Computation of equilibria in models of flue gas washer plants

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Abstract

A flue gas washer is a plant for the absorption of noxious components of industrial gas output. The type of flue gas washers that is considered here is a spray tower. In the tower gas is raising upward and water drops with suspended limestone fall downward. Thereby sulfur dioxide is washed out of the gas. In industrial practice, the plant is operated under long term constant gas and water input. In this article, a mathematical model for equilibrium states of the plant is set up from basic principles. Its discretized version consists of a large system of algebraic equations for the relevant chemical concentrations in the two phases. An efficient iterative algorithm for solving this system is proposed and shown to converge in a wide range of examples. A satisfactory validation of the model is given by comparison with experimental data from existing literature. It is demonstrated that the model along with the algorithm is applicable for the design of spray towers.

Introduction

A flue gas washer is a plant for the absorption of noxious components of industrial gas output. The type of flue gas washers that is considered here is a spray tower designed by Austrian Energy & Environment in Graz, Austria. Leaving out technical details, a spray tower is a big vertical cylinder where gas continuously enters at the bottom, raises upward, and exits the cylinder at the top. At the same time water drops are sprayed into the tower at the top, fall down, and exit the cylinder at the bottom for further processing and reuse. Typical dimensions of a cubic tower is a sidelength of 14 m with a throughput of 400 m3 gas and 6 m3 water per second.

We consider the case of flue gas from a power plant, where sulfur dioxide is to be dissolved in the water drops and thereby washed out of the gas. To enhance the process, the pH value in the water is stabilized by adding ground limestone CaCO3 (prior to spraying the water into the tower). The overall effect of the liquid/gas mass transfer and the chemical reactions in the drops is the replacement of SO2 in the gas by CO2; correspondingly, the carbon in CaCO3 is replaced by sulfur to form innoxious calcium sulfit and calcium sulfate (gypsum) that is separated from the water in a basin below the tower.

In industrial practice, the rate of gas inflow and its chemical composition is constant over a long period of time, and the tower is operated by steady input of water with a fixed content of suspended limestone. Therefore, all physical and chemical processes in the plant converge to an equilibrium state, where all state variables become independent of time. We make the simplifying assumption that the gas and the drops everywhere move upward and downward, respectively, with given average net velocities. Then, such an equilibrium state is described by chemical concentrations that vary only with the vertical position in the tower.

In order to compute approximations to these functions of one variable, we discretize the tower by slicing it into N horizontal cells of equal height. In an equilibrium state, the flow of a chemical component into the cell plus its production by reactions and liquid/gas mass transfer within the cell is equal to the flow of the same component out of the cell plus its loss by reactions and liquid/gas mass transfer within the cell. This stationary mass balance has to hold for each cell, for each chemical component and for each of the two phases, gas and water drops. Thereby we arrive at a mathematical model that consists of a system of 9N nonlinear coupled algebraic equations for the concentrations of the various chemical components in each cell.

We present an algorithm for the numerical computation of a solution of this system. It proceeds iteratively, starting with gas and water inflow concentrations for all cells. Based on the current concentrations in the drops the gas concentrations are updated cell by cell from bottom to top. Then, based on the current gas concentrations, the water concentrations are updated cell by cell from top to bottom. In course of this half loop, the solution of a highly nonlinear electroneutrality equation is required for each cell, which is done by Newton’s method. The up–down iterations are repeated with given inflow concentrations of the gas and the water in the bottom and top cell, respectively. The iteration is stopped when the updates in all cells become smaller than a chosen relative error tolerance. The resulting concentrations solve the system of the 9N balance equations up to a small residual error.

To insure relevance for realistic industrial plants, all model parameters are taken from real world spray towers; the most important parameter is the transfer coefficient for SO2. By computing the dependence of the efficiency of SO2 removal on various parameters, it is demonstrated that the algorithm can be used for the design of spray tower plants. Furthermore, good quantitative agreement with an empirical formula for the efficiency is found. Finally, we present optimization results with respect to tower height and water flow rate. With our algorithm, the numerical solution of the model equations typically is a matter of a few seconds of computation time, using a conventional personal computer.

This work was done in close cooperation with Austrian Energy & Environment. More details can be found in Horn (2001). We express our thanks to D. Lang and A. Glasner (Austrian Energy & Environment) for many helpful and stimulating discussions.

Section snippets

Chemical reactions and mass transfer

Following Lang (1995), this section contains a description of the relevant chemical processes in the spray tower. The solubility of SO2 and CO2 in water depends on its pH value, pH=logm(H+), m(H+) being the concentration of H+ ions. For an effective solubility of SO2, the pH value should be large, whereas for small solubility of CO2 it should not be too large. An efficient substitution of SO2 by CO2 requires a pH value of about 7 in the water. When SO2 is dissolved in water, it dissociates to

The discrete equilibrium system

In case of states that are independent of time, the model reduces to steady-state balance equations. Furthermore, we discretize the vertical space coordinate by considering N horizontal cells that are numbered from top (n=1) to bottom (n=N), see Fig. 2 . We assume that the net average flow of gas is everywhere directed upward and the net average flow of water is everywhere directed downward. In order to formulate the balance equations we introduce the following notation

    c(X)(n)

    concentration of X

Evaluation and application of the model

This section presents outcomes of the iterative procedure. The procedure, henceforth called the up/down algorithm, produces 9N numbers c(SO2)(n),c(CO2)(n),c(O2)(n),m(CaCO3)(n),m(Ca2+)(n), QC(n), TS(n), TO(n), m(H+)(n), that approximately solve the 9N equations (19)–(23),(14). Eq. (19) is valid for X=SO2,CO2,O2. Eq. (14) has to be solved for m(H+)(n) in each cell n, i.e. in (14) m(H+),m(Ca2+), QC, TS, TO are replaced by m(H+)(n),m(Ca2+)(n), QC(n), TS(n), TO(n). Thus, there are 9N coupled

Conclusions

Based on basic physical and chemical principles a mathematical model for the equilibrium states of a spray tower plant was derived. According to a one-dimensional spatial discretization by N cells, the model consists of a system of 9N algebraic equations for the chemical concentrations in the two phases, gas and water drops, in the tower. An efficient numerical algorithm for solving this system was presented. It is based on the opposite flow directions of the phases and converged in a wide

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