Uniform Open Systems

The previous sections were concerned with isolated chemical reaction systems but most of the results are extendable to other uniform closed systems. Open systems on the other hand present a distinct behavior as 

The conservation equations for the chemical species can be written immediately as 

We wish to show that the initial value problem (1.8.1), (1.8.2), (1.8.4) has a solution for all times. The existence proof is easily carried out by the fixed point method of Section 1.3. The first task is to obtain a priori bounds for c, . c, T Multiplying Eq.(1.8.1) by yn, a quantity defined by Eq. (1.2.9), and summing over i there is obtained 

The last equation shows that the concentrations are subject to 

To apply the fixed point method we consider the Banach space of vector valued functions u ci(tX. ..,CN(t),T t)) continuous in some interval [0, with norm 

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The solutions of Eqs.(2.1.34), (2.1.35) with appropriate boundary conditions such as Eqs. (2.4.3), (2.4.4) will be called the steady states of the system. Various properties of the steady states, such as the invariant manifolds and a priori bounds, the existence and uniqueness of solutions, the asymptotic behavior, and the stability will be treated in the sections that follow. There is a strong similarity in the properties of the uniform open systems investigated in Sections 1.8,1.9 and the distributed systems to be studied now. In both types of systems the interplay between reaction and transport rates (or flow rates) creates the possibility of multiple steady states for certain types of reaction kinetics. Furthermore, the conditions for uniqueness and stability of the steady state have a common mathematical and physical basis. 

As in the case of a single reaction, Section 2.3, we shall show that when the reaction rates are small compared to the transport rates the steady state is unique. In this respect, the situation is analogous with that of the uniform open systems of Section 1.9. 

It is hoped that the monograph will be of interest to both the theoretical engineer and the applied mathematician. The former will notice that chemical reactions have been considered in the framework of three specific systems, the uniform isolated system (batch reactor), a uniform open system (stirred-tank reactor), and a distributed system (catalyst pellet). However, most of the methods presented should be useful in analyzing other systems such as fixed bed reactors. The complex systems... 

An amazing paradox The lawyer who was responsible for hiding millions of dollars in assets, for obscuring the true owner of the Jasco patents in the United States, was dreaming up a uniform patent system under which the fruits of Europe s scientific labors would be open to all. 

When the subsurface materials are uniform and isotropic (no gradational changes or confining layers) the airflow pathways are also uniform. The airflow paths developed for an open system and a covered system are shown in Figure 10.5. Selection of covered or uncovered is determined by the air paths necessary to contact the contamination. At some sites, inlet vent wells are installed to ensure air entry at specific locations. 

Chemical reactions with autocatalytic or thermal feedback can combine with the diffusive transport of molecules to create a striking set of spatial or temporal patterns. A reactor with permeable wall across which fresh reactants can diffuse in and products diffuse out is an open system and so can support multiple stationary states and sustained oscillations. The diffusion processes mean that the stationary-state concentrations will vary with position in the reactor, giving a profile , which may show distinct banding (Fig. 1.16). Similar patterns are also predicted in some circumstances in closed vessels if stirring ceases. Then the spatial dependence can develop spontaneously from an initially uniform state, but uniformity must always return eventually as the system approaches equilibrium. 

The final possibility, a uniformly interesting movie, would have to depict a process with thousands or millions of critical steps occuring in a definite order, each step necessary to understand the next, as in an industrial process, the functioning of a digital computer, or the development of an embryo. Enzymes, having been optimized by natural selection, may be expected to have somewhat complex mechanisms of action, perhaps with several equally important critical steps, but not with thousands of them. There is reason to believe that processes with thousands of reproducible non-trivial steps usually occur only in systems that are held away from thermal equilibrium by an external driving force. They thus belong to the realm of complex behavior in continuously dissipative open systems, rather than to the realm of relaxation processes in closed systems. 

There are central embracing characteristics for the engineering approach deployed during the development of the following basic plant concepts. Examples are presented of plants with a uniform appearance, standardized pitch dimensions, interfaces, etc., either as closed systems or as open systems with interfaces to other suppliers. This strategy is completely different from the hybrid plant concepts. 

Equilibrium in a multiphase system implies thermal, mechanical, and material equilibrium. Thermal equilibrium requires uniformity of temperature throughout the system, and mechanical equilibrium requires uniformity of pressure. To find the criterion for material equilibrium, we treat a two-phase system and consider a transfer of dn moles from phase p to phase a. First, we regard each phase as a separate system. Because material enters or leaves these phases, they are open systems and we must use Eq. (4) to write their change in internal energy ... 

Transport problems in discontinuous (heterogeneous) system discuss the flows of the substance, heat, and electrical energy between two parts of the same system. These parts or phases are uniform and homogeneous. The two parts make up a closed system, although each individual part is an open system, and a substance can be transported from one part to another. There is no chemical reaction taking place in any part. Each part may contain n number of substances. For example, thermal diffusion in a discontinuous system is usually called thermal osmosis. If the parts are in different states of matter, there will be a natural interface. However, if both parts are in liquid or gas phases, then the parts are separated by a porous wall or a semi-permeable membrane. 

We note the at the outset that equilibrium between two or more phases, considered as coexisting open systems with no rigid partitions, requires minimally the uniformity of temperature and of pressure throughout the entire system. This makes it apposite to deal with the Gibbs free energy as the function of state for such a system. We also restrict ourselves to mechanical work the generalization to other types of work is dealt with in Exercise 2.1.5. 

Let us find djS in such an open system where certain processes cause changes in the chemical composition of the system. If uniformity and equilibrium in temperature and pressure distribution (but not the chemical composition) are achieved inside the system, and the exchange processes with the surrounding medium proceed in an equilibrium manner, then... 

It was the work of Josiah Willard Gibbs that introduced the concept of the thermodynamics of multi-component systems and applied the ideas to the behavior of chemical systems. A homogenous system is one in which the system properties are uniform throughout. An open system is one in which mass may be transferred between phases. We can then write the fundamental equation defining the Gibbs free energy function, G, for this system. 

The extraction and preparation of petroleum and gas, which begin at the opening of the oil wells and end at the preparation units, follow a uniform technological system. There are many technological schemes of petroleum preparation. However, they are usually considered together with the petroleum extraction systems at the oil wells. 

When (4-5) is applied to a closed system consisting of two or more phases in equilibrium at uniform temperature and pressure, where each phase is an open system capable of mass transfer with another phase ... 

Closed systems provide for complete separation between the environment in which personnel (uniformly accepted as the primary source of contamination in aseptic environments) are located from that in the materials are processed. Theoretically, if a sterile BPC could be processed in its entirety within closed systems, there would no possibility of microbial contamination. In marked contrast to the closed system is the open system , perhaps best defined by what it is not. Essentially, an open system lacks one or more of the features of a closed system, thus leaving it vulnerable to the potential ingress of contamination. One substantial issue associated with these definitions is establishing that a system remains closed over the length of the production campaign. 

In order to define the energy-momentrun tensor, matter is assruned to be uniformly distributed, as in an ideal gas, with stellar velocities orders of magnitude less than the velocity of light. However, a stable Boltzmann distribution of this type cannot persist in an open system from which entire stars may escape. Einstein therefore proposed a spatially closed continuum of constant curvature. The cosmological constant served to define both the mean density of the equilibrium distribution and a radius of the closed... 

We now consider the well-stirred open system of Section 2.8 with the continuous phase vector represented by the spatially uniform Y t) in the domain Q, and at the entrance region Integrating Eq. (2.9.1) over the region and recognizing that the diffusive flux Jy must vanish everywhere, we obtain the equation... 

We shall consider here a population of particles distinguished from one another by a finite dimensional vector x of internal coordinates and distributed uniformly in space. Further, we shall be concerned with the open system of Section 2.8 whose behavior is dictated by the population balance equation (2.8.3). Thus the number density in the feed,/i jn(x), may be assumed to be Nff x) where Nf is the total number density in the feed stream and /(x) is probability density of particle states in it. It will also be assumed that the continuous phase plays no role in the behavior of the system. Relaxing this assumption does not add to any conceptual difficulty, although it may increase the computational burden of the resulting simulation procedure. 

For close systems with instantaneous diffusion, steady states are either stable or unstable depending on whether the solid phase consists in a single grain or in N grains (N>1). For open systems and uniform liquid phases, steady states are unstable saddle-points. For close systems with diffusion, results are similar to the first case ones. For open systems with diffusion, steady states are unstable and uniform. 

Nevertheless, this self-organization does not contradict the second law of thermodynamics because the total entropy of the open system keeps increasing, but this increase is not uniform throughout disorder. In fact, such dissipative structures are the islands (fluctuations) of order in the sea (background) of disorder, maintaining (and even increasing) their order at the expanse of greater disorder of their environment. 

The other approach is to use a closed slot draw system. The best example of this is in Reifenhauser lines. In this approach the system is essentially sealed or closed from the spinneret to the laydown table. Flow through the slot comes from the quench above and the low pressure created by the vacuum box below the belt. This system generally results in fabrics with excellent uniformity once the air flows are balanced correctly. However, the filament velocities that can be obtained are much less than with the open system. 

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