Kinetic energy loss

The pressure drop in a cyclone will be due to the entry and exit losses, and friction and kinetic energy losses in the cyclone. The empirical equation given by Stairmand (1949) can be used to estimate the pressure drop ... 

The theoretical solar conversion efficiency of a regenerative photovoltaic cell with a semiconductor photoelectrode therefore depends on the model used to describe the thermodynamic and kinetic energy losses. The CE values, which consider all the mentioned losses can generally only be estimated the full line in Fig. 5.65 represents such an approximation. Unfortunately, the materials possessing nearly the optimum absorption properties (Si, InP, and GaAs) are handicapped by their photocorrosion sensitivity and high price. 

This equation is plotted as curve C in Figure 4.1. The form of equation 4.21 is somewhat similar to that of equations 4.16 and 4.17, in that the first term represents viscous losses which are most significant at low velocities and the second term represents kinetic energy losses which become more significant at high velocities. The equation is thus applicable over a wide range of velocities and was found by Ergun to correlate experimental data well for values of Rei/(l — e) from 1 to over 2000. 

The first term represents the pressure loss due to viscous drag (this is essentially the Carman-Kozeny equation) whilst the second term represents kinetic energy losses, which are significant at higher velocities (kinetic energy being proportional to velocity squared). Equation 1.43 is valid in the range 1 < Re < 2000 where the Reynolds number is defined by... 

The simplest theory of impact, known as stereomechanics, deals with the impact between rigid bodies using the impulse-momentum law. This approach yields a quick estimation of the velocity after collision and the corresponding kinetic energy loss. However, it does not yield transient stresses, collisional forces, impact duration, or collisional deformation of the colliding objects. Because of its simplicity, the stereomechanical impact theory has been extensively used in the treatment of collisional contributions in the particle momentum equations and in the particle velocity boundary conditions in connection with the computation of gas-solid flows. 

It is clear that the collision between two elastic but frictional spheres is inelastic due to the inevitable sliding at contact which yields the kinetic energy loss by frictional work. Furthermore, the preceding analyses of both Hertzian collision and frictional collision can also be applied to the particle-wall collision, where the radius of the wall is simply set to be infinitely large. 

Once the relative impact velocity between two colliding spheres is higher than the critical yield velocity, plastic deformation must occur. Heat loss is another phenomenon often coupled with such collisions. Collisions with plastic deformation are referred to as inelastic collisions. All the energy transfer in the form of plastic deformation and heat loss in an inelastic collision is considered as a kinetic energy loss. 

Verify that, for a collinear collision of two frictionless, nonspinning, rigid spherical particles, the total kinetic energy loss can be expressed by Eq. (2.6). 

For simplicity, it is assumed that the impact is a Hertzian collision. Thus, no kinetic energy loss occurs during the impact. The problem of conductive heat transfer due to the elastic collision of solid spheres was defined and solved by Sun and Chen (1988). In this problem, considering the heat conduction through the contact surface as shown in Fig. 4.1, the change of the contact area or radius of the circular area of contact with respect to time is given by Eq. (2.139) or by Fig. 2.16. In cylindrical coordinates, the heat conduction between the colliding solids can be written by... 

The term fa) is the volume-averaged dissipation function for the energy dissipated by the viscous force, which is irreversible dissipation of mechanical work into thermal energy or heat. For the solid-particle phase, the kinetic energy loss by attrition or inelastic collision may be included in this term. 

For kinetic energy loss at high flow rates, the theory of Burke and Plummer [Ergun and Oming, 1949] assumes that the total resistance of the packed bed can be treated as the sum of the resistances of the individual particles. For the fully developed turbulent flow, the drag force acting on an isolated spherical particle is... 

For viscous energy loss, from Kozeny s equation, the pressure drop is proportional to the square of the specific surface area of solids So- For kinetic energy loss, from Burke and Plummer s relation, the pressure drop is proportional to So- So is related to the particle diameter by Eq. (5.351) for spherical particles for nonspherical particles, the dynamic diameter (see 1.2) may be used for the particle diameter. The general form of the pressure drop can be expressed as... 

In the MEB equation, kinetic energy losses can be calculated easily provided that the kinetic energy correction factor a can be determined. In turbulent flow, often, the value of a = 2 is used in the MEB equation. When the flow is laminar and the fluid is Newtonian, the value of a = 1 is used. Osorio and Steffe (1984) showed that for fluids that follow the Herschel-Bulkley model, the value of a in laminar flow depends on both the flow behavior index ( ) and the dimensionless yield stress ( o) defined above. They developed an analytical expression and also presented their results in graphical form for a as a function of the flow behavior index ( ) and the dimensionless yield stress ( o)- When possible, the values presented by Osorio and Steffe (1984) should be used. For FCOJ samples that do not exhibit yield stress and are mildly shear-thinning, it seems reasonable to use a value of a = 1. 

The use of monoliths as catalytic reactors focuses mainly on applications where low pressure drop is an important item. When compared to fixed beds, which seem a natural first choice for catalytic reactors, monoliths consist of straight channels in parallel with a rather small diameter, because of the requirement of a comparably large surface area. The resulting laminar flow, which is encountered under normal practical circumstances, does not show the kinetic energy losses that occur in fixed beds due to inertia forces at comparable fluid velocities. Despite the laminar flow, monolith reactors still may be approached as plug-flow reactors because of the considerable radial diffusion in the narrow channels [1]. 

This derivation neglects the correction of kinetic energy loss due to nonuniformity of flow in both cross sections and the frictional degradation of energy during passage through the constriction. This is corrected by the introduction of a numerical coefficient, Cd ... 

Once all the projectile impact parameter grid has been generated as described in Section 3.2. one obtains the projectile center of mass kinetic energy loss for each trajectory as... 

We have studied the reaction dynamics of the collision Nag+ + Na for a fixed collision geometry (with impact parameter 6=0) but in a wide range of impact energies Ecm. In Fig. 1, the total kinetic energy loss (tkel) AE =Eem -Ecm(t -> +oo), with Ecm(t +oo) the final kinetic energy of the relative motion between cluster-projectile and atomic target in the centre-of-mass system is shown for Ecm = 0.2 eV... 1 MeV. 

Kinetic energy loss loss of effective pressure because of the kinetic energy retained by the stream of liquid which issues from the active part of the viscometer. 

Off-design conditions in an impulse blade typical corrections for kinetic energy losses... 

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