Linear momentum balance equations

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. ... 

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... 

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... 

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... 

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... 

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. 

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 main advantage of the separation is the reduction of the computational effort. Another aspect is the fact that the two sets of equations can be solved with different numerical methods and on different numerical grids. Due to the nature of the non-linearities in material and momentum balance equations, they usually require different grid refinements in different areas of the computational domain. Additionally, if the assumption of a stationary flow field is valid, the simulation of the coupled set of equations would be unnecessarily slowed down by solving the momentum equations. The material balance to be solved for each of the reacting components k reads... 

To close the system of equations for the fluid motion the tangential stress boundary condition and the force balance equation are used. The boundary condition for the balance of the surface excess linear momentum, see equations (8) and (9), takes into account the influence of the surface tension gradient, surface viscosity, and the electric part of the bulk pressure stress tensor. In the lubrication approximation the tangential stress boundary condition at the interface, using Eqs. (17) and (18), is simplified to... 

Starting with Newton s second law we will develop the integral momentum-balance equation for linear momentum. Angular momentum will not be considered here. Newton s law may be stated The time rate of change of momentum of a system is equal to the summation of all forces acting on the system and takes place in the direction of the net force. 

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... 

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]. 

A linear stability analysis is performed in terms of normal modes. For illustrative purpose and so as to be able to proceed analytically we here restrict ourselves to growth under zero gravity and assume that the principle of the exchange of stabilities holds. The viscoelastic effects appear in the momentum balance equation (Eq. 5] and in the Laplace condition [Eq. 12]. The latter contribution has been previously neglected [6,7]. 

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 ... 

It is now possible to eliminate the velocity gradients and C from the angular momentum balance equations (5.211) and (5.212) by using the linear momentum... 

The problems experienced in drying process calculations can be divided into two categories the boundary layer factors outside the material and humidity conditions, and the heat transfer problem inside the material. The latter are more difficult to solve mathematically, due mostly to the moving liquid by capillary flow. Capillary flow tends to balance the moisture differences inside the material during the drying process. The mathematical discussion of capillary flow requires consideration of the linear momentum equation for water and requires knowledge of the water pressure, its dependency on moisture content and temperature, and the flow resistance force between water and the material. Due to the complex nature of this, it is not considered here. 

General equations of momentum and energy balance for dispersed two-phase flow were derived by Van Deemter and Van Der Laan (V2) by integration over a volume containing a large number of elements of the dispersed phase. A complete system of solutions of linearized Navier-Stokes equations... 

For the computation of compressible flow, the pressure-velocity coupling schemes previously described can be extended to pressure-velocity-density coupling schemes. Again, a solution of the linearized, compressible momentum equation obtained with the pressure and density values taken from a previous solver iteration in general does not satisfy the mass balance equation. In order to balance the mass fluxes into each volume element, a pressure, density and velocity correction on top of the old values is computed. Typically, the detailed algorithms for performing this task rely on the same approximations such as the SIMPLE or SIMPLEC schemes outlined in the previous paragraph. 

This chapter gave an overview of how to simplify complex processes sufficiently to allow the use of analytical models for their analysis and optimization. These models are based on mass, momentum, energy and kinetic balance equations, with simplified constitutive models. At one point, as the complexity and the depth of these models increases by introducing more realistic geometries and conditions, the problems will no longer have an analytical solution, and in many cases become non-linear. This requires the use of numerical techniques which will be covered in the third part of this book, and for the student of polymer processing, perhaps in a more advanced course. 

Momentum Balance Since momentum is a vector quantity, the momentum balance is a vector equation. Where gravity is the only body force acting on the fluid, the linear momentum principle, applied to the arbitrary control volume of Fig. 6-3, results in the following expression (Whitaker, ibid.). 

Balance equations for angular momentum, or moment of momentum, may also be written. They are used less frequently than the linear momentum equations. See Whitaker (Introduction to Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1968, Krieger, Huntington, N.Y., 1981) or Shames (Mechanics of Fluids, 3d ed., McGraw-Hill, New York, 1992). 

By taking xf = mv, we can obtain the local form of the balance equation of the linear momentum as... 

From the preceding equations we conclude that the pressure is a function of coordinate z only. Consequently, in the last equation the left-hand side is a function of z only, whereas the right-hand side is a function of y only. This is only possible if both equal a constant. Thus we conclude that the pressure gradient is constant, that is, pressure rises (or drops) linearly with z, and that the shear stress, in the presence of a pressure gradient, is a linear function of y, and in the absence of a pressure gradient it is constant across the gap. These observations follow from the momentum balance, and, they are therefore, true for all fluids, Newtonian and non-Newtonian alike. 

The generic equations of balance are statements of truth, which is a priori self-evident and which must apply to all continuum materials regardless of their individual characteristics. Constitutive relations relate diffusive flux vectors to concentration gradients through phenomenological parameters called transport coefficients. They describe the detailed response characteristics of specific materials. There are seven generic principles (1) conservation of mass, (2) balance of linear momentum, (3) balance of ro-... 

Solution of Equation (10.2.1) provides the pressure, temperature, and concentration profiles along the axial dimension of the reactor. The solution of Equation (10.2.1) requires the use of numerical techniques. If the linear velocity is not a function of z [as illustrated in Equation (10.2.1)], then the momentum balance can be solved independently of the mass and energy balances. If such is not the case (e.g., large mole change with reaction), then all three balances must be solved simultaneously. 

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