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By R. T. Lahey Jr., D. A. Drew (auth.), Jeffery Lewins, Martin Becker (eds.)

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THREE-DIMENSIONAL CONSERVATION EQUATION 49 FLOW Figure 5. One-dimensional Eulerian Control Volume A. One-Dimensional Mass Conservation Equation (107) becomes (145) where we have defined the axial speed as (146) In addition, since we have taken the limit as L3-+O, the volume (V), which appears in the nume~ator and denominator of the gradient term in Equation (23), has been replaced by the cross sectional area (A ) • x-s B. One-Dimensional Momentum Conservation In a similar manner, Equation (108) becomes 50 R.

Let us now tabulate the one dimensional, space-and timeaveraged, conservation equations. THREE-DIMENSIONAL CONSERVATION EQUATION 49 FLOW Figure 5. One-dimensional Eulerian Control Volume A. One-Dimensional Mass Conservation Equation (107) becomes (145) where we have defined the axial speed as (146) In addition, since we have taken the limit as L3-+O, the volume (V), which appears in the nume~ator and denominator of the gradient term in Equation (23), has been replaced by the cross sectional area (A ) • x-s B.

X,t) ~k /:,~d k. 1 (m" /:'~k. ::'k dS, (65d) 1 ~k W (65c) dS, ~k 1- V a. (x,t) ~k (65b) dS, . q" V a. (x,t) ~k If 1 -1 ff 1 /:, 1 d /:, r\ -k. v. 1 w . ,9;"k dS ' . i . ::'k dS k/P k ) dS (65e) (65f) (65g) THREE-DIMENSIONAL CONSERVATION EQUATION 21 The conservation equations tabulated in Equations (36), (52) and (64) are instantaneous volume-averaged equations. It is well known that, even for single-phase turbulent flows, one must time-average these equations to achieve a tractable set. As an example, consider the velocity of phase-k, ~k.

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