Experimental study of mass transfer limited reaction—Part I: Use of fibre optic spectrometry to infer asymmetric mass transfer coefficients

Chemical Engineering Science 60 (2005) 2879 – 2893 www.elsevier.com/locate/ces

Experimental study of mass transfer limited reaction—Part I: Use of fibre optic spectrometry to infer asymmetric mass transfer coefficients Kiran B. Deshpande, William B. Zimmerman∗ Department of Chemical and Process Engineering, University of Sheffield, Newcastle Street, Sheffield S1 3JD, UK Received 9 November 2004; received in revised form 8 January 2005; accepted 20 January 2005 Available online 16 March 2005

Abstract Mass transfer coefficients, often quantified using empirical correlations in chemical engineering, are useful in predicting the final concentration of the reactants [(Fogler 1992). In: Amundson (Ed.)., Element of Chemical Reaction Engineering, International Series in the Physical and Chemical Engineering, Princeton-Hall]. In the present work, we propose an inverse methodology to estimate mass transfer coefficients from the experimentally measured concentrations of the reactants. The potential of the inverse methodology is that asymmetric mass transfer coefficients of all the premixed reactants can be inferred, which has not been reported so far in literature. In general, only the slowest transfer coefficient estimate has been feasible before. We first review the potential reactions satisfying the required criteria and then discuss multi-component spectrum analysis to decipher the concentrations of the reactants from the absorbance spectrum of the reaction mixture using the principle of additivity. We show that linear additivity does not hold if there is any interaction between the reacting species. In that case, we use a non-linear calibration for the concentrations of the reactants. The mass transfer coefficients inferred using the inverse methodology are validated by solving the forward problem and could be potentially used to study transport limited characteristics of heterogeneous reactions. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Transport processes; Mass transfer; Multiphase flow

1. Introduction Heterogeneous reactions are very common in the chemical industry. In some heterogeneous reactions, the reactants are present in two different phases and reaction occurs at the interface. Hence, the reactants are initially separated in such a reaction. Mass transfer limited reactions are well studied for such systems (Astarita, 1967). There are other systems where the two reactants are dissolved in a continuous phase, but do not react with each other in that phase. Instead, the two reactants diffuse into a dispersed phase that is introduced, where the reaction takes place, e.g. manufacturing of propylene oxide (Warnecke et al., 1999) and proton exchange reactions. In the present work, we are interested in the latter case. ∗ Corresponding author. Tel.: +44 114 222 7517; fax: +44 114 222 7501.

E-mail address: [email protected] (W.B. Zimmerman). 0009-2509/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2005.01.016

Mass transfer is one of the most important unit operations in chemical industry and is often quantified in terms of mass transfer coefficients. It is general chemical engineering practice to evaluate mass transfer coefficients using empirical correlations. Fogler (1992) studied mass transfer limited reactions in a packed bed for a binary reaction where axial variation in concentration of the reactants can be represented as

a  CA A = exp − z . (1) CA0 U The above formulation was proposed by Fogler (1992) to evaluate the final concentration of the reactant from the initial concentration, with the mass transfer coefficient evaluated using an appropriate empirical correlation, which is a common chemical engineering practice for species with unknown mass transfer coefficient. The experiment can be conducted and kA inferred from Eq. (1). In this work, we

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can measure the initial and the final concentration of the reactants; the active surface area of the dispersed phase per unit volume of the reactor can also be calculated using the phase fraction of the dispersed phase. Hence we can use Eq. (1) to evaluate the mass transfer coefficient. It can be seen that Fogler (1992) assumed that the surface concentration of both the reactants is equal to zero. This assumption is valid for mass transfer limited reaction with only one reactant transported from one phase to other. But, it is not valid for a system with two premixed reactants with asymmetric transport rates. This is because such a system with asymmetric transport rates potentially results in a crossover phenomenon, where there is a switch in concentration of the reactants (Mchedlov-Petrossyan et al., 2003) from the dispersed phase being populated preferentially by one then the other reactant. Hence, in this work, we propose an inverse methodology to infer mass transfer coefficients from experimentally measured concentrations of the reactants for transport limited reactions. This article is organized as follows: the reaction we are dealing with is mass transfer limited and hence, the transport of the two reactants from the continuous phase to the dispersed phase is the controlling parameter. In Section 2, potential reactions with the possible asymmetric transport rates are reviewed. Since the reaction is almost instantaneous, reaction kinetics do not play a crucial role. It is very difficult to measure the concentration of reactants in such a fast reaction. The experimental set-up is discussed in detail for an in situ concentration measurement (spectroscopic) technique in Section 3. There are various issues in extracting the concentration of the reactants from the absorbance spectrum of the reaction mixture, such as, the principle of linear additivity and non-linear additivity. We address these issues in the multicomponent spectrum analysis, Section 4. We propose an inverse methodology to infer mass transfer coefficients from the experimentally measured concentrations of the reactants. Its potential is discussed in Section 5 followed by the conclusions in Section 6. Mass transfer coefficients inferred using the inverse methodology can be potentially useful to study mass transfer limited reactions leading to cross-over phenomenon, which is a complex field itself and hence, is communicated in Part II (Deshpande & Zimmerman, 2004).

2. Selection of the reaction We are interested in inferring asymmetric mass transfer coefficients of the reactants transporting from the continuous phase to the dispersed phase. In this section, the criteria required for a potential reaction are discussed followed by a review of all the potential reactions with their advantages and disadvantages. Different chemicals considered for a desirable reaction are discussed in the later section, concluding with the most appropriate reaction chosen for the present experimental work.

2.1. Criteria The selection of the reaction is the crucial step in the present work, since the reaction should satisfy the following criteria: (1) The two reactants dissolved in the continuous phase do not react with each other. It would be interesting to see why the reactants do not react in the continuous phase though they are soluble in it. The solvent effect in catalysis is a possible explanation and is discussed later in detail. (2) Both the reactants diffuse into the dispersed phase and the reaction takes place in the dispersed phase. (3) Since the objective is to study mass transfer limited characteristics, the reaction should be almost instantaneous. This assumption assures that the rate limiting step is the mass transfer of the reactants from one phase to the other. (4) A colour change could be an added advantage since it facilitates visual observation and on-line spectrometry. An extensive search was carried out to find the most appropriate reaction that satisfies the above mentioned criteria. Four potential reactions were found to be relevant to the required criteria and are summarized in Table 1. Weak acid–weak base equilibrium reaction is found to satisfy all the required criteria and, hence, is chosen for the present work. Selection of chemicals for the above reaction is found to be equally difficult task. Various solvents, solutes and reactive indicators are considered for the above reaction and are summarized in Table 2. 2.2. Selected reaction Among the four potential reactions summarized in Table 1, the first reaction does not satisfy the required criteria. The second reaction is also rejected because of the difficulty in concentration measurement, since the reaction is a free radical reaction and the life span of free radicals is very small. The third reaction, the reaction between gaseous chlorine and gaseous propylene, which takes place in a dispersed aqueous phase, is a very fast chemical reaction and hence, the rate determining step is gas–liquid mass transfer. Though the third reaction is found to satisfy all the required criteria, because of its lower sensitivity, concentration measurements of gaseous reactants using spectral analysis would be difficult. The fourth reaction, acid–base equilibrium reaction, is found to be very attractive because a colour change that occurs during the reaction facilitates concentration measurements. Various organic solvents, organic acids and indicators were reviewed before choosing a reaction mixture containing nicotinic acid and bromophenol blue dissolved in 1-chlorobutane. We performed pH measurements for the individual species dissolved in the aqueous phase and the organic phase, in

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Table 1 The potential reactions considered in this work are summarized along-with the required criteria Potential reactions

Phase transfer catalysis reaction (Dehmlow & Dehmlow, 1993; Weber and Gokel, 1977) Selective non-catalytic reduction reaction (Park et al., 1996; Sun et al., 2001; Miller and Klippenstein (2000)) Simultaneous absorption of two gases (Warnecke et al., 1999; Ramachandran & Sharma, 1971) Weak acid–weak base equilibrium reaction

Criteria

Remarks

1

2

3

4

N

N

Y

Y

Y

Y

Y

N

Does not satisfy required criteria

Complex free radical reaction Y

Y

Y

N

Y

Y

Y

Y

Low sensitivity of gaseous reactants for spectroscopic measurement Satisfies all the required criteria

Weak acid–weak base equilibrium reaction is found to be the most appropriate reaction. In the above table, Y indicates that respective criterion is satisfied while N indicates that respective criterion is not satisfied.

Table 2 Various solvents, organic acids and indicators are reviewed for weak acid–weak base equilibrium reaction before finalising nicotinic acid and bromophenol blue dissolved in 1-chlorobutane system Remarks Selection of solvent MIBK Diethyl ether 1-Chlorobutane

Most commonly used, cut-off wavelength is 330 nm Cut-off wavelength is 330 nm, considerably polar Cut-off wavelength is 220 nm, relatively non-polar Remarks

Selection of solute Acetic acid Benzoic acid Nicotinic acid

Water is considerably soluble in acetic acid is from In the presence of acidic water transport the aqueous phase to the organic phase Considerably soluble in water and hence, transport is from the aqueous phase to the organic phase even in the presence of acidic aqueous phase Remarks

Selection of indicator Methyl red Ferric chloride Bromophenol blue

Two absorbance peaks could be deceptive, sparingly soluble in water Reacts with diethyl ether increasing cut-off wavelength to 450 nm Useful indicator as changes colour from blue (pH
4.7) to yellow (pH  3.1)

order to understand why nicotinic acid and bromophenol blue do not react in the organic phase, and pH values are reported in Table 3. It can be seen that both nicotinic acid and bromophenol blue are acidic in the organic phase (pH < 4), hence, do not react with each other in the organic phase. However, nicotinic acid is acidic in water and bromophenol blue is mildly basic; this acid–base reaction is possible in the aqueous phase which is almost instantaneous. The spectroscopic properties of nicotinic acid, bromophenol blue and 1-chlorobutane allow us to measure absorbance of all the

Table 3 The pH measurements of the individual species in the aqueous and the organic phases for nicotinic acid–bromophenol blue system Sample 0.001 M 0.001 M 0.001 M 0.001 M

pH NA in the aqueous phase NA in the organic phase BP in the aqueous phase BP in the organic phase

4.31 3.9–4.0 7.23 3.3–3.4

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Inlet Reactants Inlet (Dispersed Phase) Pressure Head Distributor Collar Light Source

Fibre Optic Probe

Support Spectrometer

Outlet Fig. 1. Schematic of the experimental setup.

species in the reaction mixture. Consequently, it is the most appropriate reaction to study mass transfer limited aspects of two phase, dispersed reactions.

3. Experimental setup The schematic of the experimental setup, as shown in Fig. 1, consists of an experimental rig and a concentration measuring device, which is discussed in detail in this section. 3.1. Experimental rig The experimental rig consists of a tubular reactor, a distributor, a constant pressure head on the top of the distributor, two outlet valves to maintain constant flow rate, and two peristaltic pumps to feed the reaction mixture and the aqueous phase, respectively. The main body of the experimental rig is a tubular reactor which is fabricated from the quartz material and is 5 cm in diameter and 60 cm in height, with a flange attached to the top of the column. The top flange offers support to the distributor which is used to generate water drops into the reaction mixture. The bottom end of the column is sealed with an outlet stream which is connected to a string of two valves. The top valve is operated from a fully closed mode to a fully opened mode, while the bottom valve is partially opened to maintain the desirable flow rate. This arrangement ensures that outlet flow rate is uniformly maintained for various experimental runs. A constant pressure is maintained at the top of the column, where the aqueous phase is maintained at uniform head to ensure uniform droplet size. The aqueous phase and the reaction mixture are stored in separate reservoirs and are con-

Fig. 2. Photograph of the actual experimental rig. Various components of the rig, as labelled in the above photograph, are discussed later in detail using a schematic diagram where appropriate.

tinuously fed into the quartz column using peristaltic pumps. Outlet flow is maintained at a particular rate in order to facilitate a steady-state operation. The actual experimental rig is shown in Fig. 2. 3.1.1. Design of the distributor In the study of transfer limited heterogeneous reaction, surface area is a key parameter since transport of the reactants is strongly dependent on the available surface area. At constant phase fraction, the smaller the drop size, the larger is the number of droplets; the larger is the surface area available per unit volume of the reactor and the more stable is the dispersed drop in the continuous phase. Ideally, the distributor should produce small and mono-dispersed (uniform sized) drops. In order to achieve this, we reviewed two types of diffusers, namely a slot diffuser (Li et al., 1994) and a perforated plate (Chen et al., 2001). A slot diffuser has been successful in generating monodispersed bubbles via the Rayleigh– Taylor instability mechanism which is induced due to density difference between the dispersed phase and the continuous phase. However, in the present work we are dealing with a liquid– liquid system where density difference is not as significant as in an

Kiran B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 60 (2005) 2879 – 2893 36 holes φ = 1mm

5 cm

Reaction mixture

Aqueous drops Annular slot 1 mm width

Fig. 3. New design of the distributor.

air–water system. It is not clear how useful it would be to disperse water drops of approximately uniform size into another immiscible phase (organic phase in the present case). The perforated plate, which is easy to construct, is commonly used as a distributor particularly for bubble formation (Chen et al., 2001). The dispersed phase (water) is stored at a particular level to maintain the pressure drop and to achieve the precisely desired flow rate of the dispersed phase. The reaction mixture is introduced through a single inlet at the top of the column, which may induce jet formation. The water drops formed at the perforated plate may experience non-uniform operating conditions and the drops may deform in the region of the jet. Dead zones may also occur near the wall of the column. Hence, this design is modified in order to study mass transfer characteristics. Increasing the inlet points for the reaction mixture could reduce the above mentioned effect but would complicate the design of the distributor. In order to overcome the difficulties with a slot diffuser and a perforated plate, an innovative design of the distributor is used as shown in Fig. 3. The distributor consists of two solid circular plates, both drilled with concentric and uniformly distributed holes of diameter 0.5 mm each. The bottom plate is filed smoothly to give an annular slot in between the concentric holes. Two more holes are drilled on the top plate which has an opening in the annular slot and serves as an inlet for the reaction mixture. The purpose of introducing the reaction mixture through a slot is to avoid jet formation which may occur in the case of a single inlet point. The dispersed phase is introduced in a similar way as discussed for the perforated plate. The design proposed is thus a combination of a slot diffuser and a perforated plate. 3.2. Concentration measuring techniques The concentration of the reactants and the product is a key parameter in quantifying mass transfer with chemical reaction, since concentration gradient is the driving force for mass transfer. It would be desirable to measure the concentration of the reactants or the product on-line to analyse the real time behaviour of the system. There are various con-

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centration measurement techniques available in the literature such as conductimetry, FTIR analysis, NMR spectroscopy and UV–visible spectrometry. In conductimetry, direct measurement of conductivity is performed by inserting a cell and is later calibrated in terms of concentration (Jeffery et al., 1989). This invasive technique also puts an additional constraint on the selection of the reaction because of the requirement of using ionic species. In Fourier transform infra-red (FTIR) reaction analysis, the probe is immersed in the reaction mixture and measures the absorption in the middle infrared region (2.5–25
m) and hence the relative concentration changes of the reacting species can be captured during the reaction. An FTIR analyser is an invasive technique which would not be able to measure the concentration of the reactants in the dispersed phase. Nuclear magnetic resonance (NMR) can be applied to estimate the spatial variation of chemical conversion of a species in a packed bed reactor (Yuen et al., 2002; Buckley et al., 2003; Gladden, 2003). The extent of conversion is estimated by measuring the chemical shift of the reaction mixture and that of the individual species present in the reaction mixture. NMR spectroscopy could be used to measure the concentration of the reactants in a single phase or in the presence of catalyst particles, but its potential to measure the concentration of the reactants both in the continuous phase and the dispersed phase is not known at the moment. Considering the cost effectiveness, we now discuss another potential way of in situ concentration measurement, called UV–visible spectroscopy, in the next subsection. 3.2.1. UV–visible spectrometry Spectroscopy is the study of the interaction between light and matter and is often used as the basis for chemical analysis. All the spectrochemical methods depend in some way on the absorption of light energy by atoms or molecules. A UV spectrum can be obtained directly from instruments such as a fibre optic spectrometer. Molecular absorption of light in the UV (200–380 nm) and visible region (380–780 nm) of the spectrum is dependent on the electronic structure of molecules. Light consists of energy packets called photons, each with energy c E = h = h , 

(2)

where h is Planck’s constant, c is the speed of light,  is the frequency of light, and  is the wavelength of light. The absorbance spectrum is a plot of the absorbance against photon wavelength. The absorbance of any chemical species is defined by   I A = − log = − log(T ), (3) I0 where I0 is the intensity of the incident light, I is the intensity of the light after it passes through the sample, and the

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3.2.2. Fibre optic spectrometer A fibre optic spectrometer, used for on-line concentration measurement, consists of a light source (laser) and a fibre optic probe which analyses the absorbance of the solution present in the optical path. The spectrometer can be connected to a computer to display and store the absorption spectrum of the sample solution. The light source and the fibre optic probe are mounted on a perspex collar, as shown in Figs. 1 and 2, to measure the concentration of the reactants at different locations along the length of column. 3.2.3. Multi-channel spectroscopy Multi-channel spectroscopy is carried out with more than one spectrometer device connected together. Spectrometer multiplexing can connect up to 8 synchronous spectrometers that all run from a single analogue-to-digital converter. Thus, using multi-channel spectroscopy, simultaneous analysis of the reactants can be done at multiple locations. In order to study the transport limited characteristics of a heterogeneous reaction, it is essential to carry out on-line and non-invasive concentration measurements which would avoid the problem of disruption of the flow pattern inside the column due to inserted electrodes or probes. On-line measurement would also eliminate the need to eject the samples for analysis giving the real-time behaviour of transport of reactants. Although multi-channel spectroscopy is found to be very attractive, it is useful only if the concentration of the reactants can be measured in the presence of moving droplets, which is a very difficult task. We will discuss the difficulties in measuring concentration of the reactants in the presence of moving droplets in the next section. Hence, a fibre optic spectrometer with movable light source and fibre optic probe is chosen for on-line concentration measurement. 4. Multicomponent spectrum analysis In the present experimental work, we are interested in inferring overall mass transfer coefficients across falling droplets for a mass transfer limited heterogeneous reaction. Firstly, we need to measure the concentration of the individual species from the absorbance spectrum of the reaction mixture. The concentration of any species can be evaluated from the absorbance of that species using Beer’s Law, which states that A =  bC A ,

(4)

where A is the absorbance of species A at the wavelength , which is peak wavelength for species A,  is the molar absorptivity of A at the wavelength , b is the optical path length which is equal to the diameter of the quartz column

Absorbance

transmittance, T, is equal to I /I0 . The difference between these intensities is due to absorption of light by the chemical species and can be measured by a spectrophotometer.

slope = ε b

Beer’s Law limit Concentration Fig. 4. Qualitative representation of Beer’s Law, which is also known as Beer’s plot, indicating a region where linear dependence of Beer’s Law holds true.

in the present work (5 cm), and CA is the concentration of species A. The linear dependence of absorbance on concentration of the species, as considered in Beer’s Law, is valid only over a particular concentration range, and beyond that absorbance is no longer linearly dependent on concentration. The dependence of absorbance on concentration can be qualitatively represented as shown in Fig. 4. The molar absorptivity,  , is sometimes referred to as the extinction coefficient and it varies with the wavelength. The molar absorptivity can be evaluated from the absorbance–concentration plot, which is also called a Beer’s Law plot. Since the absorbance is linearly proportional to the concentration within the Beer’s Law limit, the slope of the absorbance–concentration plot is equal to  b. Thus, given the molar absorptivity, the optical path length and the absorbance, an unknown concentration of the species can be evaluated using Beer’s Law. In the present work, the reaction mixture contains nicotinic acid and bromophenol blue dissolved in 1chlorobutane. Since we have more than one absorbing species present in the reaction mixture, we can use the principle of additivity to extract the concentration of each species present in the mixture and this is discussed in detail in the next section. 4.1. Linear additivity The principle of linear additivity is that the absorbance of the reaction mixture at any wavelength is the sum of the absorbance of each component in the reaction mixture at that wavelength. This principle is a simple technique to extract the concentration of each species present in the reaction mixture from the absorbance of the reaction mixture. We will illustrate the principle of additivity for a two component mixture, since the reaction mixture considered

in the present work consists of two absorbing species. Let us assume that we have two absorbing species, A and B, present in the reaction mixture and they have peak wavelengths A and B , respectively. The absorbance of the reaction mixture at any wavelength, , is the sum of the absorbance of each species present in the mixture at that wavelength, , and can be written as

Absorbance

Kiran B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 60 (2005) 2879 – 2893

B A = AA  + A ,

(a)

0.3 0.2 0.15 0.1 0.05 0

(5)

0.0002 0.0004 0.0006 0.0008 0.0012 0.0012 Concentration

Absorbance

0.05

(6)

y = 25.556x + 0.0173

0.04 0.03 0.02 0.01 0

and B AB = A B bC A + B bC B .

y = 194.17x + 0.0619

0.25

0

B where A , AA  and A are the absorbance of the reaction mixture, the species A, and the species B, respectively, at wavelength . Beer’s Law can be applied to the two species A and B at peak wavelengths A and B , respectively, and can be written as B AA = A A bC A + A bC B

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0 (b)

(7)

The concentrations of the two species, A and B, in the reaction mixture can be evaluated by solving the above set of simultaneous algebraic equations (Eqs. (6,7)). In the above equations, AA and AB are the absorbance values of the reaction mixture at the wavelengths A and B , respectively, and can be obtained experimentally and b is the optical path length which can be measured. The molar absorptivity, , varies with the wavelength, as it is dependent on molecular electronic structure which results in light of some wavelengths being absorbed more strongly than other wavelengths. The molar absorptivity of species A at wavelengths A and B can be calculated by varying the concentration of species A and measuring the absorbance values at wavelengths A and B , respectively. Thus, the absorbance–concentration plot (Beer’s Law plot) can be obtained at the wavelengths A and B for species A. The slopes of the above plots for A species A are equal to A A b and B b, respectively. The same procedure can be repeated for species B to evaluate B A b and B b. B The principle of additivity, discussed in this section, is applied for the reaction mixture considered in the present work and its validity is investigated in the following section.

0.0005

0.001

0.0015

Concentration

Fig. 5. The peak absorbance is plotted against the concentration of nicotinic acid dissolved in 1-chlorobutane at the wavelengths, 269 nm (a) and 297.6 nm (b), to evaluate the molar absorptivity. The symbols represent the experimental values and the solid line is the linear fit. Concentration is measured in M units.

and (b). The slope of the absorbance–concentration plot at the wavelengths 269 and 297.6 nm gives the molar absorptivity of nicotinic acid at the two different wavelengths which NA are represented as, NA 269 and 297.6 , and are found to be equal −1 to 38.834 and 5.1112 M cm−1 , respectively. Similarly, the absorbance values are measured by varying the concentration of bromophenol blue in 1-chlorobutane at the wavelengths 269 and 297.6 nm and the respective absorbance–concentration plot are shown in Figs. 6(a) and (b). The slopes of the above plots give information about the molar absorptivity of bromophenol blue at the two different wavelengths which are represented as, BP 269 and BP , respectively, and are found to be equal to 6.2426 and 297.6 10.7064 M−1 cm−1 , respectively. The equations of linear additivity (Eqs. (6,7)) can be written for nicotinic acid and bromophenol blue as BP A269 = NA 269 bC NA + 269 bC BP

(8)

and 4.1.1. Nicotinic acid–bromophenol blue in 1-chlorobutane system Nicotinic acid and bromophenol blue dissolved in 1chlorobutane is the reaction mixture chosen for the present experimental work, for the various reasons discussed in the previous section. The peak absorbance wavelength of nicotinic acid is 269 nm and that of bromophenol blue is 297.6 nm. The absorbance values are measured for different concentrations of nicotinic acid dissolved in 1-chlorobutane at the wavelengths 269 and 297.6 nm, respectively, and are plotted against the concentrations, as shown in Figs. 5(a)

BP A297.6 = NA 297.6 bC NA + 297.6 bC BP .

(9)

The above set of equations can be solved simultaneously for the concentrations of nicotinic acid and bromophenol blue, CNA and CBP , by feeding experimentally measured values of absorbance, A269 and A297.6 . The optical path length, b, which is equal to 5 cm and the molar absorptivity values calculated earlier are used to solve the above set of simultaneous equations. The potential of the principle of additivity is investigated by measuring the absorbance for six sets of known

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Kiran B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 60 (2005) 2879 – 2893 2.5 y = 31.213x-0.0419

0.5

Absorbance

Absorbance

0.6 0.4 0.3 0.2 0.1 0 0.00E+00 5.00E-03 1.00E-02 1.50E-02 2.00E-02

(a)

Concentration

y = 10918x + 0.0285

1.5 1 0.5 0 0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04 Concentration

(a)

1.2

0.08

1

y = 53.532x - 0.0486

0.8 0.6 0.4 0.2 0 0.00E+00 5.00E-03 1.00E-02 1.50E-02 2.00E-02

(b)

Absorbance

Absorbance

2

(b)

Table 4 The potential of the principle of linear additivity is investigated by comparing the measured and calculated values of absorbance at different wavelengths for nicotinic acid–bromophenol blue system CNA (M)

CBP (M)

Measured A269

Measured A297.6

Calculated A269

Calculated A297.6

0.001 0.0007 0.0006 0.0004 0.0002 0.0001

0.011378 0.007946 0.006347 0.004699 0.004633 0.001523

1.5891 1.0389 0.8102 0.6365 0.4032 0.1672

0.6937 0.4803 0.3872 0.3108 0.2956 0.1167

0.5493 0.3839 0.3146 0.2243 0.1834 0.067

0.635 0.4433 0.355 0.2618 0.2531 0.0841

concentrations. The measured values of absorbance and the values of absorbance calculated using the principle of additivity are represented in Table 4. It is found that the measured values of absorbance are very different than those calculated using the principle of additivity at the wavelength equal to 269 nm. At this wavelength the absorbance values are not linearly additive, and this could be attributed to possible interaction between nicotinic acid and bromophenol blue, such as hydrogen bonding, resulting in a non-linear contribution to the absorbance. We will discuss the method we used to extract concentrations from the absorbance values for such a nonlinearly additive system in the next section. At the wavelength equal to 297.6 nm, nicotinic acid has a negligible contribution to the absorbance value and it is found to be nearly linearly additive. To investigate the potential of the principle of additivity further, we applied the above formulation for the reaction

y = 525.93x - 0.0325

0.04 0.02 0 -0.02 -0.04 0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04

Concentration

Fig. 6. The peak absorbance is plotted against the concentration of bromophenol blue dissolved in 1-chlorobutane at the wavelengths, 269 nm (a) and 297.6 nm (b), to evaluate the molar absorptivity. The symbols represent the experimental values and the solid line is the linear fit. Concentration is measured in M units.

0.06

Concentration

Fig. 7. The peak absorbance is plotted against the concentration of benzoic acid dissolved in 1-chlorobutane at the wavelengths, 280 nm (a) and 297.6 nm (b), to evaluate the molar absorptivity. The symbols represent the experimental values and the solid line is the linear fit. Concentration is measured in M units.

mixture containing benzoic acid and bromophenol blue dissolved in 1-chlorobutane. 4.1.2. Benzoic acid–bromophenol blue in 1-chlorobutane system We have seen that the peak absorbance wavelength of bromophenol blue in 1-chlorobutane is 297.6 nm, while that of benzoic acid in the same solvent is 280 nm. We apply the principle of additivity, which was discussed in detail in the previous section, for the reaction mixture containing benzoic acid and bromophenol blue dissolved in 1-chlorobutane. The molar absorptivity of benzoic acid at the wavelengths 280 and 297.6 nm are evaluated by measuring the absorbance of benzoic acid in 1-chlorobutane for different concentrations of benzoic acid. The absorbance–concentration plots for benzoic acid at the wavelengths 280 and 297.6 nm are shown in Figs. 7(a) and (b), respectively. The molar absorptivity of benzoic acid at the two different wavelengths, which BA are represented as BA 280 and 297.6 , respectively, are found to be equal to 2183.6 and 105.186 M−1 cm−1 , respectively. Similarly, the absorbance–concentration plots are obtained for bromophenol blue in 1-chlorobutane at the wavelengths 280 and 297.6 nm as shown in Figs. 8(a) and (b), respectively. The molar absorptivity of bromophenol blue at the two different wavelengths, which are represented as BP BP 280 and 297.6 , respectively, are found to be equal to 7.7542 and 10.7064 M−1 cm−1 , respectively. To illustrate the principle of additivity (Eqs (6,7)), absorbance spectra are obtained for known concentrations

Kiran B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 60 (2005) 2879 – 2893

mixture, when there is an interaction between the species, in the next section.

Absorbance

0.8 0.6

y = 38.771x - 0.0403

0.4

4.2. Non-linear additivity

0.2 0 0.00E+00 5.00E-03

1.00E-02

1.50E-02

2.00E-02

Concentration

(a)

Absorbance

1.2 1

y = 53.532x - 0.0486

0.8 0.6 0.4 0.2 0 0.00E+00 5.00E-03

(b)

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1.00E-02

1.50E-02

2.00E-02

Concentration

Fig. 8. The peak absorbance is plotted against the concentration of bromophenol blue dissolved in 1-chlorobutane at the wavelengths, 280 nm (a) and 297.6 nm (b), to evaluate the molar absorptivity. The symbols represent the experimental values and the solid line is the linear fit. Concentration is measured in M units.

of the individual species and a reaction mixture of the same concentrations of the individual species, as shown in Fig. 9. The absorbance spectra are shown for 34.4
M benzoic acid, 15 mM bromophenol blue, and the reaction mixture containing 34.4
M benzoic acid and 15 mM bromophenol blue, respectively. We have earlier calculated the molar absorptivity of benzoic acid and bromophenol blue. The absorbance values of the reaction mixture evaluated at 280 and 297.6 nm using the principle of additivity (Eqs. (6,7)) are 0.95 and 0.8, respectively, and are in a very good agreement with the experimentally measured absorbance values. The absorbance values obtained from the principle of additivity are found to be in a good agreement with the experimentally measured absorbance values for benzoic acid–bromophenol blue system, since there is no apparent interaction between benzoic acid and bromophenol blue in 1-chlorobutane. Thus, the principle of additivity can be applied to extract the concentration of the individual species from the absorbance of the reaction mixture if the species present in the reaction mixture do not interact with each other in the solvent. Benzoic acid and bromophenol blue dissolved in 1-chlorobutane cannot be used for the present experimental work for the various reasons discussed in the selection of reaction section. Since the principle of linear additivity does not hold for the chosen reaction mixture of nicotinic acid and bromophenol blue in 1-chlorobutane, we need to find a way to measure the concentration of the individual species. We propose a methodology to extract the concentra- tion of the individual species from the absorbance of the reaction

For a better understanding of the interaction between nicotinic acid and bromophenol blue in 1-chlorobutane, we performed experiments by keeping the concentration of one of the reactants constant and then varying the concentration of the other reactant. The peak absorbance wavelength of nicotinic acid is 269 nm and that of bromophenol blue is 297.6 nm. Concentration of nicotinic acid is kept constant and the absorbance is measured at the wavelengths 269 and 297.6 nm by varying the concentration of bromophenol blue. The same procedure is repeated for different concentrations of nicotinic acid and the data generated is represented in Table 3. The absorbance at 269 nm is plotted against the concentration of bromophenol blue for different concentrations of nicotinic acid, and is shown in Fig. 10. It is found that the molar absorptivity, which is evaluated from the slope of the absorbance–concentration plot, is different for various concentrations of nicotinic acid. This indicates that there is possible interaction (such as hydrogen bonding) between nicotinic acid and bromophenol blue, particularly at higher concentrations of nicotinic acid, resulting in the increase in the molar absorptivity value with increasing nicotinic acid concentration. Hence, the principle of linear additivity cannot be applied to extract concentration from absorbance. The absorbance at 297.6 nm is also plotted against the concentration of bromophenol blue for different concentrations of nicotinic acid, and is shown in Fig. 11. It is found that the absorbance values do not change substantially with increase in the concentration of nicotinic acid, at the wavelength equal to 297.6 nm, and hence the molar absorptivity values are nearly the same for different concentrations of nicotinic acid. This is because nicotinic acid has little contribution at 297.6 nm. Thus, we need to explore a way to extract the concentration of both the reactants, nicotinic acid and bromophenol blue, from the absorbance values of the reaction mixture using a non-linear fit. 4.2.1. Calibration of data using a non-linear fit We understand that nicotinic acid and bromophenol blue dissolved in 1-chlorobutane do not follow linear additivity because of possible interaction between them in the organic solvent. This interaction may result in formation of one or more complexes. We try to incorporate the contribution of these complexes using a non-linear fitting technique. Since we already know the absorbance values at the peak wavelength of both the reactants, which are 269 and 297.6 nm for nicotinic acid and bromophenol blue, respectively, we need to find a non-linear relationship between the absorbance and the concentration of both the reactants.

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Kiran B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 60 (2005) 2879 – 2893 2.5 0.0000344 M BA 0.015 M BP Reaction mixture

Absorbance

2

1.5

1

0.5

0 250

260

270

280

290

300

310

320

330

340

350

Wavelength Fig. 9. The potential of the principle of linear additivity is investigated by obtaining the absorbance spectrum for known concentrations of the individual species and the reaction mixture of the same concentrations of the individual species, for benzoic acid–bromophenol blue dissolved in 1-chlorobutane. Wavelength is represented in nm.

1.8 y = 91.057x + 0.5544

Absorbance at 269 nm

1.6 1.4

y = 70.79x + 0.4588 y = 60.782x + 0.4223 y = 58.04x + 0.3548 y = 64.747x + 0.2615

1.2 1 0.8

y = 54.597x + 0.1582

0.6 y = 38.462x + 0.1112

0.4 0.2 0 0

0.005

0.01

0.015

0.02

0.025

Concentration of bromophenol blue 0.0001 M NA 0.0006 M NA

0.0002 M NA 0.0007 M NA

0.0003 M NA 0.001 M NA

0.0004 M NA

Fig. 10. The absorbance of the reaction mixture at the wavelength of 269 nm is plotted against the concentration of bromophenol blue, for different concentrations of nicotinic acid. The symbols represent the experimental values and the solid lines are the respective linear fits. Concentration is measured in M units.

(10)

To obtain the functional dependence of CNA and CBP on the absorbance values A269 and A297.6 , we use a non-linear fitting function in MathematicaTM . The functional form represented by

(11)

A269 = a1 CNA + a2 CBP + a3 CNA CBP

From Table 5, we know that A269 = f (CNA , CBP ), and A297.6 = f (CNA , CBP ).

(12)

Kiran B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 60 (2005) 2879 – 2893

2889

0.8

Absorbance at 297.6 nm

0.7

y = 57.373x + 0.0239

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

Concentration of bromophenol blue 0.0001 M NA 0.0006 M NA

0.0002 M NA 0.0007 M NA

0.0003 M NA 0.001 M NA

0.0004 M NA

Fig. 11. The absorbance of the reaction mixture at the wavelength of 297.6 nm is plotted against the concentration of bromophenol blue for different concentrations of nicotinic acid. The symbols represent the experimental values and the solid line is the linear fit. Concentration is measured in M units.

Table 5 The absorbance data at the two wavelengths, 269 and 297.6 nm, for various concentrations of nicotinic acid and bromophenol blue dissolved in 1-chlorobutane, which is used to obtain the dependence of absorbance on concentration of the reactants CNA (M)

CBP (M)

A269

A297.6

CNA (M)

CBP (M)

A269

A297.6

0.0001 0.0001 0.0001 0.0001 0.0001 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004

0.001523 0.003094 0.004699 0.006339 0.008016 0.001523 0.004633 0.006221 0.007843 0.009502 0.0112 0.001531 0.003102 0.004733 0.006392 0.008088 0.009814 0.01158 0.001523 0.003094 0.004699 0.006339 0.009674 0.011371

0.1672 0.231 0.2951 0.3567 0.4166 0.2467 0.4032 0.4951 0.5887 0.6807 0.769 0.351 0.4571 0.577 0.6807 0.7965 0.9027 0.9943 0.4398 0.5257 0.6365 0.7362 0.9022 1.0182

0.1167 0.2032 0.2892 0.3806 0.4641 0.1053 0.2956 0.3856 0.472 0.57 0.6627 0.1184 0.2081 0.2991 0.385 0.4821 0.5845 0.6639 0.1234 0.2108 0.3018 0.392 0.5845 0.6783

0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 0.0007 0.0007 0.0007 0.0007 0.0007 0.0007 0.0007 0.001 0.001 0.001 0.001 0.001 0.001 0.001 — — —

0.001531 0.003102 0.004707 0.006347 0.008024 0.0097399 0.0114755 0.001523 0.0030859 0.0046823 0.006288 0.007946 0.009643 0.011399 0.001523 0.0030778 0.004666 0.0062881 0.007947 0.0096427 0.011378 — — —

0.5146 0.6051 0.7131 0.8102 0.9127 1.0167 1.1145 0.5523 0.6687 0.795 0.9264 1.0389 1.1445 1.2408 0.6887 0.8339 0.9799 1.1333 1.2862 1.4238 1.5891 — — —

0.1253 0.2069 0.2955 0.3872 0.4737 0.5669 0.6596 0.1353 0.2188 0.3103 0.3964 0.4803 0.5616 0.6432 0.1413 0.2384 0.3328 0.421 0.5135 0.61 0.6937 — — —

and A297.6 = b1 CNA + b2 CBP + b3 CNA CBP ,

(13)

is used to fit the data represented in Table 3. The coefficients, a1 , a2 and a3 are found to be 643.198, 56.6628 and 22925.0 M−2 , respectively, and b1 , b2 and b3 are found to be 77.837, 58.5786 and −5912.05 M−2 , respectively, as re-

ported in Table 6. The estimated variance is found to be very small indicating the accuracy of the non-linear fitting. The diagonal elements of the correlation matrix are found to be exactly one, which is another indicative of the accuracy of the fitting. 3-D plots of the experimentally measured absorbance values and surface plots of the non-linear fits for the absorbance

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Kiran B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 60 (2005) 2879 – 2893

Absorbance at 269 nm

2

1.5

1

0.5

0 0.015 Co

nce

1.25

0.01

ntr

atio

no

1 0.005

fb

rom

0.5

op

hen

0

ol b

0

n

ratio

cent

Con

lue

tinic

co of ni

acid

x 10

-3

Fig. 12. A 3-D surface plot of the absorbance at 269 nm against the concentration of bromophenol blue and nicotinic acid. The symbols represent the experimentally obtained absorbance values, while the surface is the non-linear fit (Eq. (12)). Concentration is measured in M units.

Absorbance at 297.6 nm

1 0.8 0.6 0.4 0.2

0 0.015 Co

nce

0.01 atio no fb

ntr

0.005

rom

op

hen

0

ol b

lue

0

0.6

0.4

0.2

n of

tratio

cen Con

1

0.8 id

c ac

tini nico

1.2

x 10

-3

Fig. 13. A 3-D plot of the absorbance at 297.6 nm against the concentration of bromophenol blue and nicotinic acid. The symbols represent the experimentally obtained absorbance values, while the surface is the non-linear fit (Eq. (13)). Concentration is measured in M units.

values against the concentration of nicotinic acid and bromophenol blue are shown in Figs. 12 and 13. The non-linear fits, represented by Eqs. (12,13) are found to be in a very good agreement with the experimental values. The accuracy of the non-linear fitting is further tested by calculating the absorbance values for those concentrations of the reactants, for which the absorbance is also experimentally measured. The calculated and experimental

values are shown in Table 7 and are found to be in a good agreement. Thus, the concentration of nicotinic acid and bromophenol blue can be extracted from the absorbance of the reaction mixture at the two wavelengths using non-linear additivity, and these concentrations will be further used to evaluate mass transfer coefficients, as discussed in the following section.

Kiran B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 60 (2005) 2879 – 2893 Table 6 The coefficients used in Eqs. (12,13) are estimated using the non-linear fitting and are reported along with estimated variance and correlation matrix Coefficients Estimated values

Estimate variance

Correlation matrix

a1 a2 a3

643.198 56.6628 22925.0

0.00274122

1.0000 −0.0047 −0.6795

−0.0047 1.0000 −0.6392

−0.6795 −0.6392 1.0000

b1 b2 b3

77.837 58.5786 −5912.05

0.00015167

1.0000 −0.0047 −0.6795

−0.0047 1.0000 −0.6392

−0.6795 −0.6392 1.0000

Table 7 The accuracy of the non-linear fitting is checked by comparing the measured and the calculated values of absorbance at the two different wavelengths, for nicotinic acid–bromophenol blue system CNA (M)

CBP (M)

Measured A269

Measured A297.6

Calculated A269

Calculated A297.6

0.001 0.0007 0.0006 0.0004 0.0002 0.0001

0.011378 0.007946 0.006347 0.004699 0.004633 0.001523

1.5891 1.0389 0.8102 0.6365 0.4032 0.1672

0.6937 0.4803 0.3872 0.3108 0.2956 0.1167

1.54875 1.11053 0.832861 0.5667 0.412401 0.154101

0.677 0.490153 0.396 0.2953 0.2815 0.0961

5. Evaluation of mass transfer coefficient In this section, we discuss a methodology to evaluate mass transfer coefficients from experimentally measured concentrations of the reactants. 5.1. Inverse methodology The concentration profiles of the two reactants can be theoretically obtained by solving the convection–diffusion– reaction equations for the two reactants, for the case of an irreversible reaction, where the effect of the concentration of the product on the dynamics of the system is decoupled. These equations are represented for the two reactants as

2891

where a is surface area of the dispersed phase per unit volume of the reactor, and k is the rate constant of the reaction. We have earlier seen that the reaction between nicotinic acid and bromophenol blue is very fast and reaction rate constant cannot be measured experimentally because of its rapid nature. The reaction rate constant is not available in literature for the above reaction. Since fast reactions are controlled by mass transfer of the reactants and are not dependent on the kinetics of the system, which is numerically illustrated in our earlier communication, the reaction rate constant k is considered to be equal to 106 M−1 s−1 in the present work. The dependence of k on the inferred mass transfer coefficients will be discussed in our next communication, in order to study cross-over phenomenon for transport limited reaction. We have seen that the formulation proposed by Fogler (1992) for transport limited reactions is not applicable in the present work. Fogler assumed that the surface concentration of both the reactants is equal to zero, which is not true in the present study because of asymmetric mass transfer coefficients of the two reactants. Hence, we propose a methodology to evaluate mass transfer coefficients using the above system of equations (Eqs. (14–17)). The forward problem in the above system evaluates the final concentration of the reactants, CA and CB , given the initial concentration, the diffusion coefficients (calculated using the Wilke–Chang equation (Reid et al., 1977)) and the mass transfer coefficients of the two reactants. Here, we are interested in evaluating the mass transfer coefficients from the initial and the final concentrations of the reactants, which is exactly opposite to the forward problem. Therefore, we propose a methodology to solve the inverse problem. A system of equations (Eqs. (14–17)) can be thought of as providing a functional representation between the concentration inputs and the predicted outputs:
 (p) (p) CA,f , CA,f = f (CA,0 , CA,0 ;
A ,
B ), (18) where (p) refers to model prediction. The error E is defined as
 
(p) (p) (m) 2 (m) 2 E = CA,f − CA,f + CB,f − CB,f ,

(19)

where (m) refers to experimentally measured final concentrations. The parameter estimates of
A and
B are found by solution to the optimization problem

U

jC A j2 CA −
A a(CA − CA,s ), = DA jz jz 2

(14)

U

jC B j2 C B −
B a(CB − CB,s ), = DB jz jz 2

(15)


A ,
B

U

jCA,s =
A a(CA − CA,s ) − kC A,s CB,s jz

(16)

where
A and
B are varied over the physically achievable range. The above methodology (Eqs. (18–20)) is implemented as follows:

(17)

• In the above system of equations (Eqs. (14–17)), the forward problem is: given the initial concentrations of the

and U

jCB,s =
B a(CB − CB,s ) − kC A,s CB,s , jz

min E,

(20)

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Kiran B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 60 (2005) 2879 – 2893

reactants, velocity at which reactants are convected, diffusion coefficients and mass transfer coefficients, estimate the final concentrations of the reactants. We know all the parameters except mass transfer coefficients of the two reactants. Hence, we first solve the forward problem (Eqs. (14–17)) for an arbitrary guess of the mass transfer coefficients in FEMLAB (Zimmerman, 2004) and then MATLAB m-file function is generated for the same. We obtain the final concentrations of the reactants A and B for the guessed mass transfer coefficients, after solving the forward problem. • From the experimental protocol, we evaluate the final concentration of the reactants from the absorbance of the reaction mixture. We write a MATLAB script file to evaluate the error, E (Eq. (19)), between the model predicted final concentration and the experimentally measured final concentration of the reactants. • We then use a MATLAB in-built optimization (Eq. (20)) function called “fminsearch” that iteratively calls the error script file and the script file for the forward problem until the desired tolerance (which is defined as 10−4 in the present study) is achieved. • The above methodology can be applied using different initial guesses to check uniqueness of the estimated mass transfer coefficients. 5.1.1. Potential of the inverse methodology Functionally, the inverse problem is well posed if the operator f in Eq. (18) is a bijection. The inverse problem so described defines f −1 such that
 (m) (m) (
A ,
B ) = f −1 CA,f , CB,f ; CA,0 , CA,0 . (21) To check the accuracy of the proposed inverse methodology and the above dependence (Eq. (21)), we performed numerical experiments for known mass transfer coefficients of the two reactants. We assume the initial concentration of nicotinic acid and bromophenol blue to be equal to 1 and 0.4 M, respectively, and the
NA a of nicotinic acid and the
BP a of bromophenol blue to be equal to 0.2 and 1 s−1 , respectively. The forward problem is solved for the known initial concentrations and mass transfer coefficients in FEMLAB and the final concentration of nicotinic acid and bromophenol blue are found to be equal to 0.672 and 0.0803 M, respectively. We then check the accuracy of the proposed methodology by solving the inverse problem for the known initial and final concentration of the two reactants, for various guesses of the mass transfer coefficients, and the results obtained are reported in Table 8. The mass transfer coefficients evaluated using the inverse methodology are found to be exactly the same as those used for the forward problem. Different initial guesses are found to converge to the same solution indicating that only one solution exists for the given problem. As the inverse methodology is found to be very accurate for the model problem, we extend the same to study mass transfer limited characteris-

Table 8 The accuracy of the inverse problem is checked by implementing the methodology for known concentrations and mass transfer coefficients of nicotinic acid and bromphenol blue Guessed
NA a (s−1 )

Guessed
BP a (s−1 )

Predicted
NA a (s−1 )

Predicted
BP a (s−1 )

Error E

0.001 0.01 0.1 0.2 0.3 0.5

0.001 0.05 1 1 2 2

0.2001 0.2000 0.2001 0.2001 0.2001 0.2001

0.9998 0.9999 0.9998 0.9998 0.9998 0.9998

4.3135e−5 6.7103e−5 3.2873e−5 3.2077e−5 2.6020e−5 2.7278e−5

The error calculated for the various guesses is also reported.

tics of transport limited reaction in a tubular reactor, in order to explore the possible existence of cross-over phenomenon in our next communication. 6. Conclusion One of the most important tasks in the present experimental study of mass transfer limited reactions is to select a reaction where the two reactants are dissolved in the continuous phase, but are not reacting with each other in the continuous phase. Instead, they diffuse into the dispersed phase and react therein. We explored various possible reactions and chose the most appropriate reaction satisfying all the requirements including limitations in spectroscopic analysis. The reaction mixture containing nicotinic acid and bromophenol blue dissolved in 1-chlorobutane is chosen to study transport limited characteristics in two phase flow. We have discussed the principle of additivity to obtain the concentration of the individual reactants from the absorbance spectrum of the reaction mixture in multicomponent spectrum analysis section. The principle of linear additivity is not valid when there is interaction between the two reactants. We proposed the technique of non-linear additivity to extract the concentration of the two reactants from the absorbance spectrum of the reaction mixture, when there is a considerable interaction between the two species. The accuracy of the non-linear additivity is checked by measuring the absorbance spectrum of the reaction mixture with known concentrations of the reactants. Non-linear additivity is found to be a useful way of extracting concentration of the individual species from the reaction mixture. Mass transfer limited reactions are theoretically well studied in the literature. Fogler (1992) has reported a model to study transport limited reactions in a packed bed. Since there is kinetic asymmetry in the transport rates of the two reactants and hence, the potential cross-over in dispersed phase concentration of the reactants in the present work, the model proposed by Fogler is not applicable here. We proposed an inverse methodology to evaluate the mass transfer coefficients of the two reactants with asymmetric transport

Kiran B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 60 (2005) 2879 – 2893

rates from experimentally measured initial and final concentration of the reactants. The mass transfer coefficients evaluated are validated by solving the forward problem and inverse problem for various different initial guesses which gives a unique solution illustrating the robustness of the inverse methodology. A system with asymmetric mass transfer coefficients potentially indicates the existence of cross-over phenomenon which helps in optimizing the length of the tubular reactor and is reported in a separate communication. Acknowledgements KBD would like to thank Dr. M. Pitt, Dr. P. Styring, Dr. D. Brown and Prof. A.E. Wraith for very useful discussions. WBZ would like to thank the EPSRC (GR/A01435, GR/S67845, GR/R72754 and GR/N20676) for financial support. References Astarita, G., 1967. Mass Transfer with Chemical Reaction. Elsevier Publishing Company, Amsterdam. Buckley, C., Hollingsworth, K.G., Sederman, A.J., Holland, D.J., Johns, M.L., Gladden, L.F., 2003. Applications of fast diffusion measurement using difftrain. Journal of Magnetic Resonance 161, 112–117. Chen, J., Li, F., Degaleesan, S., Gupta, P., Al-Dahhan, M.H., Dudukovic, M.P., Tosel, B.A., 2001. Fluid dynamic parameter in bubble columns with internals. Chemical Engineering Science 54, 2187–2197. Dehmlow, E.V., Dehmlow, S.S., 1993. Phase Transfer Catalysis. VCH, Weinheim. Deshpande, K.B., Zimmerman, W.B., 2004. Esperimental study of mass transfer limited reaction: Part II: Existence of cross-over phenomenon. Chemical Enginerring Science, in press. 10.1016/j.ces.2005.01.034.

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