Ozone mass transfer in water treatment: hydrodynamics and mass transfer modeling of ozone bubble columns

2001 ◽  
Vol 1 (2) ◽  
pp. 123-130 ◽  
Author(s):  
M.G. El-Din ◽  
D.W. Smith
Keyword(s):  
Mass Transfer ◽  
Steady State ◽  
Liquid Phase ◽  
Plug Flow ◽  
Axial Dispersion ◽  
Bubble Columns ◽  
Algebraic Equations ◽  
Mixed Flow ◽  
Back Flow ◽  
Linear Algebraic Equations

Most of the mathematical models that are employed to model the performance of bubble columns are based on the assumption that either plug flow or complete mixing conditions prevail in the liquid phase. Although due to the liquid-phase axial dispersion, the actual flow pattern in bubble columns is usually closer to being mixed flow rather than plug flow, but still not completely mixed flow. Therefore, the back flow cell model (BFCM), that hypothesises both back flow and exchange flow to characterise the liquid-phase axial dispersion, is presented as an alternative approach to describe the hydrodynamics and mass transfer of ozone bubble columns. BFCM is easy to formulate and solve. It is an accurate and reliable design model. Transient BFCM consists of NBFCM ordinary-first-order differential equations in which NBFCM unknowns (Yj) are to be determined. That set of equations was solved numerically as NBFCM linear algebraic equations. Steady-state BFCM consists of 3 × NBFCM non-linear algebraic equations in which 3 × NBFCM unknowns (qG,j, Xj, and Yj) are to be determined. Those non-linear algebraic equations were solved numerically using Newton–Raphson technique. Steady-state BFCM was initially tested using the pilot-scale experimental data of Zhou. BFCM provided excellent predictions of the dissolved ozone profiles under different operating conditions for both counter and co-current flow modes.

The Transonic Aerofoil Problem with Embedded Shocks

1973 ◽  
Vol 24 (2) ◽  
pp. 129-138 ◽  
Author(s):  
Helge Nørstrud
Keyword(s):  
Flow Problem ◽  
Equation Approach ◽  
Algebraic Equations ◽  
Initial Value ◽  
Mixed Flow ◽  
Linear Algebraic Equations ◽  
Integral Equation Approach ◽  
Supercritical Flows ◽  
Method Of Steepest Descent ◽  
Finite Difference Techniques

SummaryThe integral equation approach to the mixed flow problem of infinite wings at high subsonic speeds is adopted for non-circulatory and circulatory (lifting) flows. The solutions are determined from a system of non-linear algebraic equations and, to ensure always unique solutions, the method of differentiation with respect to a parameter has been applied. The resulting Cauchy problem is then solved with the linearised flow solution as the initial value vector. For the case of embedded shocks in the flow field, the method of steepest descent has been added to the calculation scheme. Results for subcritical and supercritical flows past aerofoils are given and compared with solutions obtained by finite-difference techniques.

Unsteady double-diffusive buoyancy-induced flow in a triangular solar collector shape enclosure with corrugated bottom wall

2017 ◽  
Vol 27 (6) ◽  
pp. 1282-1303 ◽  
Author(s):  
M.M. Rahman ◽  
Sourav Saha ◽  
Satyajit Mojumder ◽  
Khan Md. Rabbi ◽  
Hasnah Hasan ◽  
...  
Keyword(s):  
Mass Transfer ◽  
Solar Collector ◽  
Linear Equations ◽  
Bottom Wall ◽  
Algebraic Equations ◽  
Weighted Residual Method ◽  
Content Type ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Induced Flow

Purpose The purpose of this investigation is to determine the nature of the flow field, temperature distribution and heat and mass transfer in a triangular solar collector enclosure with a corrugated bottom wall in the unsteady condition numerically. Design/methodology/approach Non-linear governing partial differential equations (i.e. mass, momentum, energy and concentration equations) are transformed into a system of integral equations by applying the Galerkin weighted residual method. The integration involved in each of these terms is performed using Gauss’ quadrature method. The resulting non-linear algebraic equations are modified by the imposition of boundary conditions. Finally, Newton’s method is used to modify non-linear equations into the linear algebraic equations. Findings Both the buoyancy ratio and thermal Rayleigh number play an important role in controlling the mode of heat transfer and mass transfer. Originality/value Calculations are performed for various thermal Rayleigh numbers, buoyancy ratios and time periods. For each specific condition, streamline contours, isotherm contours and iso-concentration contours are obtained, and the variation in the overall Nusselt and Sherwood numbers is identified for different parameter combinations.

How to Deal with Several Reservoirs

1991 ◽  
Author(s):  
James C. G. Walker
Keyword(s):  
Steady State ◽  
Euler Method ◽  
Coupled Systems ◽  
Algebraic Equations ◽  
Coupled Equations ◽  
Linear Algebraic Equations ◽  
Linear Differential ◽  
Surface Reservoir

The previous chapter showed how the reverse Euler method can be used to solve numerically an ordinary first-order linear differential equation. Most problems in geochemical dynamics involve systems of coupled equations describing related properties of the environment in a number of different reservoirs. In this chapter I shall show how such coupled systems may be treated. I consider first a steady-state situation that yields a system of coupled linear algebraic equations. Such a system can readily be solved by a method called Gaussian elimination and back substitution. I shall present a subroutine, GAUSS, that implements this method. The more interesting problems tend to be neither steady state nor linear, and the reverse Euler method can be applied to coupled systems of ordinary differential equations. As it happens, the application requires solving a system of linear algebraic equations, and so subroutine GAUSS can be put to work at once to solve a linear system that evolves in time. The solution of nonlinear systems will be taken up in the next chapter. Most simulations of environmental change involve several interacting reservoirs. In this chapter I shall explain how to apply the numerical scheme described in the previous chapter to a system of coupled equations. Figure 3-1, adapted from Broecker and Peng (1982, p. 382), is an example of a coupled system. The figure presents a simple description of the general circulation of the ocean, showing the exchange of water in Sverdrups (1 Sverdrup = 106 m3/sec) among five oceanic reservoirs and also the addition of river water to the surface reservoirs and the removal of an equal volume of water by evaporation. The problem is to calculate the steady-state concentration of dissolved phosphate in the five oceanic reservoirs, assuming that 95 percent of all the phosphate carried into each surface reservoir is consumed by plankton and carried downward in particulate form into the underlying deep reservoir. The remaining 5 percent of the incoming phosphate is carried out of the surface reservoir still in solution.

scholarly journals Ozone absorption in a mechanically stirred reactor

2007 ◽  
Vol 72 (8-9) ◽  
pp. 847-855 ◽  
Author(s):  
Ljiljana Takic ◽  
Vlada Veljkovic ◽  
Miodrag Lazic ◽  
Srdjan Pejanovic
Keyword(s):  
Mass Transfer ◽  
Liquid Phase ◽  
Transfer Coefficient ◽  
Mass Transfer Coefficient ◽  
Plug Flow ◽  
Operating Conditions ◽  
Volumetric Mass Transfer Coefficient ◽  
Pseudo First Order Reaction ◽  
Ozone Absorption ◽  
Stirred Reactor

Ozone absorption in water was investigated in a mechanically stirred reactor, using both the semi-batch and continuous mode of operation. A model for the precise determination of the volumetric mass transfer coefficient in open tanks without the necessity of the measurement the ozone concentration in the outlet gas was developed. It was found that slow ozone reactions in the liquid phase, including the decomposition of ozone, can be regarded as one pseudo-first order reaction. Under the examined operating conditions, the liquid phase was completely mixed, while mixing in a gas phase can be described as plug flow. The volumetric mass transfer coefficient was found to vary with the square of the impeller speed. .

Diffusion Transport in Electrochemical Systems: A New Approach to Determining of the Membrane Potential at Steady State

2009 ◽  
Vol 283-286 ◽  
pp. 487-493 ◽  
Author(s):  
Robert Filipek ◽  
Krzysztof Szyszkiewicz ◽  
Bogusław Bożek ◽  
Marek Danielewski ◽  
A. Lewenstam
Keyword(s):  
Steady State ◽  
Electrical Potential ◽  
Original System ◽  
Potential Difference ◽  
Constant Field ◽  
Electrical Potential Difference ◽  
Algebraic Equations ◽  
Liquid Junction ◽  
Electrochemical Systems ◽  
Linear Algebraic Equations

Ionic concentrations and electric field space profiles in one dimensional membrane are described using Nernst-Planck-Poisson (NPP) equations. The usual assumptions for the steady state NPP problem requires knowledge of the boundary values of the concentrations and electrical potential difference. In analytical chemistry the potential difference may not be known and its theoretical prediction from the model is desirable. The effective methods of the solution to the NPP equations are presented. The Poisson equation is solved without widely used simplifications such as the constant field or the electroneutrality assumptions. The first method uses a steady state formulation of NPP problem. The original system of ODEs is turned into the system of non-linear algebraic equations with unknowns fluxes of the components and electrical potential difference. The second method uses the time-dependent form of the Nernst-Planck-Poisson equations. Steady-state solution has been obtained by starting from an initial profiles, and letting the numerical system evolve until a stationary solution is reached. The methods have been tested for different electrochemical systems: liquid junction and ion selective electrodes (ISEs). The results for the liquid junction case have been also verified with the approximate solutions leading to a good agreement. Comparison with the experimental results for ISEs has been carried out.

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