Pressure drop acceleration

The material that is fed into the pipeline will essentially have zero velocity at the feed point and so will have to be accelerated to its terminal velocity at the end of the pipeline. The pressure drop for this can be approximated with  

The density and velocity terms are those of the air at the end of the pipeline. The approximations lie in the fact that the air will have an initial velocity, and the terminal velocity of the material will be below that of the air, the actual velocity depending upon particle size, shape and density. 

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Pressure Drop. The pressure drop across a two-phase suspension is composed of various terms, such as static head, acceleration, and friction losses for both gas and soflds. For most dense fluid-bed appHcations, outside of entrance or exit regimes where the acceleration pressure drop is appreciable, the pressure drop simply results from the static head of soflds. Therefore, the weight of soflds ia the bed divided by the height of soflds gives the apparent density of the fluidized bed, ie... 

Yang, W. C., and Keaims, D. L., Estimating the Acceleration Pressure Drop and the Particle Acceleration Length in Vertical and Horizontal Pneumatic Transport Lines, Pneumotransport 3, pp. D7-89 (1976a)... 

It should be remembered that these correlations as originally devised by Lockhart and Martinelli were based almost entirely on experimental data obtained for situations in which accelerative effects were minor quantities. The Lockhart-Martinelli correlation thus implies the assumption that the static pressure-drop is equal to the frictional pressure-drop, and that these are equal in each phase. The Martinelli-Nelson approach supposes that the sum of the frictional and accelerational pressure-drops equals the static pressure-drop (hydrostatic head being allowed for) and that the static pressure-drop is the same in both phases. When acceleration pressure losses become important (e.g., as critical flow is approached), they are likely to be significantly different in the gas and liquid phases, and hence the frictional pressure losses will not be the same in each phase. In these circumstances, the correlation must begin to show deviations from experiment. 

Acceleration pressure drop. When the gas density or the vapor mass fraction changes, there is an acceleration pressure drop calculated from... 

In their second method, as in the case of a vertical riser, the total pressure drop along the spout height is composed of (i) a solids static head equivalent to the dispersed-solids bulk density, (ii) an acceleration pressure drop, and (iii) a solids friction loss due to relative motion of the particles with respect to the gas and to the spout wall. Thus,... 

However, in heated channels with large heat and mass fluxes the acceleration pressure drop can be considerable and no longer assumes negligible values. In an adiabatic, two-phase flow, the acceleration pressure drop only exists as a result of further evaporation or expansion of the vapour or gas phase, and is small. A simple rule for adiabatic two-phase flow in refrigeration plants say that the acceleration pressure drop does not play a role as long as Apt/ps < 0.2 holds for the frictional pressure drop. If this is not the case, then the acceleration pressure drop has to be determined, according to (4.123), over the channel length between inlet 1 and outlet 2 ... 

The model suggested by Yang (1977) for calculating the pressure drop in vertical pneumatic conveying lines can be applied here to estimate the acceleration pressure drop. The acceleration length can be calculated from numerical integration of the equation... 

Yang WC, Keairns DL. Estimating the acceleration pressure drop and the particle acceleration length in vertical and horizontal pneumatic transport lines. Proceedings of the Pneumotransport 3, Bedford (England) BHRA Fluid Engineering, 1977, pp D7-89-D7-98. 

Gas contraction pressure drop from freeboard to cyclone inlet, cm H2O = Solids acceleration pressure drop, cm H2O = Cyclone barrel friction pressure drop, cm H2O = Cyclone exit pressure drop, cm H2O = Gas viscosity, kg/(m-s)... 

Since pipeline bore comes in incremental sizes, fine tuning and spare capability need to be considered in terms of reserve pressure available. With first approximation values for pressure and pipeline bore, the available conveying data can be scaled more precisely to take account of differences between pipeline geometries. Conveying air velocities and the solids loading ratio can be evaluated so that differences between air only pressure drop and acceleration pressure drop values can also be taken into account. This is an iterative process, as there are many inter-dependent variables, and so in the initial stages approximations can be made. 

Using this model it was possible, for any given conditions of flow rates of solids and air, to determine an initial strand velocity for use in the acceleration / pressure drop model. 

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