Control volume within

Conservation is a general concept widely used in chemical engineering systems analysis. Normally it relates to accounting for flows of heat, mass or momentum (mainly fluid flow) through control volumes within vessels and pipes. This leads to the formation of conservation equations, which, when coupled with the appropriate rate process (for heat, mass or momentum flux respectively), enables equipment (such as heat exchangers, absorbers and pipes etc.) to be sized and its performance in operation predicted. In analysing crystallization and other particulate systems, however, a further conservation equation is... 

A differential balance written for a vanishingly small control volume, within which t A is approximately constant, is needed to analyze a piston flow reactor. See Figure 1.4. The differential volume element has volume AV, cross-sectional area A and length Az. The general component balance now gives... 

The strategy for solving fluid dynamics problems begins by putting a control volume within the fluid that matches the symmetry of the macroscopic boundaries, and balancing the forces that act on the system. The system is defined as the fluid that is contained within the control volume V, which is completely surrounded by surface S. Since a force is synonymous with the time rate of change of momentum as prescribed by Newton s laws of motion, the terms in the force balance are best viewed as momentum rate processes. The force balance for an open system is stated without proof as l = 2- -3H-4- -5, where... 

If one chooses a different control volume within the fluid medium and performs a force balance, the same integral expression is obtained because the original control volume was chosen arbitrarily. However, different limits of integration are needed. There is only one way that (8-22) can be satisfied with several different choices for the integration limits—the integrand must vanish. Hence, the microscopic force balance at the continuum level is... 

Since there are several choices for this arbitrarily chosen control volume within the fluid, one can change the limits of each three-dimensional integral to coincide with the boundaries of the system. Equation (e) must be satisfied for each choice of integration limits. This is possible only if one equates the integrands, which yields the microscopic or differential form of the eqnation of continuity. 

The following discussion represents a detailed description of the mass balance for any species in a reactive mixture. In general, there are four mass transfer rate processes that must be considered accumulation, convection, diffusion, and sources or sinks due to chemical reactions. The units of each term in the integral form of the mass transfer equation are moles of component i per time. In differential form, the units of each term are moles of component i per volnme per time. This is achieved when the mass balance is divided by the finite control volume, which shrinks to a point within the region of interest in the limit when aU dimensions of the control volume become infinitesimally small. In this development, the size of the control volume V (t) is time dependent because, at each point on the surface of this volume element, the control volnme moves with velocity surface, which could be different from the local fluid velocity of component i, V,. Since there are several choices for this control volume within the region of interest, it is appropriate to consider an arbitrary volume element with the characteristics described above. For specific problems, it is advantageous to use a control volume that matches the symmetry of the macroscopic boundaries. This is illustrated in subsequent chapters for catalysts with rectangular, cylindrical, and spherical symmetry. 

Mixing time techniques all work on the principle of adding material to the vessel which has different properties from the bulk. Measurements are then made (usually in a controlled volume within the vessel) that show the presence of the added material. The decay of material property fluctuations is used to measure the mixing time for the system. Many of the devices used for measurement of the added liquid are intrusive, and the experimenter must try to minimize the number of probes installed in the vessel in order to ... 

In order to analyze flames and other reacting systems in terms of basic flow principles and chemical kinetics, it is necessary to consider the conditions in a differential control volume within the reacting fluid. To this control volume we must apply the four fundamental conservation principles which are the basis of all physical and chemical problems conservation of mass, momentum, energy, and atoms. For each of these, the fundamental continuity equation for a volume element states that the rate of accumulation of the quantity within the element is equal to the rate of gain due to flow plus the rate of gain due to reaction, i.e., for a quantity denoted by the subscript /c, and referring to unit volume... 

The process is internally reversible within the control volume. 

Elementary single-component systems are those that have just one chemical species or material involved in the process. Filling of a vessel is an example of this kind. The component can be a solid liquid or gas. Regardless of the phase of the component, the time dependence of the process is captured by the same statement of the conservation of mass within a well-defined region of space that we will refer to as the control volume. 

The net rate of mass accumulation within a control volume is equal to the rate at which mass enters the control volume by any process minus the rate at which it leaves the control volume by any process. 

Therefore, the sum of the component balances is the total material balance while the net rate of change of any component s mass within the control volume is the sum of the rate of mass input of that component minus the rate of mass output these can occur by any process, including chemical reaction. This last part of the dictum is important because, as we will see in Chapter 6, chemical reactions within a control volume do not create or destroy mass, they merely redistribute it among the components. In a real sense, chemical reactions can be viewed from this vantage as merely relabeling of the mass. 

The PC version runs comparatively slow on large problems. FIRAC can perform lumped parameter/control volume-type analysis but is limited in detailed multidimensional modeling of a room or gaa dome space. Diffusion and turbulence within a control volume is not modeled. Multi-gas species are not included in the equations of state. 

Fig. 1.2 shows a gas turbine power plant operating on a closed circuit. The dotted chain control surface (F) surrounds a cyclic gas turbine power plant (or cyclic heat engine) through which air or gas circulates, and the combustion chamber is located within the second open control surface (Z). Heat (2b is transferred from Z to Y, and heat (2a is rejected from Y. The two control volumes form a complete power plant. 

The lost work due to irreversibility within the control volume CV is = [(Wcv)rev]x [(W )cvJx... 

In Equation 2-114, is the rate of entropy production within the control volume symbols with dots refer to the time rate of change of the quantity in question. The second law requires that the rate of entropy production be positive. 

In reactor design, we are interested in chemical reactions that transform one kind of mass into another. A material balance can be written for each component however, since chemical reactions are possible, the rate of formation of the component within the control volume must now be considered. The component balance for some substance A is... 

In order to increase the accuracy of the approximation to the convective term, not only the nearest-neighbor nodes, but also more distant nodes can be included in the sum appearing in Eq. (37). An example of such a higher order differencing scheme is the QUICK scheme, which was introduced by Leonard [82]. Within the QUICK scheme, an interpolation parabola is fitted through two downstream and one upstream nodes in order to determine O on the control volume face. The un-... 

Figure 2.9 Determination of the field values on the control volume faces by interpolation within...

The present state of the art in blood pH measurements allows for rapid (1 minute) determination of pH between 6.4 and 8.0 to within at least 0.005 units for whole blood sample volumes < 100 microliters. The temperature of the electrodes and sample is generally controlled to within 0.1 °C for this level of precision and frequent calibration is carried out (in some cases a one point calibration for each sample). The electrodes require (both the glass and external reference) some maintenance due to protein fouling, however this procedure is largely automated. The useful life of an electrode is one year or less and the cost is well over 100 (U.S.) each. New technologies, both electrochemical and non-electrochemical, must compete with this attractive performance and provide for lower operating costs in order to be successful. 

The basic conservation laws, as well as the transport models, are applied to a system (sometimes called a control volume ). The system is not actually the volume itself but the material within a defined region. For flow problems, there may be one or more streams entering and/or leaving the system, each of which carries the conserved quantity (e.g., Q ) into and out of the system at a defined rate (Fig. 1-2). Q may also be transported into or out of the system through the system boundaries by other means in addition to being carried by the in and out streams. Thus, the conservation law for a flow problem with respect to any conserved quantity Q can be written as follows ... 

Fig. 20.1. Control volume of an aquifer, showing the origin of divergence principle. Volume dimensions are dx x dy x dz. The rate at which a chemical component i accumulates within the volume depends on the divergence of the mass fluxes i.e., the rate at which the component s mass is transported into the volume along x and y, less the rate it is transported out.

The input and output terms of equation 1.5-1 may each have more than one contribution. The input of a species may be by convective (bulk) flow, by diffusion of some kind across the entry point(s), and by formation by chemical reaction(s) within the control volume. The output of a species may include consumption by reaction(s) within the control volume. There are also corresponding terms in the energy balance (e.g., generation or consumption of enthalpy by reaction), and in addition there is heat transfer (2), which does not involve material flow. The accumulation term on the right side of equation 1.5-1 is the net result of the inputs and outputs for steady-state operation, it is zero, and for unsteady-state operation, it is nonzero. 

The control volume depicted in Figure 1.3 is for one fixed in position (i.e., fixed observation point) and of fixed size but allowing for variable mass within it this is often referred to as the Eulerian point of view. The alternative is the Lagrangian point of view, which focuses on a specified mass of fluid moving at the average velocity of the system the volume of this mass may change. 

In further considering the implications and uses of these two points of view, we may find it useful to distinguish between the control volume as a region of space and the system of interest within that control volume. In doing this, we consider two ways of describing a system. The first way is with respect to flow of material ... 

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