by the frictional drag is directly proportional to velocity for low speed flow. This is the familiar Darcy 's law which relates the pressure drop and velocity in an unbounded porous medium. Darcy 's law can not account for the no-slip boundary condition at the interface between the porous media and the solid wall. The modification of the velocity profile in the near-wall region due to the no-slip boundary condition may have noticeable effects on the heat transfer. Furthermore at higher velocities inertia force is no longer negligible in comparision to the viscous force. In this case which inertial effects are prevalent, the thermal dispersion effect is also expected to become important. Consequently because of these effects Darcy 's law is no longer valid. Momentum Equation The continuity and momentum equations are derived by volume-averaging the Navier-St okas equations and equating the additional terms to empirical relations. The steady governing equations are 2 £ < u >. V. < u > = - < u > - p. C. J< u )? K > !<, U.> - V. < P > + p. g + : V. < u > where u is the average velocity. In the above equation, fj andp are the fluid density and viscosity, £ the porosity and K and C the permeability and inertial coefficient. The porosity s is assumed to vary exponentially with the distance y from the wall s =? s C 1 + a.C- b. y /d 3 3 t» where s is the free-stream value, a and b are empirical constants which depend on the packing of the spheres near the solid boundary and d is the sphere diameter. IX Energy Equation The porous medium is considered homogeneous and isotropic and is saturated with a fluid which is in local thermodynamic equilibrium with the solid matrix. The thermophysical properties of the fluid are assumed to be constant. Employing these assumptions, the homogeneous energy equation can be written as u >. V. < T > = 7.Cût.7.< T > } 6» where a is the effective thermal diffusivity, which is assumed to have two components. Hence "usivity is defined as the effective molecular is the thermal diffusivity due to the transverse thermal dispersion. Following the linear model proposed by Plumb, the dispersion diffusivity is assumed to be proportional to the streamwise velocity component, that is ot = j-".u.d, where y is a constant which may be expected to vary from approximately 1/7 to 1/3. In most packed-sphere beds analysis the dispersion coefficient, y, is assumed a value of 0.1. However, since the geometry of the beds has not been taken into account this constant coefficient assumption may not be valid. For this reason in the packed-sphere bed a theoretical model of the transverse dispersion is established. Since there are two regions which have different bed geometries, the transverse dispersion is described separetely in as the core region and the wall region. In the core regi on, the dispersion coefficient varies slightly with the porosity variations and is practically a constant. The results on the wall region shows that the dispersion coefficient can be expressed as y - mCy/d Z> The dispersion coefficient in the care region as well as in the wall region are matched by introducing a wall function which appears the form of y = y\.CI - exp C- B.Cy/dD2 3 3 It is shown that the dispersion coefficient is only a function of geometry and is independent of the flow. When the dispersion effects are important, the temperature gradient is greatly increased in a v&ry small region near the wall. Therefore the heat transfer is increased due to the dispersion effect. However, in most parts of the thermal boundary layer, the temperature gradient is decreased due to better mixing by the mechanical dispersion effect. This phenomenon is very similar to that for a turbulent flow. The results for the high permeability material show a higher heat transfer rate due to dispersion. The wall -channeling and the dispersion effects tend to enhance the heat transfer, while the wall and inertia effects decrease the heat transfer. Whether the heat transfer will be increased or decreased as compared to the Darcy flow depends on the competition among these mechanisms. Both the no-slip wall and nonhomogenei ty effects are more pronounced near the leading edge and decrease with increasing distance downstream. The nonhomogeni ty effect tends to increase heat transfer while the no-slip effect decreases it. At high Grashof numbers, both inertia and dispersion effect become important for natural convection. The inertia effect decreases the heat transfer while the dispersion effect increases it. The no-slip wall effect is negligible for low-porosity media. However, for small apparatus with high permeability, this effect may not be neglected. The effective thermal conductivity is one of the parameter to describe the dispersion effect. Like the effective thermal diffusivity, the effective thermal conduct! vi ty, k, can be written as = k + k o D where k is the stagnant thermal conductivity and k is the dispersion conductivity. The stagnant conductivity, k.depends on the porosity and conductivities of the fluid and solid. Many investigators have always encountered problems in predicting accurately the stagnant thermal conductivity of the working medium. Therefore empiricial correlations are employed to present the stagnant thermal conductivity of the fluid-saturated porous bed. But it is very difficult to choose one Xi or the other correlation for any particular application. In this thesis, heat transfer in porous media is generally considered. For boundary layer flow and heat transfer in porous media, momentum and energy equations are defined and the effects of no- slip wall, inertia force and nonhomogenei ty are discussed in Chapter II. The other effect, dispersion which increse heat transfer is considered in Chapter III. In this chapter a theoretical model for transverse dispersion in packed-sphere beds is presented. In Chapter IV, the effective thermal conducti vi ty is defined and empirical correlations for the effective stagnant conductivity are given. Laminar flow through a porous channel bounded by two parallel plates maintained at a constant and equal temperature is considered in Chapter V. In the last part of this study the results for the effects on the heat transfer in porous media are explained with the figures.