Energy dissipation average

As may be expected, turbulent flow (9,11) is more efficient for droplet formation in low viscosity Hquids. With the average amount of energy dissipated per unit time and volume equal to Z and mass density equal to p, the larger eddies are characterized by a velocity gradient equal to... 

Turbulent velocity fluctuations ultimately dissipate their kinetic energy through viscous effects. MacroscopicaUy, this energy dissipation requires pressure drop, or velocity decrease. The ener dissipation rate per unit mass is usually denoted . For steady ffow in a pipe, the average energy dissipation rate per unit mass is given by... 

What about the micro-scale phenomena These are dependent primarily on the energy dissipation per unit volume, although one must also be concerned about the energy spec tra. In general, the energy dissipation per unit volume around the impeller is approximately 100 times higher than in the rest of the tank. Tnis results in an rms velocity fluc tuation ratio to the average velocity on the order of 10 I between the impeller zone and the rest of the tank. 

It should be emphasized that under conditions of energy dissipation the definition of the heat transfer coefficient as k dT/dr)/ T — T ), where T is the average fluid temperature and 7 is the wall temperature, does not characterize the acmal heat transfer properly (Kays and Crawford 1993 Schlichfing 2000). 

The system could be energy conservative if the atoms moved smoothly over the potential field. In that case, an atom, when traveling over one period of the potential, would experience a symmetrically distributed lateral force so that its time average and the net work done by the force would be zero. In reality, however, this is not going to happen that way. The author will demonstrate in the following how the system becomes unstable which inevitably leads to energy dissipation and friction. 

The example demonstrates that the instability and consequent energy dissipation, similar to those in the Tomlinson model, do exist in a real molecule system. Keep in mind, however, that it is observed only in a commensurate system in which the lattice constants of two monolayers are in a ratio of rational value. For incommensurate sliding, the situation is totally different. Results shown in Fig. 21(b) were obtained under the same conditions as those in Fig. 21 (a), but from an incommensurate system. The lateral force and tilt angle in Fig. 21(b) fluctuate randomly and no stick-slip motion is observed. In addition, the average lateral force is found much smaller, about one-fifth of the commensurate one. 

Fig. 22. Dependency of average particle diameter dp on maximiun energy dissipation of impeller systems with baffles by stirring of biological and model particle systems explanations see Tables 3 and 4...

Fig. 23. Average particle size dp after t = 120 h stirring for various impeller types and working conditions (left hand diagram data from [60]) and correlation with the maximum energy dissipation 8 (right hand diagram) stirred bioreactor with 4 baffles V = 6L D = 0.2m H/D = 0.96 zi=l...

The ratio Zt/Zp is in technical reactors much higher than 1. It becomes, e.g. also for a small scale reactor of V-IOOL (H/D = 2 D = 0.4 m) equipped with three turbines (d/D = 0.3) and working at a average impeller power per mass of only = lmVs in media with water like viscosity to Zt/Zp>36...72. The maximal energy dissipation in the impeller zones, required for the calculation of length scale of turbulence here taken from Eq. (20). 

In all of the above equations, is assumed to be constant and uniform throughout the flow field. In most items of bioprocess equipment, however, there is a spatial distribution of energy dissipation. The definition of an average or a maximum energy dissipation rate is notoriously difficult in the case of bioprocess equipment such as high pressure homogenisers, centrifuges, pumps and microfiltration units which all have complex flow fields. 

Fig. 7. Variation in cell death rates with average energy dissipation rate for capillary and jet flows. From reference [59], redrawn with permission. 1998, Wiley-Liss, Inc, a subsidiary of John Wiley Sons, Inc...

The blend time increases significantly with reactor size. To keep the average energy dissipation constant one has to decrease the speed of rotation. However, as the stirrer diameter increases, the tip speed becomes greater despite the lower rotation speed. In this situation energy dissipated in the vicinity of the stirrer increases with reactor size. 

Fig. 9. Average efficiency of stretching of material elements (e.) in a simple shear flow with random reorientation after an average length stretch ym. pgives the with of the distribution of length stretch about the mean value (ym). Results for a random distribution (top) and a normal distribution (bottom) of length stretch are shown. The maximum in the efficiency corresponds to the maximum length stretch for a fixed amount of energy dissipated and occurs at an average stretch of about 5 per period (Khakhar and Ottino, 1986a).

It can easily be shown that the average energy dissipation per unit of time, as caused by the presence of one rigid dumb-bell, reads ... 

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