Momentum balance transfer

The numerical solution of the energy balance and momentum balance equations can be combined with flow equations to describe heat transfer and chemical reactions in flow situations. The simulation results can be in various forms numerical, graphical, or pictorial. CFD codes are structured around the numerical algorithms and, to provide easy assess to their solving power, CFD commercial packages incorporate user interfaces to input parameters and observe the results. CFD... 

Tlie equation of momentum transfer - more commonly called die equation of motion - can be derived from niomentmii consideradons by applying a momentum balance on a rate basis. The total monientmn witliin a system is uiicluinged by an c.xchaiige of momentum between two or more masses of the system. This is known as die principle or law of conservation of monientmn. This differendal equation describes the velocity distribution and pressure drop in a moving fluid. 

Momentum balance equations are of importance in problems involving the flow of fluids. Momentum is defined as the product of mass and velocity and as stated by Newton s second law of motion, force which is defined as mass times acceleration is also equal to the rate of change of momentum. The general balance equation for momentum transfer is expressed by... 

While the mass balances given above are relatively straightforward (assuming that a suitable closure can be derived for the mass-transfer terms), the momentum balances are significantly more complicated. In their simplest forms, they can be written as follows ... 

Our discussion of multiphase CFD models has thus far focused on describing the mass and momentum balances for each phase. In applications to chemical reactors, we will frequently need to include chemical species and enthalpy balances. As mentioned previously, the multifluid models do not resolve the interfaces between phases and models based on correlations will be needed to close the interphase mass- and heat-transfer terms. To keep the notation simple, we will consider only a two-phase gas-solid system with ag + as = 1. If we denote the mass fractions of Nsp chemical species in each phase by Yga and Ysa, respectively, we can write the species balance equations as... 

One assumes a propulsion engine operated in the atmosphere, as shown in Fig. 1.3. Air enters in the front end i, passes through the combustion chamber c, and is expelled from the exit e. The heat generated by the combustion of an energetic material is transferred to the combustion chamber. The momentum balance to generate thrust F is represented by the terms ... 

Writing a momentum balance for the gas phase in a two-phase mixture and assuming no mass transfer [see Lamb and White (L3)], we have... 

When writing the boundary conditions for the above pair of simultaneous equations the heat transferred to the surroundings from the reactor may be accounted for by ensuring that the tube wall temperature correctly reflects the total heat flux through the reactor wall. If the reaction rate is a function of pressure then the momentum balance equation must also be invoked, but if the rate is insensitive or independent of total pressure then it may be neglected. 

The reactor is modeled by three partial differential equations component balances on A and B [Eqs. (6.1) and (6.2)] and an energy balance [Eq. (6.3) for an adiabatic reactor or Eq. (6.4) for a cooled reactor]. The overall heat transfer coefficient U in the cooled reactor in Eq. (6.4) is calculated by Eq. (6.5) and is a function of Reynolds number Re, Eq. (6.6). Equation (6.7) is used for pressure drop in the reactor using the friction factor /given in Eq. (6.8). The dynamics of the momentum balance in the reactor are neglected because they are much faster than the composition and temperature dynamics. A constant... 

As the rate of momentum transfer is equal to a force, momentum balances are equivalent to force balances. 

This chapter has been concerned with flows in wb ch the buoyancy forces that arise due to the temperature difference have an influence on the flow and heat transfer values despite the presence of a forced velocity. In extemai flows it was shown that the deviation of the heat transfer rate from that which would exist in purely forced convection was dependent on the ratio of the Grashof number to the square of the Reynolds number. It was also shown that in such flows the Nusselt number can often be expressed in terms of the Nusselt numbers that would exist under the same conditions in purely forced and purely free convective flows. It was also shown that in turbulent flows, the buoyancy forces can affect the turbulence structure as well as the momentum balance and that in turbulent flows the heat transfer rate can be decreased by the buoyancy forces in assisting flows whereas in laminar flows the buoyancy forces essentially always increase the heat transfer rate in assisting flow. Some consideration was also given to the effect of buoyancy forces on internal flows. 

The flow of materials is accounted for with two balances conservation of mass and conservation of momentum transfer. The most important is a momentum balance, which is also called the equation of motion. The mass balance (also called the continuity equation) makes sure that mass is conserved. 

As we do for all mass transfer problems, we must satisfy the differential equation of continuity for each species as well as the differential momentum balance. Since we are dealing with a porous medium having a complex and normally unknown geometry, we choose to work in terms of the local volume averaged forms of these relations. Reviews of local volume averaging are available elsewhere (23-25). 

The setting up of the constitutive relation for a binary system is a relatively easy task because, as pointed out earlier, there is only one independent diffusion flux, only one independent composition gradient (driving force) and, therefore, only one independent constant of proportionality (diffusion coefficient). The situation gets quite a bit more complicated when we turn our attention to systems containing more than two components. The simplest multicomponent mixture is one containing three components, a ternary mixture. In a three component mixture the molecules of species 1 collide, not only with the molecules of species 2, but also with the molecules of species 3. The result is that species 1 transfers momentum to species 2 in 1-2 collisions and to species 3 in 1-3 collisions as well. We already know how much momentum is transferred in the 1-2 collisions and all we have to do to complete the force-momentum balance is to add on a term for the transfer of momentum in the 1-3 collisions. Thus,... 

While the film and surface-renewal theories are based on a simplified physical model of the flow situation at the interface, the boundary layer methods couple the heat and mass transfer equation directly with the momentum balance. These theories thus result in anal3dical solutions that may be considered more accurate in comparison to the film or surface-renewal models. However, to be able to solve the governing equations analytically, only very idealized flow situations can be considered. Alternatively, more realistic functional forms of the local velocity, species concentration and temperature profiles can be postulated while the functions themselves are specified under certain constraints on integral conservation. Prom these integral relationships models for the shear stress (momentum transfer), the conductive heat flux (heat transfer) and the species diffusive flux (mass transfer) can be obtained. 

Basically, the processes taking place in a chemical reactor are chemical reaction, and mass, heat and momentum transfer phenomena. The modeling and design of reactors are therefore sought from emplo3dng the governing equations describing these phenomena [1] the reaction rate equation, and the species mass, continuity, heat (or temperature) and momentum balance equations. 

Microscale fluid turbulence is, by deflnition, present only when the continuous fluid phase is present. The coefficients Bpv describe the interaction of the particle phase with the continuous phase. In contrast, Bpvf models rapid fluctuations in the fluid velocity seen by the particle that are not included in the mesoscale drag term Ap. In the mesoscale particle momentum balance, the term that generates Bpv will depend on the fluid-phase mass density and, hence, will be null when the fluid material density (pf) is null. In any case, Bpv models momentum transfer to/from the particle phase in fluid-particle systems for which the total momentum is conserved (see discussion leading to Eq. (5.17)). 

The mesoscale models for momentum transfer between phases differ quite substantially depending on the multiphase system under investigation, and different semi-empirical relationships have been developed for different systems. Since the nature of the disperse phase is particularly important, the available mesoscale models are generally divided into those valid for fluid-fluid and those valid for fluid-solid systems. The main difference is that in fluid-fluid systems the elements of the disperse phase are deformable particles (i.e. bubbles or droplets), whereas in fluid-solid systems the disperse phase is constituted by particles of constant shape. Typical fluid-fluid systems for which the mesoscale models reported below apply are gas-liquid, liquid-liquid, and liquid-gas systems. The mesoscale models reported for fluid-solid systems are valid both for gas-solid and for liquid-solid systems. As a general rule, the mesoscale model for Afp should be derived starting from a single-particle momentum balance ... 

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