Flows homogeneous

For homogeneous flow in a pipe of diameter D, the differential form of the Bernoulli equation (6-15) rearranges to... 

Frictional Pressure Drop Usually this does not have a significant effect on the reaclor size, except perhaps when the flow is two-phase. Some approximate relations fbe cited that are adequate for pressure-drop calculations of homogeneous flow reactions in pipelines. The pressure drop is given by... 

The same procedure may be applied in principle to design of forced-recirculation reboilers with shell-side vapor generation. Little is known about two-phase flow on the shell side, out a reasonable estimate of the fric tion pressure drop can be made from the data of Diehl and Unruh [Pet Refiner, 36(10), 147 (1957) 37(10), 124 (1958)]. No void-fraction data are available to permit accurate estimation of the hydrostatic or acceleration terms. Tnese may be roughly estimated by assuming homogeneous flow. 

The momentum balance for homogeneous flow can be factored to a form which enables integration as ... 

For homogeneous flow, the equivalent specific volume is the same as the homogeneous specific volume and the enthalpy is given by ... 

The flows of gas and solid particles are assumed to be parallel. This means that the velocity of gas f and the velocity of solid particles c are both in the direction of the pipeline. On the other hand, because we do not assume a homogeneous flow at this stage of the theory, the absolute values of v and c may vary across the cross-section of the pipe. 

The way in which the force /j j is modeled clearly determines the type of the pneumatic flow this has been discussed earlier in Section 14.2.2, where we considered the classification of different types of flow. In the following we will give a detailed description for the force in a way that suits a particular type of flow. This approach will be adequate for so-called dilute-phase flow or, more generally speaking, for homogeneous flow where the particles move separately. 

Modeling of the force was actually based on the force balance of a single solid particle extending the result to cover a larger set of solid particles, we obtained Eq. (14.105). This method of modeling is adequate for a homogeneous flow where the particles move separately, not as a new kind of solid-phase structure. Weber has written the friction force in the form... 

Homogeneous flow may be described using volumetric gas quality. 

Fig. 5.22a,b Comparison of measured void fractions by Triplett et al. (1999b) for circular test section with predictions of various correlations (a) homogeneous flow model (b) Lockhart-Martinelli-Butterworth (Butterworth 1975). Reprinted from Triplett et al. (1999b) with permission... 

In the homogeneous flow model, pa is the homogenous mixture density defined... 

The two-phase pressure drop was measured by Kawahara et al. (2002) in a circular tube of d = too pm. In Fig. 5.30, the data are compared with the homogeneous flow model predictions using the different viscosity models. It is clear that the agreement between the experimental data and homogeneous flow model is generally poor, with reasonably good predictions (within 20%) obtained only with the model from Dukler et al. (1964) for the mixture viscosity. 

Fig. 5.30 Comparison of the two-phase frictional pressure gradient between micro-channel data and homogeneous flow model predictions using different viscosity formulations. Reprinted from Kawahara et al. (2002) with permission...

Similar void fraction data can be obtained in micro-channels and conventional size channels, but the micro-channel void fraction can be sensitive to the inlet geometry and deviate significantly from the homogeneous flow model. 

A homogeneous flow basis must be used when thermodynamic equilibrium is assumed. For furtl er simplification it is assumed there will be no reaction occurring in the pipeline. The vapor and liquid contents of the reactor are assumed to be a homogeneous mass as they enter the vent line. The model assumes adiabatic conditions in the vent line and maintains constant stagnation enthalpy for the energy balance. 

The treatment of mixing of immiscible fluids starts with a description of breakup and coalescence in homogeneous flows. Classical concepts are briefly reviewed and special attention is given to recent advances—satellite formation and self-similarity. A general model, capable of handling breakup and coalescence while taking into account stretching distributions and satellite formation, is described. 

Bubbly flow. In bubbly flow, the holdup is generally known and/or near homogeneous flow condition exists, and the frictional pressure drop can be correlated through similarity analysis (Dukler et al., 1964). The development shows that the frictional loss is expressed by a Fanning-type equation,... 

In open rod bundles, transverse flow between subchannels is detectable by variations in hydraulic conditions, such as the difference in equivalent diameter in rod and shroud areas (Green et al., 1962 Chelemer et al., 1972 Rouhani, 1973). The quality of the crossflow may be somewhat higher than that of the main stream (Madden, 1968). However, in view of the small size of the crossflow under most circumstances, such variation generally will not lead to major error in enthalpy calculations. The homogeneous flow approximation almost universally used in subchannel calculations appears to be reasonable (Weisman, 1973). The flow redistribution has a negligible effect on the axial pressure drop. 

Later, Weisman et al. (1978) also found that assuming homogeneous flow everywhere provided nearly as good a correlation of the data as the slip flow model. The total pressure drop across a contraction can be approximated by... 

For uniformly distributed homogeneous flows, the fluid properties can be described in terms of averages over the flow cross section. Such flows can be described as one dimensional, as opposed to separated or heterogeneous flows, in which the phase distribution varies over the cross section. 

The presence of slip also means that the acceleration term in the general governing equation [Eq. (15-45)] cannot be evaluated in the same manner as the one for homogeneous flow conditions. When the acceleration term is expanded to account for the difference in phase velocities, the momentum equation, when solved for the total pressure gradient, becomes... 

The pressure drop over a given length of pipe must be determined by a stepwise procedure, as described for homogeneous flow. The major additional complication in this case is evaluation of the holdup (ipm) or the equivalent slip ratio (S) using one of the above correlations. 

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