Random Square Waves

The quantification of overall mass transfers in gas-liquid systems depends on the spatial evolution of the relevant variables close to the interface of the two phases. When turbulence is present (in the present study the turbulence is considered in the liquid phase), the methods of treating the problem consider the differential form of the momentum and mass conservation equations. The continuous hypothesis that underlies these equations in principle allows verifying the limiting trends very close to the interface. Because the theoretical concepts of turbulence are defined using statistical tools, the mentioned verification depends on the intrinsic definitions used in the statistical approach. In this study the turbulent mass transfer parameters are calculated for the thin region close to the interface based on the tool of random square waves (RSW). Theoretical results are obtained and analyzed in the context of existing experimental data and conceptual discussions of the literature, using a constant 'reduction function', a parameter defined in this methodology. The results of the present analysis show that the RSW method allows obtaining functional trends, as well as indicate the adequacy of using a variable reduction function to better represent reality.

The quantification of overall mass transfers in gas-liquid systems depends on the spatial evolution of the relevant variables close to the interface of the two phases. When turbulence is present (in the present study the turbulence is considered in the liquid phase), the methods of treating the problem consider the differential form of the momentum and mass conservation equations. The continuous hypothesis that underlies these equations in principle allows verifying the limiting trends very close to the interface. Because the theoretical concepts of turbulence are defined using statistical tools, the mentioned verification depends on the intrinsic definitions used in the statistical approach. In this study the turbulent mass transfer parameters are calculated for the thin region close to the interface based on the tool of random square waves (RSW). Theoretical results are obtained and analyzed in the context of existing experimental data and conceptual discussions of the literature, using a constant 'reduction function', a parameter defined in this methodology. The results of the present analysis show that the RSW method allows obtaining functional trends, as well as indicate the adequacy of using a variable reduction function to better represent reality.

The quantification of overall mass transfers in gas-liquid systems depends on the spatial evolution of the relevant variables close to the interface of the two phases. When turbulence is present (in the present study the turbulence is considered in the liquid phase), the methods of treating the problem consider the differential form of the momentum and mass conservation equations. The continuous hypothesis that underlies these equations in principle allows verifying the limiting trends very close to the interface. Because the theoretical concepts of turbulence are defined using statistical tools, the mentioned verification depends on the intrinsic definitions used in the statistical approach. In this study the turbulent mass transfer parameters are calculated for the thin region close to the interface based on the tool of random square waves (RSW). Theoretical results are obtained and analyzed in the context of existing experimental data and conceptual discussions of the literature, using a constant 'reduction function', a parameter defined in this methodology. The results of the present analysis show that the RSW method allows obtaining functional trends, as well as indicate the adequacy of using a variable reduction function to better represent reality.

The quantification of overall mass transfers in gas-liquid systems depends on the spatial evolution of the relevant variables close to the interface of the two phases. When turbulence is present (in the present study the turbulence is considered in the liquid phase), the methods of treating the problem consider the differential form of the momentum and mass conservation equations. The continuous hypothesis that underlies these equations in principle allows verifying the limiting trends very close to the interface. Because the theoretical concepts of turbulence are defined using statistical tools, the mentioned verification depends on the intrinsic definitions used in the statistical approach. In this study the turbulent mass transfer parameters are calculated for the thin region close to the interface based on the tool of random square waves (RSW). Theoretical results are obtained and analyzed in the context of existing experimental data and conceptual discussions of the literature, using a constant 'reduction function', a parameter defined in this methodology. The results of the present analysis show that the RSW method allows obtaining functional trends, as well as indicate the adequacy of using a variable reduction function to better represent reality.

Mean profiles of scalar properties close to moving gas-liquid interfaces subjected to turbulence are quantified using Random Square Waves (RSW). The condition of stationary turbulent transfer allows reducing the third order nonlinear governing equation successively to a second order and to a first order equation, which admits theoretical integration, furnishing the set of solutions analyzed in the present study. It is shown that different solutions may apply to different parts of the calculation domain (physical domain). It is also shown that the stationary problem admits a linear profile close to the gas-liquid interface, while a nonlinear function describes the mean properties apart from the interface.

The quantification of overall mass transfers in gas-liquid systems depends on the spatial evolution of the relevant variables close to the interface of the two phases. When turbulence is present (in the present study the turbulence is considered in the liquid phase), the methods of treating the problem consider the differential form of the momentum and mass conservation equations. The continuous hypothesis that underlies these equations in principle allows verifying the limiting trends very close to the interface. Because the theoretical concepts of turbulence are defined using statistical tools, the mentioned verification depends on the intrinsic definitions used in the statistical approach. In this study the turbulent mass transfer parameters are calculated for the thin region close to the interface based on the tool of random square waves (RSW). Theoretical results are obtained and analyzed in the context of existing experimental data and conceptual discussions of the literature, using a constant 'reduction function', a parameter defined in this methodology. The results of the present analysis show that the RSW method allows obtaining functional trends, as well as indicate the adequacy of using a variable reduction function to better represent reality.

The one-dimensional mass transfer in turbulent flows is considered. The closure problem related to the mean product between concentration and velocity fluctuations is treated by using random square waves. This approximation allows us to represent the statistical variables of turbulent mass transfer as depending on a finite set of basic parameters. The number of equations needed is then limited by the number of basic parameters used. The analysis is applied, in this study, to the interfacial mass transfer at air-water interfaces, generating a closed set of three equations involving three unknown functions. The resulting differential equations are nonlinear. A simplified example is solved.

The mathematical treatment of phenomena that oscillate randomly in space and time, generating the so called “statistical governing equations”, is still a difficult task for scientists and engineers. Turbulence in fluids is an example of such phenomena, which has great influence on the transport of physical proprieties by the fluids, but which statistical quantification is still strongly based on ad hoc models. In turbulent flows, parameters like velocity, temperature and mass concentration oscillate continuously in turbulent fluids, but their detailed behavior, considering all the possible time and space scales, has been considered difficult to be reproduced mathematically since the very beginning of the studies on turbulence. So, statistical equations were proposed and refined by several authors, aiming to describe the evolution of the “mean values” of the different parameters (see a description, for example, in Monin & Yaglom, 1979, 1981). The governing equations of fluid motion are nonlinear. This characteristic imposes that the classical statistical description of turbulence, in which the oscillating parameters are separated into mean functions and fluctuations, produces new unknown parameters when applied on the original equations. The generation of new variables is known as the “closure problem of statistical turbulence” and, in fact, appears in any phenomena of physical nature that oscillates randomly and whose representation is expressed by nonlinear conservation equations. The closure problem is described in many texts, like Hinze (1959), Monin & Yaglom (1979, 1981), and Pope (2000), and a general form to overcome this difficulty is matter of many studies. As reported by Schulz et al. (2011a), considering scalar transport in turbulent fluids, an early attempt to theoretically predict RMS profiles of the concentration fluctuations using “ideal random signals” was proposed by Schulz (1985) and Schulz & Schulz (1991). The authors used random square waves to represent concentration oscillations during mass transfer across the air-water interface, and showed that the RMS profile of the concentration fluctuations may be expressed as a function of the mean concentration profile. In other words, the mean concentration profile helps to know the RMS profile. In these studies, the authors did not consider the effect of diffusion, but argued that their equation furnished an upper limit for the normalized RMS value, which is not reached when diffusion is taken into account. The random square waves were also used by Schulz et al. (1991) to quantify the so called “intensity of segregation” in the superficial boundary layer formed during mass transport, for which the explanations of segregation scales found in Brodkey (1967) were used. The time constant of the intensity of segregation, as defined in the classical studies of Corrsin (1957, 1964), was used to correlate the mass transfer coefficient across the water surface with more usual parameters, like the Schmidt number and the energy dissipation rate. Random square waves were also applied by Janzen (2006), who used the techniques of Particle Image Velocimetry (PIV) and Laser Induced Fluorescence (LIF) to study the mass transfer at the air-water interface, and compared his measurements with the predictions of Schulz & Schulz (1991) employing ad hoc concentration profiles. Further, Schulz & Janzen (2009) confirmed the upper limit for the normalized RMS of the concentration fluctuations by taking into account the effect of diffusion, also evaluating the thickness of diffusive layers and the role of diffusive and turbulent transports in boundary layers. A more detailed theoretical relationship for the RMS of the concentration fluctuation showed that several different statistical profiles of turbulent mass transfer may be interrelated. Intending to present the methodology in a more organized manner, Schulz et al. (2011a) showed a way to “model” the records of velocity and mass concentration (that is, to represent them in an a priori simplified form) for a problem of mass transport at gas-liquid interfaces. The fluctuations of these variables were expressed through the so called “partition, reduction, and superposition functions”, which were defined to simplify the oscillating records. As a consequence, a finite number of basic parameters was used to express all the statistical quantities of the equations of the problem in question. The extension of this approximation to different Transport Phenomena equations is demonstrated in the present study, in which the mentioned statistical functions are derived for general scalar transport (called here “scalar-velocity interactions”). A first application for velocity fields is also shown (called here “velocity-velocity interactions”). A useful consequence of this methodology is that it allows to “close” the turbulence equations, because the number of equations is bounded by the number of basic parameters used. In this chapter we show 1) the a priori modeling (simplified representation) of the records of turbulent variables, presenting the basic definitions used in the random square wave approximation (following Schulz et al., 2011a); 2) the generation of the usual statistical quantities considering the random square wave approximation (scalar-velocity interactions); 3) the application of the methodology to a one-dimensional scalar transport problem, generating a closed set of equations easy to be solved with simple numerical resources; and 4) the extension of the study of Schulz & Johannes (2009) to velocity fields (velocity-velocity interactions). Because the method considers primarily the oscillatory records itself (a priori analysis), and not phenomenological aspects related to physical peculiarities (a posteriori analysis, like the definition of a turbulent viscosity and the use of turbulent kinetic energy and its dissipation rate), it is applicable to any phenomenon with oscillatory characteristics.

The quantification of overall mass transfers in gas-liquid systems depends on the spatial evolution of the relevant variables close to the interface of the two phases. When turbulence is present (in the present study the turbulence is considered in the liquid phase), the methods of treating the problem consider the differential form of the momentum and mass conservation equations. The continuous hypothesis that underlies these equations in principle allows verifying the limiting trends very close to the interface. Because the theoretical concepts of turbulence are defined using statistical tools, the mentioned verification depends on the intrinsic definitions used in the statistical approach. In this study the turbulent mass transfer parameters are calculated for the thin region close to the interface based on the tool of random square waves (RSW). Theoretical results are obtained and analyzed in the context of existing experimental data and conceptual discussions of the literature, using a constant “reduction function”, a parameter defined in this methodology. The results of the present analysis show that the RSW method allows obtaining functional trends, as well as indicate the adequacy of using a variable reduction function to better represent reality.