Transient axial profiles

A theoretical and experimental study of multiplicity and transient axial profiles in adiabatic and non-adiabatic fixed bed tubular reactors has been performed. A classification of possible adiabatic operation is presented and is extended to the nonadiabatic case. The catalytic oxidation of CO occurring on a Pt/alumina catalyst has been used as a model reaction. Unlike the adiabatic operation the speed of the propagating temperature wave in a nonadiabatic bed depends on its axial position. For certain inlet CO concentration multiplicity of temperature fronts have been observed. For a downstream moving wave large fluctuation of the wave velocity, hot spot temperature and exit conversion have been measured. For certain operating conditions erratic behavior of temperature profiles in the reactor has been observed. 

Unsteady-State Analysis Including Axial Dispersion. As in the previous unsteady-state analysis, the effects of placental barrier tissue oxygen consumption are neglected in this study. For the unsteady-state analysis of the model in which axial dispersion was included, one study was conducted. This study involved placing a step change on the maternal blood velocity to a new maternal blood velocity of 0.125 times the normal in an attempt to determine the effects of axial dispersion on the system at low maternal blood velocities. The discussion of this study is divided into the following two parts first, the effect of axial dispersion on the response of the fetal blood end capillary oxygen concentration, and second, the effect on the transient axial profiles. 

Figure 14.3.6.A-1 illustrates the transient calculation of the meso-scale fluctuations around a statistically stationary state and shows the a posteriori timeaveraging period. Figure 14.3.6.A-2 shows radially averaged axial profiles of the concentration of A in the gas phase and in the liquid phase. The comparison with the conventional ideal flow models shows a behaviour typically between the extremes of plug flow and complete mixing, illustrating the importance of accounting for the details of the flow pattern.

Figure 28 shows comparisons of the transient gas and solid axial temperature profiles for a step-input change with the full model and the reduced models. The figure shows negligible differences between the profiles at times as short as 10 sec. Concentration results (not shown) show even smaller discrepancies between the profiles. Additional simulations are not shown since all showed minimal differences between the solutions using the different linear models. Thus for the methanation system, Marshall s model reduction provides an accurate 2Nth-order reduced state-space representation of the original 5/Vth-order linear model. 

Fig. 28. Transient axial temperature profiles during start-up, type I conditions.

Consider a long circular duct in which an incompressible, constant-property fluid is initially at rest. Suddenly a constant pressure gradient is imposed. The axial momentum equation that describes the transient response of the velocity profile for this situation is... 

Fig. 4.9 Transient nondimensional axial velocity profiles in a long circular duct, responding to a suddenly imposed pressure gradient. The fluid is initially at rest.

Axial dispersion can affect measurements of decay and growth rates of transients of interest. In Figure 5 is sketched the concentration of a transient, initially formed as a square wave by a light pulse of uniform intensity from — L < x < L and zero elsewhere. As shown in Figure 6, at later times the profile becomes smoothed by diffusion. As the purge flow pushes reactive species past the pinhole at x = 0, the spatial dependence of the concentration becomes a time-varying concentration that will contribute to any time variation caused by kinetics. 

A model for transient simulation of radial and axial composition and temperature profiles In pressurized dry ash and slagging moving bed gasifiers Is described. The model Is based on mass and energy balances, thermodynamics, and kinetic and transport rate processes. Particle and gas temperatures are taken to be equal. Computation Is done using orthogonal collocation In the radial variable and exponential collocation In time, with numerical Integration In the axial direction. 

The transient behavior of a continuous countercurrent multicomponent system was considered in detail by Rhee, Aris and Amimdson [22,23] from the perspective of the equilibrium theory, i.e., assuming that axial dispersion and the mass transfer resistances are negligible and that equilibrium is established everywhere, at every time along the colinnn. The final steady-state predicted by the equilibrium theory is simply a uniform concentration throughout the colimm, with a transition at one end or the other. Therefore, the equilibriinn theory analysis is of lesser practical value for a coimtercurrent system, which normally operates rmder steady-state conditions, than for a fixed-bed (i.e., an SMB) system, which normally operates under transient conditions. The equilibrium theory analysis, however, reveals that, under different experimental conditions, several different steady-states are possible in a coimtercurrent system. It shows how the evolution of the concentration profiles may be predicted in order to determine which state is obtained in a particular case. 

We consider the material to be in a visco-elastic state. A transient stress distribution will therefore occur after each change of the applied stress and/or temperature profile. Only very small local deformations and, thus, short times are necessary to adjust local stresses to the general continuity condition. After the transition, the whole specimen will creep in tension under the action of a radial distribution of axial stresses o(r) which assures, respecting the creep rate equation, an equal creep rate for the whole specimen. From the viewpoint of continuum mechanics, a chemically homogeneous specimen with a radial temperature gradient is indeed a "graded material" inasmuch as each coaxial shell offers a different resistance to the applied stress and has a different time constant for relaxation. We may speak of a "thermally graded material". 

The boundary conditions are zero velocity at the walls and zero slope at any planes of symmetry. Analytical solutions for the velocity profile in square and rectangular ducts are available but cumbersome, and a numerical solution is usually preferred. This is the reason for the transient term in Equation 16.7. A flat velocity profile is usually assumed as the initial condition. As in Chapter 8, is assumed to vary slowly, if at all, in the axial direction. For single-phase flows, u can vary in the axial direction due to changes in mass density and possibly to changes in cross-sectional area. The continuity equation is just AcUp = constant because the cross-channel velocity components are ignored. 

The false-transient method can be applied to convective diffusion equations in a manner similar to that used for velocity profiles. Finite-difference approximations are written for the spatial derivatives. Second-order approximations can be used for first derivatives since they involve only the same five points needed for the second derivatives. The result is a set of simultaneous ODEs with (false) time as the independent variable. The computational template of Figure 16.3 is unchanged. The next two examples illustrate its application to problems where axial diffusion is negligible. Such problems are also readily solved by the method of lines as described in Chapter 8. Cases with significant axial diffusion are troublesome for the method of lines and require special boundary conditions for the method of false transients. They are treated in Section 16.2.4. 

Figure 23. Axial oxygen concentration profiles for the fetal channel for transient conditions after step change V2 = 0.125 normal for case including axial gradients...

Figure 24 shows the axial oxygen concentration profiles for the fetal channel for the same transient conditions that were just discussed but for the case which neglected axial dispersion. These may be compared directly with the profiles in Figure 23 which represent the case which includes axial dispersion. As in the previous discussion, comparison of these figures shows that at this reduced maternal blood velocity the results are essentially the same, and the effects of axial diffusion are negligible. 

The peculiar aspect of this study is represented by the in-situ measurement of thermal effects produced throughout the reactor under real operating conditions. In this respect, it is important to verify that the thermal map of the surface of the catalyst bed, obtained by thermography, describes reliably the phenomena occurring in the bulk of the bed. For this purpose, the external temperature profile was compared with the profile obtained by an axial multiple thermocouple placed inside the catalyst bed. It was observed that both in the steady and in the transient state, the two profiles have similar shape, although the temperatures are not identical due to axial gradient. 

Knowing the concentration profile as given in eq. (12.2-6), the transient flux across any plane perpendicular to the capillary axial direction is calculated from the Knudsen flux equation (12.2-2), that is ... 

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