Variable cross-section

Diffuser A device of variable cross-sectional area used to spread airflow into a space. 

Variable volume (e.g., a constant-pressure reactor), V t) Variable cross section, A (z)... 

The emphasis in this chapter is on the generalization of piston flow to situations other than constant velocity down the tube. Real reactors can closely approximate piston flow reactors, yet they show many complications compared with the constant-density and constant-cross-section case considered in Chapter 1. Gas-phase tubular reactors may have appreciable density differences between the inlet and outlet. The mass density and thus the velocity down the tube can vary at constant pressure if there is a change in the number of moles upon reaction, but the pressure drop due to skin friction usually causes a larger change in the density and velocity of the gas. Reactors are sometimes designed to have variable cross sections, and this too will change the density and velocity. Despite these complications, piston flow reactors remain closely akin to batch reactors. There is a one-to-one correspondence between time in a batch and position in a tube, but the relationship is no longer as simple as z = ut. 

FIGURE 3.1 Differential volume elements in piston flow reactors (a) variable cross section (b) constant cross section. 

The units on A are mol/(m s). This is the convective flux. The student of mass transfer will recognize that a diffusion term like —3>Adaldz is usually included in the flux. This term is the diffusive flux and is zero for piston flow. The design equation for the variable-density, variable-cross-section PFR can be written as... 

Example 3.1 Find the fraction unreacted for a first-order reaction in a variable density, variable-cross-section PER. 

Repeat the numerical solution in Example 3.2 for a reactor with variable cross section, Ac = Ainia exp(Bz). Using the numerical values in that example, plot the length needed to obtain 50% conversion versus B for —1 < B < 1 (e.g. 2 = 0.3608 for B = 0). Also plot the reactor volume V versus B assuming = 1. 

For completeness, axial dilfusion and variable cross-section terms were included in Equations (8.74) and (8.75). They are usually dropped. Also, the variations in and X are usually small enough that they can be brought outside the derivatives. The primary utility of these equations, compared with Equations (8.12) and (8.52), is for gas-phase reactions with a signihcant pressure drop. 

SRJ Effective reaction rate for a tubular reactor with variable cross section 3.8... 

The two BCs of the TAP reactor model (1) the reactor inlet BC of the idealization of the pulse input to tiie delta function and (2) the assumption of an infinitely large pumping speed at the reactor outlet BC, are discussed. Gleaves et al. [1] first gave a TAP reactor model for extracting rate parameters, which was extended by Zou et al. [6] and Constales et al. [7]. The reactor equation used here is an equivalent form fi om Wang et al. [8] that is written to be also applicable to reactors with a variable cross-sectional area and diffusivity. The reactor model is based on Knudsen flow in a tube, and the reactor equation is the diffusion equation ... 

In the capillaries of variable cross section or the lattice of the interconnected sites and bonds of the various size the situation is essentially different. At the beginning, the mercury is at the external surface of particles of a sample, and only the pores that are directly contiguous to external surface can be filled according to the considered model of a bunch of capillaries. The cavities with windows of size rw (F>HgX which are adequate to an equilibrium condition but inside the bulk of a sample, can be filled only under a condition of their connection to an external surface through a circuit of cavities with windows of size V > rwP already filled with mercury. Therefore, the condition for the filling of a cavity with a window of the size rm can be expressed as the requirement of a direct contact of a considered cavity with mercuiy. Accordingly, under each pressure PHg, all windows of size rWl are only poten-... 

Energy transfer along the polynucleotide chain may be one factor in the variable cross section for reaction of the sites on the chain. A poly... 

The aim of this study is to show that the law of density variation in detonating material may accelerate the propagation of the deton wave, raise the temp pressure behind the wave front, and greatly influence die motion of the deton gases. Similar effects are bound to occur in tubes of variable cross-section with a const density of detonating material... 

Note Disregard the variable bathymetry of the lake. Assume that the cross-sectional area of the lake is A0 at all depths. See Problem 22.4 for variable cross-sectional area. 

Fig. 7.9 Cylindrical duct with a variable cross-sectional area.

The plug-flow problem may be formulated with a variable cross-sectional area and heterogeneous chemistry on the channel walls. Although the cross-sectional area varies, we make a quasi-one-dimensional assumption in which the flow can still be represented with only one velocity component u. It is implicitly assumed that the area variation is sufficiently small and smooth that the one-dimensional approximation is valid. Otherwise a two- or three-dimensional analysis is needed. Including the surface chemistry causes the system of equations to change from an ordinary-differential equation system to a differential-algebraic equation system. 

Fig. 16.3 Illustration of a plug-flow channel with variable cross sectional area.

The critical nucleus of a new phase (Gibbs) is an activated complex (a transitory state) of a system. The motion of the system across the transitory state is the result of fluctuations and has the character of Brownian motion, in accordance with Kramers theory, and in contrast to the inertial motion in Eyring s theory of chemical reactions. The relationship between the rate (probability) of the direct and reverse processes—the growth and the decrease of the nucleus—is determined from the condition of steadiness of the equilibrium distribution, which leads to an equation of the Fourier-Fick type (heat conduction or diffusion) in a rod of variable cross-section or in a stream of variable velocity. The magnitude of the diffusion coefficient is established by comparison with the macroscopic kinetics of the change of nuclei, which does not consider fluctuations (cf. Einstein s application of Stokes law to diffusion). The steady rate of nucleus formation is calculated (the number of nuclei per cubic centimeter per second for a given supersaturation). For condensation of a vapor, the results do not differ from those of Volmer. 

The flux is slightly asymmetric, and this can be attributed to lack of perfection in shaping the lead. More important is the clear contribution from backscattering which has fashioned the shape of the electron flux so that there is a relatively large area of uniformity. Silhouettes of typical fruit sizes have been indicated so that one can visualize why uniformity can be achieved. Obviously, the pattern at other positions in this variable cross-section chute is different. 

Figure 1. Scheme of liquid evaporation a) a simple cylinder capillary connected to an infinite reservoir b) 2D capillary of variable cross-section c) a group of interconnected capillaries. 

To augment the effective heat transfer, fins (extended surfaces) are often used in practical applications. To understand the heat flow through a fin requires a knowledge of the temperature distribution in the fin. Consider the variable cross section fin shown in Fig. 3.9. 

Equation (3.53) is the one-dimensional fin equation for fins with variable cross section. This special case occurs when A is constant. Let this constant be equal to a = Px, where P is the perimeter. In this case, da/dx = P. The one-dimensional fin equation then becomes... 

An aqueous solution with a specific gravity of 1.12 flows through a channel with a variable cross section. Data taken at two axial positions in the channel are shown here. 

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