This essay has been submitted by a student. This is not an example of the work written by our professional essay writers.
A little work has been done on the mass transfer with chemical reaction in jet ejector. The principal focus of research in this chapter is to develop some significant experimental and modeling techniques for efficient design of multi nozzle jet ejectors. To achieve this goal the recent literature on mass transfer with and without chemical reaction along with experimental data developed for different gas liquid contactors are required to be adopted. An attempt is made for necessary modification and development to existing theories which can be used for our study.
On the basis of the variety of generated experimental data, multi nozzle-single nozzle, laboratory scale-industrial scale, horizontal installation-vertical installation etc., were utilized to develop general mathematical models. These developed models may be utilized to design multi nozzle jet ejector without experimental data on the basic available physicochemical properties like diffusivities, rate constant, equilibrium constant, Henerys' law constant etc.
Â In this chapter the mathematical model for absorption of chlorine in aqueous sodium hydroxide system were developed by making use of software like STATGRAPHICS and MATLAB. With the help of recent mathematical techniques the problem of gas-liquid mass transfer reduces to mathematical problem and solved by good mathematical models.
4.1 Prediction of absorption rate and reaction rate constant of chlorine into aqueous sodium hydroxide solution
The rate constant of chlorine absorption in aqueous solution reported by different investigators has been summarized and presented in table 4.1.1 given below:
Table 4.1.1 : Rate constant for reaction as reported by different investigators
Ashour et al. (1996) studied the absorption of into aqueous bicarbonate and aqueous hydroxide solutions both experimentally and theoretically. They estimated the reaction rate coefficient of reaction between and over the temperature range of 293 - 312K:
It is observed that there is disagreement in the literature about the value of the forward rate coefficient of absorption of into aqueous solution of .
In this section, correlation to estimate the rate constant for absorption of in to aqueous solution of sodium hydroxide in jet ejector is developed using penetration model. The results obtained by this model are compared with the experimental values. Apart from this a mathematical model is also developed to estimate rate of absorption and enhancement factor which may be utilized to estimate further interfacial area in the section 4.4.
4.1.1 Model for the absorption of chlorine into aqueous solution based on penetration theory
When is absorbed in aqueous solution, the following reactions may take place:
In this model, all reactions are assumed to be reversible. However reaction (4.1.2) and (4.1.3) have finite reaction rates, whereas reaction (4.1.4) and (4.1.5) are assumed to be instantaneous.
Here three equilibrium constants and are independent and remaining can be obtained by following equation:
The concentrations of chemical species that are present in aqueous solutions are renamed as follows:
126.96.36.199 Concentration of an individual chemical species in bulk of liquid
Assuming all the reactions are at equilibrium the following equations can be derived by overall mass balance.
where is the molar ratio of chlorine to initially.
As the reactions are at equilibrium the independent equilibrium constants are:
We have 7 unknowns and 7 algebraic independent equations.
These equations are linear system of equations and may be solved by minimal residual technique using MATLAB to get which converge to the solution).
It may be noted that in case of aqueous solution do not contain any chlorine initially (means L=0) then
188.8.131.52 Mass balance at interface applying Higbie's penetration model
In jet ejector liquid jet ensuing at high velocity from nozzles situated at top, pulls the gaseous phase in to the jet. The gaseous stream is broken into small bubbles due to high kinetic energy of liquid jet and the gas-liquid comes in contact for a short time at about 1/10 second. The mass transfer takes place from the interface of bubble to the encircled liquid. To adopt penetration model let, , be the distance from the interface of the bubble. So x = 0 denotes the gas-liquid interface. The liquid coming out of jet travel as free jet from outlet of multi nozzle to entry of throat where the surrounding gas gets entrained in it through its exposed outer surface. This liquid stream with bubbles then travels in uniform cross section of throat and at the end it passes through a conical diffuser section. Let Z be the length along the axis of liquid jet Z = 0 at the out let of nozzles and at the outlet of the jet ejector. Let , be the time of and may be computed by
To compute volume between outlets of nozzles to inlet of throat, it is assumed that fluid travels through cylindrical passage having diameter equal to inside diameter of nozzle and length
By assuming all reactions reversible, the following reaction rate expression may be written.
The reactions (4.1.4) and (4.1.5) are instantaneous having large values of rate of reaction therefore are eliminated. We also assume that:
Reactions are at equilibrium.
The diffusivity of ionic spices are equal.
The fluxes of the nonvolatile species at interface are equal to zero.
By considering the mass balance following differential equation are derived:
Total chlorine balance
As it is assumed all reactions are reversible hence the instantaneous reactions are also at equilibrium and their equilibrium constant may be written as follow:
There are 6 unknowns and 6 partial differential equations/algebraic equations which can be solved for the concentrations of all chemical species.
Initial condition and boundary conditions
the concentration of chemical species are equal to bulk concentrations in liquid.
Boundary conditions at interface
At the interface of gas-liquid
For non volatile species
for all except =1 ()
For volatile species (), the rate of absorption per unit interfacial area may be written as
Here in our system there is only one volatile species i.e. chlorine and hence So we may write and =
The equation (4.1.27) may be re-written as
The value of the true gas side mass transfer coefficient, for chlorine required in equation (4.1.27) was predicted from the correlation reported by Lydersen (1983, pp. 129) which is about 0.000432.
Where is the physical equilibrium constant (Henry's law constant) of
For chlorine-aqueous solution the equation (4.1.28) may be re-written as
Therefore is negligible.
The boundary condition for pure at the gas liquid inference reduces to
184.108.40.206 Numerical solution and its implementation
Equation (4.1.19 ) - (4.1.29) represents a mathematical model to obtain the values of It is not possible to obtain analytical solution and therefore we have used Finite Difference Method (FDM) to transform each partial differential equation of the model into the system of ordinary differential equations in
We choose following finite difference expressions to approximate the partial derivatives:
where refer to the chemical species, refers to the spatial node number and
. Typical values for the initial nodal spacing at the gas-liquid interface are about. The transformed system of ordinary differential equation in t can be solved by MATLAB software by using ODE15 solver with preconditioning technique and with special Jacobi pre-conditioner.
This simulation gives and simultaneously and which gives the average rate of absorption of per unit interfacial area (flux). This may be written as
The gas liquid exposure time for the jet ejector may be stated as
Similarly, the enhancement factor of may be determined from the following equation
where are interfacial and bulk concentrations of in the liquid respectively and is the liquid phase mass transfer coefficient for physical absorption of and is given by
To solve the mathematical model the diffusivities of different species and Henry's law constant of are required which are tabulated in Table (4.1.2). We also need the equilibrium constants and the forward rate coefficients of all chemical reactions (4.1.2) through (4.1.5), which are tabulated in Table (4.1.3).
Table 4.1.2: Henery's law constant of and diffusion coefficients of (), (), and () (Ashour et al., 1996)
* rate of reactions and are instantaneous having large value of and are eliminated.
Table 4.1.3: Values for Equilibrium constants of Reactions (4.1.2) and (4.1.5) at various temperatures, (Ashour et al., 1996)
4.1.2 Results and discussion
The penetration model has been used to develop mathematical model for absorption of in to aqueous solution of sodium hydroxide. The mathematical model to predict absorpaiton rate is presented by equation (4.1.19 ) - (4.1.29).
To solve this model the value of and were required. The value of was determined by the correlation given by Brian et al. (1966)
The reaction and are instantaneous hence eliminated.
There is large variation in the value of in the literature as clear from Table (4.1.1). Hence attempt has been made to estimate value of by using data obtained for absorption of in aqueous sodium hydroxide solution in the jet ejector. The value of was adjusted until the theoretically predicted rate of absorption was within 1% of the experimentally measured rate of absorption of
Thus following correlation is developed to predict the value of :
The predicted value of from equation (4.1.1) reported by Ashour et al. (1996) and predicted value of from equation (4.1.37) by proposed model, along with the value obtained from the experimental result of present work are presented in table (4.1.4) and plotted in figure (4.1.1) and (4.1.1b).
Table 4.1.4 : The values of rate constants for reaction 4.1.3, , at atmospheric pressure for -Aqueous system
Figure 4.1.1 : Comparison of obtained by Ashour et al. (1996), proposed mathematical model and experimental result by Ashour et al. (1996) with present experimental result over the temperature range of 293-312 K
Figure 4.1.1a : Detailed view of Figure 4.1.1 at a temperature T-1 =3.3x10-3
Figure 4.1.2 : Error estimates for k2 and Ashour et al. (1996)
The experimental value and value predicted by present model(equation 4.1.37) are comparable. The value estimated by Ashour et al. (1996) are a little higher for which error is estimated. The error in-general may be defined as the absolute value of difference between estimated or measured value and actual value. Here error is defined as follow:
The error estimates for Ashour et al. (1996) and proposed model along with error between proposed model and present experimental value are presented in figure (4.1.2). The error between proposed model and Ashour et al. (1996) are less than 5.2 x 10-4 and the error between proposed model and experimental value is about 0.8 x 10-4 . As the error is very less it may be concluded that the proposed model is good.
The figure (4.1.3) presents vs , ( as parameter). The values obtained by experiment and by proposed model are in good agreement.Thus the chemical absorption mechanism proposed in the present work may be considered to be correct.
Figure 4.1.3 : Comparison between the values for rate of absorption obtained experimentally and predicted by proposed model with respect to at different
The value obtained by experiment and predicted from the proposed model are in good agreement. Hence, the proposed mathematical model may be used to predict the value of reaction rate, These values may be further utilized to predict enhancement factor and interfacial area using following co-relations:
The correlation obtained to estimate rate constant for forward part of absorption of chlorine in aqueous is
4.2 Effect of the diffusivities on absorption of chlorine into aqueous sodium hydroxide solution
The absorption of chlorine into aqueous sodium hydroxide solutions is one of the important systems having industrial importance and also is of theoretical interest.
Danckwerts (1950a and 1950b) and Sherwood and Pigford (1952) showed that absorption rate could be predicted by the penetration theory for absorption accompanied by an instantaneous irreversible reaction of the type
Spalding (1962) studied the absorption rate of into water and aqueous solutions of and using liquid-jet column. They have also established that the absorption rate of will be affected by the reactions (4.2.1) and /or (4.2.2):
depending upon the value of the solution.
Further, they have observed that when value was higher than 12.6 (i.e. concentration more than the forward part of reaction (4.2.2) was rate-controlling and the effect of this reaction on the absorption rate could be predicted by the penetration theory for absorption accompanied by an instantaneous irreversible reaction.
Brian et al. (1965) studied gas absorption accompanied by a two-step chemical reaction,
followed by. They have considered both steps irreversible and of finite reaction rates and presented the theoretical analysis based on both, the film theory and the penetration theory, with numerical solutions for the enhancement factor,.
Takahashi et al. (1967) used two different types of absorbers viz. liquid-jet column and a
stop-cock type absorber to study the absorption rates of into aqueous (. The predicated absorption rate using penetration theory was in good agreement with experimental results.
Hikita et al. (1972) studied gas absorption of two-step chemical reaction, followed by , accompanied by They have studied the effect of ratio of chemical equilibrium constants, P (which is defined as ), on enhancement factor, Î². They have developed mathematical models for finite value and âˆž, for equal diffusivity and unequal diffusivities of species on the basis of penetration theory.
Hikita et al. (1973) stated that in case of strong hydroxide solution the forward part of reaction (4.2.2) is not only reaction which governs the absorption rate of but the rapid reaction
also affects the absorption rate of as the equilibrium constant of this reaction is very large.
In this section, the author has developed a mathematical model to study the effect of diffusivities on the enhancement factor and analyzed the experimental data obtained by him and Hikita et al. (1973) on the basis of the penetration theory for gas absorption accompanied by a two step instantaneous chemical reaction. In this work, the rate of absorption in the jet ejector is studied by using - aqueous system at .
4.2.1 Mechanism of chemical absorption
Spalding (1962) mentioned that, when concentration is more than the forward part of reaction (4.2.2) is rate-controlling which affects the absorption rate.
Hikita et al. (1973) stated that (hypochlorous acid) formed by reaction (4.2.2) can react again with ions results in rapid reaction (4.2.3) having equilibrium constant
Hence considering two step mechanism of the reaction (absorption) between and an aqueous hydroxide solution, may be written as follows:
The values of the equilibrium constants of reactions (4.2.2) and (4.2.3) are and . They are stated as follows:
The values of and at is given by (Connick et al., 1959) and
(Morris, 1966) respectively.
The hydrolysis of dissolved with water takes place according to the reaction
The equilibrium constant of this reaction is given by
and having value (Connick et al.,1959) at .
The values of is very low compared to the value of , and hence reaction (4.2.1) will not have a significant contribution to the total reaction rate of
Therefore, the absorption of into aqueous hydroxide solutions can be considered as an instantaneous two-step reaction, which is equivalent to say that reaction (4.2.4) followed by reaction (4.2.5) and having over all reaction (4.2.6).
4.2.2 Mathematical models related to absorption
Enhancement factor, Î²
The rate of absorption of reactant (gas) with instantaneous chemical reaction can be predicted by summing
(a) Amount of diffuse away unreacted and
(b) Amount of reacted (in the form of ) diffuse away from the gas-liquid interface.
The integration of equation (4.2.7) from time zero to the total exposure timegives the average absorption rate and which may be written as:
The average absorption rate of in absence of the chemical reaction is given by the well-known Higbie equation
It is known that the enhancement factor is the ratio of average rates of absorption with chemical reaction and without chemical reaction. Therefore from equation (4.2.8) and (4.2.9):
Desorption of the
Lahiri et al. (1983) studied the process of desorption of the intermediate product hypochlorous acid, , during the process of absorption of in aqueous alkaline hydroxides. They gave correlation for the rate of desorption of the product:
Therefore, average rate of desorption over a total exposure time, is given by
= enhancement factor for desorption
= true liquid side mass transfer coefficient without chemical reaction.
The equation (4.2.10b) will become:
One reaction-plane model
Hikita et al. (1972) developed the model based on penetration theory for absorption with instantaneous chemical reaction and found that for there exists only one reaction plane, where the over-all reaction of reactions and is , proceeds irreversibly. The average rate of absorption of the solute gas can be calculated by following equations which were derived by Danckwerts (1950) and Sherwood and Pigford (1952).
Where is root of the equation
This absorption mechanism is called a "one reaction-plane model".
Two reaction plane model
Hikita et al. (1972) developed a two reaction plane model when and established the fact that two reaction planes are formed within the liquid, which are as follows:
The reaction which is the sum of forward part of the first-step reaction
) and the backward part of second step reaction take place irreversibly at the first reaction plane (which is located closer to the gas-liquid interface)
The reaction take place irreversibly at the second reaction plane.
Further, the absorption rate may be calculated by equation (4.2.11) and following equation:
where is the root of following equations.
Two reaction plane model for absorption ofinto aqueous solution
The reaction scheme for the system studied in this work is similar to the work of Hikita et al. (1972). The present reaction system (Equation (4.2.4) and (4.2.5)) may be described in the form
The present system is different than the work of Hikita et al. (1972) due to presence of which was not present in Hikita et al. (1972). However the species is non-reactive. Hence it does not affect reactive mechanism.
Figure 4.2.1: Concentration profiles for absorption of into aqueous solution
Since, the equilibrium constant ratio is very high, so we can apply the two reaction plane model to the present system.
The diffusion coefficients of all species, based on the penetration theory is modeled by partial differential equations. The concentration profile for each species which will be derived by solving the developed model will be similar to that as shown in Figure 4.2.1.
with the following initial and boundary conditions:
where and are the locations of the first and the second reaction planes, respectively, and and represent the values of when approaches from region 2 and from region 3, respectively.
This proposed model is more general.
Specifically, when the effect of are negligible and hence removed from the equations, the proposed model will reduce to model of Hikita et al. (1973)
The analytical solution of this problem (i.e. concentration profile in the liquid) was given by Hikita et al. (1973) and is as follows:
Proposed mathematical model I: This model is represented by equations (4.2.17 - 4.2.27), when are non zero.
Proposed mathematical model II: This model is represented by equations (4.2.17 - 4.2.22) when effect of are negligible with modifying boundary conditions which are as follows:
In proposed mathematical model II, we have consider the effect of jump at and The two plane model theory suggest us that at there is a instantaneous reaction between species () and () and causes jump in concentration of species . Similarly, at there is a sudden reaction between species () and () and causes jump in concentration of species (). This jump values are defined by the last term of the Equation (4.2.42) and Equation (4.2.43) respectively.
The absorption rate for proposed mathematical model I can be calculated by equation (4.2.11) and
where the constant can be determined by solving the following pair of simultaneous equations.
Similarly the absorption rate for proposed mathematical model II can be calculated by equation (4.2.11) and
where the constant can be determined by solving the following pair of simultaneous equations.
The equations (4.2.45), (4.2.46) (4.2.47) (4.2.48) (4.2.49) and (4.2.50) are solved by the trial and error procedure based on Newton - Raphson technique to evaluate and . In this technique the first guess values of and was calculated by considering equal diffusivities i.e. .
There are several numerical methods like finite difference method, finite volume method, finite element method etc. to solve proposed mathematical model I and proposed mathematical Model II. Looking to the nature of the mathematical model (time dependent and in one dimension) FDM is the best suitable technique. Other methods are expensive from time point of view. Hence, we have used the numerical technique, finite difference method, to solve the model using Matlab software.
4.2.3 Discussion of results
The effect of the diffusivity ratio on enhancement factor
Figure 4.2.2 shows the plot of the enhancement factor versus the concentration ratio for different diffusivities ratio of with constant and The value of are taken to be 2.43, 1 and 0.1. In this figure the plots of Hikita (1972), proposed mathematical model I [equations (4.2.17 - 4.2.27) with are non zero] and proposed mathematical model II [equations (4.2.17 - 4.2.22 and 4.2.40 - 4.2.44) with are zero] are presented.
Figure 4.2.2 : Variation in enhancement factor with respect to at different , 1, 0.1 and constant and for absorption of into aqueous solution
The following results may be drawn from figure (4.2.2).
The lines in the figure having higher are at higher position for the same
. This indicates that at higher ratio of diffusivities of reactants (liquid and gas), the enhancement factor is higher. It can be concluded that higher the diffusivity of liquid reactant with respect to gaseous reactant, higher is the enhancement factor. This may because as is higher the thickness of film reduces due to travels faster toward the interface then
For same plots shows that enhancement factor increases with increase in the reactant ratio. The increase in enhancement factor is steeper at initial increase of. After that the rate of rise in enhancement factor with respect to rate of rise in reactant ratio is reducing and after certain value of there is hardly any rise is enhancement factor with respect to reactant ratio. It can be concluded that there is increase in enhancement factor with increase in liquid reactant concentration up to certain limits. This may be interpreted that at higher enhancement factor is higher. The reduction of at higher may be due to high viscosity of solution at higher.
For the same ratio of the value of enhancement factor derived from proposed mathematical model II is higher than the value from proposed mathematical model I. The value derived from Hikita (1972) is the lowest. The predicted values from proposed mathematical model I and proposed mathematical model II are higher than Hikita (1972) as the effect of diffusing out have been considered. Moreover, the effect of have been considered in the proposed mathematical model I.
Figure 4.2.3 : Error estimates between experimental data and proposed
mathematical model at different , 1, 0.1 and constant and .
Figure 4.2.3 shows the error estimates between Hikita (1972) model and proposed mathematical model I. Here error is defined as follow:
It is observed that the lower value of for different, the proposed mathematical model I and Hikita (1972) model are comparable. However, for higher values of , the comparison shows that there is a numerical instability in Hikita (1972) model (higher value of error). Therefore, we conclude that the proposed model I is well-posed.
Comparison of experimental results with simulated results
Figure 4.2.4 is a comparison of predicted by the simulated results of Hikita (1973), proposed mathematical model I and proposed mathematical model II with experimentally determined values ( for) at actual value of diffusivity ratio: and (Table A 3.4). It may be observed that the values obtained by experiment, Hikita (1973) model and proposed model I are comparable. Equations (4.2.17) to (4.2.22) and (4.2.45) to (4.2.50) indicate that is a function of rate of reactions and three diffusivity (in liquid) ratios, and The predicted values by proposed mathematical model I are higher to some extent than Hikita (1973) model which is due to the effect of reaction on , have been consider in mathematical model I. It may be make out that the influence of rate of reaction are marginal that may be because being instantaneous reaction diffusivity ratio of species are rate controlling.
Figure 4.2.4 : Comparison of experimental observation and proposed
The values predicted by proposed mathematical model II for are higher than experimental values. It may be concluded that the effect of jumping in the concentration of and which have been considered in the model at the interface 1 and 2 are not appreciable. Hence the values predicted by model II are higher. So model II is not appropriate under operating conditions of the experiment.
The enhancement factor depends on the five independent dimensionless parameters i.e., three diffusivity ratios, and , and two concentration ratios and
The enhancement factor increases as the value of with respect to increases.
The value of enhancement factor increases as the value increases, and the effect becomes large at low values and low at high values
The proposed mathematical model I is more appropriate to experimental results at operating conditions i.e. at 300C.
4.3 Numerical model of rate of absorption in multi nozzle jet ejector (chlorine- aqueous solution)
High velocity jet from the nozzles entrains the gas and due to very high turbulence in the throat, gas is split into bubbles. In the diffuser section partial separation of the gas and liquid may occur. The high interfacial area formed by bubbles is desirable for increasing rate of mass transfer. Different researchers, including Kuznetsov and Oratovskii (1962), Boyadzhiev (1964), Volgin et al. (1968) and Beg and Taheri (1974), attempted to simulate the operation of the jet ejector for gas absorption.
Johnstone et al. (1954) reported a jet ejector study in which was absorbed in solution, and the amount of sulfur dioxide absorbed in the liquid, was measured at various distances from the point of liquid injection. It was found that the mass transfer increased substantially as the liquid injection rate increased.
Kuznetsov and Oratovskii (1962) developed a mathematical model for predicting absorption of by reacting with solution in the throat and the divergent section of a venturi scrubber.
removal efficiency of a jet ejector was investigated by Talaie et al. (1997) using a three-dimensional mathematical model based on a non uniform droplet concentration distribution predicted from a dispersion model in the gas flow where the gas-phase mass transfer coefficient was calculated by empirical equations.
Mandal et al (2003) studied the jet ejector followed by bubble column and developed two simple correlations of and as a function of superficial gas velocity. This correlation can be combined to calculate liquid-side mass transfer coefficient.
Utomo et al. (2008) investigated the influence of operating conditions and ejector geometry on the hydrodynamics and mass transfer characteristics of the ejector by using three-dimensional CFD modeling. The CFD results were validated with experimental data.
Taheri et al. (2010) studied the three-dimensional mathematical model, based on annular two-phase flow model for the prediction of the amount of removed in a venturi scrubber.
We have made an attempt to predict mass transfer characteristics by numerical modeling. Here, the author has described the mathematical model for the prediction of the amount of chlorine removed in jet ejector. The results of simulation are compared with the experimental data.
4.3.1 Mathematical modeling
In this study the model developed by Taheri et al. (2010) is modified to suit the jet ejector used in the present work. Taheri et al. (2010) developed a three-dimensional mathematical model based on annular two-phase flow model in rectangular geometry of the venturi scrubber. They develop a model to predict interfacial area by predicting drop size and droplet concentration. Instead in this work an attempt is made to predict the change in concentration of reactant,, through the ejector.
The concentration of bubbles has been assumed uniform across the cross section of the scrubber.
The continuity equation of bubbles is solved to obtain bubble concentration distribution considering the effect of gas turbulence.
For developing the model, the pollutant concentration distribution in gas phase was obtained by the following model using mass balance.
The general equation can be obtained by writing differential mass balance for pollutants over a differential control volume.
The rate of reaction of pollutant per unit volume at time for constant volume system may be written as
The may be calculated by using rate of mass transfer per unit area as
where = interfacial area per unit volume
= (number of bubble / volume) x (interfacial area / bubble)
may be computed from velocity in case of moving gas as
equation (4.3.1) may be written as
Boundary conditions for Equation (4.3.1) are as follows:
The value of may be estimated by using the following equation (Ogawa et al., 1983)
Substituting equation (4.3.3) in equation (4.3.1) it will reduce to
In order to evaluate the bubble concentration distribution, in the above equations, the following one-dimensional dispersion equation, expressing material balance for bubble in a differential control volume, must be solved:
with the boundary conditions of:
In Equation (4.3.3), the bubbles are convected in the x direction.
It is assumed that for each nozzle the source of bubbles is limited to one element. The source strength, S, is the number of bubbles generated per unit volume per unit time. Bubbles are carried from element to element and are dispersed by convection and eddy diffusion effects. Number of bubbles per second is defined by the following equation:
where is the total gas flow rate.
Substituting equation (4.3.3) and (4.3.7) in equation (4.3.4) it will reduce to
The bubble velocity can be obtained by solving the following equation. This is obtained by writing a force balance for bubbles.
The modified drag coefficient, , can be calculated by using the following expressions given by Taheri et al. (2010) adopted for bubbles:
Here can be obtained by the formula given by Tahari et al. (2010) adopted for bubbles:
Substituting Equation (4.3.11) (4.3.12) and (4.3.13) in Equation (4.3.10) it will reduce to
The gas velocity is computed by the following equation:
The equation (4.3.1) can be solved simultaneously with equation (4.3.3), (4.3.5), (4.3.7), (4.3.9), (4.3.16) and (4.3.17).
The mass transfer rate, , in each element can be evaluated by model developed in previous section presented by equation 4.1.32.
When pollutants undergo a very fast reaction into the liquid phase such as absorption of into aqueous solution, the bulk concentration of gas in the liquid phase can be considered equal to zero.
4.3.2 Results and discussions
Figure 4.3.1 is a plot of variation of gas phase concentration along the axis of ejector for different values of initial gas concentration for nozzle N1 having number of orifice1. For comparison of the experimental results and predicted results obtained by the proposed model are plotted in the same figure. From both the profiles shown in the figure, it is clear that the proposed model is in good agreement with experimental results.
Figure (4.3.2) shows the variation of gas phase concentration along the axis of the ejector for different nozzles N5 (no. of orifice 1), N6 (no. of orifice 3) and N7 (no. of orifice 5). The results predicted by the model are in good agreement with the experimental results. Thus the model is applicable for multi nozzle jet ejector.
It is also shows that the conversion in the jet ejector first increases then decreases and finally becomes almost constant. The number of orifice in the nozzle affects the gas conversion in the jet ejector with three orifice (N6) the conversion obtained is maximum with five orifice (N7) minimum and with one orifice (N5) in between maximum and minimum.
Figure 4.3.1 : Variation of gas phase concentration along the axis of ejector for different values of initial gas concentration at (comparison between proposed model and experimental value)
Figure 4.3.2 : Variation of gas phase concentration along the axis of ejector for different nozzles N5 (no. orifice 1), N6 (no. of orifice 3) and N7 (no. of orifice 5) for setup 3 at and initial gas concentration (comparison between proposed model and experimental value)
The figure 4.3.3 shows the variation of bubble velocity along the axis of the ejector. It indicates that the bubble velocity suddenly increases to a maximum value and then it remains constant.
Figure 4.3.3 : Velocity profiles of gas and droplet along axial direction
The proposed model is in good agreement with experimental values for single nozzle as well as for multi nozzles. Hence the proposed model may be used for designing the industrial ejectors.
The number of nozzle (orifice) affects the gas conversion. In present work the maximum conversion is obtained for no. of nozzle 3 (N6).
4.4 Mass transfer characteristics in multi nozzle jet ejector
In this section a mathematical model to predict hold up, mass transfer coefficient and interfacial area has been proposed for multi nozzle jet ejector and compared with experimental data obtained.
Many researchers have published their work on jet ejectors (Jackson, 1964; Volmuller and Walburg, 1973; Nagel et al., 1970; Hirner and Blenke, 1977; Zehner, 1975; Pal et al., 1980; Ziegler et al., 1977) because of the high energy efficiency in gas liquid contacting.
The kinetic energy of a high velocity liquid jet is used for getting fine dispersion and intense mixing between the phases in the jet ejectors.
Zlokamik, (1980) has reported that oxygen absorption efficiency is as high as 3.8 kg O2/kwh in ejectors as compared to 0.8 kg O2/kwh in a propeller mixer. The higher gas dispersion efficiency of the ejector type can be understood from the well known fact : "gas dispersion is possible only if the fraction of micro turbulence is high" (Schugerl, 1982).
Radhakrishnan et al. (1984) have used a vertical column fitted with a multi jet ejector for gas-dispersion for studying the pressure drop, holdup and interfacial area.
Agrawal (1999) has reported interfacial area, about 13000 m2/m3 in horizontal single nozzle jet ejector. The measured values of the interfacial area in the jet ejector are in the range of 3000 to 13000 m2/m3.
4.4.1 Hold up
Yamashita and Inoue (1975), Koetsier et al. (1976) and Mandal (2004) reported the holdup characteristics with respect to gas flow rate in the jet ejector. At lower range of gas flow rate, gas hold up increases with increase in gas flow rate but at higher range of gas flow rates the increase in gas flow rate decreases the gas hold up or it remain constant depending on the height of liquid in the follow up column is high or low respectively. At lower gas flow rates small bubbles produced are in large number and at higher gas flow rate due to coalescence the bubbles of larger size are produced which lead to decrease in number of bubbles.
Hills (1976) has reported that the holdup is not affected by liquid flow rate. Mandal (2004) observed that for the same gas flow rate the increase in liquid flow rate decreases the gas hold up.
The variables and affect the liquid holdup in a jet ejector.
Radhakrishnan et al. (1984) obtained following correlation by applying multi linear regressions analysis on their experimental data:
A new mathematical model has been attempted to predict the gas hold up as follows:
It is assumed that the model is of the form:
Using experimental data and multi linear regression analysis the values of and were obtained. The values obtained are and
Figure 4.4.0 : Comparison of liquid holdup predicted by Radhakrishnan (1984), present model and experimental value at different ratio.
Thus mathematical model for liquid hold up is as follows.
Liquid holdup may be determined by following equation.
The results predicted from Radhakrishnan (1984) model and present model (equation - 4.4.3) is compared with actual experimental value at different in figure (4.4.0).
4.4.2 New model to predict mass transfer characteristics, and
To predict mass transfer characteristics the value of and are required to be predicted. Here a mathematical model is developed to predict the value of and using chlorine-aqueous sodium hydroxide solution.
Doraiswamy and Sharma (1984) have reported that if
then the reaction is considered to be pseudo first order and in the fast reaction regime.
As absorption of in aqueous solution of studied in the present work satisfy the condition (4.4.4) and (4.4.5), it is treated as pseudo first order fast reaction.
Levenspiel (1999) presented a simplified solvable pseudo first order rate expression as a replacement for second order reaction rate equation when the value of is so high that it do not change appreciably, which is presented here as follows:
Substituting equation (4.4.7), (4.4.8) in equation (4.4.6) we have:
By rearranging equation (4.4.9) and integrating between will yield the equation:
Sharma and Danckwerts, (1970) stated that when
Then gas phase resistance is negligible.
But for chlorine-aqueous system
Hence, to predict interfacial area few experiments were carried out for -aqueous system.
-aqueous system satisfies the condition as of equation (4.4.12). Therefore gas phase resistance is negligible. Hence equation (4.4.11) will turn to
And equation (4.4.10) may be written for calculating interfacial area as follows:
True gas side mass transfer coefficient for chlorine
For aqueous system
# Only a sample is shown. Run No. 110
Table 4.4.1 : Typical range of experimental values
That implies that gas phase resistance controls the rate of reaction (Levenspiel, 1999).
Therefore the rate of absorption of may be written as
The true gas side mass transfer coefficient, , is given by,
The model presented by the equations (4.4.10), (4.4.14) and (4.4.17) are the models to predict the value of and
4.4.3 New mathematical model related to interfacial area for multi nozzle ejectors
Radhakrishnan et al. (1986) have suggested the following correlations to estimates interfacial area i.e.
Mandal et al. (2003) have suggested the following estimates for interfacial area of system i.e.
, where is gas superficial velocity
In this work a new model has been proposed for estimation of '' This model is easy to apply and require minimum input data.
is determined experimentally and is equal to .
can be estimated by the model developed in section 4.1 given by expression (4.1.32).
So can be determined
This mathematical model is employed for multi nozzle ejector with number of orifice 3, 5 and 7. The dimensions of multi nozzle ejector are given in chapter 3. The results obtained with this model for multi nozzle ejector are compared with experimental data.
4.4.4 Results and discussions
A new mathematical model has been proposed as per experiment (4.4.16a), (4.4.17) and (4.4.20) to predict mass transfer characteristics by determining the value of and . The predicted values by using this proposed model is presented graphically in the figures (4.4.1) to (4.4.16). The figures show the effect of on predicted value of, and for different nozzles and different values of. The variation in the rate of absorption of with chemical reaction in aqueous solution is discussed below.
220.127.116.11 Factors affecting rate of absorption () in liquid jet ejector
is higher for higher (figure - 4.4.13). This is because the liquid jet spray area in the free jet section is higher for more number of nozzles and will result in more entrainment of gas and high rate reaction due to high interfacial area. The high liquid jet exposed area counters the effect of increase in viscosity of aqueous solution due to increase in its concentration.
Figure (4.4.1), (4.4.5), (4.4.9) and (4.4.13) show the effect of on using different nozzles. The following conclusions can be derived from the study of the figures.
A common trend has emerged that as increases the also increases in all setups for all nozzles. This is because rate of reaction is function of concentration of both reactants i.e. chlorine () and ().
decreases with increase in.
The concentration of aqueous solution in is kept high to maintain pseudo first order condition. So that rate of reaction is independent of viscosity of . The reduction in absorption rate is due to increase in aqueous solution with increase in concentration, which leads to less diffusivity coefficient (Stokes-Einstein equation) and also reduction of physical solubility of (Krevelen and Hoftijzer theory, 1948).
This is due to (i) the increase in viscosity of aqueous solution, (ii) reduction of physical solubility of and (iii) diffusivity of in aqueous solution when concentration of is increased. But for lower concentration of, the value of is maximum for nozzle N6 (three nozzle). The value of is minimum for nozzle N5 (single nozzle).
The maximum absorption obtained is in vertical installation having three nozzle that is setup 3 having nozzle N6.
18.104.22.168 Effect of different parameters on mass transfer characteristics ( and ) in jet ejectors.
Effect of on and
Volumetric mass transfer coefficient (): Figure (4.4.2), (4.4.6), (4.4.10) and (4.4.11) are the plots of predicted by proposed model versus for setup 1, 2 and 3. It is observed that by increasing keeping other parameters constant the value of volumetric mass transfer coefficient increases.
Interfacial Area (): Figure (4.4.3), (4.4.7) and (4.4.11) are the plots of interfacial area, predicted by proposed model against different values of for setup 1 and 3. These figures show that interfacial area decrease with increase in. The reduction in interfacial area is due to higher viscosity of the solution at higher concentration.
The higher concentrations have adverse effect on diffusivity and physical solubility of in solution.
Mass transfer coefficient (): The figures (4.4.4), (4.4.12) and (4.4.16) are the plots of predicted by proposed model against different values of for setup 1, 2 and 3.
The mass transfer coefficient () is higher at higher exception for the setup 1 (with nozzle N1) the highest value of is observed for lower () with .
Effect of number of nozzles on and
Volumetric mass transfer coefficient (): Variation of with number of nozzles for different, in jet ejector are shown in figures (4.4.6) and (4.4.14). It is observed in figure (4.4.14) that is highest for nozzle N7. (no. of orifice 5) for all . is less for nozzle N6 (no. of orifice 3) than nozzle N7 (no. of orifice 5). is minimum for N5 (no. of orifice 1). In figure (4.4.) similar pattern is shown for nozzle N2 and N3 in set up 2 i.e. is higher for nozzle N3 (no. of orifice (3) than nozzle N2 (no. of orifice 1).
Hence obtained is higher for large number of nozzles for the same total flow area.
So we may deduce that as the number of nozzle are more the value of is more.
Interfacial area '':
The interfacial area generated in jet ejector by varying for â€¦â€¦â€¦â€¦â€¦.. number of nozzles (orifice ) and at different are presented in figure (4.4.7) and (4.4.15).
Interfacial area produced in jet ejector decreases with increase in number of nozzles (orifices) for all values of that were studied. This seems to be due to coalescence of bubbles at the junction where jets meets near the throat.
Mass transfer coefficient (): Figure (4.4.8) and (4.4.16) are the comparison of values obtained by proposed model for for different nozzles (N2, N3, N5, N6 and N7) at different.
In the setup 3 where industrial multi nozzle ejector was used for experiment shows that the value of is more for the nozzles having more orifice only a small variation is seen in figure (4.4.16a) where at higher the value is less for nozzle N7 (5 orifice) then for nozzle N6 (3 orifice).
Similar trend is observed in figure (4.4.8) plotted for setup 2, where the value of for nozzle N3 (no. of orifice 3) is higher than for nozzle N2 (no. of orifice 1).
So we may conclude that the as number of nozzle increases the value of increases.
The predicted by the proposed model for different nozzles (N2, N3, â€¦. & N7) at different are presented in figures (4.4.8) and (4.4.16). The is higher for higher number of nozzles (orifices) having the same flow area. Both setup 2 and setup 3 shows similar treach.
Effect of on and
The variation of and is shown in figure (4.4.1) to (4.4.16).
Volumetric mass transfer coefficient (): The effect of on at different conditions is shown in figures (4.4.2), (4.4.6), (4.4.10) and (4.4.14). All the figures are similar qualitatively i.e. as the value of increases the value of decreases. The decreases is very sharp at initial values of . Afterwards the decrease in with respect to is reduced. This is because of reduction in value of higher
Interfacial area '': The effect of on interfacial area for different nozzles and at different. It is shown in figures (4.4.3), (4.4.7), (4.4.11) and (4.4.15). It is observed that there is only a little variation in interfacial area with change in. This is because the interfacial area generated depends on viscosity of the liquid and â€¦â€¦to gas ratio. In the present experiment the liquid to gas ratio is kept constant and very low concentration of has been used. Under these conditions the viscosities of liquid and gas will not change significantly.
From figures it is seen that at higher the value of reduces with increase in . While at lower (0.525 & 0.11) there is not much variation in with respect to increase in . This is because of higher the viscosity of liquid is more and that causes the reduction in diffusivity and physical solubility of chlorine.
Mass transfer coefficient (): The effect of on is shown in figures (4.4.4), (4.4.8), (4.4.12) and (4.4.16). From the figures it is seen that at higher (0.79) and orifice 1 and 3 the value of reduces with respect to . This is because at higher viscosity of solution is higher.
Correlations for prediction of the holdup and interfacial area in a multi-jet ejector contactor system have been proposed by the equation (4.4.3) and (4.4.14).
Two models have been proposed for the estimation of interfacial area. Hence, a new model is developed by equation (4.4.14) and (4.4.20) to predict interfacial area. The results are presented in figure (4.4.17). The predictions are well fitted with experimental data.
The figures (4.4.1) to (4.4.16) are plotted on the basis of the prediction from proposed new model presented by equations (4.4.20).The behavior of are shown against different initial concentration of gases for different nozzles and in these figures .The results may be analyzed as summarized in the following table.
Table 4.4.2 : Summary of analysis of results for different and nozzles
4.5 Removal efficiency of chlorine in jet ejector (Chlorine aqueous solution)
The major factors which affect the efficiency of jet ejector are liquid flow rate, gas flow rate, the concentration of absorbing liquid and the concentration of the solute in the gas.
Ravindram and Pyla (1986) proposed a theoretical model for the absorption of and in dilute based on simultaneous diffusion and irreversible chemical reaction for predicting the amount of gaseous pollutant removed.
Many researchers (Volgin et al., 1968; Ravindram and Pyla, 1986; Cramers et al., 1992, 2001; Gamisans et al., 2001, 2002; Mandal, 2003, 2004, 2005; Balamurugan et al., 2007, 2008; Utomo et al., 2008; Yadav, 2008; Li and Li, 2011.) have reported different theories and correlations to predict scrubbing efficiency of jet ejectors.
Uchida and Wen (1973) developed a mathematical model to predict the removal efficiency of 2 into water and alkali solution. The simulated results of their model were compared with experimental results and they found that there is a good agreement with the experimental results. They have also found enhancement factor to predict rate of the chemical absorption.
Gamisans et al. (2001) evaluated the suitability of an ejector-venturi scrubber for the removal of two common stack gases, sulphur dioxide and ammonia. They studied the influence of several operating variables for different geometries constructions of venturi tube. A statistical approach was presented by them to characterize the performance of scrubber by varying several factors such as gas pollutant concentration, gas flow rate and liquid flow rate. They carried out the computation by multiple regression analysis making use of the method of the least squares method. They have used commercial software package, STATGRAPHICS, to determine the multiple regression coefficients.
Less attention has been paid in the area of mathematical and statistical modeling. The statistical models have edge over other models due to their capacity to handle random data correctly. There are several techniques available to relate the controllable factors and experimental facts.
In this chapter, we have made an attempt to develop statistical model based on non-linear non linear quadratic multiple regression analysis to predict removal efficiency of jet ejector for -aqueous system.
4.5.1 Statistical modeling
We have used the non linear quadratic relation between independent variables and dependent variables and is as follows:
Here, is a response variable, is the main factor;is the constant value of the regression; is the linear coefficient;is the quadratic coefficient and is the interaction coefficient. When.; and .
The computation was carried out by non linear regression analysis making use of the generalized minimal residual method.
The non linear regression coefficients determined by computation with the software package, STATGRAPHICS Plus 4.0, were used to determine the optimal model fitting.
4.5.2 Results and discussions
The factors which affect the absorption efficiency are gas concentration and the scrubbing liquid concentration. In this work the jet ejector is operated on critical value of liquid flow rate. For a given geometry, reduction in the liquid flow rate will lead to reduction of induced gas flow rate. Therefore, in the present work the liquid flow rate is kept constant. Effect of and on the removal efficiency) of the ejector have been investigated in this work.
The experimental values for the operating variables used in the present work are presented in Table 4.5.1 and the experimental data are tabulated in Table 4.5.2, 4.5.3.
Table 4.5.1 : Codification of the operating variables for the statistical analysis
Table 4.5.2 : Experimental matrix for chlorine removal efficiency using setup - 1
Table 4.5.3 : Experimental matrix for chlorine removal efficiency using setup - 3
22.214.171.124 Statistical analysis
STATGRAPHICS Plus 4.0 is used to predict the removal efficiency (Y) using statistical model (4. 5. 1) for the nozzles N1, N5, N6 and N7.The results are summarized in table 4.5.4 and 4.5.5. Table 4.5.4 demonstrates the parameters as outcome of simulated results of STATGRAPHICS plus 4.0. The regression coefficients of fitted models are summarized in table 4.5.5.
The analysis of variance (ANOVA) for the operational variables and indicate that removal efficiency is well described by nonlinear quadratic models. The convergence is obtained successfully after 4 iterations for estimation of regression coefficients.
Furthermore, the statistical analysis showed that both factors ( and) had significant effects on the response () and the liquid concentration is more significant between two.
It may be observed that fitted models do not contain the independent term (). This implies that the removal efficiency () is a function of the factors considered only.
Tests are run to determine the goodness of fit of a model and how well the non linear regression plot approximates the experimental data. As the results are multi numerical they are presented in figure (4.5. ) and table (4.5. ). Statistical tests like R-squared, R-squared (adjusted for d.f.), standard error of estimate, mean absolute error and Durbin-Watson statistic are covered. The tables containing confidence interval, analysis of variance (ANOVA) and residual analysis are also reported.
Results of statistical analysis in STATGRAPHICS Plus 4 for different nozzles:
126.96.36.199 Interpretation of the results of statistical analysis in STATGRAPHICS Plus 4 for different nozzles
The results of fitted model, R-squared test, R-squared (adjusted for d.f.) test, standard error of estimates, mean absolute error and Durbin-Watson statistic test are summarized in table (4.5.6) and may be interpreted as follow.
The R-Squared statistic indicates that the model as fitted explains 85.84% , 86.55 %, 88.27% and 48.59% of the variability in Y for N1, N5, N6 and N7 respectively.
The adjusted R-Squared statistic which is more suitable for comparing models with different numbers of independent variables are 77.75%,78.86%, 80.46% and 0.0% for N1, N5, N6 and N7 respectively
The standard error of the estimate shows the standard deviation of the residuals to be 9.79, 7.69, 7.08 and 32.60 for N1, N5, N6 and N7 respectively. This value can be used to construct prediction limits for new observations.
The mean absolute error (MAE) of 6.18, 5.15, 4.43 and 17.94 is the average value of the residuals for N1, N5, N6 and N7 respectively
Table 4.5.6 : Summary of statically results
The Durbin-Watson (DW) statistic tests the residuals to determine if there is any significant correlation based on the order in which they occur. Since, the DW value is less than 1.4 for N1 there may be some indication of serial correlation. Similarly, since, the DW value is greater than 1.4 for N5, N6, N7 there is probably not any serious autocorrelation in the residuals.
The output also shows asymptotic 95.0% confidence intervals for each of the unknown parameters.
After analysis the results it may be concluded that the model developed is not fit for nozzle N7.
188.8.131.52 Interpretation of figure (graph)
For each set of experiment a mathematical model describing the effect of related variables on removal efficiency were derived and plotted in the figures (4.5.1) to (4.5.20).These figures may be analyzed as follows:
Figures (4.5.2), (4.5.7), (4.5.12) and (4.5.17) show the response surfaces for the removal of chlorine with variation in gas concentration initially and the scrubbing liquid concentration. The response surface shows removal efficiency varies from 50% to maximum value of 96%. It is observed that the effect of liquid concentration is greater than the gas concentration on.
Dependency of removal efficiency () on gas concentration () and on initial concentration of liquid ()
Figures (4.5.1), (4.5.6), (4.5.11) and (4.5.16) are demonstrative curve of the fitted model showing the effect of on at constant.The similar curve may be obtained and plotted for other value of.
Figures (4.5.3), (4.5.8), (4.5.13) and (4.5.18) show the contours of estimated response surface for nozzle N5, N6, N7 and N1 respectively. The presentation of contours is for visualization of the best region where the is maximum.
A common trend (except small variation for nozzle N6) may be observed that at higher concentration of there is decrease of with increase in initial concentration of. But a reverse trend is observed at lower i.e. is increasing with increase in. The reason for this behavior is that at higher the viscosity of liquid increases. The higher viscosity has adverse effect on diffusivity and physical solubility. And this effect becomes more appreciable at higher because of higher scrubbing load due to higher initial concentration of ().
The figures (4.5.3), (4.5.8), (4.5.13) and (4.5.18) showing contours have significance that they show the correlations of all the parameters .The counters are useful to identify the regions of maximum efficiency The following table shows the regions of the maximum efficiency.
Analysis of the contours suggests the following regions to have maximum.
Table 4.5.7 : Summary of analysis of contours for removal efficiency
Observed versus Predicted
The figures (4.5.4), (4.5.9), (4.5.14) and (4.5.19) show the observed versus predicted plot for N1, N5, N6 and N7 respectively. The Y axis shows the observed value of and X axis show the predicted value by fitted model of . It may be observed that the points are randomly scattered around the diagonal line indicating that model fits well. It may also be observed that the plot is straight line having no curve that means no need to try for higher order polynomial.
Residuals versus Predicted
The figures (4.5.5), (4.5.10), (4.5.15) and (4.5.20) show of the residual analysis. The Y axis shows Studentized residual and X axis shows the predicted from the fitted models. It may be observed that there is uniformity in variability with change in mean value shown by line in the center.
The models developed as shown in table (4.5.4) for nozzles N1, N5, N6 and N7 to predict by using STATGRAPHICS considering variation with respect to and are well fitted.