Linear momentum balance

The Macroscopic Linear Momentum Balance and the Equation of Motion, 32... 

THE MACROSCOPIC LINEAR MOMENTUM BALANCE AND THE EQUATION OF MOTION... 

The jet is governed by four steady-state equations representing the conservation of mass and electric charges, the linear momentum balance, and Coulomb s law for the E field [9]. Mass conservation requires that... 

CLM.IO PIPE FORCE REQUIREMENT 57 A linear momentum balance in the vertical direction results in... 

Consider an inelastic collision between two smooth identical spherical particles 1 and 2, of mass m and diameter dp = d 2 [32] [60]. If J12 is the impulse of the force exerted by particle 1 on particle 2, the linear momentum balances over a collision relate the velocity vectors of the center of each sphere just before and after the collision through ... 

Though the integral form of linear momentum balance written above as eqn (2.26) serves as the starting point of many theoretical developments, it is also convenient to cast the equation in local form as a set of partial differential equations. The idea is to exploit the Reynolds transport theorem in conjunction... 

We have adopted the convention that the position vector be written in component form as r = XiCj. Now we recognize that upon factoring out Xj, three of these terms are a restatement of linear momentum balance, and are thus zero. In addition, the term involving vj also clearly yields zero since it is itself symmetric and is contracted with the antisymmetric Levi-Cevita symbol, The net result is that we are left with = 0 which immediately implies that the stress tensor itself... 

During our discussion of linear momentum balance in chap. 2, we noted that the fundamental governing equations of continuum mechanics as embodied in eqn (2.32) are indifferent to the particular material system in question. This claim is perhaps most evident in that eqn (2.32) applies just as well to fluids as it does to solids. From the standpoint of the hydrodynamics of ordinary fluids (i.e. Newtonian fluids) we note that it is at the constitutive level that the distinction... 

In the macro level, it is assumed that the material obeys the basic laws of continuum mechanics, i.e. linear momentum balance and energy balance ... 

The flow model is obtained by combining the equations of linear momentum and mass balance for the liquid phase. Neglecting inertial effects, the equation of linear momentum balance can be written as. 

Table 11.5 contains the simulation parameters of the simplified 0-order axisym-metrical volume-average ferrohydrodynamic model. Note that three superficial velocities are chosen such that the ferrofluid flow is i) completely controlled by inertia, i.e. Burke-Plummer type of flow (Uq = 0.2 m/s) ii) mostly dominated by viscous forces, i.e. Darcy flow (Uo = 2x 10 m/s), iii) mixed with comparable inertial and viscous contributions, i.e. Forchheimer flow (Uq = 2 x 10 m/s). The Reynolds number values. Re, corresponding to these three cases are, respectively, pUodp/f] = 2110,0.211 and 21.1. The consensual values for the laminar and inertial Ergun constants are used, i.e. respectively, 150 and 1.75 (see Eqs. (11.56)-(11.58) linear momentum balance. Table 11.4). 

Otrly those properties Z of a system, that are conserved properties, may enter into an equation of change. Energy, hnear momentum, and angular momentum are conserved quantities and the respective equations of change are named energy balance, linear momentum balance, and angular momentum balance. 

The differential linear momentum balance for a multicomponent mixture can be derived from a shell balance approach or from a more sophisticated analysis using continuum mechanics. Such formulations are tabulated in many texts, for example, that by Bird et al., and will not be presented here in any detail. By way of illustration the x component of the differential monrantum balance in rectangular coordinates would be... 

Substituting Equations (2.8-2), (2.8-6), and (2.8-7) into (2.8-3), the overall linear momentum balance for a control volume becomes... 

Newton s law is applied in order to derive the linear momentum balance equation. Newton s law states that the sum of all forces equals the rate of change of linear momentum. The derivation/application of this law ultimately provides information critical to any meaningful analysis of most chemical reactors. Included in this information are such widely divergent topics as flow mechanisms, the Reynolds number, velocity profiles, two-phase flow, prime movers such as fans, pumps and compressors, pressure drop, flow measurement, valves and fittings, particle dynamics, flow through porous media and packed beds, fluidization— particularly as it applies to fluid-bed and fixed bed reactors, etc. Although much of this subject matter is beyond the scope of this text, all of these topics are treated in extensive detail by Abulencia and Theodore. ... 

The basic governing equations for the stable jet region are the equation of continuity, conservation of electric charges, linear momentum balance, and electric field equation. As main source for these flow equations we refer to Refs. [1, 62, 69]. 

To obtain the linear momentum balance, one proceeds from Eq. 1-2 to Eq. 1-59 and uses the continuity equation to obtain the conservative form on the left hand side leading to... 

Whether this completely general form of the linear momentum balance was available to the hydraulicians of the nineteenth century remains a question. However, it is certain that the fixed control volume, steady flow form of Eq. 2-5 was in use. As an example we note that Rouse and Ince (page 168, 1957) cite the work of Bresse who correctly anal3rzed the hydraulic Jump and the equation for gradually varied flow in an open channel. 

While the Navier-Stokes equations and Bernoulli s equation, along with the macroscopic mass and linear momentum balances, were clearly available prior to 1888, it seems likely that the general macroscopic mechanical energy balance was not. Nevertheless, the fixed control volume... 

Equations (11) and (12) are the linear momentum balances in the flow direction. Here the total pressure head has been partitioned into partial pressure components through the use of the solid volume distribution function . Equation (13) is the balance of linear momentum for the solid phase (the fluid phase is the same but of opposite sign) in the y direction. In this equation, we have allowed for an interphase pressure effect (incorporated in the parameter E) to ensure that the mixture remains saturated at all times [8]. Equations (14) and (15) account for the angular momentum balance in the binormal direction for the two constituents. Finally, the primes imply differentiation with respect to 3 ... 

We state for the sake of completeness, as in the angular momentum case, that if the director inertial contribution is incorporated then the two forms for the linear momentum balance equation given by equations (4.106) and (4.114) are replaced by, respectively [168, Eqns.(57),(58)],... 

It then follows from equations (5.530) to (5.541) that the linear momentum balance equations (5.527) and (5.529) finally reduce to, respectively, the two simultaneous equations... 

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