POROUS MEDIUM. where. Q = Total discharge of the fluid. A = Cross sectional area of the porous medium = Viscosity = Density. h s.

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1 POROUS MEDIUM The phenomena of transport in porous media are encountered in many engineering disciplines. Civil engineering deals, for example, with the flow of water in aquifers, the movement of moisture through and under engineering structures, transport of pollutants in aquifers and the propagation of stresses under foundations of structures. Agricultural engineering deals, for example, with the movement of water and solutes in the root zone in the soil. Heat and mass transport in packed - bed reactor columns and drying processes are encountered in chemical engineering. Reservoir engineering deals with the flow of oil, water and gas in petroleum reservoirs. A porous medium can be defined as a material consisting of solid matrix with an interconnected void, like soil, sand, fissured rock, karstic lime stone, ceramics, bread, lungs and kidneys, aquifers from which ground water is pumped, reservoirs which yield oil and / or gas, sand filters for purifying water, packed beds in the chemical engineering industry and the root zone in agriculture are examples of porous medium domains. The definition of the porosity of the porous medium can be given as the ratio of pore volume to the total volume of a given sample of material. Permeability of the porous medium is an important property of the solid material and it is independent of the density and viscosity of the fluid. The permeability k can be defined as where Q k A Q = Total discharge of the fluid h ( ) g s A = Cross sectional area of the porous medium = Viscosity = Density 1 1

2 g = Acceleration due to gravity h s = The hydraulic gradient in the direction of the flow. The physical principles determining the behavior of the flow through porous medium must fundamentally be the same as governing the motion of viscous fluids in consists which are expressed in Navier Stokes equations. DARCY LAW In 1856 Darcy made an experimental study in analyzing the flow characteristics of sand filters and gave the Darcy law which states that the rate of flow of water through the filter bed was directly proportional to the area of the sand and to the difference h between the fluid heads at the inlet and outlet of the bed and inversely proportional to the thickness L of the bed, given as Q h CA L where C is a constant characteristic of the sand. His studies into the hydrology of fountains and his experiments on steady state unidirectional flow in a uniform medium showed proportionality between flow rate and the applied pressure difference, that is expressed as q k p x where p is the pressure gradient in the flow direction and is the dynamic x viscosity of the fluid. The coefficient k is independent of the nature of the fluid but it depends on the geometry of the medium. It has the dimension (length) and is called the permeability of the medium. p V k

3 Typical values of k for soils, in terms of the unit m are : clean gravel , clean sand , peat , stratified clay , and unweathered clay In other words, the mean filter velocity, called Darcy s velocity q, is proportional to the sum of the gradient of the pressure and the gravitational force. constant q ( p g) Later Muskat (1937) proved that the constant should be related to the permeability of the porous material and is of the form k q p where k is the permeability of the porous medium which is to be determined experimentally. BRINKMAN MODEL An alternative to Darcy s equation widely known as Brinkman s equation, proposed by Brinkman given by ( P) g u u (1) k This equation is valid when the permeability k is very high. In general the particles of the porous media are loosely packed so that k is small. Hence there exists a boundary layer very near to the surface. Beavers and Joseph examined the nature of the tangential flow in the boundary region of a permeable interface and proposed that Darcy s law be retained for the interior flow, but at the boundary, the conditions are modified so as to match the free fluid flow by a slip condition of the form du dy 1 1 ( u K 1 U) 3

4 Taylor, Richardson and Saffman (1959) further verified the above condition experimentally and theoretically. Saffman (1959) has further shown that if the limit K - 0, then the slip condition at the fluid porous interface, can be written as k du1 u1 o( k) dy 1 Since the direct effect of the flow with in a porous solid on the exterior flow is 0(k), the flow inside the porous medium can be ignored when determining the exterior flow. Brinkman proposed the extension to the Darcy s law to describe the flow near a porous surface. The extended Darcy s law is given by p 0 K The basic field equations incorporating the convective terms were investigated by Yamamoto and Yoshida (1974). FORCHHIEMER EXTENDED DARCY EQUATION Darcy s equation is linear in the velocity V. This equation holds good when the Reynolds number of the flow, based on a typical pore or particle diameter, is of order unity or smaller, as the velocity increases the transition to the non linear drag is still smooth and there is no sudden transition as this Reynolds number is increased in the range This transition is not one from laminar to turbulent flow, since at such comparatively small Reynolds number the flow in the pores still continuous to be laminar. However the breakdown of the linearity is observed to be the result of the form drag due to the solid obstacles which is comparable with the surface drag due to internal friction. Incorporating this drag the Darcy s equation is modified as p v C f k f v v k 4

5 NON DARCY LAW In many practical problems, the flow through porous media is curvilinear and the curvature of the path yields the inertia effect, so that the streamlines become more distorted and the drag increases more rapidly. Lapwood was the first to suggest for the inclusion of convective inertial term ( q. )q in the momentum equation. Subsequently many research articles have been appeared on the non-darcy model (1937, 1947, 1956, and 1967). 1 Including the usual inertia term written as ( q. ) q, the governing equation can be 1 f ( q. ) q p g q ( q) e k However, equation () does not take care of possible unsteady nature of velocity. If the is unsteady then equation () becomes 1 q 1 f [ ( q. ) q] p g q ( q) e t k (3) This equation is known as Darcy-Lapwood-Brinkman equation. For anisotropic porous medium equation (3) takes the form 1 q 1 [ ( q. ) q] p g Q ( q) f e t (4) () BOUNDARY CONDITIONS NO SLIP CONDITION We consider the case where the region y < 0 is occupied by a porous medium, and there is a boundary at y =0, with respect to the Cartesian coordinates x, y, z. If the boundary is impermeable then the velocity component must vanish at y = 0. This is called the no slip condition 5

6 FREE SURFACE BOUNDARY CONDITION The appropriate condition is that the pressure is constant along the boundary. If Darcy s law is applicable and the fluid is incompressible then the velocity gradient vanishes at y = 0. BEAVERS AND JOSEPH SLIP CONDITION If the porous medium is adjacent to clear fluid identical to that which saturates the porous medium and if there is unidirectional flow in the direction of the fluid flow then the appropriate boundary condition which is the empirical relationship, given by u f y BJ K u f u m where u f is the velocity in the fluid and u m is the seepage in the porous medium. It is because when a fluid flows over a permeable surface, it is necessary to specify some condition on the tangential component of the velocity of the free fluid at the permeable interface. In this case, there will be a migration of fluid tangential to the boundary within the permeable surface. At the permeable interface. In this case, there will be a migration of fluid tangential to the boundary within the permeable surface. That is, there will be a net tangential drag due to transfer of forward momentum across the permeable interface. The velocity inside the permeable bed will be different from the velocity of the fluid past/over the permeable bed. These two velocities are to be matched at the nominal boundary (surface) of the permeable bed. The nominal boundary of a permeable bed is defined as a smooth geometric surface with the assumption that the outermost perimeters of all surface pores of the permeable material are in this surface. Thus if the surface is filled with solid material to the level of their respective perimeters, a smooth rigid boundary of the assumed shape results. 6

7 MAGNEOTOHYDRODYNAMICS We can describe scientifically the interaction of electromagnetic fields and fluids by the proper application of the principles of the special theory of relativity. The practical application of these principles, in physical Engineering, Astrophysics, Geophysics etc., have become an important in recent years. The study of three applications to continuum is known as Magneto hydrodynamics or Magneto fluid dynamics. The study of magneto hydrodynamics (MHD) plays an important role in agriculture, engineering and petroleum industries. MHD has won practical applications, for instance, it may be used to deal with problems such as cooling of nuclear reactors by liquid sodium and induction flow water which depends on the potential difference in the fluid direction perpendicular to the motion and goes to the magnetic field. The study of magneto-hydro-dynamics (MHD) of viscous conducting fluids is playing a significant role, owing to its practical interest and abundant applications, in astro-physical and geo-physical phenomenon. Astro-Physicists and geo-physicists realized the importance of MHD in stellar and planetary processes. The main impetus to the engineering approach to the electromagnetic fluid interaction studies has come from the concept of the magneto hydro dynamics, direct conversion generator, ion propulsion study of flow problems of electrically conducting fluid, particularly of ionized gases is currently receiving considerable interest. Such studies have made for years in convection with astro-physical and geo-physical problems such as Sun spot theory, motion of the interstellar gas etc., Recently, some engineering problems need the studies of the flow of an electrically conducting fluid, in ionized gas is called plasma. Many names have been used in referring to the study of plasma phenomena. Hartman (1937) called it mercury dynamics, as he worked with mercury. Astrophysicist called it comical electro dynamics, and some called it 7

8 magneto-hydro-dynamics. Physicist and electrical enginers commonly use the term plasma dynamics. The aerodynamicists has spoken of magneto-hydrodynamics. BASIC MHD EQUATIONS When an electromagnetic body force acts on the fluid, the motion of the fluid may generate in induced electromagnetic field in the presence of the electromagnetic field and alters the field. Due to the interaction of the fluid forces and the external electromagnetic force, the velocity field in the fluid and electric and magnetic fields inside the fluid get disturbed. Now to discuss the magneto fluid dynamics, we have to combine the basic concepts of fluid dynamics and electromagnetism. The equations governing motion will be the Navier-Stokes equation of fluid and Maxwell s equations which are usually coupled and must be solved simultaneously. The electro-magnetic body force is usually non-conservative (rotational) and not derivable from a scalar potential function. Only under rare circumstances it may be approximately conservative and derivable from a scalar potential function. In such cases the electromagnetic body force alter simply the pressure, provided the boundary conditions on velocity and pressure remain the same. However this happens in exceptional cases and the force is rotational in general and does alter the flow. The Maxwell s equations in RMKS units are. D (Coulombs law) (5). B O (Absence of free magnetic poles) (6) B. E (Faraday s law) (7) t D H J (Ampere s law) (8) t Ohm s law is E q B q J (9) 8

9 The current conservation equation is. J 0 t The continuity equation is (10) q.( q) 0 (11) t The equation of motion (Navier - Stokes equation) is Dq P q F fe (1) Dt These equations hold in any frame of reference. MHD APPROXIMATIONS In addition to the non-relativistic approximation, certain additional simplifications can be made for flow which is quasi-steady (Steady of low frequency oscillatory), and in which the electric field is of the order of magnitude of the induced quantity q B. The following assumptions are made under MHD approximation. 1. q c This allows the quantity q 1 to be taken as unity and use c the non-relativistic Newtonian form of equations of motion, where c is the velocity of the light.. The electric field E is of the same order of magnitude as the induced electric field q B. This assumption allows us to assume that the induced magnetic field is magnetic field much smaller than the externally applied 3. The problems of very high frequency are not considered, so that the displacement current D t is neglected compared to J, the conduction current. Thus the equations of electromagnetic fields (10) and (11) become 9

10 E 0 (13). J 0 (14) Equations (13) and (14) are called pre-maxwell s equations for electromagnetic fields. 4. The electric energy is negligible compared to the magnetic energy. Since the displacement current and electric field energy are neglected, the main interaction is between the magnetic field and the fluid. 5. The current density vector J is approximately the same in any inertial frame. This means, using Ohm s law J ( E q B) (15) 6. The magnetic force density is represented by f e E J B (16) In comparison with magnetic term, the electric term is negligible. For this reason E is often neglected on MHD. 7. In using. D, great care must be taken. If the equation. J 0 t is used in metals (Even in alternating current problem under the MHD approximation), the e t mentioned although it is identically zero. term may be dropped and ρ e need not be Also. J 0 is valid even for high frequency phenomena as long as the fluid velocity q is very small compared to c. EQUATIONS UNDER THE MHD AOOROXIMATIONS The basic equation under the MHD approximation takes the following form H J (17) B E t (18) 10

11 . H 0 (19). J 0 (0) and the Ohm s law is J ( E q B) (1) The equation of conservation of mass is.( q) 0 () t The equation of motion is q Dt D P q J B Where Ψ is the Gravitational Potential. Magnetic diffusion equation is B t (3). B ( q B) (4) 1 Where is magnetic diffusivity (or) magnetic viscosity. e HALL EFFECT In 1879 Edwin Herbert discovered the Hall Effect while he was working for his Ph. D. The strength of the applied magnetic field is very strong, we cannot neglect the effect of Hall currents, due to gyration and drift of charged particles, the conductivity parallel to the electric field is reduced and the current is induced in the direction normal to both electric and magnetic fields. This phenomenon is known as Hall Effect. The Hall effect is the production of a voltage difference across the electric conductor, transverse to the electric current in the conductor and a magnetic field perpendicular to the current. Current consists of the movement of many small charge carriers. Moving charges experience the Lorentz force, when a magnetic field is present that is perpendicular to their motion. But when a magnetic field is applied perpendicularly, their paths between collisions are curved that the moving 11

12 charges accumulate on one face of the material. This leaves equal and opposite charges exposed on one face of the material. This leaves equal and opposite charges exposed on the other face, where there is scarcity of mobile charges. The result is an asymmetric distribution of charge density across the Hall element that is perpendicular to both the line of sight path and the applied magnetic field. The separation of charge establishes an electric field that opposes the migration of further charge, so a steady electrical potential builds up for as long as the charge is flowing. For a simple metal where there is only one type of charge carrier the Hall voltage V H is given by V H IB ned The Hall coefficient is defined as the ratio of the induced electric field to the product of the current density and the applied magnetic field. It is a characteristic of the material from which the conductor is made. The Hall coefficient is defined as R H E y J B x As a result, the Hall Effect is very useful as a means to measure either the carrier density or the magnetic field. The Hall effect can be used to measure the average drift velocity of the charge carriers by mechanically moving the Hall probe at different speeds until the Hall voltage disappears, showing that the charge carriers are now not moving with respect to the magnetic field. Other types of investigations of carrier behavior are studied in the quantum Hall Effect. The effect of Hall currents on the fluid with variable concentration has lot of applications in MHD power generators, astrophysical and meteorological studies as well. 1

13 HEAT TRANSFER A detailed knowledge of principles of heat transfer and a thorough understanding of its back mechanism is necessary to design efficient and economical heat exchangers such as boilers, condensers and combustion engines etc. Design problems in nuclear reactors also demand the knowledge of temperature distribution and hence heat transfer has become essential in aeronautics for the durability and proper functioning of space vehicles to avoid structural failures and also for safety considerations. Construction of dams and other large multistoried buildings need the study of heat transfer. A keen analysis of heat flow is necessary in heating, air conditioning of buildings to estimate the amount of insulation and to reduce the heat losses or gains. Activities like heating, cooling, evaporation, sublimation in chemical operations in research laboratories involve heat transfer. Many metallurgical and industrial mechanisms or operations like extracting metals involve the study of heat transfer. Considerable knowledge of heat transfer is necessary in the controlling or solving or lessening the environmental problems. To develop better variety of seeds, breeders have to deal with heat transfer problems. Food processing needs lot of knowledge of heat transfer. Heat transfer can be defined as the transmission of energy from on region to another as a result of temperature difference between them. There are three different modes of heat transfer Conduction Convection Radiation CONDUCTION Heat conduction may be stated as the transfer of internal energy between the molecules. Heat flows from a region of higher temperature to a region of 13

14 lower temperature by kinetic motion or direct impact of molecules whether the body is at rest or in motion. Heat conduction is due to the property of matter which allows the passage of heat energy even if a physical body is impermeable to any kind of reaction. CONVECTION Heat convection is due to the capacity of moving matter to carry heat energy such as transporting a load from one place to another. Heat transfer due to convection involves the energy exchange between a solid surface and an adjacent fluid. Convection is a mechanism in which heat flows or transferred between a fluid and a solid surface because of motion of fluid particles relative to the solid surface when there exists a temperature gradient. Convection of heat transfer is classified as Forced Convection and Free convection FORCED CONVECTION If the heat transfer between the fluid and the solid surface occurs by fluid motion induced by external agencies (or) forces then the mode of heat transfer is termed as Forced Convection in all types of heat. Exchangers, nuclear reactors, air conditioning apparatus are examples of devices functioning based on forced convection. FREE CONVECTION A free convection flow field is a self-sustained flow driven by the presence of a temperature gradient (as opposed to a forced convection flow where external means are used to provide the flow). As a result of the temperature difference, the density field is not uniform also. Buoyancy will 14

15 induce a flow current due to the gravitational field and the variation in the density field. In general, a free convection heat transfer is usually much smaller compared to a forced convection heat transfer. It is therefore important only when there is no external flow exists. The density difference is due to the temperature difference and it can be characterized by their volumetric thermal expansion coefficient Free convection results from the buoyancy forces imposed on the fluid, whether it is gas or a liquid, when its density in the neighborhood of the heat transfer surface is reduced as a result of heating process. Free convection flow also arises when a heated object is placed in a fluid, otherwise at rest, the density of which varies with temperature. Heat is transferred from the surface of the object to the fluid layers in its neighborhood. The density decrease in a normal fluid due to temperature increase causes these layers to rise and create the free convection flow which now transfers away from the object. Physically such flow is described by stating that it is caused by body force. The example of a practical device which transfers heat by free convection is a radiator used for heating a room. Free convection flow occurs frequently in Nature. It occurs not only due to temperature, but also due to concentration difference or combination of these two. E.g., in atmospheric flows, there exist differences in the H 0 concentration and hence the flow is affected by such concentration difference. Flows in bodies of water driven through the comparable effects upon density of temperature, concentration of dissolved materials and suspended particulate matter and many transport processes exist in nature. In industrial applications the simultaneous heat and mass transfer occur as a result of combined buoyancy effects of diffusion of chemical species. 15

16 RADIATION All substances (solids, liquids and gases) at normal and especially at elevated temperatures emit energy in the form of radiation and are also capable of absorbing such energy. This shows that all heat transfer processes are accompanied by heat exchange through radiation. However, in some cases heat exchange by radiation may be very small fraction of the total quantity of heat exchanged; as such it may be neglected. In case significant amount of heat transfer occurs by radiation, then use may be made of the various laws of radiation. The relative importance of the various modes of heat transfer differs considerably with the temperature. Heat transfer by conduction and convection depends basically on the temperature difference and is little affected by the temperature level. For example, other factors remaining constant, heat transfer by conduction or convection from a body at c to a body at 00 0 c remains the same as that from same body at c to a body at c. In case of radiation, this however, does not hold good. There may be about 35% more heat transfer at higher temperature even for the same temperature difference assuming all other factors as constant. Another difference between the radiation and the other modes of heat transfer lies in the fact that radiation heat transfer does not require any intermediate medium where as in case of conduction, and convection, medium for heat transfer is essentially required. Moreover, in case of conduction, heat flows from a body at high temperature to a body at low temperature if a third body, colder than either of the two, is interposed at any point between the two bodies, then heat will flow from both bodies to colder body. In case of radiation, however, this does not necessarily hold good. Radiative heat transfer may occur from a hot body, through a cold non-absorbing medium leaving it unaffected, and then reach a warmer body. Several theories have been proposed to explain the transport of energy by radiation. Whichever theory is used, radiant energy is the same type of 16

17 wave motion as radio waves, X-rays and light waves except for the wave length. In fact, there is a whole spectrum of electromagnetic radiation in which the various arbitrary divisions are referred to by names reflecting the methods of origin or some characteristic quality. All forms have the same velocity of propagation but different wave lengths and sources of origin. All forms produce heat when absorbed. Nevertheless, it is only the electromagnetic radiation produced by virtue of the temperature of the emitter that we call thermal radiation. Table 1 gives the approximate ranges of wave length of some forms of radiation. The amount of thermal radiation emitted by a body depends on its temperature and surface condition. Radiant energy emitted by a hot body is not confined to the visible range of wave length. Table.1 Characteristic wave lengths of Radiation Name Wave length Range in Microns * Cosmic rays upto (10-6 ) Gamma rays 1(10-6 ) to (140x10-6 ) X-rays 6(10-6 ) to 100,000(10-6 ) Ultraviolet rays to 0.4 Visible or light rays 0.4 to 0.8 Infrared rays 0.8 to 400 Radio 10(10 6 ) to 30,000(10 6 ) *1 micron =10-6 metre But extends itself on both sides somewhat beyond this region. A thermometer placed in the dark or invisible region beyond the red end of a solar spectrum will detect a temperature rise. 17

18 BOUSSINESQ APPROXIMATION Variations in the properties of the fluid, namely density and viscosity are caused by the variations in the temperature within a convective flow. Hence, to analyze these effects, we require some approximation methods one such method is the famous Boussinesq approximation. For thermal convection to occur, the density of the fluid must be a function of the temperature and hence we need an equation of state in addition to the equations of mass, momentum and energy. The simplest equation of state is T 0 1 T 0 where ρ 0 is the fluid density at some reference temperature T 0 and β is the coefficient of thermal expansion. In the processing simplication of the analysis, we use Boussinesq approximation. The approximation consists of all properties of the fluid viz. the kinematic viscosity and thermal diffusivity are independent of temperature and hence constant except that the buoyancy term involving β which is retained in the momentum equation. HEAT FLUX The heat transfer per unit area is called Heat Flux. If q is the amount of heat transfer and A is the area normal to the direction of the heat flow, then the heat flux is q Q A FOURIER LAW OF HEAT CONDITION The fundamental law of heat conduction states that the heat flux by conduction in a direction is proportional to the temperature gradient in that direction. And is expressed as Q x dt dx 18

19 Q x k dt dx dt where Q x is the heat flux in x direction, dx is the temperature gradient in x direction, opposite to the direction of heat flow, and k is the thermal conductivity of the material. VISCOUS DISSIPATION The heat generated by internal friction with in the fluid element of the fluid per unit time is called viscous dissipation. MASS TRANSFER The involvement and application of mass transfer process goes to greater lengths in numerous fields of science, engineering and technology. Mass transfer operations quite often occur in the fields of electric engineering, civil engineering, aeronautics, metallurgy, environmental engineering, refrigeration, air conditioning, biological and industrial processes. Mass transfer broadly occurs in biological, chemical, physical and engineering fields. It involves in biological functions or process like respiratory mechanisms, oxygenation (or) purification of blood, kidney functions, osmosis and assimilation of food and drugs. Evaporation of clouds, smoke formation, dispersion of fog, distribution of temperature and moisture over agricultural fields and groves of fruit trees, damages of crops due to freezing and pollution of the Nature. Mass transfer finds its place in ablative coding transpiration and film cooling of rocket and jet engines. Mass transfer applications are widely found in chemical engineering process like distillation, absorption of gases, interaction of solids and liquids from their mixtures, crystallization adsorption (solid taking up vapor on its surface) and chromatography processes like air humidification, cooling of water, ion exchange involve mass transfer. 19

20 Mass transfer occurs by two mechanisms. 1. Diffusion mass transfer. Convective mass transfer. DIFFUSION MASS TRANSFER In diffusion mass transfer the transfer of matter occurs by the movement of molecules or species or particles of one component to another. Diffusion mass transfer may occur either due to concentration gradient or temperature gradient or pressure gradient CONVECTIVE MASS TRANSFER Convective mass transfer is mechanism in which mass is transferred between the fluid and the solid surface as a result of movement of matter from the fluid to the solid surface or fluid. Convection mass transfer is again classified into a) Natural or free convective mass transfer b) Forced convection mass transfer In natural convection mass transfer, the transfer of mass occurs by the motion of species due to the density differences resulting from temperature or concentration differences or mixture of varying composition. In forced convection mass transfer, mass is transferred due to forced circulation of species by some external agency. MASS FLUX The amount of mass transfer per unit area of the flow is called Mass Flux. If m is the amount of mass flow and A is the area normal to the direction m of mass flow, then the mass flux is G A 0

21 SORET EFFECT When heat and mass transfer occur simultaneously in a moving fluid, the relations between the fluxes and the driving potentials are of more intricate in nature. Mass fluxes can be created by temperature gradients and this is the Soret effect or the thermo-diffusion effect. The Soret effect dramatically lowers the thermal convection threshold, since concentration gradients release much more slowly than temperature gradients due to the disparate values of the mass diffusion coefficient and of the thermal diffusivity. The name Soret effect is usually attributed to mass separation induced by temperature gradients. The effect was discovered in 1879 by the Swiss scientist Charles Soret who noticed that a salt solution contained in a tube with two ends at different temperatures did not remain uniform in composition. The salt was more concentrated near the cold end than near the hot end of the tube. Charles Soret concluded that a flux of salt was generated by a temperature gradient resulting, in steady state conditions, in a concentration gradient. Although the German C. Ludwig described the same phenomenon several years before in 1856 in a short communication, the phenomenon bears his name because Soret studied the effect rather in detail and formulated the fundamental equations describing the phenomenon. The Soret effect plays an important role in the operation of solar ponds, biological systems and the microstructure of the world oceans. In biological systems, mass transport across biological membranes induced by small thermal gradients in living matter is an important factor. One of the challenges in optimizing exploitation of oil reservoirs is a good knowledge of the fluid physics in crude oil reservoirs. Today, the modeling methods are based on pressure temperature equilibrium diagrams and on gravity segregation of the different components of crude oil. However, improved models which more 1

22 accurately predict the concentration of the different components are necessary. The concentration distribution of the different components in hydrocarbon mixtures is mainly driven by phase separation and diffusion and the Soret effect plays an important role. NON DIMENSIONAL PARAMETERS Dimensional analysis of any problem enables to know the qualitative behavior of the physical problem. The dimensionless parameters enable to understand the physical significance of a particular phenomenon associated with the problem. There are usually two general methods for obtaining the dimensionless parameters. (i) The inspection analysis (ii) The dimensionless analysis In this thesis we adopt the dimensional analysis to obtain certain characteristic values in the problem. Consequently certain non-dimensional numbers appear as the coefficient of various terms in the equations. Some of the nondimensional numbers used in this thesis are given below. GRASHOF NUMBER (Gr) It plays a significant role in free convection heat and mass transfer. It is the ratio of the product of the inertial force and the buoyant force to the square of viscous force in the convection Grashof number in free convection in analogous to Reynolds number in forced convection. G r 3 g T L where g = Acceleration due to gravity = Coefficient of viscosity

23 β = Coefficient of volume expansion T = Temperature difference PRANDTL NUMBER (Pr) It is an important dimensional parameter dealing with the properties of a fluid. It refers to or relates the relative thickness of velocity boundary layer and thermal boundary layers. It is defined as the ratio of kinematic viscosity (v) to thermal diffusivity (k) of a fluid. P r C k p where = Coefficient of viscosity C p = specific heat at constant pressure k = Coefficient of thermal conductivity Prandtl number physically means or signifies the relative speed with which the momentum and heat energies are transmitted through a fluid, it thus associates the velocity and temperature fields of a fluid for gasses Prandtl number is of unit order and varied over a wide range in case of liquids. RAYLEIGH NUMBER (Ra) The ratio of the apparent conductivity to the true molecular conductivity is a function, which is the product of Grashof and Prandtl numbers. This function is referred as the Rayleigh number. gth Ra 3 where α is the thermal expansion coefficient of the liquid and T is the temperature difference, g is acceleration due to gravity, is coefficient of viscosity, β is coefficient of volume expansion 3

24 ECKERT NUMBER (Ec) Eckert Number is the ratio of kinetic energy and enthalpy change Ec a(aa)c p SCHMIDT NUMBER (Sc) The ratio of molecular diffusivity of momentum to the mass molecular diffusivity is given by Schmidt number. It plays a major role in convective mass transfer. Sc where ν = Coefficient of kinematic viscosity v D D = Molecular diffusivity NUSSELT NUMBER (Nu) It is defined as the ratio of the conductive thermal resistance to the convective thermal resistance of the fluid is called Nusselt number. N u h ( Tw T ) y y0 where h is some characteristic length, T w T ) is the difference between the ( temperature of the wall and the fluid. This number gives a measure of the heat transfer to the rate at which heat flow rate by convection under unit temperature gradient to the heat flow rate by conduction processes under unit temperature gradient through a stationary thickness. 4

25 SHERWOOD NUMBER (S h ) The dimensionless quantity S h is known as Sherwood number and is defined as S h h ( Cw C) y y0 where h is some characteristic length, y is a mass concentration gradient C w, C are reference concentrations HARTMAN NUMBER Hartman Number is the ratio of the Lorentz force to the viscous force. H M e 0 a POROUS PARAMETER The Porous parameter is defined as the ratio of Darcy resistance to viscous force D 1 a k Physically, this represents the scale factors, which describes the extent of division of porous structure (permeability) as compared to the vertical extant of the porous layer. When the permeability is very high the resistance to the flow becomes effectively controlled by ordinary viscous resistance. In that case, the convection phenomenon is similar to that in an ordinary fluid layer. But in most of the problems, either the viscous force is negligible or is of comparable order to the Darcy resistance. 5