For compressible flow

Equation (6-95) is valid for incompressible flow. For compressible flows, see Benedict, Wyler, Dudek, and Gleed (J. E/ig. Power, 98, 327-334 [1976]). For an infinite expansion, A1/A2 = 0, Eq. (6-95) shows that the exit loss from a pipe is 1 velocity head. This result is easily deduced from the mechanic energy balance Eq. (6-90), noting that Pi =pg. This exit loss is due to the dissipation of the discharged jet there is no pressure drop at the exit. 

Figure 2-33. Reynolds number for compressible flow, steel pipe. By permission, Crane Co., Technical Paper 410, Engineering Div., 1957. Also see 1976 edition.

Figure 2-38B. Net expansion factor, Y, for compressible flow through nozzles and orifices. By permission, Crane Co., Technical Paper 410, Engineering Div., 1957. Also see 1976 edition and Fluid Meters, Their Theory and Application, Part 1, 5th Ed., 1959 and R. G. Cunningham, Paper 50-A-45, American Society of Mechanical Engineers.

The investigation shows agreement between the standard laminar incompressible flow predictions and the measured results for water. Based on these observations the predictions based on the analytical results of Shah and London (1978) can be used to predict the pressure drop for water in channels with as small as 24.9 pm. This investigation shows also that it is insufficient to assume that the friction factor for laminar compressible flow can be determined by means of the well-known analytical predictions for its incompressible counterpart. In fact, the experimental and numerical results both show that the friction factor increases for compressible flows as Re is increased for a given channel with air. 

As a simplification, the term in Eq. (10) that accounts for the kinetic energy of the gas jets emerging from the gas distributor is based on the expression ( 9goVl/2, which is valid for incompressible flow. Experimental investigations show [27], that for relatively low gas velocities it is possible to represent the empirically determined loss coefficients q as accurately with this simplification as by the use of expressions for compressible flow. 

The flow phenomena described by the Navier-Stokes equation fall into two classes discriminated by the nature of the compressibility effects to be taken into account. For compressible flow, the Navier-Stokes equation [Eq. (1)] has to be solved in com-... 

For compressible flow where AP > 10% P1 either break into sections where AP < 10% P or use... 

Omega Method Model for Compressible Flows The factored momentum balance, Eq. (23-42), can be analytically integrated after first relating the dimensionless specific volume 8 to the dimensionless pressure ratio r. A method to do this, designated the omega method, was suggested by Leung (1986) ... 

Due to the change in the average velocity w, it is more convenient in calculations for compressible flow in pipes of constant cross-sectional area to work in terms of the mass flux G. This is the mass flow rate per unit flow area and is sometimes called the mass velocity. If the mass flow rate is constant, as will usually be the case, then G is constant when the area is constant. The relationship between G and u is given by... 

Pressure profiles for compressible flow through a convergent nozzle... 

Using the standard k-e model for compressible flows [5], the turbulent fluxes are modeled in the following way ... 

The proposed model consists of a biphasic mechanical description of the tissue engineered construct. The resulting fluid velocity and displacement fields are used for evaluating solute transport. Solute concentrations determine biosynthetic behavior. A finite deformation biphasic displacement-velocity-pressure (u-v-p) formulation is implemented [12, 7], Compared to the more standard u-p element the mixed treatment of the Darcy problem enables an increased accuracy for the fluid velocity field which is of primary interest here. The system to be solved increases however considerably and for multidimensional flow the use of either stabilized methods or Raviart-Thomas type elements is required [15, 10]. To model solute transport the input features of a standard convection-diffusion element for compressible flows are employed [20], For flexibility (non-linear) solute uptake is included using Strang operator splitting, decoupling the transport equations [9],... 

In order for a model to be closured, the total number of independent equations has to match the total number of independent variables. For a single-phase flow, the typical independent equations include the continuity equation, momentum equation, energy equation, equation of state for compressible flow, equations for turbulence characteristics in turbulent flows, and relations for laminar transport coefficients (e.g., fJL = f(T)). The typical independent variables may include density, pressure, velocity, temperature, turbulence characteristics, and some laminar transport coefficients. Since the velocity of gas is a vector, the number of independent variables associated with the velocity depends on the number of components of the velocity in question. Similar consideration is also applied to the momentum equation, which is normally written in a vectorial form. 

Care is needed when modeling compressible gas flows, flows of vapor-liquid mixtures, slurry flows, and flows of non-Newtonian liquids. Some simulators use different pipe models for compressible flow. The prediction of pressure drop in multiphase flow is inexact at best and can be subject to very large errors if the extent of vaporization is unknown. In most of these cases, the simulation model should be replaced by a computational fluid dynamics (CFD) model of the important parts of the plant. 

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