Momentum principle

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 hnear momentum principle, apphed to the arbitraiy control volume of Fig. 6-3, results in the following expression (Whitaker, ibid.). 

When the loading duration is short compared with the member s natural period, td /1 < 0.1, the shape of the load-time function becomes insignificant. The maximum response can be calculated using the impulse-momentum principle. The ductility demand, pj, can be determined in terms of the impulse, I0l and the maximum resistance of the member ... 

Applying the conservation of momentum principle for the x-direction to the control volume gives ... 

As was mentioned above, the Navier-Stokes equations are obtained by the appli-cation of the conservation of momentum principle to the fluid flow. The same control volume that was introduced above in the discussion of the continuity equation is considered and the conservation of momentum in each of the three coordinate directions is separately considered. The net force acting on the control volume in any of these directions is then set equal to the difference between the rate at which momentum leaves the control volume in this direction and the rate at which it enters in this direction. The net force arises from the pressure forces and the shearing forces acting on the faces of the control volume. The viscous shearing forces for two-dimensional flow (see later) are shown in Fig. 2.3. They are expressed in terms of the velocity field by assuming the fluid to be Newtonian and are then given by [4],[5] ... 

In order to find the relation between the wall shearing stress, rw, and the friction factor, /, for the fully developed flow, the conservation of momentum principle is applied to the control volume shown in Fig. 7.3. 

Most equipment used in the chemical, petroleum, and related industries is designed for the movement of fluids, and an understanding of fluid flow is essential to a chemical engineer. The underlying discipline is fluid mechanics,t which is based on the law of mass conservation, the linear momentum principle (Newton s second law), and the first and second laws of thermodynamics. 

The application of thermodynamics to flow processes is also based on conservation of mass and on the first and second laws. The addition of the linear momentum principle makes fluid mechanics a broader field of study. The usual separation between thermodynamics problems and fluid-mechanics problems depends on whether this principle is required forsolution. Those problems whose solutions depend only On conservation of mass and on the laws of thermodynamics are commonly set apart from the study of fluid mechanics and are treated in courses on thermodynamics. Fluid mechanics then deals with the broad spectrum of problems which require application of the momentum principle. This division is arbitrary, but it is traditional and convenient... 

On the other hand, if one has only incomplete knowledge of the initial or] final state of the gas, then more detailed information about the process is needed ] before any calculations are made. For example, the exit pressure of the gas may not be specified. In this case, one must apply the momentum principle of fluid] mechanics, and this requires an empirical or theoretical expression for the shear-stress at the pipe wall. 

The linear momentum principle for species s states that the rate of change of momentum is balanced by the forces acting on the system and the rate of production of momentum of species s [101] [78] ... 

Although there is no immediately useful information that we can glean from (2-56), we shall see that it provides a constraint on allowable constitutive relationships for T and q. In this sense, it plays a similar role to Newton s second law for angular momentum, which led to the constraint (2 41) that T be symmetric in the absence of body couples. In solving fluid mechanics problems, assuming that the fluid is isothermal, we will use the equation of continuity, (2-5) or (2-20), and the Cauchy equation of motion, (2-32), to determine the velocity field, but the angular momentum principle and the second law of thermodynamics will appear only indirectly as constraints on allowable constitutive forms for T. Similarly, for nonisothermal conditions, we will use (2-5) or (2-20), (2-32), and either (2-51) or... 

MACROSCOPIC MOMENTUM BALANCE. A momentum balance, similar to the overall mass balance, can be written for the control volume shown in Fig. 4.3, assuming that flow is steady and unidirectional in the x direction. The sum of all forces acting on the fluid in the x direction, by the momentum principle, equals the increase in the time rate of momentum of the flowing fluid. That is to say, the sum of forces acting in the x direction equals the difference between the momentum leaving with the fluid per unit time and that brought in per unit time by the fluid, or... 

Since the layer is in steady flow with no acceleration, by the momentum principle the sum of all forces on the control volume is zero. The possible forces acting on the control volume in a direction parallel to the flow are the pressure forces on the ends, the shear forces on the upper and lower faces, and the component of the force of gravity in the direction of flow. Since the pressure on the outer surface is atmospheric, the pressures on the control volume at the ends of the volume are equal and oppositely directed. They therefore vanish. Also, by assumption, the shear on the upper surface of the element is neglected. The two forces remaining are therefore the shear force on the lower surface of the control volume and the component of gravity in the direction of flow. Then... 

There exists a rather remarkable type of wall-effect which, rather than vanishing, approaches a finite limit as a//->0—that is, as the fluid becomes unbounded relative to the particle. When a particle settles in an otherwise quiescent fluid confined within a vertical duct of constant cross section, a dynamic pressure difference is set up, the pressure being greatest on that end of the duct towards which the particle is moving. The vector AP points in the direction of diminishing pressure. It is natural to expect that when ajl -> 0 the vertical container walls can play no role. In this event, elementary momentum principles require that the external pressure-drop force, P A, exerted on the fluid be exactly equal to F, where A is the cross sectional area of the duct and F is the hydrodynamic force on the particle (equal and opposite to the net gravity force on the latter). Detailed theoretical analysis (B13, B17, B28) reveals that such is not the case. Rather, in this limit, one obtains for a particle of any shape in a duct of any cross section (B17)... 

Momentum transfer is introduced in this section by reviewing the units and dimensions of momentum, time rate of change of momentum, and force. A momentum balance (also termed the impulse-momentum principle) is important in chemical reactor problems where forces, velocities, pressure drop, and prime movers, need to be determined. This analysis is inherently more complicated than those previously presented (i.e., forces possess both magnitude and direction), because the force, F, and momentum, M, are vectors. In order to describe force and momentum vectors, both direction and magnitude must be specified for mass and energy, only the magnitude is required. 

From the impulse-momentum principle, it follows that... 

The conservation of momentum principle is a common approach to a system composed of an arbitrary (differential) cubical volume within any flow field. By accounting for convection of momentum throughout the surface, all possible stress components on any and all surfaces, and any other forces (e.g. gravity), a general microscopic form of momentum equation can be derived, as shown in Eq. 3.9, which is valid at all points within any fluid. 

Consider the data given in Figure 7.4. A scatter plot of the data is shown in Figure 7.5. From the plot, it does not appear that a straight line can represent the data adequately. In the absence of a physical system model (conservation of mass, energy, or momentum principles), the usual procedure is to simply seek a function that will represent the data in an appropriate manner (this can be highly subjective—even very approximate models are better than a pure guess about the function). 

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