Balance or Conservation Equations

The balance principle, that is well known from the previous discussion, underlie any fundamental formulation of the total energy balance or conservation equation. Starting out from fluid dynamic theory the resulting partial differential equation contains unknown terms that need further consideration. We need a sound procedure for the formulation of closure laws. 

There are also relationships between inrtermediates giving balance or conservation equations. For example, such an equation could relate the total concentration of intermediate species bound to enzyme. The enzyme could be then either in free form or bound to substrate and the sum of these two forms is equal to the initial concentration of enzyme that was introduced in the system. For heterogeneous catalytic reactions a balance equation relates the surface coverage of adsorbed... 

The four balance or conservation principles can all be represented in terms of a general equation of balance written in integral form as... 

Therefore, the total energy balance, or the equation for the conservation of energy, becomes... 

Under conditions of constant current, namely, under steady-state operation, reactants must be supplied continuously and at a constant rate to precisely balance the rate of reactant consumption in electrode reactions. The coupled fluxes of electrons, protons, and gaseous reactants are subdued to two fundamental conservation laws, which complete the description of the fuel cell principle conservation of charge and mass. These laws allow balance or continuity equations to be written among all involved... 

Equation (2.33) says that the time rate of change of momentum in the i direction (mass times velocity in the i direction) is equal to the net sum of the forces pushing in the z direction. It can be thought of as a dynamic force balance. Or more eloquently it is called the conservation ofinomentum. 

The number of equations and unknowns must balance. Thus, one can calculate the appropriate number of needed relationships from the degrees of freedom of a system, as shown for various systems by Newman. In terms of the relations, the equations can be broken down into five main types. The first are the conservation equations, the second are the transport relations, the third are the reactions, the fourth are equilibrium relationships, and the fifth are the auxiliary or supporting relations, which include variable definitions and such relations as Faraday s law. 

The gas channels contain various gas species including reactants (i.e., oxygen and hydrogen), products (i.e., water), and possibly inerts (e.g., nitrogen and carbon dioxide). Almost every model assumes that, if liquid water exists in the gas channels, then it is either as droplets suspended in the gas flow or as a water film. In either case, the liquid water has no affect on the transport of the gases. The only way it may affect the gas species is through evaporation or condensation. The mass balance of each species is obtained from a mass conservation equation, eq 23, where evaporation/condensation are the only reactions considered. 

There are not many models that do transients, mainly because of the computational cost and complexity. The models that do have mainly been discussed above. In terms of modeling, the equations use the time derivatives in the conservation equations (eqs 23 and 68) and there is still no accumulation of current or charging of the double layer that is, eq 27 still holds. The mass balance for liquid water requires that the saturation enter into the time derivative because it is the change in the water loading per unit time. However, this treatment is not necessarily rigorous because a water capacitance term should also be included,although it can be neglected as a first approximation. 

In practice, of course, it is rare that the catalytic reactor employed for a particular process operates isothermally. More often than not, heat is generated by exothermic reactions (or absorbed by endothermic reactions) within the reactor. Consequently, it is necessary to consider what effect non-isothermal conditions have on catalytic selectivity. The influence which the simultaneous transfer of heat and mass has on the selectivity of catalytic reactions can be assessed from a mathematical model in which diffusion and chemical reactions of each component within the porous catalyst are represented by differential equations and in which heat released or absorbed by reaction is described by a heat balance equation. The boundary conditions ascribed to the problem depend on whether interparticle heat and mass transfer are considered important. To illustrate how the model is constructed, the case of two concurrent first-order reactions is considered. As pointed out in the last section, if conditions were isothermal, selectivity would not be affected by any change in diffusivity within the catalyst pellet. However, non-isothermal conditions do affect selectivity even when both competing reactions are of the same kinetic order. The conservation equations for each component are described by... 

With such an understanding on system complexity in mind, the DBS model is composed of two simple force balance equations, respectively, for small or large bubble classes, and one mass conservation equation as well as the stability condition serving as a variational criterion and a closure for conservative equations. For a given operating condition of the global system, six structure parameters for small and large bubble classes (their respective diameters dg, dL, volume fraction... 

Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and conservation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential conservation equations, respectively. These are often called macroscopic and microscopic balance equations. 

A key aspect of modeling is to derive the appropriate momentum, mass, or energy conservation equations for the reactor. These balances may be used in lumped systems or derived over a differential volume within the reactor and then integrated over the reactor volume. Mass conservation equations have the following general form ... 

Balancing a chemical equation requires an understanding of the Law of Conservation of Mass, which says that mass cannot be created or destroyed. The amount of mass in the reactants will be the amount of mass in the products. The credit for this discovery is given to Antoine Lavoisier, who took very careful measurements of the quantities of chemicals and equipment that he used. Conservation of mass also holds true when balancing equations. The number of atoms of each element in the reactants will be equal to the number of atoms of each element in the products. A useful mnemonic device for conservation of mass is What goes in, must come out. ... 

Both the macro and micro population balances just derived conserve the number of particles. In some cases, it is appropriate to perform balances where the particles length, area, or volume (or mass) is conserved. For example, length conservation is critical in grinding fibers and volume conservation is critical in grinding other particle shapes. Such conservation equations can also be developed under the umbrella of a population balance, but this population balance must be different than those previously derived, where particle number is conserved. The way to make them different is to couple to the population balance an appropriate conservation equation. The population based on length, area, md volume (or mass) can be derived from the population based on number as shown in Table 3.1. Let us illustrate this idea of property conservation with an example showing conservation of length. 

An unbalanced chemical equation is of limited use. Whenever you see an equation, you should ask yourself whether it is balanced. The principle that lies at the heart of the balancing process is that atoms are conserved in a chemical reaction. The same number of each type of atom must be found among the reactants and products. Also, remember that the identities of the reactants and products of a reaction are determined by experimental observation. For example, when liquid ethanol is burned in the presence of sufficient oxygen gas, the products will always be carbon dioxide and water. When the equation for this reaction is balanced, the identities of the reactants and products must not be changed. The formulas of the compounds must never be changed when balancing a chemical equation. That is, the subscripts in a formula cannot be changed, nor can atoms be added or subtracted from a formula. 

Boundary and interface conditions must be known if solutions to the conservation equations are to be obtained. Since these conditions depend strongly on the model of the particular system under study, it is difficult to give general rules for stating them for example, they may require consideration of surface equilibria (discussed in Appendix A) or of surface rate processes (discussed in Appendix B). However, simple mass, momentum, and energy balances at an interface often are of importance. For this reason, interface conditions are derived through introduction of integral forms of the conservation equations in Section 1.4. 

In the operator L, the first term represents convection and the second diffusion. Equation (44) therefore describes a balance of convective, diffusive, and reactive effects. Such balances are very common in combustion and often are employed as points of departure in theories that do not begin with derivations of conservation equations. If the steady-flow approximation is relaxed, then an additional term, d(p(x)/dt, appears in L this term represents accumulation of thermal energy or chemical species. For species conservation, equations (48) and (49) may be derived with this generalized definition of L, in the absence of the assumptions of low-speed flow and of a Lewis... 

The basis for both of these observations is the law of conservation of mass, which states that mass can neither be created nor destroyed. (We will not be concerned in this book with the almost infinitesimal conversions between mass and energy associated with chemical reactions.) Statements based on the law of conservation of mass such as total mass of input = total mass of output or (Ibm sulfui/day), = (Ibm sulfur/day)oui" are examples of mass balances or material balances. The design of a new process or analysis of an existing one is not complete until it is established that the inputs and outputs of the entire process and of each individual unit satisfy balance equations. 

This result appears to be counterintuitive, especially since we normally allow the energy to depend on mole numbers, as specified by the relation E = E S, V, N( ). However, this problem is apparent rather than real from the viewpoint of chemistry the fundamental species in any chemical reaction are the participating atoms whose numbers are strictly conserved—witness the process of balancing any chemical equation. Thus, while the arrangement or configuration of the atoms changes in a chemical process their numbers are not altered in this process. Under conditions of strict isolation the system behaves as a black box no indication of the internal processes is communicated to the outside. One should not attempt to describe processes to which one has no direct access. However, under conditions illustrated in Remark 1.21.2, even an isochoric reaction carried out very slowly in strict isolation, produces an entropy change dS = dO = 1 Hi dNi > 0. See also Eq. (2.9.3) which proves Eq. (1.21.3) under equilibrium conditions. 

For the well-mixed reservoirs, a conservation equation is written in which gain of 14C by inflow to the box (atmosphere or surface ocean) is balanced by the outflow to other boxes plus radioactive decay (see Section 2.8) of the tracer during its time in the reservoir. For the deep ocean, conservation is described by a partial differential advection-diffusion equation. The... 

Note that each chemical element is conserved. This means that ordinary chemical reactions involving the elements are merely changing partners, rather than being produced or destructed. Individual chemical species are however not really conserved since they can be generated or consumed in chemical reactions, but one can nevertheless write an appropriate balance equation or transport equation for each chemical species [134]. 

Not all of the balance equations are independent of one another, thus the set of equation used to solve particular problems is not solely a matter of convenience. In chemical reactor modeling it is important to recall that all chemical species mass balance equations or all chemical element conservation equations are not independent of the total mass conservation equation. In a similar manner, the angular momentum and linear momentum constraints are not independent for flow of a simple fluid . 

In addition to the material balance Equation 18.38 that one normally encounters in design problems involving chemical reactions with no charge transfer, for the case of an electrochemical reaction, one must consider a charge balance equation that governs the distribution of the electric potential across the particle or the pores. For the particle in itself, such a conservation equation is usually presented in the form of Ohm s law given as follows ... 

Conservation Equations. Mass balance equations for the gaseous and aqueous phases are written in standard reservoir simulator form 10, 93). For the nonwetting foam or gas phase in a one-dimensional medium,... 

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