Balance of mass

The living microbial, animal, or plant cell can be viewed as a chemical plant of microscopic size. It can extract raw materials from its environment and use them to replicate itself as well as to synthesize myriad valuable products that can be stored in the cell or excreted. This microscopic chemical plant contains its own power station, which operates with admirably high efficiency. It also contains its own sophisticated control system, which maintains appropriate balances of mass and energy finxes through the links of its internal reaction network. 

By using the concept of Helmholtz s free energy per unit volume hi = Pii i — GiVi) an(J the balances of mass (2) and energy (6), we can recast the entropy equation (7) in the following form ... 

Another commonly used model is based on the general differential balance of mass and momentum [Burgers, 1948]. Consider a steady, incompressible, and axially symmetric flow in which the body forces are negligible. In cylindrical coordinates, the equation of continuity of the fluid can be given as... 

The interaction of these two processes can be described by a simple isothermal model, which is based on balances of mass and charge. The model describes the extent of the reforming and oxidation reactions along the anode channel. The essential simulation results can easily be displayed in a conversion diagram which is a phase diagram of the two dynamic state variables, namely the extents of two reactions. 

Krishnamurthy and Taylor [51] developed a so-called non-equilibrium stage model , the characteristics of which were the balance of mass and energy for each component in the two phases. These are coupled over the energy and mass flows in the boundary layers and are at equilibrium at the phase boundary. 

The first pattern in Figure 9.8 is disqualified because there is no way to achieve steady state balance of mass of A or D given this sign pattern. The second pattern is disqualified because there is no feasible chemical potential pattern associated with this flux pattern. 

Rational thermodynamics provides a method for deriving the constitutive equations without assuming local equilibrium. In this formulation, absolute temperature and entropy do not have a precise physical interpretation. It is assumed that the system has a memory, and the behavior of the system at a given time is determined by the characteristic parameters of both the present and the past. However, the general expressions for the balance of mass, momentum, and energy are still used. 

The cross-section averaged equations governing phase k are obtained defining the specific values of the generalized variables in (3.468) in accordance with Table 3.1 gives the instantaneous area-averaged equations for the balance of mass, momentum, total energy, species mass and entropy. 

The recommended approach to modeling is to create models based on fundamental balances (of mass, species, energy, population) and basic kinetics and use them to build a complete model of the precipitator, as shown in earlier sections. Such a set of equations is known as a physical or a mechanistic model. Complete physical models are difficult to create and solve because they require identification in advance of all physical and chemical subprocesses, properties, and parameters. That is why the semiempirical models of a form similar to the complete physical models (but usually simpler) and with fewer equations are often used for scaling up. Parameters of such models are often given in lumped form, some of them fitted to available experimental data obtained from the small-scale system. Such a model can be useful for scaling up, but one cannot be sure that the scale-up will be completely correct because there is no guarantee that the model contains the complete mechanism (88). However, scale-up errors should be smaller than in the case of purely empirical models. CFD codes that are based on reasonable simplifications (closures) regarding their accuracy can be placed between the physical and semiempirical models their application was demonstrated earlier. 

Dynamic flowsheeting is based on the unsteady state balance of mass and energy, and may be formulated in general terms by the following equation ... 

Fluid sub-problem. Here the balance of mass for the liquid phase is given in its volumetric representation ... 

Furthermore, the program has to interpret chemical formulas. For example if the student types "NH3OH" for the formula of ammonium hydroxide he should be told that the hydrogen is incorrect and not that he has misspelled a word. The routine F.QTJUDG was written to interpret any chemical equation. The routine controls the balancing of mass and charge in chemical equations and checks other rules for writing chemical equations, e.g. 

Therefore, if equation (1.83) determines balance of electrically charged solution ions in the process of relaxation and equation (1.21) - constancy of the basis components concentrations, then equation (1.87) controls balance of mass in individual reactions. However, they do not touch upon balance of energy of chemical reactions and for this reason do not determine either the direction or the rate of reactions and even more so the conditions of their equilibrium. This is implemented by the law of mass action. 

The expression (6.174) allows us to write the balance-of-mass equation for gas in the bubble ... 

The balance-of-mass relation for the absorbent at the plate, accounting for the recirculations and for the absorbent flowing from plate to plate, gives us the following equation for aa... 

Substituting the flux into the balance-of-mass equation (22.20) for the bubble, we obtain ... 

Denote by Jg the flux of gas from a unit surface area of the mixture. Then the balance-of-mass condition for a column of mixture of a unit cross section gives us ... 

Now the balance-of-mass condition for i-th component in the solution can be written as... 

We also want to mention the contribution to modem thermodynamics made by Muller [10, 16, 29] which lies somewhere between the extended irreversible and rational approaches as indicated in the title of one of corresponding books, coauthored by Ruggeri [30]. Particularly, the reference [ 16] can be recommended even for the very beginners in modern approaches to fundaments of thermodynamics. Although the substantial part of this book deals with the equilibrium theory Mullers reintroduce time into consideration and thermodynamics equations and treat both the equilibrium and (and least some) nonequilibrium processes within a natural, common framework. Their book contains a lot of real application examples and explains and illustrates the common basis of probably all rigorous thermodynamic approaches— the equations of balance of mass, momenrnm, and energy and equations describing the specific behavior of different material bodies (systems) which were traditionally called the equations of state and in modern terms the constitutive equations. 

Primitives and definitions are used to formulate general postulates (e.g., the First and Second Laws, balances of mass, momentum, etc.) valid for all (in fact for a broad class of) material models. Real materials are expressed through special mathematical models in the form of constitutive equations which describe idealized materials expressing features important in assumed applications. Moreover, the same real material may be described by more models with various levels of description. The levels are motivated by the observer s time and space scales— typically the time and space intervals chosen (by the observer) for description of a real material having its own... 

The whole mass m of the system is constant and this closed system, exchanging only heat and volume work with its surroundings, is supposed to be described again by fields (time function) (2.3) and (2.4) added with fields of (positive) masses /ni(t), m2(t) of both constituents 1, 2 respectively. For closed systems (cf. Sects. 1.2, 2.1) the balance of energy (2.1) and entropy inequality (2.2) are valid but now together with the balance of mass... 

To formulate the balance of mass let us consider the single (one-constituent) body in arbitrary actual configuration (in inertial frame noted above). 

Balances of Mass, Momentum, tind Moment of Momentum... 

Summary. The first three balance equations are formulated in this section. The balances are necessary conditions to be fulfilled not only in thermodynamics but generally (in continuum mechanics). The balance of mass was formulated locally in several alternatives—(3.62), (3.63), or (3.65). The most important consequence of the balance of momentum is the Cauchy theorem (3.72), which introduces the stress tensor. The local form of this balance is then expressed by (3.76) or (3.77). The most relevant outcome of the balance of moment of momentum is the symmetry of the stress tensor (3.93). Note that in this section also an important class of quantities— the specific quantities—was introduced by (3.66) note particularly their derivative properties (3.67) and (3.68). 

We call the fields (3.114)-(3.116) fulfilling the balances of mass (3.63), (3.65), momentum (3.76), moment of momentum (3.93), and energy (3.107) a thermodynamic process, because only these are of practical interest. Then we denote the fields (3.114) as the thermokinetic process and the fields (3.115) as the responses (we limit to the models with symmetric T (3.93) in more general models we must introduce also the torque M into responses (3.115), cf. Rems. 17, 32). The fields (3.116) are controlled from the outside (at least in principle). Just constitutive equations, which express the difference among materials, represent the missing equations and are relations between (3.114) and (3.115) [6, 7, 9, 10, 23, 34, 38, 40, 41, 44, 45], Referring to Sect. 2.1 we briefiy recall that constitutive equations are definitions of ideal materials which approximate real materials in the circumstances studied (i.e at chosen time and space scales). Constitutive equations may be proposed in rational thermodynamics using the constitutive principles of determinism, local action, memory, equipresence, objectivity, symmetry, and admissibility. 

The balance of mass for the mixture asserts that in any fixed volume V of the mixture the whole mass (sum of mass of all constituents) can be changed only through the flxed surface 9 V as a result of the flow of each constituent a by the velocity Vq,... 

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