Conservation of momentum

The rate of change of momentum is equivalent to a force as stated by Newton s second law. 

Therefore when applying the conservation principle to momentum, we will have three component balances corresponding to the three mutually orthogonal directions in Euclidian three space. 

There are three specific surface forces that are important in chemical engineering problems. These are pressure forces, shear or viscous forces, and structural or resultant forces. The pressure force fxp is the x directed force caused by the integral of the pressure acting on the surface of the control volume, 

The structural force, or resultant of the forces acting on the system is rx- This force appears when the control volume cuts through solid objects such as the containing wall of a pipe. 

The conservation principle for the rate of change of the x component of momentum therefore becomes 

The conservation of momentum or Newton s second law applies to a particle or fixed set of particles, namely a system. The velocity used must always be defined relative to a fixed or inertial reference plane. The Earth is a sufficient inertial reference. Therefore, any control volume associated with accelerating aircraft or rockets must account for any differences associated with how the velocities are measured or described. We will not dwell on these differences, since we will not consider such noninertial applications. 

The law in terms of a control volume easily follows from Equations (2.12) and (3.12) for the bulk system with / selected as pvx, taking vv as the component of v in the x direction  

Example 3.3 Determine the force needed to hold a balloon filled with hot air in place. Consider the balloon to have the same exhaust conditions as in Example 3.1 and the hot air density, p, is constant. The cold air density surrounding the balloon is designated as Poo. Assume the membrane of the balloon to have zero mass. 

Let us now consider the sum of the forces upwards. F is designated as the force, downwards, needed to hold the balloon in place. Another force is due to the pressure distribution of the cold atmosphere on the balloon. Assuming the atmosphere is motionless, fluid statistics gives the pressure variation with height x as 

The pressure acts normal everywhere on the membrane, but opposite to n. This force can therefore be expressed as 

The conservation of momentum equation is derived from Newton s second law of motion, which expresses proportionality between applied force and resulting acceleration of a particle. Momentum equation for Newtonian viscous fluid is given by the Navier-Stokes equation 

For Cartesian coordinate, the Navier-Stokes equation is expressed as follows x-momentum  

For incompressible fluid flow and for constant fluid viscosity (p), the Navier-Stokes equation reduces to 

Performing a momentum balance over a differential volume of a homogenous material leads to the Cauchy equation of motion, 

If we assume that the body force is only due to gravity and use the definition r = —prL +1 where pr is the isotropic resin pressure, / is the unit tensor, and r is the deviatoric stress as well as assuming a constant density and define a new pressure Pr —pr + prgh, Equation 5.21 simplifies to, 

It is more convenient to use an intrinsic phase average for the pressure drop because it is measured that way experimentally. Replacing (Pr) by er(PrY and using the fact that [16], 

At this point, we need to develop an expression for fd that contains only the averaged field variables. This has been done vigorously by Slattery [18,19], and it has shown that in absence of resin inertia 

As mentioned earlier, the expression forfd is obtained under conditions of no inertia. If we further assume the resin is Newtonian (i.e., r = p[V Ur + V / ])) and the fiber phase is stationary, then Equation 5.25 can be simplified to the well-known Brinkman equation [22], 

The momentum equation (the Navier-Stokes equation) for fluid flow (De Groot and Mazur, 1962) is complicated and difficult to solve. It is the subject of fluid mechanics and dynamics and is not covered in this book. When fluid flow is discussed in this book, the focus is on the effect of the flow (such as a flow of constant velocity, or boundary flow) on mass transfer, not the dynamics of the flow itself. 

For flow in a porous medium, Darcy s law describes the flow rate  

Mass balances must be used with the flow pattern (knowm a priori from theory or experimental measurements) to establish species concentrations and fluxes at any point in the fuel cell. When the flow pattern is a priori unknown, conservation-of-momentum equations (also called equations of motion) must be used with mass balance equations to establish the velocity and concentration profiles. Conservation of momentum for gases leads to the following equations (Navier-Stokes equations), in which k represents one of the three orthogonal directions in the coordinate sj stem (x, y, and z)  

Here P is the pressure, g is the acceleration due to gravity, and Pe is the effective viscosity. The term Xj represents other than Newtonian viscous losses and may be 

Equation (3) provides details of gas flow movements. The full treatment requires a rigorous computational fluid dynamics (CFD) tool. Startup and transient processes as well as variations in certain operating parameters may have a sizeable effect on flow and concentration profiles, but the effect on overall electrochemical performance of the cell is not necessarily of the same order. Sometimes it is desirable to make a simplification such as assuming laminar flow to reduce the computation cost and allow quick estimates of certain flow properties. For example, the pressure drop of a laminar flow through a channel can be estimated as 

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If a beam of monoenergetic ions of mass A/, is elastically scattered at an angle 6 by surface atoms of mass Mg, conservation of momentum and energy requires that... 

The equation of motion is based on the law of conservation of momentum (Newton s second law of motion). This equation is written as... 

The relationship between the two conditions is estabUshed by conservation of energy and by conservation of momentum across the shock front. 

The conservation of mass gives comparatively Httle useful information until it is combined with the results of the momentum and energy balances. Conservation of Momentum. The general equation for the conservation of momentum is... 

Direct and Indirect Energy Gap. The radiative recombination rate is dramatically affected by the nature of the energy gap, E, of the semiconductor. The energy gap is defined as the difference in energy between the minimum of the conduction band and the maximum of the valence band in momentum, k, space. Eor almost all semiconductors, the maximum of the valence band occurs where holes have zero momentum, k = 0. Direct semiconductors possess a conduction band minimum at the same location, k = O T point, where electrons also have zero momentum as shown in Eigure la. Thus radiative transitions that occur in direct semiconductors satisfy the law of conservation of momentum. 

Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and con-servation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential consei vation equations, respectively. These are often called macroscopic and microscopic balance equations. 

Cauchy Momentum and Navier-Stokes Equations The differential equations for conservation of momentum are called the Cauchy momentum equations. These may be found in general form in most fliiid mechanics texts (e.g., Slatteiy [ibid.] Denu Whitaker and Schlichting). For the important special case of an incompressible Newtonian fluid with constant viscosity, substitution of Eqs. (6-22) and (6-24) lead to the Navier-Stokes equations, whose three Cartesian components are... 

Recall that Cauchy s law of conservation of momentum is given by... 

Several discrete forms of the conservation of momentum equation, (9.3), can be derived, depending on the type of mesh and underlying assumptions. As an example, assume the equation will be solved on staggered spatial and temporal meshes, in two dimensions, in rectangular geometry, and with the velocities located at the nodes. Assume one quarter of the mass from each adjacent element is associated with the staggered element as shown in Fig. 9.11. 

Conservation relations are used to derive mechanical stress-volume states from observed wave profiles. Once these states have been characterized through experiment or theory they may, in turn, predict wave profiles for the material in question. For the case of a well-defined shock front propagating at constant speed L/ to a constant pressure P and particle velocity level u, into a medium at rest at atmospheric pressure, with initial density, p, the conservation of momentum, mass, and energy leads to the following relations ... 

In kinetics, Newton s second law, the principles of kinematics, conservation of momentum, and the laws of conservation of energy and mass are used to develop relationships between the forces acting on a body or system of bodies and the resulting motion. 

Conservation of Momentum. If the mass of a body or system of bodies remains constant, then Newton s second law can be interpreted as a balance between force and the time rate of change of momentum, momentum being a vector quantity defined as the product of the velocity of a body and its mass. 

In collisions between two bodies the contact force and the duration of contact are usually unknown. However, the duration of contact is the same for both bodies, and the force on the first body is the negative of the force on the second body. Thus the net change in momentum is zero. This is called the principle of conservation of momentum. 

In fluid mechanics the principles of conservation of mass, conservation of momentum, the first and second laws of thermodynamics, and empirically developed correlations are used to predict the behavior of gases and liquids at rest or in motion. The field is generally divided into fluid statics and fluid dynamics and further subdivided on the basis of compressibility. Liquids can usually be considered as incompressible, while gases are usually assumed to be compressible. 

Applying the conservation of momentum to the control volume for a onedimensional flow conduit, it is found that [62]... 

Euler s equation (equation 9.7) may be recovered from Boltzman s equation as a consequence of the conservation of momentum, but only in the zeroth-order approximation to the full distribution function. Setting k — mvi in equation 9.52 gives, in component form. 

Since /S Tj0) = , its integral over the collision term is zero (conservation of momentum in a collision). Thus the result of multiplying the Boltzmann equation by and integrating is ... 

As developed from the equation for conservation of momentum (Ref 27), the thrust, F, on a rocket motor is... 

According to the ideal stripping model, the incident X + ion collides with a quasi-free H atom while the other H atom in the H2 molecule merely participates as idle spectator to the reaction. The conservation of momentum in the system X +-H requires the secondary ion XH + to be formed with the velocity ... 

When the product ion moved with a higher kinetic energy than predicted by the stripping model, the collision apparently was more elastic— i.e., less kinetic energy of the incident ion was used for internal excitation of the products. In an ideal elastic collision with H transfer the products carry no internal energy at all. If the secondary ion moves forward and the H atom moves backwards, conservation of momentum requires that the primary ion has a velocity ... 

The first term in constant phase of E(r. z) that arises from the refractive index of the medium through which the beams propagate. The second term vanishes because of conservation of momentum in the harmonic generating medium. The third term is the Gouy phase, which changes by (n — m) n as the beams pass through a focal point. 

Extraction of the speed distribution is achieved in an analogous manner by integrating over all angles for each speed. The speed distributions can be further transformed, using the law of conservation of momentum, into total translational energy distributions for the O3 — O2(X3S ) + 0(3Pj) dissociation. 

Mixture conservation of mass equation Mixture conservation of momentum equation Mixture conservation of energy equation Slip equation (concerning the difference in velocity)... 

Newton s law of viscosity and the conservation of momentum are also related to Newton s second law of motion, which is commonly written Fx = max = d(mvx)/dt. For a steady-flow system, this is equivalent to... 

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