Mass transfer rate momentum equations

In laminar flow with low mass-transfer rates and constant physical properties past a solid surface, as for the two-dimensional laminar boundary layer of Fig. 3.10, the momentum balance or equation of motion (Navier-Stokes equation) for the X direction becomes [7]... 

The numerical model for analysing the membrane reactor was developed using a commercial CFD code, Star-CD v3.2. The modelling procedure is almost the same as that for the single tube described above. Equations [13.1-13.35a] were also used for the mass, momentum, species and energy conservation on the reaction and permeation sides, the reaction rate, heat and mass transfer rates, pressure drop and permeability of hydrogen. 

The three-dimensional, fully parabolic flow approximation for momentum and heat- and mass-transfer equations has been used to demonstrate the occurrence of these longitudinal roll cells and their effect on growth rate uniformity in Si CVD from SiH4 (87) and GaAs MOCVD from Ga(CH3)3 and AsH3 (189). However, gas expansion in the entrance zone combined with flow obstructions, such as a steeply sloped susceptor, can also produce... 

Process-scale models represent the behavior of reaction, separation and mass, heat, and momentum transfer at the process flowsheet level, or for a network of process flowsheets. Whether based on first-principles or empirical relations, the model equations for these systems typically consist of conservation laws (based on mass, heat, and momentum), physical and chemical equilibrium among species and phases, and additional constitutive equations that describe the rates of chemical transformation or transport of mass and energy. These process models are often represented by a collection of individual unit models (the so-called unit operations) that usually correspond to major pieces of process equipment, which, in turn, are captured by device-level models. These unit models are assembled within a process flowsheet that describes the interaction of equipment either for steady state or dynamic behavior. As a result, models can be described by algebraic or differential equations. As illustrated in Figure 3 for a PEFC-base power plant, steady-state process flowsheets are usually described by lumped parameter models described by algebraic equations. Similarly, dynamic process flowsheets are described by lumped parameter models comprising differential-algebraic equations. Models that deal with spatially distributed models are frequently considered at the device... 

A flow is completely defined if the values of the velocity vector, the pressure, and the temperature are known at every point in the flow. The distributions of these variables can be described by applying the principles of conservation of mass, momentum, and energy, these conservation principles leading to the continuity, the Navier-Stokes, and the energy equations, respectively. If the fluid properties can be assumed constant, which is very frequently an adequate assumption, the first two of these equations can be simultaneously solved to give the velocity vector and pressure distributions. The energy equation can then be solved to give the temperature distribution. Fourier s law can then be applied at the surface to get the heat transfer rates. 

Engineering systems mainly involve a single-phase fluid mixture with n components, subject to fluid friction, heat transfer, mass transfer, and a number of / chemical reactions. A local thermodynamic state of the fluid is specified by two intensive parameters, for example, velocity of the fluid and the chemical composition in terms of component mass fractions wr For a unique description of the system, balance equations must be derived for the mass, momentum, energy, and entropy. The balance equations, considered on a per unit volume basis, can be written in terms of the partial time derivative with an observer at rest, and in terms of the substantial derivative with an observer moving along with the fluid. Later, the balance equations are used in the Gibbs relation to determine the rate of entropy production. The balance equations allow us to clearly identify the importance of the local thermodynamic equilibrium postulate in deriving the relationships for entropy production. 

Basically, the processes taking place in a chemical reactor are chemical reaction, and mass, heat and momentum transfer phenomena. The modeling and design of reactors are therefore sought from emplo3dng the governing equations describing these phenomena [1] the reaction rate equation, and the species mass, continuity, heat (or temperature) and momentum balance equations. 

Moving up into the reactor level, effects of convection, dispersion and generation are described in the conservation equations for mass and energy. The momentum balance describes the behavior of pressure. The interface between the reactor and the catalyst level is described by the external mass transfer conditions, most often represented in a Fickian format, i.e., a linear dependence of the rate of mass transfer on the concentration gradient. In cases where an explicit description of mixing and hydrodynamic patterns is required, the simultaneous integration of the Navier-Stokes equations is also conducted at this level. I f the reaction proceeds thermally, the conversion of mass and the temperature effect as a result of it are described here as well. 

Consider die simple case in which a low rate of mass transfer normal to die surface does not influence die velocity profile (vy 0), The velocity profile vj,y) in die film is determined by solving the z component of die eqnetion of motion for a Newtonian fluid in steady flow owing to iha ection of gravity. If the solid surface is inclined at an angle a with respect to die horizontal the momentum equation is... 

They are needed to describe the rate of mass, energy, and momentum transfer between a system and its surroundings. These equations are developed in courses on transport phenomena. 

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. 

Equations (16.14), (16.18), and (16.22) govern the fluid velocity and temperature in the lower atmosphere. Although these equations are at all times valid, their solution is impeded by the fact that atmospheric flow is turbulent (as opposed to laminar). It is difficult to define turbulence instead we can cite a number of the characteristics of turbulent flows. Turbulent flows are irregular and random, so that the velocity components at any location vary randomly with time. Since the velocities are random variables, their exact values can never be predicted precisely. Thus (16.14), (16.18), and (16.22) become partial differential equations whose dependent variables are random functions. We cannot therefore expect to solve any of these equations exactly rather, we must be content to determine some convenient statistical properties of the velocities and temperature. The random fluctuations in the velocities result in rates of momentum, heat, and mass transfer in turbulence that are many orders of magnitude greater than the corresponding rates due to pure molecular transport. 

Many researches adopted one of the aforanentioned approaches and modified it to include various aspects of the pneumatic drying process. Andrieu and Bressat [16] presented a simple model for pneumatic drying of polyvinyl chloride (PVC), particles. Their model was based on elementary momentum, heat and mass transfer between the fluid and the particles. In order to simplify their model, they assumed that the flow is unidirectional, the relative velocity is a function of the buoyancy and drag forces, solid temperature is uniform and equal to the evaporation temperature, and that evaporation of free water occurs in a constant rate period. Based on their simplifying assumptions, six balance equations were written for six unknowns, namely, relative velocity, air humidity, solid moisture content, equilibrium humidity, and both solid and fluid temperatures. The model was then solved numerically, and satisfactory agreanent with their experimental results was obtained. A similar model was presented by Tanthapanichakoon and Srivotanai [24]. Their model was solved numerically and compared with their experimental data. Their comparison between the experimental data and their model predictions showed large scattering for the gas temperature and absolute humidity. However, their comparisons for the solid temperature and the water content were failed. 

As discussed in Section 2.3 for the general molecular transport equation, all three main types of rate-transfer processes—momentum transfer, heat transfer, and mass transfer— are characterized by the same general type of equation. The transfer of electric current can also be included in this category. This basic equation is as follows ... 

Algebraic equations (14.3) correspond to constitutive equations, which are generally based on physical and chemical laws. They include basic definitions of mass, energy, and momentum in terms of physical properties, like density and temperature thermodynamic equations, through equations of state and chonical and phase equilibria transport rate equations, such as Pick s law for mass transfer, Fourier s law for heat conduction, and Newton s law of viscosity for momentum transfer chemical kinetic expressions and hydraulic equations. 

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