Piston-flow region

The New Model, Mechanistic Cyclic Form In this form we follow each slug, the wake of the slug, and the piston-flow region from formation at the bottom of the fluidized bed to the top of the bed where the slug bursts. Thus the steady-state is approached from an initial state in which the bed is fluidized with non-reacting gas and then reacting gas is admitted at zero time. Only the final steady-state solutions are presented here for comparison with the time-averaged version, the two-phase model of Hovmand and Davidson and the experimental data. 

Figure 1. Slugging fluidized bed. (A) well-mixed wake region (B) piston-flow region.

Piston-Flow Region of Particulate Phase Since piston-fl( obtains, accumulation terms can, in this case, be ignored -although qualifications would be made to this statement if space permitted. However there remains a distance - velocity lag which needs to be taken into account. If z is the length dimension and z = 0 at the bottom of the we I I-mixed wake. 

Material Balances on Cel I 1 The slug-balance is the same as equation (18). The wake and piston-flow regions are no longer of constant dimensions but of dimensions which vary with time. The wake begins to form immediately and when it has grown to full size, the piston-flow region commences to form. 

Piston-flow Region This region commences to form at time wt and the region extends from z = 0 to z = C(t/T)-w]TD. Equation (21) applies with this proviso. 

Kqp = volumetric rate of gas exchange between cloud-wake and piston-flow region per unit slug volume, s ... 

In addition to a near-shock and an acoustic region, Deshaies and Clavin (1979) distinguished a third—a near-piston region—where nonlinear effects play a role as well. As already pointed out by Taylor (1946), the near-piston flow regime may be well approximated by the assumption of incompressibility. For each of these regions, Deshaies and Clavin (1979) developed solutions in the form of asymptotic expansions in powers of small piston Mach number. These solutions are supposed to hold for piston Mach numbers lower than 0.35. 

Example 9.6 Compare the nonisothermal axial dispersion model with piston flow for a first-order reaction in turbulent pipeline flow with Re= 10,000. Pick the reaction parameters so that the reactor is at or near a region of thermal runaway. 

The values of Sherwood number fall below the theoretical minimum value of 2 for mass transfer to a spherical particle and this indicates that the assumption of piston flow of gases is not valid at low values of the Reynolds number. In order to obtain realistic values in this region, information on the axial dispersion coefficient is required. 

Whether the conditions for the maximum and minimum values of /(C R) correspond with the practical mixing phenomena is considered. The condition under which 1(0, R) has the maximum value is realized when the substance in a region does not disperse to any other region or flow out to another region without dispersion. In other words, the condition for the latter case occurs when the piston flow is established. On the other hand, the condition under which the mutual entropy assumes the minimum value is realized when the substance in a region flows out equally to all regions. In other words, this condition occurs when perfect mixing flow is established. Therefore, the... 

In the cross-flow or piston exchange model (PE) the liquid phase is assumed to be devided into a plug flow region and stagnant region in contact with the flowing liquid with which mass can be exchanged 6. The physical interpretation can be shown as Case I and Case II in Fig.l. 

So far as gas-exchange piston-tlow region is concerned, this is considered to be determined by the solids movement, allowance being made for gas flow with respect to stationary solids. 

With the intake valve open, the piston movement to the right creates a low pressure region in the cylinder, which causes air and fuel to flow through the intake valve to fill the cylinder. 

For different regions in the flow field in front of an expanding piston, separate solutions in the form of asymptotic expansions may be developed. An overall solution can be constructed by matching these separate solutions. This mathematical technique was employed by several authors including Guirao et al. (1976), Gorev and Bystrov (1985), Deshaies and Clavin (1979), Cambray and Deshaies (1978), and Cambray et al. (1979). 

To prevent this kink, Gorev and Bystrov (1985) suggested a correction by a properly chosen coordinate transformation. The substitution was chosen in such a way that the equations after linearization describe the desired behavior in the near-shock region during the period when the influence of the correction fades gradually towards the piston. In this way, Gorev and Bystrov (1985) obtained an approximate solution which holds for the entire flow field. 

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