Fluid properties momentum balance

Here, the superficial velocity, v, represents a fluid state, and the density, p, a fluid property which, for a compressible fluid, can be related to the pressure through an equation of state. The porosity, (p, which is defined as the void fraction within the media, is a macroscopic property of the porous material. Sources and/or sinks located within the physical system are represented using y/. Volume averaging the differential momentum balance for the same physical situation yields Darcy s law ... 

We have considered the situation of only one phase for any mixture composition this means that there is no surface tension and the fluid behavior is completely characterized by the turbulent flow described by the mass and momentum balance equations. To solve these equations, one needs to model the diffusional mixing of the species present in the system and to identify local values of the thermodynamic and transport properties, as considered in Section 3.2. Here we just point out that once the methods for predicting local values of fluid density and viscosity have been worked out, one should be able to integrate Eqs. (10) and (11). 

The flow behavior of fluids is governed by the basic laws for conservation of mass, energy, and momentum coupled with appropriate expressions for the irreversible rate processes (e.g., friction loss) as a function of fluid properties, flow conditions, geometry, etc. These conservation laws can be expressed in terms of microscopic or point values of the variables, or in terms of macroscopic or integrated average values of these quantities. In principle, the macroscopic balances can be derived by integration of the microscopic balances. However, unless the local microscopic details of the flow field are required, it is often easier and more convenient to start with the macroscopic balance equations. 

Two-phase pressure drop can typically be correlated with two models, i.e. homogeneous or separated. Homogeneous fluid models are well suited to emulsions and flow with negligible surface forces, where the two-phase mixture can be treated as a single fluid with appropriately averaged physical properties of the individual phases. Separated flow models consider that the two phases flow continuously and separated by an interface across which momentum can be transferred (Angeli and Hewitt 1999). The simplest patterns that can be easily modelled are separated and annular flow (Brauner 1991 Rovinsky et al. 1997 Bannwart 2001). In this case, momentum balances are written for both phases with appropriate interfacial and wall friction factors. 

The balance equations for mass, momentum and energy describe the entire flow situation. The continuity assumption of smooth fluid properties and no-shp flow conditions at the wall hold for most cases in microprocess engineering, hence the change in density p with time is correlated with the velocity vector w as... 

The analysis of many flow problems can be simplifled by considering only one component of the equation of motion, the one in the direction of flow. Further simplifying assumptions are often necessary in order to solve the problem. In the analysis of isothermal processes, only two balance equations are needed, the mass and momentum balance. In order to solve the problem, additional information is required. This is information on how the fluid deforms under application of various stresses. This information is described by the constitutive equation of the fluid see also Section 6.2 on melt flow properties. 

One typical example that can be addressed in this way is the start-up of a flow through a pipe. We can consider a one dimension problem for the profile in the radial direction. We evaluate now the unsteady fluid flow through a long pipe with constant fluid properties (p, p). The momentum balance is given by the following equation [5] ... 

In this section we first review general modeling principles, emphasizing the importance of the mass and energy conservation laws. Force-momentum balances are employed less often. For processes with momentum effects that cannot be neglected (e.g., some fluid and solid transport systems), such balances should be considered. The process model often also includes algebraic relations that arise from thermodynamics, transport phenomena, physical properties, and chemical kinetics. Vapor-liquid equilibria, heat transfer correlations, and reaction rate expressions are typical examples of such algebraic equations. 

The major goal of The direct quadrature method of moments (DQMOM) was to derive transport equations for the weights w and abscissas that can be solved directly and which yield the same moments nk without resorting to the ill-conditioned PD algorithm. Another novel concept imposed is that each phase can be characterized by a weight w and a property vector )i, thus the DQMOM can be employed solving the multi-fluid model describing multi-phase systems. Moreover, since each phase has its own momentum balance in the multi-fluid model, the nodes of the DQMOM quadrature approximation are convected with their own velocities. The DQMOM was proposed by Marchisio and Fox [143] and Fan et al. [53] in order to handle poly-dispersed multi-variate systems. 

Shear thinning or pseudoplastic behavior is an important property that must be taken into account in the design of polymer processes. However, it is not the only property, and in Chapter 3 models that describe the viscoelastic response of polymeric fluids will be discussed. However, first we would like to solve some basic one-dimensional isothermal flow problems using the shell momentum balance and the empiricisms for viscosity described in this section. 

It is necessary to associate mathematical quantities with each type of momentum transfer rate process that is contained in the vector force balance. The fluid momentum vector is expressed as p, which is equivalent to the overall mass flux vector. This is actually the momentum per unit volume of fluid because mass is replaced by density in the vectorial representation of fluid momentum. Mass is an extrinsic property that is typically a linear function of the size of the system. In this respect, mv is a fluid momentum vector that changes magnitude when the mass of the system increases or decreases. This change in fluid momentum is not as important as the change that occurs when the velocity vector is affected. On the other hand, fluid density is an intrinsic property, which means that it is independent of the size of the system. Hence, pv is the momentum vector per unit volume of fluid that is not affected when the system mass increases or decreases. The total fluid momentum within an arbitrarily chosen control volume V is... 

At the end, we summarize the results of the model of a reacting mixture of fluids with linear transport properties from Sects. 4.5 and 4.6 (properties such as kinematics, stoichiometry and balances of mass, momentum and their moment, energy and entropy inequality are as in Sects. 4.2, 4.3 and 4.4). Constitutive equations, their properties and final form of entropy production are given in the end of Sect. 4.5 (from Eq. (4.156)), further thermodynamic quantities and properties are given at the... 

The strength of I-Shih Liu method, therefore, manifests itself at the more complicated constraints [24, 26]. The most complicated case in our book—the reacting mixture with linear transport properties—with the use of entropy inequality and all balances (of mass, momentum, energy) as (A.IOO), (A.99), would be laborious. Therefore, to demonstrate the application of the I-Shih Liu s Theorem A.5.5, we choose relatively simple examples of the uniform fluid model B from Sect. 2.2 and the simple thermoelastic fluid from the end of Sect. 3.6. ... 

Based on continuum mechanics, the motion of fluid particles is modeled as a two-phase flow with phase boundary represented by a sharp interface. More precisely, we consider isothermal flows of two fluids of constant material properties each, which are immiscible on the molecular scale. Within each phase, the balance equations of mass and momentum read... 

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