Fluid media rate equations

The approach is based on the universal transformation of solutions of rate equations for constant concentration conditions to those of variable concentration conditions as published earlier [93,94]. The isothermic case of Fick s diffusion in a fluid mixture consisting of N components is considered for any geometry of the sorbing medium, e.g. NS crystals, at variable surface concentration. The model is described by the following equations and initial conditions [94] ... 

Equation (3.24) thus gives the prediction from transition-state theory for the rate of a reaction in terms appropriate for a supercritical fluid. The rate is seen to depend on (1) the pressure (except for unimolecular reactions and as a consequence of defining the rate coefficient in terms of mole fractions) and some universal constants, (2) the equilibrium constant for the activated-complex formation in an ideal gas and (3) a ratio of fugacity coefficients, which express the effect of the supercritical medium. 

For the moment, we shall not go into great detail about the compatibility conditions which the vectorial quantities must satisfy. In the above equations, the interface is considered to be a two-dimensional fluid medium. This consideration is by no means obligatory, though. We could also envisage interfaces with the behavior of an elastic surface, for example. This exploits the product pressure tensor by the strain rate tensor (see Chapter 3 of [PRU 12]). We retain the option of bringing these tensors into play in applications when it becomes necessary to do so. However, in the demonstrations given below, so as not to complicate the discussion, we shall suppose that the interface behaves like a fluid. 

The governing flow equation describing flow through as porous medium is known as Darcy s law, which is a relationship between the volumetric flow rate of a fluid flowing linearly through a porous medium and the energy loss of the fluid in motion. 

Equation 10.4, which describes the mass transfer rate arising solely from the random movement of molecules, is applicable to a stationary medium or a fluid in streamline flow. If circulating currents or eddies are present, then the molecular mechanism will be reinforced and the total mass transfer rate may be written as ... 

This expression applies to the transport of any conserved quantity Q, e.g., mass, energy, momentum, or charge. The rate of transport of Q per unit area normal to the direction of transport is called the flux of Q. This transport equation can be applied on a microscopic or molecular scale to a stationary medium or a fluid in laminar flow, in which the mechanism for the transport of Q is the intermolecular forces of attraction between molecules or groups of molecules. It also applies to fluids in turbulent flow, on a turbulent convective scale, in which the mechanism for transport is the result of the motion of turbulent eddies in the fluid that move in three directions and carry Q with them. 

This equation defines the permeability (K) and is known as Darcy s law. The most common unit for the permeability is the darcy, which is defined as the flow rate in cm3/s that results when a pressure drop of 1 atm is applied to a porous medium that is 1 cm2 in cross-sectional area and 1 cm long, for a fluid with viscosity of 1 cP. It should be evident that the dimensions of the darcy are L2, and the conversion factors are (approximately) 10 x cm2/darcy C5 10-11 ft2/darcy. The flow properties of tight, crude oil bearing, rock formations are often described in permeability units of millidarcies. 

The basis for connecting the stress and strain-rate tensors was postulated first by G. G. Stokes in 1845 for Newtonian fluids. He presumed that a fluid is a continuous medium and that its properties are independent of direction, meaning they are isotropic. His insightful observations, itemized below, have survived without alteration, and are an essential underpinning of the Navier-Stokes equations ... 

As seen clearly from Equation (4.81), the settling rate will be zero when the density difference between the particles and the dispersing medium is zero (i.e., po - p = 0). This method can only be applied to systems with smaller density differences because of the limitation to increase the density of the continuous medium by dissolving some inert simple molecules (e.g., sugar and water-miscible solvents) in Newtonian fluids. In addition, even if the matched density is obtained at one temperature, it cannot be maintained at other temperatures. 

Using the procedure outlined in this chapter for using the boundars laser equations to find the-forced convective heat transfer rate from a circular cylinder buried in a saturated porous medium, investigate the heat transfer rate from cylinders with an elliptical cross-section with their major axes aligned with the forced flow. The surface velocity distribution should be obtained from a suitable book on fluid mechanics. 

For the production of chemicals, the rate of the reaction is a key parameter for the productivity defined in Equation (5) as the number of molecules produced per time. In homogeneous systems, the reaction rate depends on temperature, pressure, and composition [1]. In the case of solarthermal cycles, a metal oxide is used for the C02-splitting reaction rendering the reaction medium a heterogeneous two-phase system consisting of a solid (metal, metal oxide) and a fluid (CO2, CO, or carrier gas with O2). Therefore, the reaction kinetics becomes much more complex. Whereas microscopic kinetics only deals with time-dependent progress of the reaction, macroscopic kinetics additionally takes the heat- and mass-transport phenomena in heterogeneous systems into account. The transfer of species from one phase to the other must be considered in the overall mass balance [1]. The reaction of a gas with a porous solid consists of seven steps ... 

Here, the terms on the right side of the above equation show the entropy production due to heat transfer and fluid friction, respectively hence, the entropy production expression has the following basic form , Sprod =, Yprod Ar +, S prod A/,. The volumetric entropy production rate is positive and finite as long as temperature or velocity gradients are present in the medium. 

In actual use for mobility control studies, the network might first be filled with oil and surfactant solution to give a porous medium with well-defined distributions of the fluids in the medium. This step can be performed according to well-developed procedures from network and percolation theory for nondispersion flow. The novel feature in the model, however, would be the presence of equations from single-capillary theory to describe the formation of lamellae at nodes where tubes of different radii meet and their subsequent flow, splitting at other pore throats, and destruction by film drainage. The result should be equations that meaningfully describe the droplet size population and flow rates as a function of pressure (both absolute and differential across the medium). 

By analogy to single-phase flow, under steady-state conditions, the flow rate of each fluid should be directly proportional to the applied pressure gradient and the cross-sectional area of the medium and inversely proportional to the fluid viscosity. Therefore, an equation analogous to equation 1 can be written for each fluid ... 

In the case of highly mobile interface between dispersed phase and dispersion medium (as in foams and emulsions) the condition of zero fluid flow velocity at interface (non-slip condition), determining the validity of Reynolds equation, may not be obeyed. In this case the decrease in the film thickness occurs at a greater rate. However, in foam and emulsion films stabilized by surfactant adsorption layers the conditions of fluid outflow from an interlayer are close to those of outflow from a gap between solid surfaces even in cases when surfactant molecules do not form a continuous solid-like film. This is the case because at surfactant adsorption below Tmax the motion of fluid surface leads to the transfer of some portions of surfactant adsorption layer from central regions of film to peripherical ones, adjacent to the Gibbs-Plateau channels. As a result, the value of adsorption decreases in the center of film, but increases at the periphery, which stipulates the appearance of the surface... 

A porous medium consists of a packed bed of solid particles in which the fluid in the pores between particles is free to move. The superficial fluid velocity V is defined as the volumetric flow rate of the fluid per unit of cross-sectional area normal to the motion. It is the imbalance between the pressure gradient (VP) and the hydrostatic pressure gradient (pg) that drives the fluid motion. The relation that includes both viscous and inertial effects is the Forscheimer equation [47]... 

Absolute and Effective Permeability. The permeability of a porous medium is a measure of the fluid conductivity of the medium that determines the flow rate at a given pressure gradient. The permeability of a porous medium can be determined using Darcy s equation as follows. The porous medium, usually a core, is saturated under vacuum with a fluid of a known viscosity. The fluid is then injected into the core in a linear mode at a constant flow rate under isothermal flow conditions. If the porous medium is homogeneous, isotropic, the core is placed in horizontal direction, and Reynolds number of the injected fluid is less than unity, then Darcy s law can be used to determine permeability. Darcy s law is... 

Washburn Equation An equation describing the extent of displacement of one fluid by another in a capillary tube or cylindrical pore in a porous medium. If h is the depth of penetration of invading fluid and dh/dt is the rate of penetration, then dh/dt = yr cos 6/(4rjh), where y is the interfacial tension, r is the capillary radius, 6 is the contact angle, and rj is the viscosity of the invading fluid. It is used in the evaluation of porosimetry data and may be used to provide information about contact angles, capillary radii, and pore radii, depending on the experiments conducted. 

The principles of conservation of momentum, energy, mass, and charge are used to define the state of a real-fluid system quantitatively. The conservation laws are applied, with the assumption that the fluid is a continuum. The conservation equations expressing these laws are, by themselves, insufficient to uniquely define the system, and statements on the material behavior are also required. Such statements are termed constitutive relations, examples of which are Newton s law that the stress in a fluid is proportional to the rate of strain, Fourier s law that the heat transfer rate is proportional to the temperature gradient. Pick s law that mass transfer is proportional to the concentration gradient, and Ohm s law that the current in a conducting medium is proportional to the applied electric field. 

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