**8. Appendix II: Solving equation (51) using mathematical tables**

Equation (51) (or equation (AI-10)) of this chapter is a third order nonlinear ordinary differential equation, for which adequate numerical methods must be applied. Some methods were considered to solve it.

A first attempt was made using the second order Finite Differences Method and the solver device from the Microsoft Excel® table, intending to solve the problem with simple and practical tools, but the results were not satisfactory. It does not imply that the Finite Differences Method does not apply, but only that we wanted more direct ways to check the applicability of equation (51).

The second attempt was made using Runge-Kutta methods, also furnished in mathematical tables like Excel ®, maintaining the objective of solving the one-dimensional problem with simple tools. In this case, the results were adequate, superposing well the experimental data.

The Runge-Kutta methods were developed for ordinary differential equations (ODEs) or systems of ODEs. Equation (AI-10) is a nonlinear differential equation, so that it was necessary to first rewrite it as a system of ODEs, as follows

$$\frac{d\mathbf{n}}{dz} = \mathbf{j} \tag{A11-1}$$

$$\frac{d^2n}{dz^2} = w \tag{A11-2}$$

$$\frac{dw}{dz} = (f\_1 + f\_2) / f\_3 \tag{A1I-3}$$

in which

30 Hydrodynamics – Advanced Topics

<sup>2</sup>

2 2

(1 2 ) 2 11 2 1 <sup>1</sup>

*d z*

*d z* :

2

2

3 3

*d z d z*

2

2 2

<sup>2</sup>

 

*n dn n A An n <sup>n</sup> d z d n <sup>A</sup>*

2

*d z*

3 3

1 1

1

(1 2 ) 2 11

2 2

*AA n dn d z*

2

Equation (51) (or equation (AI-10)) of this chapter is a third order nonlinear ordinary differential equation, for which adequate numerical methods must be applied. Some

*n*

*n*

2 2 2 2 (AI-8)

(AI-9)

(AI-10)

2

*d z*

2 11

*n A d n*

2

2 2 2 2

*d z*

2

*d z d n d z dn*

*n dn n A S An n <sup>n</sup>*

2 11 <sup>1</sup> <sup>1</sup>

*dn n A A dn*

*dz dz dn d z*

2

and simplifying *dn*

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

*dn d n SA n d z d z*

2 12 1

(1 2 ) 2 1

 

*dn n dn S An n d z d z*

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

1

*<sup>A</sup> d n dn S n A d z d z*

2 11

*dn n A n A*

2

*n dn S An n*

Rearranging (after multiplying the equation by A and using *S*=1/ߢ(:

2

2 1 1

<sup>3</sup> 11 12 2

**8. Appendix II: Solving equation (51) using mathematical tables** 

*dn A n AA n n*

 

*A d z*

(1 2 ) 2 1

*n dn dn A An n*

(1 2 ) 2 1

12 12 1

Equation (AI10) is the equation (51) presented in the text.

2 *dn d z* 

1

2

*d z*

0

methods were considered to solve it.

2

Multiplying by

$$f\_1 = -A \begin{bmatrix} -\left[ 2An(1-n) + \frac{(1-2n)}{2} \right] w + \kappa (1-n) \left[ \frac{2n(A-1) + 1}{2} \right] + \\\\ + \frac{\left\{ 1 + 2A \left[ A \left( 1 - 2n \right) - 1 \right] \right\}}{A} j^2 \end{bmatrix} w \tag{A11-4}$$

$$f\_2 = -\kappa \left[ (A - 1)(1 - n) - A \left[ A(1 - 2n) - \left( \frac{3}{2} - 2n \right) \right] \right] j^2 \tag{A11-5}$$

$$f\_3 = A \left[ 2An(1-n) + \frac{(1-2n)}{2} \right] j \tag{A11-6}$$

Figure 6 shows that 3th, 4th and 5th orders Runge-Kutta methods were applied to obtain numerical results for the profile of *n*. This Appendix shows a summary of the use of the 5th order method. Of course, similar procedures were followed for the lower orders. As usual in this chapter, equations (AII-1) up to (AII-3) use the nondimensional variable *z* without the star "\*" (that is, it corresponds to z\*). Considering "*y*" the dependent variable in a given ODE, the of 5th order method, presented by Butcher (1964) appud Chapra and Canale (2006), is written as follows

$$y\_{k+1} = y\_k + \frac{\Delta x}{90} (7\nu\nu\_1 + 32\nu\nu\_3 + 12\nu\nu\_4 + 32\nu\nu\_5 + 7\nu\nu\_6) \tag{A1I-7}$$

One Dimensional Turbulent Transfer

Reading, Massachusetts.

J., 10(6), pp. 870-877.

Washington, U.S.A., 383 p.

Verlag & Studio, 918 p.

N.8, pp. 2005-2017.

São Carlos. 299p.

**9. References** 

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Brodkey, R.S. (1967) The phenomena of Fluid Motions, *Addison–Wesley Publishing Company*,

Butcher, J.C. (1964). On Runge-Kutta methods of high order. J.Austral. Math. Soc.4, p.179-194. Chapra, S.C.; Canale, R.P. (2006). Numerical methods for engineers. McGraw-Hill, 5th ed., 926 p. Corrsin, S. (1957) Simple theory of an idealized turbulent mixer, AIChE J., 3(3), pp. 329-330. Corrsin, S. (1964) The isotropic turbulent mixer: part II - arbitrary Schmidt number, AIChE

Donelan, M.A., Drennan, W.M., Saltzman, E.S. & Wanninkhof, R. (2002) Gas Transfer at

Jähne, B. & Monahan, E.C. (1995) Air-Water Gas Transfer, Selected papers from the Third

Janzen, J.G. (2006) Fluxo de massa na interface ar-água em tanques de grades oscilantes e

Janzen J.G, Schulz H.E. & Jirka GH. (2006) Air-water gas transfer details (portuguese).

Janzen, J.G., Schulz, H.E. & Jirka, G.H. (2011) Turbulent Gas Flux Measurements near the Air-

Monin, A.S. & Yaglom, A.M. (1981), Statistical Fluid Mechanics: Mechanics of Turbulence,

Schulz, H.E. (1985) Investigação do mecanismo de reoxigenação da água em escoamento e

Schulz, H.E.; Bicudo, J.R., Barbosa, A.R. & Giorgetti, M.F. (1991) Turbulent Water Aeration:

Schulz, H.E. & Janzen, J.G. (2009) Concentration fields near air-water interfaces during

Schulz, H.E., Lopes Junior, G.B. & Simões, A.L.A. (2011b) Gas-liquid mass transfer in

Pope, S.B. (2000), Turbulent Flows, Cambridge University Press, 1st ed., UK, 771p.

eds. Air Water Mass Transfer, ASCE, New York, pp.142-155.

Hinze, J.O. (1959), Turbulence, Mc. Graw-Hill Book Company, USA, 586 p.

Revista Brasileira de Recursos Hídricos; 11, pp. 153-161.

Volume 1, the MIT Press, 4th ed., 769p.

Volume 2, the MIT Press, 2th ed., 873p.

*Braz. J. Chem. Eng.* vol.26, n.3, pp. 527-536.

Water Surfaces, Geophysical Monograph Series, American Geophysical Union,

International Symposium on Air-Water Gas Transfer, Heidelberg, Germany, AEON

detalhes de escoamentos turbulentos isotrópicos (Gas transfer near the air-water interface in an oscillating-grid tanks and properties of isotropic turbulent flows – text in Portuguese). Doctoral thesis, University of Sao Paulo, São Carlos, Brazil. Janzen, J.G., Herlina,H., Jirka, G.H., Schulz, H.E. & Gulliver, J.S. (2010), Estimation of Mass

Transfer Velocity based on Measured Turbulence Parameters, AIChE Journal, V.56,

Water Interface in an Oscillating-Grid Tank. In Komori, S; McGillis, W. & Kurose, R. Gas Transfer at Water Surfaces 2010, Kyoto University Press, Kyoto, pp. 65-77. Monin, A.S. & Yaglom, A.M. (1979), Statistical Fluid Mechanics: Mechanics of Turbulence,

sua correlação com o nível de turbulência junto à superfície - 1. (Investigation of the roxigenation mechanism in flowing waters and its relation to the turbulence level at the surface-1 – text in Portuguese) MSc dissertation, University of São Paulo, Brazil

Analytical Approach and Experimental Data, In Wilhelms, S.C. and Gulliver, J.S.,

interfacial mass transport: oxygen transport and random square wave analysis.

turbulent boundary layers using random square waves, 3rd workshop on fluids

in which

$$\begin{cases} \boldsymbol{\nu}\_{1} = \boldsymbol{f}\left(\boldsymbol{x}\_{k}, \boldsymbol{y}\_{k}\right) \\ \boldsymbol{\nu}\_{2} = \boldsymbol{f}\left(\boldsymbol{x}\_{k} + \frac{1}{4}\Delta\mathbf{x}, \boldsymbol{y}\_{k} + \frac{1}{4}\boldsymbol{\nu}\_{1}\Delta\mathbf{x}\right) \\ \boldsymbol{\nu}\_{3} = \boldsymbol{f}\left(\mathbf{x}\_{k} + \frac{1}{4}\Delta\mathbf{x}, \boldsymbol{y}\_{k} + \frac{1}{8}\boldsymbol{\nu}\_{1}\Delta\mathbf{x} + \frac{1}{8}\boldsymbol{\nu}\_{2}\Delta\mathbf{x}\right) \\ \boldsymbol{\nu}\_{4} = \boldsymbol{f}\left(\mathbf{x}\_{k} + \frac{1}{2}\Delta\mathbf{x}, \boldsymbol{y}\_{k} - \frac{1}{2}\boldsymbol{\nu}\_{2}\Delta\mathbf{x} + \boldsymbol{\nu}\_{3}\Delta\mathbf{x}\right) \\ \boldsymbol{\nu}\_{5} = \boldsymbol{f}\left(\mathbf{x}\_{k} + \frac{3}{4}\Delta\mathbf{x}, \boldsymbol{y}\_{k} + \frac{3}{16}\boldsymbol{\nu}\_{1}\Delta\mathbf{x} + \frac{9}{16}\boldsymbol{\nu}\_{4}\Delta\mathbf{x}\right) \\ \boldsymbol{\nu}\_{6} = \boldsymbol{f}\left(\mathbf{x}\_{k} + \Delta\mathbf{x}, \boldsymbol{y}\_{k} - \frac{3}{7}\boldsymbol{\nu}\_{1}\Delta\mathbf{x} + \frac{2}{7}\boldsymbol{\nu}\_{2}\Delta\mathbf{x} + \frac{12}{7}\boldsymbol{\nu}\_{3}\Delta\mathbf{x} - \frac{12}{7}\boldsymbol{\nu}\_{4}\Delta\mathbf{x} + \frac{8}{7}\boldsymbol{\nu}\_{5}\Delta\mathbf{x}\right) \end{cases} \tag{A1-8}$$

In the system of equations (AII-8), generated from equations (AII-4) through (AII-6), *x* = *z* and *y* = *n* , following the representation used in this chapter.

The system of equations (AII-1) through (AII-6) was solved using a spreadsheet for Microsoft Excel®, available at www.stoa.usp.br/hidraulica/files/. Two initial values were fixed and one was calculated. Note that in the present study it was intended to verify if the method furnishes a viable profile, so that boundary or initial values obtained from the experimental data were assumed as adequate. The first was *n*(0)=1. The second was *n'*(0)=-3, corresponding to the experiments of Janzen (2006). The third information did not constitute an initial value, and was *n*(1)=0 or 0<*n*(1)<0.01 (threshold value corresponding to the definition of the boundary layer). As the Runge-Kutta methods need initial values, this information was used to obtain *n''*(0), the remaining initial value needed to perform the calculations. With the aid of the Newton (or quasi-Newton) method, it was possible to obtain values for *n''*(0) that satisfied the third condition imposed at *z* = 1.

The derivative of *n* at *z*=0 is generally unknown in such mass transfer problems. In this case, solutions must be found considering, for example, *n*(0)=1, 0<*n*(1)<0.01 and *n'*(1)=0 (three reasonable boundary conditions), for which another scheme must be developed to calculate the first and second derivatives at the origin. As mentioned, the aim of this study was to verify the applicability of the method. The details of solutions for different purposes must be considered by the researchers interested in that solution.

The construction of the spreadsheet is described in the following steps:


The spreadsheet available at www.stoa.usp.br/hidraulica/files/ presents some suggestions that simplify some items of the above described steps (some manual work is simplified). The estimate of n"(0), for example, is obtained following simplified procedures.
