Equations energy dissipation

In other words, if we look at any phase-space volume element, the rate of incoming state points should equal the rate of outflow. This requires that be a fiinction of the constants of the motion, and especially Q=Q i). Equilibrium also implies d(/)/dt = 0 for any /. The extension of the above equations to nonequilibriiim ensembles requires a consideration of entropy production, the method of controlling energy dissipation (diennostatting) and the consequent non-Liouville nature of the time evolution [35]. 

Similarly, the rate of energy dissipation in Eq. (2.6) has units energy volume time, so the dimensions of that equation are... 

An alternative point of view assumes that each repeat unit of the polymer chain offers hydrodynamic resistance to the flow such that f-the friction factor per repeat unit-is applicable to each of the n units. This situation is called the free-draining coil. The free-draining coil is the model upon which the Debye viscosity equation is based in Chap. 2. Accordingly, we use Eq. (2.53) to give the contribution of a single polymer chain to the rate of energy dissipation ... 

In this equation, represents the rate of energy dissipation per unit mass of fluid. In pulsed and reciprocating plate columns the dimensionless proportionahty constant K in equation 38 is on the order of 0.3. In stirred tanks, the proportionaUty constant has been reported as 0.024(1 + 2.5 h) in the holdup range 0 to 0.35 (67). The increase of drop si2e with holdup is attributed to the increasing tendency for coalescence between drops as the concentration of drops increases. A detailed survey of drop si2e correlations is given by the Hterature (65). 

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... 

Here 4 = F,Jfn is the energy dissipation per unit mass. This equation has been called the engineering Bernoulli equation. For an incompressible flow, Eq. (6-15) becomes... 

Using turbulenee models, this new system of equations ean be elosed. The most widely used turbulenee model is the k-e model, whieh is based on an analogy of viseous and Reynolds stresses. Two additional transport equations for the turbulent kinetie energy k and the turbulent energy dissipation e deseribe the influenee of turbulenee... 

The energy dissipated per unit mass F is then given by equation 3.19 ... 

In considering the flow in a pipe, the differential form of the general energy balance equation 2.54 are used, and the friction term 8F will be written in terms of the energy dissipated per unit mass of fluid for flow through a length d/ of pipe. In the first instance, isothermal flow of an ideal gas is considered and the flowrate is expressed as a function of upstream and downstream pressures. Non-isothermal and adiabatic flow are discussed later. 

We can estimate the effect of energy dissipation on liquid heating and values of flow parameters corresponding to arising oscillations in the flow. We assume that the density of the fluid and its thermal conductivity are constant. Then, the energy equation attains the form... 

Hetsroni et al. (2005) evaluated the effect of inlet temperature, channel size and fluid properties on energy dissipation in the flow of a viscous fluid. For fully developed laminar flow in circular micro-channels, they obtained an equation for the adiabatic increase of the fluid temperature due to viscous dissipation ... 

Consider the mass, thermal and momentum balance equations. The key assumption of the present analysis is that the Knudsen number of the flow in the capillary is sufficiently small. This allows one to use the continuum model for each phase. Due to the moderate flow velocity, the effects of compressibility of the phases, as well as mechanical energy, dissipation in the phases are negligible. Assuming that thermal conductivity and viscosity of vapor and liquid are independent of temperature and pressure, we arrive at the following equations ... 

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 ... 

The value of this latter parameter is proportional to the energy dissipated as heat per cycle, and is known as the loss modulus. The former quantity, Gj, is proportional to the recoverable energy, and is called the storage modulus. The two are combined to form the complex modulus, G related by the equation... 

Consider a deformation consisting of repeated sinusoidal oscillations of shear strain. The relation between stress and strain is an ellipse, provided that the strain amplitude is small, and the slope of the line joining points where tangents to the ellipse are vertical represents an effective elastic modulus, termed the storage modulus /r. The area of the ellipse represents energy dissipated in unit volume per cycle of deformation, expressed by the equation... 

Equation 18.2 reveals the importance of energy dissipation in the fracmre of mbbery materials, that is, the strength of rubbers, whether filled or unfilled, is governed by the hysteresis energy dissipated in deforming the system. 

FIGURE 26.78 Theoretical acceleration (force) distribution function in a computer simulation, the resulting energy dissipation multiphed by the frequency of occurrence and the expected abrasion loss, using Equation 26.18 with parameter for a passenger tread compound. 

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 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). 

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. 

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 ... 

Figures 12 and 13 show the effects of agitation and time of exposure on suspensions of biological materials in bioreactors [61]. In turbulent flow the energy dissipation rate per unit mass, e, of a stirred bioreactor is normally expressed by the following equation ...

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