Liquid momentum

As noted previously, mixing in highly viscous liquids is slow both at the molecular scale, on account of the low values of diffusivity, as well as at the macroscopic scale, due to poor bulk flow. Whereas in low viscosity liquids momentum can be transferred from a rotating impeller through a relatively large body of fluid, in highly viscous liquids only... 

The pressure force term in the axial component of the gas momentum equation is discretized in the same way as described when considering the liquid momentum equation in sect C.4.5. 

L Liquid momentum equation. This considers the acceleration of the droplets as the vapor velocity increases along the channel. The difference in velocity between the vapor and the drops is an important parameter in heat transfer, and it is necessary to include this calculation. 

According to Prandtl (95), the particle diffusion coefficient is related to the liquid momentum eddy diffusivity e/ defined in terms of the turbulent shear stress (r0) and the time-average strain rate (7 )... 

This situation is idealized and outlet driven, in that plug flow is initiated at the inlet in some manner. This is certainly not the case for any inlet reported. As inlets have to handle an incoming liquid momentum from the inlet nozzle, the incoming fluid will enter the vessel in a defined region. Depending on the length and diameter of the vessel, the flow pattern will be partly driven by the inlet and outlet boundaries of the flow, as imposed by the inlet and outlet geometries/velocities. The best tool to look at these effects... 

This equation can also be written in terms of the gas to liquid momentum ratio qOh >lQ, where... 

By increasing the gas velocity, the shear stress at the interface is augmented consequently breakup is accelerated and the frequency is increased. By increasing the liquid velocity, the liquid momentum is increased which accelerates the rush of the liquid toward the axis of the nozzle, hence increasing the gas-liquid interaction and accelerating the liquid breakup and increasing the fi-equency. This self-induced pulsation is one of the major sources of noise in twin-fluid nozzles. 

Also, changing the entrance angle increases pulsation. However, the amplitude of the pulsation is substantially decreased. The reason for this behavior lies in the gas-liquid interaction. As the entrance angle is reduced, liquid surface exposed to steam and consequently shear stresses at the interface are increased. Therefore, an increase in pulsation is observed. However, as the liquid direction is more aligned with the gas flow, the effect of liquid momentum opposing gas flow is reduced and as a result the amplitude of the pulsation is noticeably reduced. When the entrance dent is removed this effect becomes so pronounced that pulsation virtually disappears. 

The simulation also provides some information about the bubble particle interaction as shown in Fig. 25. As the bubble rises in the liquid-solid fluidized bed, an interaction between the bubble and particles takes place. In the simulation model, the bubble-particle interaction is accounted for by adding a surface-tension-induced force to the particle motion equation. This force is also added to the source term of the liquid momentum equation for the liquid elements in the interfacial area to account for the particle effect on the interface. The particle movement is determined from the resulting total force acting on the particle. From the simulation results, it is seen that most particles contacting the bubble do not penetrate the bubble only one or two particles penetrate. Instead, they pass around the bubble surface. When the particles penetrate the bubble, they fall through quickly to the bubble base because of the low viscosity and density of the gas phase. 

If the fermentation contains sohds, then special precautions may be required to prevent sedimentation. One is a fluidized bed, where a dense slurry phase in the bottom of the tank is kept in gentle motion via upflow of the hquid or some induced gas/liquid momentum. The upflow anaerobic sludge bed (UASB) reactor is an example. In biogas installations, sohds may sediment and bacterial consortia may take several months to convert ah the material into methane and CO2. Because of the slow reaction rates, these instahations can be extremely large, up to 20 000 m. Other devices are packed-bed bioreactors, where ceUs are immobihzed on a solid support and the liquid is percolated or trickled down to supply nutrients and remove products. These can be found in specific antibiotic or food fermentations. Because of the lack of free fluid flow, the performance characteristics of such vessels are usually poor compared to suspensions. 

Themelis NJ, Tatasof P, Szekely J (1969) Gas-liquid momentum transfer in a copper converter, Trans Metall Soc AIME 245 2425-2433... 

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