Function of axial position

Most of this chapter assumes that the mass flow rate down the tube is constant i.e., the tube wall is impermeable. The reactor cross-sectional area Ac is allowed to vary as a function of axial position, Ac = Adz). Figure 3.1 shows the system and indicates the nomenclature. An overall mass balance gives... 

The method of lines formulation for solving Equation (8.52) does not require that T aii be constant, but allows T aiiiz) to be an arbitrary function of axial position. A new value of T aii may be used at each step in the calculations, just as a new may be assigned at each step (subject to the stability criterion). The design engineer is thus free to pick a T au z) that optimizes reactor performance. 

Dispersed flow model. To calculate the actual quality, vapor temperature, and wall temperature, or heat flux, as functions of axial position beyond dryout... 

A second equation is needed to determine the surface tension as a function of axial position. We adopt the quasistatic assumption that a is a unique equilibrium function of the surface excess concentration, T, even during dynamic events (17). A surface species continuity balance dictates how T varies along the interface. Upon neglect of surface diffusion and for h <1, the steady state form of this balance is... 

To evaluate the effect of pressure drop on performance, differential equations for the pressure drop (15.2-11), material balance (15.2-4), and energy balance (15.2-10) must be integrated simultaneously to solve for P,fA, and T as functions of axial position, x ... 

Solution of the system of equations in (3) and (4), subject to ary constraints imposed ty (2). The solution in same cases may be possible analytically, but, mare generally numerical methods are required. The solution may give, for example, Ft as a function of axial position or volume, or C as a function of time. 

The differential equation for dispersion in a cylindrical bed of voidage e may be obtained by taking a material balance over an annular element of height SI, inner radius r, and outer radius r + Sr (as shown in Figure 4.5). On the basis of a dispersion model it is seen that if C is concentration of a reference material as a function of axial position /, radial position r, time t, and DL and DR are the axial and radial dispersion coefficients, then ... 

Figure 1. Collection efficiency (f)for a denuder of annular geometry in which only the inner surface of the annulus is a sink as a function of axial position X (see equation 14). The curves are based on numerical data in reference 54. From top to bottom, the traces correspond to dj/d0 = 1.0, 0.5, 0.25, 0.1, 0.05,...

Figure 5. Effect of varying inlet pressure on liquid holdup as a function of axial position in a trickle bed.

In order to complete the description of the system, the pressure within the reactor must be known. In the present state of knowledge, it is a sufficient approximation to consider the pressure in a cross section of the reactor as constant, and to calculate it as a function of axial position... 

The value of for Catalyst C indicates that thiophene conversion will be much less than for Catalyst A , and Figure 4 shows that the conversion of thiophene is only about 50% on the fresh catalyst. In fact, the disappearance of thiophene and the active site concentration as a function of axial bed position for Catalyst C is nearly linear. Indeed, Figure 5 suggests a fairly uniform profile for active sites as a function of axial position. 

Figure 4.13 Electron density (cm ) as a function of axial position and methane flow rate in low pressure cascade arc torch (LPCAT) 8.00 A, 2000 seem argon, and 560mtorr (75 Pa).

For automotive exhaust gas treatment. Oh [42] presented a transient, one-dimensional single-channel model that predicts the temperature and species concentrations as a function of axial position and time in both solid and gas phase. Uniform flow distribution was... 

Fig. 5.15. Coverage of the noble metals in the three-way catalyst by various surface species as a function of axial position in the monolith at steady-state conditions and a gas inlet temperature of 500 K. In the middle part, where acetylene and CO have largely been converted, NO dissociation results in a high coverage of N-atoms. O-atoms become dominant when the pollutant conversion is complete (from Marin and Hoebink [57]).

The same tendency as with longer reactors is observed when lowering the gas velocity. Fig 11.10 shows dry hydrogen purity as function of axial position in a 4 meter long reactor at t = 200 s, and the conversion is strongly dependent of the gas velocity. The C02-capture kinetics are dependent on the fractional conversion of the C02-acceptor. Thus, the sorption kinetics will be slower with time. This lead to lower conversion in the reactor as shown in Fig 11.11. 

Fig. 11.11. Dry hydrogen mole fraction as a function of axial position at different...

Fiiiure 6. Optical lluorescencc spectroscopy results showinj the sum of the principal stresses us a function of axial position along the surface of the specimen, at a fixed depth of 52 Mm. ... 

The hot spot in a packed-bed reactor normally occurs at the centerline. Figure 9.3 shows the maximum centerline temperature as a function of wall temperature. The results are remarkably insensitive to the inlet temperature, and it is clear that Twaii rather than Tin must be used for control. Figure 9.4 shows the centerline temperature as a function of axial position. The maximum centerline temperature is insensitive to the inlet temperature, varying by about 3 K for inlet temperatures that vary by 200 K. The conversions for the two cases, 1 - flout/flin, are essentially identical as well, 92.0 and 92.5%, respectively. 

FIGURE 12.3 (a) variation of meniscus curvature as a function of axial position [7] (b) typical liquid and vapor pressure distributions in a heat pipe [7],... 

The first consideration is the very general one where both the gas-phase flow rate, G, and the solute concentration, Cy, are functions of axial position, but plug flow prevails. This is roughly the situation when the solute is transferred from a concentrated gas stream. The following governing equations can be written. 

Step 1. Use inlet conditions as a starting point to generate graphs of temperature and conversion as a function of axial position z, or trx. 

Then, separate variables and integrate the ODE for Ca, bulk as a function of axial position z through the tubular duct ... 

The PEP technique enables one to measure concentration profiles of labelled molecules (reactants and/or products) within reactors as a function of axial position and time. Once these profiles have been obtained, they provide substantial information on the reaction kinetics and involved intermediates. We will illustrate this with our research on the understanding of the mechanism of the ammonia oxidation on platinum based catalysts. 

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