Isothermal system

The equilibrium ratios are not fixed in a separation calculation and, even for an isothermal system, they are functions of the phase compositions. Further, the enthalpy balance. Equation (7-3), must be simultaneously satisfied and, unless specified, the flash temperature simultaneously determined. 

Fig. XI-11. Relation of adsorption from binary liquid mixtures to the separate vapor pressure adsorption isotherms, system ethanol-benzene-charcoal (n) separate mixed-vapor isotherms (b) calculated and observed adsorption from liquid mixtures. (From Ref. 143.)...

As a result of the discussion in Chapters 1 to 6, we are now in a position to formulate certain conditions which must be satisfied by any acceptable model for the gaseous phase fluxes in a porous medium. These are very useful, as It turns out that they are sufficiently restrictive to determine completely the formulation of certain problems, without the need to appeal to any particular flux model. All the following conditions refer to isothermal systems. 

When the mean free paths are long compared with all pore diameters, condition (i) above determines the form of the flux relations completely in isothermal systems, and no further modelling is required if is regarded... 

For an isothermal system the simultaneous solution of equations 30 and 31, subject to the boundary conditions imposed on the column, provides the expressions for the concentration profiles in both phases. If the system is nonisotherm a1, an energy balance is also required and since, in... 

Equilibrium Theory. The general features of the dynamic behavior may be understood without recourse to detailed calculations since the overall pattern of the response is governed by the form of the equiUbrium relationship rather than by kinetics. Kinetic limitations may modify the form of the concentration profile but they do not change the general pattern. To illustrate the different types of transition, consider the simplest case an isothermal system with plug flow involving a single adsorbable species present at low concentration in an inert carrier, for which equation 30 reduces to... 

Solution of the model equations shows that, for a linear isothermal system and a pulse injection, the height equivalent to a theoretical plate (HETP) is given by... 

Gas—solids fluidization is the levitation of a bed of solid particles by a gas. Intense soflds mixing and good gas—soflds contact create an isothermal system having good mass transfer (qv). The gas-fluidized bed is ideal for many chemical reactions, drying (qv), mixing, and heat-transfer appHcations. Soflds can also be fluidized by a Hquid or by gas and Hquid combined. Liquid and gas—Hquid fluidization appHcations are growing in number, but gas—soHds fluidization appHcations dominate the fluidization field. This article discusses gas—soHds fluidization. 

Tanks cool, contents partially freeze, and solids drop to bottom or rise to top. This case requires a two-step calculation. The first step is handled as in case 1. The second step is calculated by assuming an isothermal system at the freezing point. It is possible, given time and a sufficiently low ambient temperature, for tank contents to freeze solid. 

Nontrace isothermal systems give the adsorption effect (i.e., significant change in fluid velocity because of loss or gain of solute). Criteria for the existence of simple waves, contact discontinmties, and shocks are changed somewhat [Peterson and Helfferich, J. Phy.s. Chem., 69, 1283 (1965) LeVan et al., AIChE J., 34, 996 (1988) Frey, AJChE J., 38, 1649(1992)]. 

Normally when a small change is made in the condition of a reactor, only a comparatively small change in the response occurs. Such a system is uniquely stable. In some cases, a small positive perturbation can result in an abrupt change to one steady state, and a small negative perturbation to a different steady condition. Such multiplicities occur most commonly in variable temperature CSTRs. Also, there are cases where a process occurring in a porous catalyst may have more than one effectiveness at the same Thiele number and thermal balance. Some isothermal systems likewise can have multiplicities, for instance, CSTRs with rate equations that have a maximum, as in Example (d) following. 

When the two liquid phases are in relative motion, the mass transfer coefficients in eidrer phase must be related to die dynamical properties of the liquids. The boundary layer thicknesses are related to the Reynolds number, and the diffusive Uansfer to the Schmidt number. Another complication is that such a boundaty cannot in many circumstances be regarded as a simple planar interface, but eddies of material are U ansported to the interface from the bulk of each liquid which change the concenuation profile normal to the interface. In the simple isothermal model there is no need to take account of this fact, but in most indusuial chcumstances the two liquids are not in an isothermal system, but in one in which there is a temperature gradient. The simple stationary mass U ansfer model must therefore be replaced by an eddy mass U ansfer which takes account of this surface replenishment. 

Guichardon etal. (1994) studied the energy dissipation in liquid-solid suspensions and did not observe any effect of the particles on micromixing for solids concentrations up to 5 per cent. Precipitation experiments in research are often carried out at solids concentrations in the range from 0.1 to 5 per cent. Therefore, the stirred tank can then be modelled as a single-phase isothermal system, i.e. only the hydrodynamics of the reactor are simulated. At higher slurry densities, however, the interaction of the solids with the flow must be taken into account. 

Stability. The first consideration is stability. Is there a stable steady state The answer is usually yes for isothermal systems. 

The steady-state design equations (i.e., Equations (14.1)-(14.3) with the accumulation terms zero) can be solved to find one or more steady states. However, the solution provides no direct information about stability. On the other hand, if a transient solution reaches a steady state, then that steady state is stable and physically achievable from the initial composition used in the calculations. If the same steady state is found for all possible initial compositions, then that steady state is unique and globally stable. This is the usual case for isothermal reactions in a CSTR. Example 14.2 and Problem 14.6 show that isothermal systems can have multiple steady states or may never achieve a steady state, but the chemistry of these examples is contrived. Multiple steady states are more common in nonisothermal reactors, although at least one steady state is usually stable. Systems with stable steady states may oscillate or be chaotic for some initial conditions. Example 14.9 gives an experimentally verified example. 

The results in this chapter are restricted in large part to steady-state, homogeneous, isothermal systems. More general theories can be developed. The next few sections briefly outline some extensions of residence time theory. 

According to irreversible thermodynamics, the entropy production per unit volume S for an isothermal system can be written... 

For an isothermal system without velocity gradients and chemical reactions, Eq. (36) reduces to... 

The body is basically an isothermal system fine-tuned to 37°C (98.6°F). The skin has major responsibility in temperature maintenance. When the body is exposed to chilling temperatures that remove heat faster than the body s metabolic output can replace it, changes take place in the skin to conserve heat. Conversely, when the body becomes overheated, physiological processes come into play that lead to cooling. 

Thus, in an isothermal system, the mass flow rate depends on the difference in pressures of the gas across the orifice and does not depend upon the thickness of the plate. One may define an area-normalized resistance, R, for mass transfer through the orifice using a generalization of Ohm s law, i.e., Resistance = force/ flux. For Knudsen flow, the force is the pressure difference (analogous to voltage difference in Ohm s law) and the flux is the mass flow per unit area of the hole (analogous to the electrical current density in Ohm s law). Thus, we have... 

Three main flow patterns exist at various points within the tube bubble, annular, and dispersed flow. In Section I, the importance of knowing the flow pattern and the difficulties involved in predicting the proper flow pattern for a given system were described for isothermal processes. Nonisother-mal systems may have the added complication that the same flow pattern does not exist over the entire tube length. The point of transition from one flow pattern to another must be known if the pressure drop, the holdups, and the interfacial area are to be predicted. In nonisothermal systems, the heat-transfer mechanism is dependent on the flow pattern. Further research on predicting flow patterns in isothermal systems needs to be undertaken... 

In order to evaluate this integral it is necessary to express all terms on the right in terms of a single variable. For isothermal systems k is a constant, and this equation can be written as... 

Adiabatic operation implies that there is no heat interaction between the reactor contents and their surroundings. Isothermal operation implies that the feed stream, the reactor contents, and the effluent stream are equal in temperature and have a uniform temperature throughout. The present chapter is devoted to the analysis of such systems. Adiabatic and other forms of non-isothermal systems are treated in Chapter 10. 

For an isothermal system separation of variables and integration gives... 

For isothermal systems, it is occasionally possible to eliminate the external surface concentrations between equations 12.6.1 and 12.6.2 and arrive at a global rate expression involving only bulk fluid compositions (e.g., equation 12.4.28 was derived in this manlier). In general, however, closed form solutions cannot be achieved and an iterative trial and error procedure must be employed to determine thq global rate. One possible approach is summarized below. 

For isothermal systems this equation, together with an appropriate expression for rv, is sufficient to predict the concentration profiles through the reactor. For nonisothermal systems, this equation is coupled to an energy balance equation (e.g., the steady-state form of equation 12.7.16) by the dependence of the reaction rate on temperature. 

Work can be completely converted to heat, but—and this is important—a complete conversion of heat to work is not possible in an isothermal system. This problem is dealt with by the second law of thermodynamics, with its statement on entropy The entropy of an isolated system not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium. ... 

For a trace, isothermal system, we have c = 1, and using the dimensionless system variables for concentrations [Eq. (16-10)], Eq. (16-123) becomes... 

Usually when closed, isothermal systems (N,V,T) are studied, the canonical distribution function is chosen ... 

Up to this point and in the following sections and as long as the contrary is not specified, all the discussion will refer to the study of closed, isothermal systems (N,V,T). Though in the applications of Monte Carlo method to the study of solutions... 

For the more general case of non-isothermal systems, the S VD of Y can still be used to partition the chemical species into reacting and conserved sub-spaces. Thus, in addition to the dependence on c, the transformed chemical source term for the chemical species, S, will also depend on /. The non-zero chemical source term for the temperature,, SY, must also be rewritten in terms of c in the transport equation for temperature. 

The chemical time scales can be defined in terms of the eigenvalues of the Jacobian matrix of the chemical source term.27 For example, for an isothermal system the K x K Jacobian matrix of the chemical source term is given by d S... 

In non-isothermal systems, the Jacobian is usually most sensitive to the value of temperature T. Indeed, for low temperatures, the components of the Jacobian are often nearly zero, while at high temperatures they can be extremely large. 

In order to determine the errors that may be introduced by the Zeldovich model, Miller and Bowman [6] calculated the maximum (initial) NO formation rates from the model and compared them with the maximum NO formation rates calculated from a detailed kinetics model for a fuel-rich (isothermal system was assumed and the type of prompt NO reactions to be discussed next were omitted. Thus, the observed differences in NO formation rates are due entirely to the nonequilibrium radical concentrations that exist during the combustion process. Their results are shown in Fig. 8.1, which indicates... 

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