Viscous dissipation turbulence

Power is the external measure of the mixer performance. The power put into the system must be absorbed through friction in viscous and turbulent shear stresses and dissipated as heat The power requirement of a system is a function of the impeller shape, size, speed of rotation, fluid density and viscosity, vessel dimensions and internal attachments, and posidon of the impeller in this enclosed system. 

For freely suspended bioparticles the most likely flow stresses are perceived to be either shear or normal (elongation) stresses caused by the local turbulent flow. In each case, there are a number of ways of describing mathematically the interactions between turbulent eddies and the suspended particles. Most methods however predict the same functional relationship between the prevailing turbulent flow stresses, material properties and equipment parameters, the only difference between them being the constant of proportionality in the equations. Typically, in the viscous dissipation subrange, theory suggests the following relationship for the mean stress [85] ... 

In turbulent flow, there is direct viscous dissipation due to the mean flow this is given by the equivalent of equation 1.98 in terms of the mean values of the shear stress and the velocity gradient. Similarly, the Reynolds stresses do work but this represents the extraction of kinetic energy from the mean flow and its conversion into turbulent kinetic energy. Consequently this is known as the rate of turbulent energy production ... 

The turbulent energy, extracted from the mean flow, passes through the energy cascade and is ultimately converted into internal energy by viscous dissipation. 

While the form of this term is the same as the viscous-dissipation term in the conditional acceleration, the modeling approach is very different. Indeed, while the velocity field in a homogeneous turbulent flow is well described by a multi-variate Gaussian process, the scalar fields are very often bounded and, hence, non-Gaussian. Moreover, joint scalar... 

The smallest size for turbulent eddies is given by the Kolmogorov microscale A. Energy loss below this size only occurs via viscous dissipation. Here also, several scales have been introduced in the framework of turbulence theory, depending whether velocity or concentration fluctuations are considered, namely Ak> Ag, and Ac (see Table I). In liquids, Ak is typically between 10 and 100 ym. The Kolmogorov microscale Ak is frequently used in the in-... 

Turbulent boundary layer flow with viscous dissipation. 

Eq.(2-200) is familiar in Chemical Engineering in turbulent flow. For example, the energy equation for one-dimensional flow [24, p.377] for a fluid of constant properties, in the absence of viscous dissipation effects and for xq = to = 0, reads ... 

The Ergun equation (described subsequently) assumes this fimctionality and incorporates explicit expressions for the two coefficients. For Re 1, turbulent flow is observed. However, due to the distribution of length scales and velocities, and the large surface area for viscous dissipation, the onset of turbulence is gradual as Re is increased, rather than sudden. 

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