Averaging theorem

Whitaker, S, A Simple Geometrical Derivation of the Spatial Averaging Theorem, Chemical Engineering Education Winter, 18, 1985. 

Since a multiphase flow usually takes place in a confined volume, the desire to have a mathematical description based on a fixed domain renders the Eulerian method an ideal one to describe the flow field. The Eulerian approach requires that the transport quantities of all phases be continuous throughout the computational domain. As mentioned before, in reality, each phase is time-dependent and may be discretely distributed. Hence, averaging theorems need to be applied to construct a continuum for each phase so that the existing Eulerian description of a single-phase flow may be extended to a multiphase flow. 

The averaging theorem for the time derivative can be derived directly from the general transport theorem. Consider the phase k in Fig. 5.7 to be fixed in space while Ak varies as a function of time as a result of phase change. Applying Eq. (5.8) to the system in Fig. 5.7, with d/dt = 3/31, we have... 

Equations (5.116), (5.117), and (5.119) characterize the volume-averaging theorems of derivatives [Slattery, 1967b Whitaker, 1969],... 

The volume-averaged continuity equation for dispersed laminar multiphase flows is given by applying the volume-averaging theorems to Eq. (5.13) as... 

Applying the volume-averaging theorems to Eqs. (5.14) and (5.18), verify that the volume-averaged momentum and energy equations for phase k in a multiphase flow can be given by Eqs. (5.123) and (5.126), respectively. 

The first, second and third terms in (3.122) have to be reformulated using the conventional volume averaging theorems . The first theorem one makes use of relates the spatial average of a time derivative to the time derivative of a spatial average, and is called the Leibnitz rule for volume averaging ... 

Alternatively, a similar result is obtained from (3.172) by use of the complete differential (1.142). This method seems more appropriate as the derivation of the interfacial terms for example is in accordance with the standard averaging theorems, hence this method might be more rigorous. The alternative form of the averaged temperature is given in (3.204). However, the choice of model approximations should of course always be determined by comparison with physical observations. 

The time averaging theorems have been derived by [112] [47]. 

The first, second and third terms in (3.270) have to be reformulated using the extended forms of the conventional averaging theorems. 

Area averaging can be considered to be a limiting case of local volume averaging [43, 47, 189]. Thus the phrase limiting form refers to the modified forms of the averaging theorems which are applicable to the governing 3D equations to derive a set of equations valid for ID problems. 

Howes FA, Whitaker S (1985) The Spatial Averaging Theorem Revisited. Chem Eng Sci 40(8) 1387-1392... 

Whitaker S (1985) A Simple Geometrical Derivation of the Spatial Averaging Theorem. Chemical Engineering Education, pp. 18-21 and pp. 50-52 Whitaker S (1992) The species mass jump condition at a singular surface. Chem Eng Sci 47(7) 1677-1685... 

Averaging theorems give, for mass flow of the particle phase... 

For a rigid porous medium, one can use the transport theorem and the averaging theorem to express this result as... 

The volume-averaging theorem can be used with the average of the gradient in equation 1.104 in order to obtain... 

Ochoa-Tapia J.A., del Rio J.A. and Whitaker S. 1993. Bulk and surface diffusion in porous media An application of the surface averaging theorem, Chem. Eng. Sci., 48, 2061-2082. Ochoa-Tapia J.A., Stroeve P. and Whitaker S. 1994. Diffusive transport in two-phase media Spatially periodic models and Maxwell s theory for isotropic and anisotropic systems, Chem. Eng. Sci., 49, 709-726. 

The nonautonomous perturbed component in (3.33d) will not survive the KAM averaging theorem, and some of the nonautonomous terms may be suspended by applying the following Hamilton-Jacobian nonautonomous canonical transformation. 

Knowing the low-frequency and high-frequency formulas, the internal impedance formula at any frequency can be given in the following form by applying Rolle s averaging theorem [2] ... 

Whitaker S (1985) A simple geometrical derivation of the spatial averaging theorem. Chemical engineering education, pp 18-21 and pp 50-52... 

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