**2.11 The heat/mass transport example**

18 Hydrodynamics – Advanced Topics

 *<sup>n</sup> f p <sup>n</sup> d F KF F*

This equation applies to the boundary value *Fn* or, in other words, it expresses the time variation of the boundary condition *Fn* shown in figure 1. *Kf* is the transfer coefficient of *F* (mass transfer coefficient in the example). To obtain the time derivative of *F* , equations (8) and (38) are used, thus involving the partition function *n*. In this example, *n* depends on the agitation conditions of the liquid phase, which are maintained constant along the time (stationary turbulence). As a consequence, *n* is also constant in time. The time derivative of

(1 ) (1 ) *p n <sup>n</sup> F F nF n F*

*tt t*

*f pn* <sup>1</sup> *<sup>F</sup> K nF F*

Equation (40) is valid for boundary conditions given by equation (38) (usual in interfacial mass and heat transfers). As already stressed, different physical situations may conduce to

> *f F n n n n FF t t*

 

or (41)

As no velocity fluctuation is involved, only the partition function *n* is needed to obtain the

**2.10.2 Mean products between powers of the scalar fluctuations and their derivatives**  Finaly, the last "kind" of statistical quantities existing in equations (3) involve mean products

general form of such mean products is given in the sequence. From equations (14) and (15), it

(1 ) 1 *<sup>f</sup>*

*<sup>n</sup> <sup>f</sup> <sup>f</sup> nF F F F*

<sup>1</sup> 1 1 11 1 <sup>1</sup> *<sup>n</sup> pn f*

1 1 11 1 <sup>1</sup> *pn f*

 

*t*

<sup>2</sup> <sup>2</sup>

1 2 2

*z z*

obtained equations depend only on *n* and ߙ , the basic functions related to *F*.

*<sup>f</sup> Kn n n n FF*

*n*

(39)

(40)

are obtained from equation (24),

 

  , and

2 3 2 *<sup>f</sup> <sup>f</sup> z* 

(42)

. The

 

, that is, no superposition coefficient is needed. The

 , <sup>2</sup> 2 2 *<sup>f</sup> <sup>f</sup> z* 

2 2 *<sup>f</sup> <sup>f</sup> z* 

*p n f p n*

(1 ) 1

*F* in equation (8) is then given by

different equations.

> *t*

mean values of the derivatives of *f*

of fluctuations and their second order derivatives, like

1

furnishing:

follows that

From equations (38) and (39), it follows that

The time derivatives of the central moments *f*

*dt* (38)

In this section, the simplified example presented by Schulz et al. (2011a) is considered in more detail. The simplified condition was obtained by using a constant ߙ, in the range from 0.0 to 1.0. The obtained differential equations are nonlinear, but it was possible to reduce the set of equations to only one equation, solvable using mathematical tables like Microsoft Excel® or similar.
