Energy dissipation calculation

The calculation of backscatter coefficients via the approach outlined above is mathematically complex. Heidenreich 44) developed a simple empirical backscatter model which is applicable to resist exposure being based on the direct observation of chemical changes produced by backscat-tered electrons at different accelerating voltages on several substrates. The model is independent of scattering trajectory and energy dissipation calculations and is essentially a radial exponential decay of backscatter current density out to the backscatter radius determined by electron range. 

A2.3.4 It is important to note that the uncertainty of the calculations is such that errors in G of more than 20 % may result at low values of G. For this reason, it is recommended that no reliance be placed upon calculated G values of less than 3 and that ad tional energy dissipation calculated G value. 

The turbulent kinetic energy is calculated from equation 41. Equation 43 defines the rate of energy dissipation, S, which is related to the length scale via... 

Energy dissipation rate per unit mass of fluid (ranges 570 < Ns < 1420) fluid and sphere, m/s. Cq,. = drag coefficient for single particle fixed in fluid at velocity i>,.. See 5-27-G for calculation details and other applica- ... 

The classic model that describes chain scission in elastomers was proposed many years ago by Lake and Thomas [26J. The aim of the model is to calculate the energy dissipated in breaking all the polymer strands that have adjacent cross-links on either side of the crack plane. The basic assumption of this model is that all the main chain bonds in any strand that breaks must be strained to the dissociation... 

The energy dissipated in the hydraulic jump is now calculated. For a small change in the flow of a fluid in an open channel ... 

To calculate the flow fields outside the evaporating meniscus we use the onedimensional model, developed by Peles et al. (1998, 2000, 2001). Assuming that the compressibility and the energy dissipation are negligible (a flow with moderate velocities), the thermal conductivity and viscosity are independent of the pressure and temperature, we arrive at the following system of equations ... 

Having calculated the force for a particular event the slip is calculated using the bush model and hence the energy dissipation is obtained. Using the factors of the abrasion equation, determined with the LAT 100 on an alumina surface the abrasion loss for each event is calculated. The forces are different for a driven and a nondriven axle and accordingly different abrasion rates will result. 

In order to use Eqs. (3) and (4) or the data given in Fig. 1, for the calculation of maximum turbulent fluctuation velocity the maximum energy dissipation e , must be known. With fully developed turbulence and defined reactor geometry, this is a fixed value and directly proportional to the mean mass-related power input = P/pV, so that the ratio ,/ can be described as an exclusive function of reactor geometry. In the following, therefore details will be provided on the calculation of power P and where available the geometric function ,/ . 

In the case of stirred vessels the values A/riL can be calculated by the following equation using the geometry parameter d/D, H/D, the Newton number Ne, the Reynolds number Re = nd /v, the energy dissipation ratio e/e and the related macro scale A/d. For standard turbines e.g. Mockel [24] found the value A/d = 0.08 close to the impeller. Corresponding to this the maximum of the dissipation ratio ,/ has to be used which can be estimated by Eq. (20). 

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). 

Numerous researchers have studied damage to micro-organisms during flow in pipes, (Fig. 11) [87,88] Most researchers use a Fanning friction factor, f, to calculate the energy dissipation rate for fully developed flow in tubular bioreactors and capillary flow devices. There are minor differences in the equations that are used but they are generally of the following form [89,901 ... 

The uncertainty involved in the calculation of the energy dissipation rate makes it difficult to compare experimental results reported by different researchers. For the same reasons, so far it has proved difficult to assess flow induced effects in different items of process equipment using the common basis of equal energy dissipation rate. For example. Fig. 14 shows the biological response of (SF-9) insect cells as a function of the energy dissipation rates in a capillary tube and in a mechanically stirred vessel [99]. In these plots the calcula-... 

For plant cell suspensions cultivated in shake flasks, Huang et al. [45] used the energy dissipation rate as a correlating parameter for system response. Specific power input was calculated using the empirical correlation proposed by Sumino et al. [46] and subsequently employed in other applications [47,48] ... 

Power input per unit mass of the system is equal to the rate of energy dissipation per unit mass of the liquid and it is estimated by considering the permanent pressure head loss across the orifice. The rate of energy dissipation due to eddy losses is the product of the head loss and the volumetric flow rate. Frictional pressure drop at downstream of the orifice can be calculated as,... 

The first term on the right-hand side of Equation (2) describes the formation rate of k-flocs, and the second term is the disappearance rate. In the present study the flow was turbulent, and an effective shear rate was calculated as (e/v) / (19), where e is the energy dissipation, W/kg, and v is the kinematic viscosity, m /s. Equation (2) was also extended to include a collision efficiency factor, a, defined as... 

The value of Am strongly depends on the size distribution of the bubble diameter in the system and the volume fraction of the bubbles. These factors are strongly dependent not only on the liquid properties, but also on the gas/liquid flow ration and the energy dissipation. Opposite to the situation with respect to kiiq it is not possible to calculate Am with theoretical equations. This is the reason that in practise almost always the product of kuq and Am is considered. This product can easily be determined experimentally. 

Integrating over the hysteresis loop between the compression and decompression curves in Figure 19 yields the amount of energy dissipated through the reversible bond formation/dissociation process. Unfortunately, it is not possible to determine the contribution of these transitions to the friction of phosphate films because such a calculation would require knowledge of the number of similar instabilities that occur per sliding distance, which is certainly beyond the limits of first-principles calculations. Nonetheless, the results do indicate that pressure- and shear-induced chemical reactions can contribute to the friction of materials. 

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