Momentum pressure losses

When a pure gas flows through a channel the accompanying fall in pressure is accounted for partly by acceleration of the flowing stream and partly by momentum transfer to the stationary walls. Since a porous medium may be regarded as an assembly of channels, similar considerations apply to flow through porous media, but in the diffusional situations of principal interest here accelerational pressure loss can usually be neglected. If more than one molecular species is present, we are also interested in the relative motions of the different species, so momentum transfers by collisions between different types of molecules are also important. 

Pressure drop. A pressure loss oeeurs in a eombustor beeause of diffusion, frietion, and momentum. The pressure drop value is 2-10% of the statie pressure (eompressor outlet pressure). The effieieney of the engine will be redueed by an equal pereent. 

To understand the difference in stagnation pressure losses between subsonic and supersonic combustion one must consider sonic conditions in isoergic and isentropic flows that is, one must deal with, as is done in fluid mechanics, the Fanno and Rayleigh lines. Following an early NACA report for these conditions, since the mass flow rate (puA) must remain constant, then for a constant area duct the momentum equation takes the form... 

The momentum pressure-drop can often be neglected, particularly when no mass transfer occurs in a system. Three equations, all approximate, are given below for calculation of these acceleration pressure losses, APa, between two sections, 1 and 2, using the mean velocities of gas and liquid,... 

The objective is to calculate the non-recoverable (frictional) pressure loss rather than the total pressure drop (which will also contain recoverable pressure drop due,to change in momentum)... 

Due to the shapes of channel cross sections, pressure losses can reach values of several bars for usual lengths. This leads to small flow velocities (some mm/s or cm/s) and low Reynolds numbers. The flow is then generally laminar or transitional. For very low Reynolds numbers (Re 1) the flow is said to be creeping and, neglecting the inertia term, the momentum equation becomes... 

To calculate flow velocity within the bed, the momentum equation is solved. The pressure loss is calculated via the Darcy equation, implying low flow velocities and a linear dependence of the pressure loss on the flow velocity. Thus the momentum equation is given by... 

Gas phase viscosity data, iTq, are used in the design of compressible fluid flow and unit operations. For example, the viscosity of a gas is required to determine the maximum permissible flow through a given process pipe size. Alternatively, the pressure loss of a given flowrate can be calculated. Viscosity data are needed for the design of process equipment involving heat, momentum, and mass transfer operations. The gas viscosity of mixtures is obtained from data for the individual components in the mixture. 

Because of the high gas velocities, solids loading ratios, and momentum loss in the collision zone, the pressure loss in ISDs is much greater than in pneumatic dryers, but it is comparable with that of fluidized and spouted bed dryers [44,45], The impinging stream configurations can, however, compete in various aspects with the classical systems for drying of particulates and pastes (Table 21.11). 

For heated horizontal channels, Martinelli and Nelson (1948) assumed the total pressure loss is given by the sum of a pressure loss due to friction and a pressure loss due to increased fluid momentum as vaporization occurs. The local two-phase friction pressure gradients in boiling water were calculated using isothermal flow pressure loss data assuming turbulent-turbulent flow. 

Hinkle s analysis assumes that particles lose momentum by collision with the pipe walls. Ilie pressure loss due to solids-wall friction is the gas pressure loss as a result of re-accelerating the solids. Thus, from Chapter 2, the drag force on a single particle is given by ... 

The COAST computer program is used to calculate the reactor coolant flow coast down transient for any combination of active and inactive pumps and forward or reverse flow in the hot or cold legs. The equations of conservation of momentum are written for each of the flow paths of the COAST model assuming unsteady one-dimensional flow of an incompressible fluid. The equation of conservation of mass is written for the appropriate nodal points. Pressure losses due to friction, and geometric losses are assumed proportional to the flow velocity squared. Pump dynamics are modelled using a head-flow curve for a pump at fiill speed and using four-quadrant curves, which are parametric diagrams of pump head and torque on coordinates of speed versus flow, for a pump at other than full speed. 

Because of the high gas velocities, solids loading ratios, and momentum loss in the collision zone, the pressure loss in ISDs is much greater than in... 

As the particle traverses a bend its velocity changes. Haag (1967) and Kovacs (1967) have analyzed the behavior of particles as they slow down in bends. The overall pressure loss in a bend may be made up of this deceleration of the particle in the bend plus the acceleration length term. Consider the horizontal bend shown in Fig. 5-4. The angle 0 varies from 0 to 90° for a right-angle turn in the horizontal plane. A single particle momentum balance on this system can be written as... 

The left-hand side of (2.24) represents the change of axial force. The first term on the right-hand side denotes the axial change of pressure force and the second term is a pressure loss term by frictional and form loss. The third term represents gravitational force and the last gives axial momentum exchange between adjacent channels. 

For given coolant channel geometries and power distribution, axial pressure loss of AP is assumed to be the same throughout all coolant channels because the transverse pressure difference between adjacent channels is considered to be much smaller than the axial pressure difference. The axial momentum equation (2.24) is solved to obtain the axial coolant velocity while adjusting the axial pressure loss. This is repeated until the total mass flow rate is converged. 

In the CFD modelling of membrane filtration process, membranes are usually modelled as a porous wall while the flow within a membrane is usually solved using both Navier-Stokes and Darqr equations (Ghidossi et al, 2006). A porous media model is widely used for determining the pressure loss during flow through packed beds, filter papers, perforated plates, flow distributors and tube banks (ANSYS, 2010). A momentum source term is added to the governing momentum equations, which creates a pressure drop that is proportional to the fluid velocity ... 

Where F ELi/Ai. It is also possible to absorb the local pressure loss coefficient, Ki into an equivalent length, Lci such that Ki=fiLci/Di. With this correction, the momentum equation can be written as ... 

In practice, the loss term AF is usually not deterrnined by detailed examination of the flow field. Instead, the momentum and mass balances are employed to determine the pressure and velocity changes these are substituted into the mechanical energy equation and AFis deterrnined by difference. Eor the sudden expansion of a turbulent fluid depicted in Eigure 21b, which deflvers no work to the surroundings, appHcation of equations 49, 60, and 68 yields... 

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