**2.10 Transforming the derivatives of the statistical equations 2.10.1 Simple derivatives**

The governing differential equations (2) and (3) involve the derivatives of several mean quantities. The different physical situations may involve different physical principles and boundary conditions, so that "particular" solutions may be found. For the example of interfacial mass transfer reported in the cited literature (e.g. Wilhelm & Gulliver, 1991; Jähne & Monahan, 1995; Donelan, et al., 2002; Janzen et al., 2010, 2011), *Fp* is taken as the constant saturation concentration of gas at the gas-liquid interface, and *Fn* is the homogeneous bulk liquid gas concentration. In this chapter this mass transfer problem is considered as example, because it involves an interesting definition of the time derivative of *Fn*.

The *pth*-order space derivative *p p F z* is obtained directly from equation (8), and is given by

$$\frac{\partial \,^p \overline{F}}{\partial z^p} = \left(F\_p - F\_n\right) \frac{\partial^p \,^n \mathbf{n}}{\partial z^p} \tag{37}$$

The time derivative of the mean concentration, *<sup>F</sup> t* , is also obtained from equation (8) and

eventual previous knowledge about the time evolution of *Fp* and *Fn*. For interfacial mass transfer the time evolution of the mass concentration in the bulk liquid follows equation (38) (Wilhelm & Gulliver, 1991; Jähne & Monahan, 1995; Donelan, et al., 2002; Janzen et al., 2010, 2011)

One Dimensional Turbulent Transfer

Excel® or similar.

2

Using the partition function *n*, we obtain the mean product

22 2

expressed as functions of *n* and ߙ only.

**2.11 The heat/mass transport example** 

equations (2), (8), (30), (37), and (40), leading to

 

<sup>1</sup> <sup>1</sup>

1

*n n*

*z*

1 1

*n n*

2 1

2

3

*Kn n n n*

*Kn n n n*

 

 

*n n*

2 1

2

1 () 1 1 <sup>1</sup> <sup>1</sup>

2

 

1

*f f*

using equations (3d), (8), (24), (32), (37), (41), and (44), leading to

11 1 1

*f f*

11 1 1

*f f*

1 1

 

1 2 2 12

 

Using Random Square Waves – Scalar/Velocity and Velocity/Velocity Interactions 19

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

*n n <sup>f</sup> f n <sup>n</sup> n n FF zz z*

 

Equation (44) shows that mean products between powers of *f* and its derivatives are

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

**2.11.1 Obtaining the transformed equations for the one-dimensional transport of F**  Equation (2) may be transformed to its random square waves correspondent using

In the same way, equation (3d) is transformed to its random square waves correspondent

2

*n n dn d K nD d z d z*

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

2 2 2

*z z*

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

<sup>1</sup> *<sup>f</sup> p n f p n*

2 2 <sup>2</sup> <sup>1</sup> 1 1

1

 

<sup>1</sup> <sup>1</sup>

2 2

*n n*

1 1

1 () 1 1

*n n n n n nn*

 

 

( 1)/2 <sup>1</sup> <sup>1</sup> 1 2

*n nnn*

*f*

 

2 1

2

2

*f*

*f*

 

/2

2

*z*

 

(43)

 

(44)

*f pn*

(45)

$$\frac{dF\_n}{dt} = K\_f \left( F\_p - F\_n \right) \tag{38}$$

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

$$\frac{\partial \overline{F}}{\partial t} = \frac{\partial \left\lfloor nF\_p + (1 - n)F\_n \right\rfloor}{\partial t} = (1 - n)\frac{\partial F\_n}{\partial t} \tag{39}$$

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

*t*

$$\frac{\partial \overline{F}}{\partial t} = K\_f \left( 1 - n \right) \left( F\_p - F\_n \right) \tag{40}$$

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 different equations.

The time derivatives of the central moments *f* are obtained from equation (24), furnishing:

$$\frac{\overline{\partial f}^{\theta}}{\overline{\partial t}} = -\theta \, n \big[ (1-n) \big] \big[ (1-n)^{\theta - 1} + (-1)^{\theta} \big( n \big)^{\theta - 1} \big] \big[ F\_p - F\_n \right)^{\theta - 1} \left( 1 - a\_f \right)^{\theta} \frac{\partial F\_n}{\partial t}$$

$$\text{or}$$

$$\frac{\partial \overline{f^{\theta}}^{\theta}}{\partial t} = -\theta K \, n \big[ (1-n) \big[ \left( 1-n \big)^{\theta - 1} + (-1)^{\theta} \big( n \big)^{\theta - 1} \big] \big[ F\_p - F\_n \right]^{\theta} \left( 1 - a\_f \right)^{\theta}$$

As no velocity fluctuation is involved, only the partition function *n* is needed to obtain the mean values of the derivatives of *f* , that is, no superposition coefficient is needed. The obtained equations depend only on *n* and ߙ , the basic functions related to *F*.

#### **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 of fluctuations and their second order derivatives, like 2 2 *<sup>f</sup> <sup>f</sup> z* , <sup>2</sup> 2 2 *<sup>f</sup> <sup>f</sup> z* , and 2 3 2 *<sup>f</sup> <sup>f</sup> z* . The

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

$$f\_1^{\theta} \frac{\partial^2 f\_1}{\partial z^2} = \left[ (1-n) \left( F\_p - F\_n \right) \left( 1 - \alpha\_f \right) \right]^{\theta} \frac{\partial^2 \left[ (1-n) \left( 1 - \alpha\_f \right) \right]}{\partial z^2} \left( F\_p - F\_n \right) \tag{42}$$

$$f\_2^{\theta} \frac{\hat{\sigma}^2 f\_2}{\hat{\sigma} z^2} = \left[ -n \left( F\_p - F\_n \right) \left( 1 - \alpha\_f \right) \right]^{\theta} \frac{\hat{\sigma}^2 \left[ -n \left( 1 - \alpha\_f \right) \right]}{\hat{\sigma} z^2} \left( F\_p - F\_n \right) \tag{43}$$

Using the partition function *n*, we obtain the mean product

$$\overline{\left(f^{\theta}\frac{\hat{\sigma}^{2}}{\hat{\sigma}z^{2}}\right)} = \left[ (1-n)^{\theta-1} \frac{\hat{\sigma}^{2}\left[ (1-n)\left(1-\alpha\_{f}\right) \right]}{\hat{\sigma}z^{2}} + (-n)^{\theta-1} \frac{\hat{\sigma}^{2}\left[ -n\left(1-\alpha\_{f}\right) \right]}{\hat{\sigma}z^{2}} \right] n \left(1-n\right) \left(1-\alpha\_{f}\right)^{\theta} \left(F\_{p} - F\_{n}\right)^{\theta+1} \tag{44}$$

Equation (44) shows that mean products between powers of *f* and its derivatives are expressed as functions of *n* and ߙ only.
