Force balance differential equations

Equation (32) is the generalized force balance differential equation for non-fluidized gas-particle flow. In this equation, for an inclined cylindrical moving bed, a = 0 for a vertical conical moving bed, 0 = 0 for vertical cylindrical moving bed, a = 0 and 0 = 0 and for a vertical cylindrical moving bed without interstitial gas flow, a = 0,0 = 0, and dp/dz = 0 simultaneously. 

If the dam and its base rock are affected only by gravity and seepage force, which are considered as body forces, and denoting that the tensile stress is positive, the balance differential equations expressed by displacements and general water heads can be given as... 

Balance Equations on a Differential Control Volume When the net forces are substituted into Eq. 2.14, the 8 V cancels from each term, leaving a differential equation. As a very brief illustration, a one-dimensional momentum equation in cartesian coordinates is written as... 

We have discussed stresses and strain rates. A critical objective is to relate the two, leading to equations of motion governing how fluid packets are accelerated by the forces acting on them. Generally, we are working toward a differential-equation description of a momentum balance, F = ma. The approach is to represent both the forces and the accelerations as functions of the velocity field. The result will be a system of differential equations in which velocities are the dependent variables and the spatial coordinates and time are the independent variables (i.e., the Navier-Stokes equations). 

In these equations dz — 1 has been dropped. On differential elements, this type of force balance was an important contribution in developing the Navier-Stokes equations. Here there is no acceleration, so J2 F = 0. That is, the pressure forces are balanced exactly by the shear forces. On an element centered at j, the force balance is... 

We consider the general balance equations of mass and energy in the absence of chemical reactions, and electrical, magnetic and viscous effects. The partial differential equations of these general balance equations represent the mathematically and thermodynamically coupled phenomena, which may describe some complex behavior due to interactions among various forces and flows within a system. 

Substituting Eq. (74) into Eq. (73) and rearranging give the differential equation of force balance ... 

In the previous analysis, we have obtained the velocity profile for fully developed flow in a circular tube from a force balance applied on a volume element, and determined the friction factor and the pressure drop. Below we obtain the energy equation by applying the energy balance on a differential volume eicineiit, and solve it to obtain tlie temperature profile for tlie constant surface temperature and the constant surface heat flux cases. 

The first achievement of the study of Holas and March [99] is to establish the differential form of the above virial theorem [their Eq. (2.15)]. Again, as in the zero field case treated above, this differential virial theorem is interpreted as a force-balance equation. The well-known Lorentz force of electromagnetism then appears quite naturally in this equation. 

The third chapter covers convective heat and mass transfer. The derivation of the mass, momentum and energy balance equations for pure fluids and multi-component mixtures are treated first, before the material laws are introduced and the partial differential equations for the velocity, temperature and concentration fields are derived. As typical applications we consider heat and mass transfer in flow over bodies and through channels, in packed and fluidised beds as well as free convection and the superposition of free and forced convection. Finally an introduction to heat transfer in compressible fluids is presented. 

Here is the mean sea level of the Baltic and gp is the net freshwater discharge. Expressing Q = Q(hi — h ) by the dynamical balance in the channel either with or without friction, we obtain a differential equation with respect to time for the sea level of the Baltic Sea forced by the sea level of the Kattegat and the net freshwater discharge into the Baltic. 

Although the expression (2-131) is a perfectly general statement of the force balance at an interface, it is not particularly useful in this form because it is an overall balance on a macroscopic element of the interface. To be used in conjunction with the differential Navier-Stokes equations, which apply pointwise in the two bulk fluids, we require a condition equivalent to (2-131) that applies at each point on the interface. For this purpose, it is necessary to convert the line integral on C to a surface integral on A. To do this, we use an exact integral transformation (Problem 2-26) that can be derived as a generalization of Stokes theorem ... 

Equation (8-62) generates three coupled linear second-order partial differential equations (PDEs). Eor complicated two-dimensional flow problems, this force balance and the equation of continuity yield three coupled linear PDEs for two nonzero velocity components and dynamic pressure. In some situations, this complexity is circumvented by taking the curl of the equation of motion ... 

Now, it is instructive to re-analyze the unsteady-state macroscopic mass balance on an isolated solid pellet of pure A with no chemical reaction. The rate of output due to interphase mass transfer from the solid particle to the liquid solution is expressed as the product of a liquid-phase mass transfer coefficient c, liquids a Concentration driving force (Ca, — Ca), and the surface area of one spherical pellet, 4nR. The unsteady-state mass balance on the solid yields an ordinary differential equation for the time dependence of the radius of the peUet. For example,... 

A general equation will be derived first for the flow configuration shown in Figure 3.16 by writing a force balance on a differential element of the fluid. 

A very long solid body having a square cross section floats at an interface with the configuration sketched. Determine h and a at equilibrium in terms of a, densities and pg, and interfadal tension Yab-(It is not necessary to solve explicitly for h and a from the final equations yon obtain.) Hint The differential equation of interfacial statics need not be solved if one is interested only in finding h and a (and not details of the interfacial profile). One can instead write a horizontal force balance for the entire fluid interface on one side of the solid. 

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