Volume-time-averaged equations

The purpose of the time averaging after the volume averaging is to express averages of products in terms of products of averages and to account for turbulent fluctuations and high-frequency fluctuations [Soo, 1989]. The volume-time averaging is presented here in a similar way to that of the Reynolds analysis of single-phase turbulent flow. 

The averaging time duration should be chosen in such a way that thf T C Ilf-The characteristic time of the high-frequency component rHF may be estimated from the reciprocal of the characteristic spectral frequency of the fluctuation, while the characteristic time of the low-frequency component Tlf may be determined from the time required to travel the characteristic dimension of the physical system at the local characteristic low-frequency speed. Thus, a time averaging after volume averaging can be defined as 

An instantaneous volume-averaged quantity is therefore expressed by two parts 

The correlation of two fluctuating components may be best formulated using Boussinesq s approach [Boussinesq, 1877] by introducing a transport coefficient. Thus, the last term in Eq. (5.153) can be expressed by 

Assuming the material density or the intrinsic averaged density of phase k to be constant (as is true for the solids and almost true for the fluid when the effect of temperature variation on the material density can be neglected), we have 

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The volume-time-averaged equations can be summarized as follows ... 

For convenience, we remove the () signs and time-averaging bars from the preceding equations. Thus, the volume-time-averaged equations can be simplified to the form... 

Sha WT, Slattery JC (1980) Local Volume-Time Averaged Equations of Motion for Dispersed, Turbulent, Multiphase Flows. NUREG/CR-1491, ANL-80-51... 

As in the volume averaged equations discussed in sect 3.4.1, three undetermined terms can be identified in the time averaged equations (3.259). The first term, —V (/3fc(pfcv V )rfc) denotes the covariance or correlation terms. The second term, Tkikj, accounts for the effects of interfacial stress. 

To save space the governing time averaged equations for each of the primary variables are not listed here as their mathematical form coincides with the volume averaged model formulation given in sect 3.4.1. Nevertheless, it is important to note that the physical interpretations of the mean quantities and the temporal covariance terms differ from their spatial counterparts. Furthermore, the conventional constitutive equations for the unknown terms are discussed in chap 5, and the same modeling closures are normally adopted for any model formulation even though their physical interpretation differ. 

Equations in Table IX are written per unit of bed volume (A )g is a time averaged, mean axial bed conductivity. A is a longitudinal diffusivity and Ai allows for particle to particle conductivity. Not all the terms in the model as given in the table are important. For example, Wu et al. (1995, 19%) and Xiao and Yuan (1996) neglect the accumulation and dispersion terms in Eq. (30) and the accumulation and conduction terms in Eq. (28). 

A similar procedure can be applied to the energy transport terms. When the volume-averaged energy equation is expressed in terms of the internal energy and time averaging... 

The mean-squared, end-to-end distance in Equation [9] is the simplest average property of interest for a polymer chain. Among other physical properties, this quantity appears in the equations of statistical mechanical theories of rubber-like elasticity. In Equation [9], the angle brackets denote the ensemble (or time) average over all possible conformations. The subscript 0 indicates that the average pertains to an unperturbed chain (theta conditions no excluded volume effects are present). (See Figure... 

Averaging methods to derive time and/or volume averaged equations for all phases... 

In Eulerian-Eulerian (EE) simulations, an effective reaction source term of the form of Eq. (5.32) can be used in species conservation equations for all the participating species. The above comments related to models for local enhancement factors are applicable to the EE approach as well. It must be noted that interfacial area appearing in Eq. (5.32) will be a function of volume fraction of dispersed phase and effective particle diameter. It can be imagined that for turbulent flows, the time-averaged mass transfer source will have additional terms such as correlation of fluctuations in volume fraction of dispersed phase and fluctuations in concentration even in the absence... 

The turbulent gas/liquid flow in baffled tanks with turbine stirrer can be predicted. A mathematical model has been developed for turbulent, dispersed G/L flow. The time-averaged two phase momentum equations were solved by using a finite volume algorithm. The turbulent stresses were simulated with a K-fi-model. The distribution of gas around the stirrer blades is predicted for the first time. This model also enables an a priori prediction of the drop in the power dissipated by the stirrer in the presence of gas. Predicted flow characteristics for the gas/liquid dispersion show good agreement with the experimental data. 

Sha WT, Chao BT, Soo SL (1983) Averaging Procedures of Multiphase Conservation Equations. Transactions of the American Nuclear Society 45 814-816 Sha WT, Chao BT, Soo SL (1983) Time Averaging of Volume-Averaged Conservation Equations of Multiphase Flow. AIChE Symposium Series 79(225) 420-426... 

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