Mass transfer/transport conservation laws

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. 

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... 

The mathematical description of simultaneous heat and mass transfer and chemical reaction is based on the general conservation laws valid for the mass of each species involved in the reacting system and the enthalpy effects related to the chemical transformation. The basic equations may be derived by balancing the amount of mass or heat transported per unit of time into and out of a given differential volume element (the control volume) together with the generation or consumption of the respective quantity within the control volume over the same period of time. The sum of these terms is equivalent to the rate of accumulation within the control volume ... 

At film condensation, the heat has to be transported through the film. If the film is laminar and the film surface is at thermodynamic equilibrium, then the heat is transferred by conduction. The transfer coefficient can be calculated, if the film thickness is known depending on the film length. In case of a laminar film the velocity profile is defined by an equihbrium between viscous and gravitational forces, see Chap. 3. Considering the conservation laws for mass and energy allows to derive the heat transfer coefficient on a theoretical basis. 

The balance (C.25) has the obvious form of a conservation law with source term. With the minus sign at the integral it means that the increase in equals the quantity transported into across the boundary plus that produced by chemical reactions, per unit time. The jk-term can be relevant at a permeable boundary imagine as liquid phase in a distillation column, thus j .n represents total mass transferred per unit area and j. n is due to molecular diffusion of species. ... 

Generally, transport processes are based on the conservation laws of momentum (flow field), energy (temperature field), and mass (concentration field) taking into account the conditions like gradients of flow velocity, temperature, and concentration at the phase boundaries of the system under investigation usually by the application of transfer coefficients. In order to describe the interaction between fluid flow, heat flow, and mass flow, at least the behavior of the following data has to be clarified ... 

The engineering science of transport phenomena as formulated by Bird, Stewart, and Lightfoot (1) deals with the transfer of momentum, energy, and mass, and provides the tools for solving problems involving fluid flow, heat transfer, and diffusion. It is founded on the great principles of conservation of mass, momentum (Newton s second law), and energy (the first law of thermodynamics).1 These conservation principles can be expressed in mathematical equations in either macroscopic form or microscopic form. 

The physical laws of conservation of mass, momentum, and energy are commonly formulated for closed thermodynamic systems,2 and for our purposes, we need to transfer these to open control volume3 formulations. This can be done using the Reynolds Transport Theorem.4... 

Local thermodynamic equilibrium in space and time is inherently assumed in the kinetic theory formulation. The length scale that is characteristic of this volume is i whereas the timescale is xr. When either L i, ir or t x, xr or both, the kinetic theory breaks down because local thermodynamic equilibrium cannot be defined within the system. A more fundamental theory is required. The Boltzmann transport equation is a result of such a theory. Its generality is impressive since macroscopic transport behavior such as the Fourier law, Ohm s law, Fick s law, and the hyperbolic heat equation can be derived from this in the macroscale limit. In addition, transport equations such as equation of radiative transfer as well as the set of conservation equations of mass, momentum, and energy can all be derived from the Boltzmann transport equation (BTE). Some of the derivations are shown here. 

Where F is the flux or rate of transfer per unit area, c is the concentration of diffusing substance, x is the space coordinate measured normal to the section, and D is the diffusion coefficient. The negative sign in equation (O 31.1) is due to the fact that diffusion occurs in the direction opposite to that of increasing concentration. Considering the transport of moisture through an element in which concentration increases, there must be a proportional decrease in flux across the distance of the element in order to conserve mass. This is expressed mathematically by Pick s second law, which is shown in equation (O 31.2) for ID diffusion. 

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