Steady drag force

For granular flow (as distinct from the classical kinetic theory for a dilute gas) the net external force acting on the particle depends on the microscopic velocity because of the phase interaction terms introduced to consider the interstitial fluid behavior. However, the net force can be divided into two t3rpes of contributions, a set of external forces Fe which are independent of c and a separate steady drag force F i. Hence, the net force exerted on a particle is re-written as ... 

The steady drag is the component of the hydrod3mamic force acting on the particle surface in the continuous phase flow direction. One might, for example, imagine a uniform velocity in the z-direction as sketched in Fig 5.1, and describe the external flow using the Cartesian coordinate system, then the steady drag force is defined by [14, 30] ... 

The virtual mass effect relates to the force required for a particle to accelerate the surrounding fluid [65, 170, 26]. When a particle is accelerated through a fluid, the surrounding fluid in the immediate vicinity of the particle will also be accelerated at the expense of work done by the particle. The particle apparently behaves as if it has a larger mass than the actual mass, thus the net force acting on the particle due to this effect has been called virtual mass or added mass force. The steady drag force model does not include these transient effects. 

The steady drag force on the impeller blade section can, as a first approach, be formulated as ... 

The laminar steady drag force on the fluid can be written as F oc pUD, where p is the fluid viscosity. Substituting U with ttND the power becomes... 

The Lagrangian steady drag force on a single particle is commonly expressed by (5.48). For Eulerian model formulations one normally employs (5.34), hence for the multifluid models the standard steady drag force parameterization 3delds ... 

The interfacial momentum transfer terms for the disturbed flow are approximated by the steady drag force ... 

In multiphase reactor simulations, the momentum, heat and species mass balances may thus contain several terms that can be placed in this framework. However, in these problem solvers, the generalized proportionality coefficient K is usually derived from the steady drag force, and the convective heat and mass transfer fluxes, respectively. 

The interphase forces considered were steady drag, added (virtual) mass and lift. The steady drag force on a collection of dispersed bubbles with a given average diameter was described by (5.48) and (5.34). The transversal lift force was determined by the conventional model (5.65), whereas the added mass force was approximated by (5.112). 

The steady drag force term is treated in the same way as described for the radial liquid velocity component. 

C.4 The 2D Axi-Symmetric Bubble Column Model 1227 Steady drag force term... 

The steady drag force term in the axial component of the gas momentum equation is treated in the same way as described when considering the liquid momentum equation, as discussed in sect C.4.5. 

For laminar flow a similar differential force balance, balancing the drag force on a blade section and the torque can be formulated. For creeping flow inertial forces are not important and viscous forces completely dominates the flow. The laminar steady drag force on the fluid can be written as F a p,UD, where p, is the fluid viscosity. Substituting U with kND the power becomes P a pN D or = K, where K is the Newton number for laminar flow. K can be rewritten as ... 

Several resent studies indicate that algebraic slip models are sufficient modeling the flow pattern in bubble columns [14, 21, 118, 157], as the pressure and steady drag forces only dominate the axial component of the gas momentum balance. Therefore, the population balance model can be merged with an algebraic slip model to reduce the computational cost required for preliminary analysis [14]. Furthermore, adopting this concept the restriction used in the two-fluid model assuming that all the particles have the same velocity can be avoided. This means that only one set... 

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