Material balance steady state

In unsteady states the situation is less satisfactory, since stoichiometric constraints need no longer be satisfied by the flux vectors. Consequently differential equations representing material balances can be constructed only for binary mixtures, where the flux relations can be solved explicitly for the flux vectors. This severely limits the scope of work on the dynamical equations and their principal field of applicacion--Che theory of stability of steady states. The formulation of unsteady material and enthalpy balances is discussed in Chapter 12, which also includes a brief digression on stability problems. 

Chapter 11. STEADY STATE MATERIAL AND ENTHALPY BALANCES IN POROUS catalyst PELLETS... 

In section 11.4 Che steady state material balance equations were cast in dimensionless form, therary itancifying a set of independent dimensionless groups which determine ice steady state behavior of the pellet. The same procedure can be applied to the dynamical equations and we will illustrate it by considering the case t f the reaction A - nB at the limit of bulk diffusion control and high permeability, as described by equations (12.29)-(12.31). 

A steady-state material balance can be carried out on a small section of length and volume (on the basis of unit cross-sectional area) ia the contactor ... 

The analysis of steady-state and transient reactor behavior requires the calculation of reaction rates of neutrons with various materials. If the number density of neutrons at a point is n and their characteristic speed is v, a flux effective area of a nucleus as a cross section O, and a target atom number density N, a macroscopic cross section E = Na can be defined, and the reaction rate per unit volume is R = 0S. This relation may be appHed to the processes of neutron scattering, absorption, and fission in balance equations lea ding to predictions of or to the determination of flux distribution. The consumption of nuclear fuels is governed by time-dependent differential equations analogous to those of Bateman for radioactive decay chains. The rate of change in number of atoms N owing to absorption is as follows ... 

Amorphous Silicon. Amorphous alloys made of thin films of hydrogenated siUcon (a-Si H) are an alternative to crystalline siUcon devices. Amorphous siUcon ahoy devices have demonstrated smah-area laboratory device efficiencies above 13%, but a-Si H materials exhibit an inherent dynamic effect cahed the Staebler-Wronski effect in which electron—hole recombination, via photogeneration or junction currents, creates electricahy active defects that reduce the light-to-electricity efficiency of a-Si H devices. Quasi-steady-state efficiencies are typicahy reached outdoors after a few weeks of exposure as photoinduced defect generation is balanced by thermally activated defect annihilation. Commercial single-junction devices have initial efficiencies of ca 7.5%, photoinduced losses of ca 20 rel %, and stabilized efficiencies of ca 6%. These stabilized efficiencies are approximately half those of commercial crystalline shicon PV modules. In the future, initial module efficiencies up to 12.5% and photoinduced losses of ca 10 rel % are projected, suggesting stabilized module aperture-area efficiencies above 11%. 

When the basic physical laws are expressed in this form, the formulation is greatly facilitated. These expressions are quite often given the names, material balance, energy balance, and so forth. To be a little more specific, one could write the law of conservation of energy in the steady state as... 

Material and energy balances are based on the conservation law, Eq. (7-69). In the operation of liquid phase reactions at steady state, the input and output flow rates are constant so the holdup is fixed. The usual control of the discharge is on the liquid level in the tank. When the mixing is adequate, concentration and temperature are uniform, and the effluent has these same properties. The steady state material balance on a reacdant A is... 

Material and energy balances of the steady-state model. 

Use of Operating Curve Frequently, it is not possible to assume that = 0 as in Example 2, owing to diffusional resistance in the liquid phase or to the accumulation of solute in the hquid stream. When the back pressure cannot be neglected, it is necessary to supplement the equations with a material balance representing the operating line or curve. In view of the countercurrent flows into and from the differential section of packing shown in Fig. 14-3, a steady-state material balance leads to the fohowing equivalent relations ... 

For a steady-state ciystallizer receiving sohds-free feed and containing a well-mixed suspension of ciystals experiencing neghgible breakage, a material-balance statement degenerates to a particle balance (the Randolph-Larson general-population balance) in turn, it simplifies to... 

The primary function of a continuous thickener is to concentrate sus-penaed solids by gravity settling so that a steady-state material balance is achieved, solids being withdrawn continuously in the underflow at the rate they are supphed in the feed. Normally, an inventory of pulp is maintained in order to achieve the desired concentration. This volume will vary somewhat as operating conditions change on occasion, this inventoiy can be used for storage of sohds when reed and underflow rates are reduced or temporarily suspended. 

Conditions at steady state are determined by heat and material balances. Such balances for a CSTR can be put in the form. 

As another example of the first interac tion, a potential parameter in the analysis of the CSTR is estimating the actual reactor volume. CSTR shown in Fig. 30-7. The steady-state material balance for this CSTR having a sin e reaction can be represented as ... 

For the simplest one-dimensional or flat-plate geometry, a simple statement of the material balance for diffusion and catalytic reactions in the pore at steady-state can be made that which diffuses in and does not come out has been converted. The depth of the pore for a flat plate is the half width L, for long, cylindrical pellets is L = dp/2 and for spherical particles L = dp/3. The varying coordinate along the pore length is x ... 

Temperature gradient normal to flow. In exothermic reactions, the heat generation rate is q=(-AHr)r. This must be removed to maintain steady-state. For endothermic reactions this much heat must be added. Here the equations deal with exothermic reactions as examples. A criterion can be derived for the temperature difference needed for heat transfer from the catalyst particles to the reacting, flowing fluid. For this, inside heat balance can be measured (Berty 1974) directly, with Pt resistance thermometers. Since this is expensive and complicated, here again the heat generation rate is calculated from the rate of reaction that is derived from the outside material balance, and multiplied by the heat of reaction. 

The material balance for steady-state operation of a perfectly mixed reactor is ... 

This program helps calculate the rate of methanol formation in mol/m s at any specified temperature, and at different hydrogen, carbon monoxide and methanol concentrations. This simulates the working of a perfectly mixed CSTR specified at discharge condition, which is the same as these conditions are inside the reactor at steady-state operation. Corresponding feed compositions and volumetric rates can be calculated from simple material balances. 

When collecting meaningful field fractionating column data, the column must not only have constant flow rates, but the flows must give a good material balance (no accumulation). In addition, a steady state condition must exist for the given flow rates. 

Consider a first order, exothermic reaction (aA —> products) in a CFSTR having a constant supply of new reagents, and maintained at a steady state temperature T that is uniform throughout the system volume. Assuming perfect mixing and no density change, the material balance equation based on reactants is expressed as uC g = +... 

A special case of the above equation applies to a continuous steady-state flow process when all of the rate terms are independent of time and the accumulation term is zero. Thus, the differential material balance for any component i in such a process is given by... 

Steady-state material balance is used for cell mass. 

At steady-state condition for chemostat operation, change of concentration is independent of time. Material balance for the fermentation vessel is ... 

At steady-state condition there is no accumulation, therefore the material balance is reduced to ... 

Reactions 2 and 3 regulate the balance of O and O3, but do not materially affect the O3 concentration. Any ozone destroyed in the photolysis step (3) is quickly reformed in reaction 2. The amount of ozone present results from a balance between reaction 1, which generates the O atoms that rapidly form ozone, and reaction 4, which eliminates an oxygen atom and an ozone molecule. Under conditions of constant sunlight, which implies constant /i and /s, the concentrations of O and O3 remain constant with time and are said to correspond to the steady state. Under steady-state conditions the concentrations of O and O3 are defined by the equations d[0]/df = 0 and d[03]/df = 0. Deriving the rate expressions for reactions 1-i and applying the steady-state condition results in the equations given below that can be solved for [O] and [O3]. 

The subsequent fate of the assimilated carbon depends on which biomass constituent the atom enters. Leaves, twigs, and the like enter litterfall, and decompose and recycle the carbon to the atmosphere within a few years, whereas carbon in stemwood has a turnover time counted in decades. In a steady-state ecosystem the net primary production is balanced by the total heterotrophic respiration plus other outputs. Non-respiratory outputs to be considered are fires and transport of organic material to the oceans. Fires mobilize about 5 Pg C/yr (Baes et ai, 1976 Crutzen and Andreae, 1990), most of which is converted to CO2. Since bacterial het-erotrophs are unable to oxidize elemental carbon, the production rate of pyroligneous graphite, a product of incomplete combustion (like forest fires), is an interesting quantity to assess. The inability of the biota to degrade elemental carbon puts carbon into a reservoir that is effectively isolated from the atmosphere and oceans. Seiler and Crutzen (1980) estimate the production rate of graphite to be 1 Pg C/yr. 

The design equations for a CSTR do not require that the reacting mixture has constant physical properties or that operating conditions such as temperature and pressure be the same for the inlet and outlet environments. It is required, however, that these variables be known. Pressure in a CSTR is usually determined or controlled independently of the extent of reaction. Temperatures can also be set arbitrarily in small, laboratory equipment because of excellent heat transfer at the small scale. It is sometimes possible to predetermine the temperature in industrial-scale reactors for example, if the heat of reaction is small or if the contents are boiling. This chapter considers the case where both Pout and Tout are known. Density and Q ut wiU not be known if they depend on composition. A steady-state material balance gives... 

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