Hydrodynamics and residence time distribution of liquid flow in tubular reactors equipped with screen-type static mixers

Chemical Engineering Journal 279 (2015) 948–963

Contents lists available at ScienceDirect

Chemical Engineering Journal journal homepage: www.elsevier.com/locate/cej

Hydrodynamics and residence time distribution of liquid flow in tubular reactors equipped with screen-type static mixers K. Abou Hweij, F. Azizi ⇑ Department of Chemical and Petroleum Engineering, Faculty of Engineering and Architecture, American University of Beirut, Lebanon

h i g h l i g h t s  Pressure drop and RTD were investigated in reactors equipped with screen mixers.  Effect of number and geometry of mixers, and operating conditions were studied.  Less energy is required to drive the flow when compared to other motionless mixers.  Axial dispersion was always minimized by the presence of screen mixers.  Choosing the proper characteristic length to calculate Re is very important.

a r t i c l e

i n f o

Article history: Received 24 December 2014 Received in revised form 24 April 2015 Accepted 26 May 2015 Available online 2 June 2015 Keywords: Static mixers Residence time distribution Axial dispersion Hydrodynamics Macromixing Deconvolution

a b s t r a c t This paper discusses the characteristics of liquid flow through tubular reactors/contactors equipped with woven mesh, screen-type static mixers from hydrodynamic and macromixing perspectives. The effect of changing the screen geometry, number of mixing elements, reactor configuration, and the operating conditions, were investigated by using four different screen types of varying mesh numbers. Pressure drop was measured over a wide range of flow rates (2300 6 Re 6 21,500) and was found to increase with a decreasing mesh opening. Friction factor values are also reported in the work, when compared to other types of motionless mixers, screen-type mixers were found to require much lower energy requirements with very low recorded Z values (1.15 6 Z 6 5) that are two to three orders of magnitude lower than those reported for other motionless mixers. Furthermore, residence time distribution experiments were conducted in the transitional and turbulent regimes (2300 6 Re 6 11,500). Using a deconvolution technique the RTD function was extracted in order to quantify the axial/longitudinal dispersion. The findings highlight that regardless of the number and geometry of the mixer, reactor configuration, and/or operating conditions, axial dispersion coefficients that are lower than those of an empty pipe were always recorded. The wire diameter and mesh opening were found to directly affect the axial dispersion in the reactor, while, the number of elements showed a minor effect. Furthermore, the choice of the characteristic parameter used to calculate the Reynolds number was found to be of paramount importance in analyzing the data. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction Multiphase flow phenomena play an important role in the chemical and process industries. Such processes include unit operations carried out in batch and continuous stirred tanks, bubble and liquid–liquid extraction columns, as well as airlift reactors, amongst others. Optimizing the performance, economy and safety of these multiphase reactors/contactors is therefore of paramount ⇑ Corresponding author. Tel.: +961 1 374374x3439; fax: +961 1 744462. E-mail address: [email protected] (F. Azizi). http://dx.doi.org/10.1016/j.cej.2015.05.100 1385-8947/Ó 2015 Elsevier B.V. All rights reserved.

importance. A proper understanding of the mixing process, if attainable, combined with an ability to predict the dynamic properties of the dispersion, such as drop/bubble size distributions and mass transfer performance, would therefore allow the introduction and implementation of major enhancements. However, due to the complex hydrodynamic conditions prevalent in most commercially available contactors/reactors, such achievements remain unreachable. Consequently, designing these units remains heavily dependent on the employment of empirical knowledge, experience, and extensive pilot-scale testing. Following the recent advances in computational fluid dynamics (CFD), the use of CFD as a design

K. Abou Hweij, F. Azizi / Chemical Engineering Journal 279 (2015) 948–963

949

Nomenclature Symbol b C(t) Cin(t) Cout(t) e ðx) C COV D Dax E(t) f fi k L M MSE N Ne p Pe

Description (unit) screen wire diameter (m) normalized concentration of tracer (–) concentration of tracer measured at the inlet (mol/L) concentration of tracer measured at the inlet (mol/L) FFT of the sampled inlet concentration (–) coefficient of variation (–) empty pipe diameter (m) axial dispersion coefficient (cm2/s) residence time distribution function (–) fanning friction factor (–) fanning friction factor calculated based on the screen open area (–) total number of sampled points (–) length of the mixing zone (m) screen mesh opening (m) mean squared error (–) number of screen elements in the system (–) Newton number (–) fitting exponent (–) Péclet number (–)

tool to predict the performance and/or suggest alternative designs for multiphase reactor/contactor has been gaining strong momentum. The validity of this approach and its accuracy remains however the subject of a plethora of investigations in the open literature. Nonetheless, plug flow reactors/contactors serve as a better choice when compared to other multiphase contactors/reactors in order to understand the complex phenomena taking place as well as providing better performance and control over mixing, breaking of drops and bubbles, as well as temperature [1–4]. The advantages of these conventional reactors lies however in their longer and controllable residence times as well as larger versatility for accommodating a variety of processing fluids. However, the growing interest in the use of tubular reactors equipped with static mixers, over conventional mixers, emanates from their inherent advantages whereby better performance can be achieved at lower capital and operating costs. Findings in the literature highlight the higher multiphase mass transfer and reaction rates that could be achieved in energy efficient manners while simultaneously handling large flow rates and achieving high heat removal or addition rates [1,3,5,6]. Furthermore, the insertion of properly-selected static mixing elements into tubular reactors allows for the introduction of the various reactants at different points along the reactor length, thereby facilitating the achievement of optimal temperature and reactant concentration profiles that are required to achieve optimal selectivity and conversion. Woven mesh screen-type static mixers have been used to repetitively superimpose an adjustable, radially-uniform, highly-turbulent field on the nearly plug flow conditions encountered in high velocity pipe flows [7]. The very high energy dissipation rates present in the thin region adjacent to the screen are particularly effective in processing multiphase systems. This not only helps in the formation of fine dispersed phase entities (bubbles and/or drops) but also considerably enhances the value of the interphase mass transfer coefficient [6–9]. This is evidenced by their ability to promote contact between different phases, where interfacial areas as high as 2200 m2/m3 [10], and volumetric mass transfer coefficients, kLa, as high as 4.1 s1 [11] could be achieved in the case of gas–liquid systems. In processing liquid– liquid systems, kLa values as high as 13 s1 [8] were attained which

Q Re Reb Rejet ReM e Sðx) t tm U V z

volumetric flow rate (L/min) empty pipe Reynolds number (–) wire Reynolds number (–) macroscopic jet Reynolds number (–) mesh Reynolds number (–) FFT of the smoothing formula (–) time (s) mean residence time calculated from RTD (s) superficial flow velocity (m/s) volume of the mixing zone (L) axial position in the pipe (m)

Greek symbols a fraction open area of the screen (–) c fitting parameter (–) DP pressure drop in the mixing zone (kPa) H dimensionless time (–) l viscosity (kg/m s) q density (kg/m3) 2 r variance (s2) s theoretical residence time in the mixing zone (s)

enabled for 99% of equilibrium conditions be achieved in less than 1 s. Additionally, narrow and easy-to-separate drop size distributions with mean diameters in the order of 40 lm have also been reported [7,12]. This can be mainly attributed to the impact that high-intensity microscale turbulence, typically encountered in this mixer configuration, can have on the mass transfer coefficient [13]. In other terms, such good performance of screen-type static mixers, which is attributed to not only the formation of very fine dispersed phase entities but also enhanced mass transfer coefficients, is credited to the very high grid-generated turbulence and the consequent elevated micro-mixing intensities generated in the regions adjacent to the screens [14]. Similarly to other static mixers, screen-type mixers also offer the flexibility of designing the reactor to meet various mixing and/or energy requirements. For example, whereas a short inter-screen spacing favors the production of fine dispersions that are typical of high mass transfer rates and fast reactions, a longer spacing would be favorable for conditions of slow reactions and/or low energy requirements [6]. However, changing the reactor configuration and/or the operating conditions will impact the performance of the reactor by affecting its mixing efficiency and consequently the characteristics of the flowing dispersed phase along its length. A frequently used technique to understand and quantify the actual flow phenomena in reactors/contactors is the residence time theory. This concept has been in place for over a hundred years [15] and became widely used after the work of Danckwerts in the early 1950’s [15,16]. Residence time distribution (RTD) is an indicator of the macro-mixing in the reactor as it measures features of ideal or non-ideal flows associated with bulk flow patterns, and knowledge concerning its characteristics would therefore offer the ability to adapt the reactor/contactor design to meet specific process requirements. RTD in static mixers, e.g. SMX, Kenics, etc. . ., was studied experimentally by several investigators for both Newtonian and non-Newtonian fluids [4,17–24]. The majority of these studies focused on measurements in the laminar flow regime and reported the data in the form of the RTD function or one of its moments. Keshav et al. [17] studied the RTD of the liquid phase in gas–liquid flow in Kenics mixer for air/Newtonian and air/non-Newtonian

950

K. Abou Hweij, F. Azizi / Chemical Engineering Journal 279 (2015) 948–963

systems. They reported that non-Newtonian fluids exhibit a lower axial dispersion compared to Newtonian fluids, which further decreases as the power-law index decreases. The axial dispersion also decreased with an increase in the liquid velocity (at constant gas flow rate) but increased with an increase in the pipe diameter. Wadley and Dawson [18] investigated the blending in three different static mixers, namely, SMV, Chemineer KM and HEV mixers in the transitional and turbulent flow regimes using a laser-induced fluorescence technique. After analyzing the mixture quality by means of the coefficient of variation, they reported results that contradicted the correlations produced by the manufacturers. Nauman [4] presented a method for analyzing the performance of static mixers as chemical reactors from RTD data and concluded that four Kenics elements can be treated as one stage of complete mixing. Pustelnik [19] studied the RTD of laminar Newtonian and non-Newtonian flow through Kenics mixers. His work focused on the effect of varying the number of mixers and the Reynolds number on the RTD function, and concluded that the former parameter has a much more pronounced effect on the axial dispersion when compared to the effect of Re in the laminar flow range. Kemblowski and Pustelnik [20] proposed a model for the RTD in Kenics mixers during laminar flow of a power-law fluid which takes into account the flattening of the velocity distribution after every Kenics element as well as longitudinal dispersion, and found good agreement between their model and their experimental data. In line with the findings of Pustelnik [19], they found that the number of Kenics elements has the most significant effect on the RTD. Using RTD measurements to validate their work, Nauman et al. [21] proposed a new class of static mixers intended to homogenize feed streams subject to variations in concentration or temperature and hence were after an increased axial and radial dispersions. Their design suggested the use of open tubes in series with static mixing elements to deliver homogeneous mixtures. Madhuranthakam et al. [22] conducted RTD experiments in a 24-element Kenics KMX reactor in a gas-polymer solution flow. Their experiments were conducted in the laminar regime and they observed an increase in the axial dispersion with the introduction of gas into the reactor while in single phase experiments, an increase in the viscosity also increased the dispersion. Recently, Rafiee et al. [23] measured the RTD in a high viscosity non-Newtonian fluid in a Kenics mixer by means of a Positron Emission Particle Tracking (PEPT) technique. Using this method, flow visualization was possible and the data was interpreted in terms of the residence time in the reactor. They observed a larger average velocity by going toward the center of the mixer, hence a lesser than average residence time. Ziolkowski and Morawski [24] studied the RTD in various types of motionless mixers including wire gauze or screen-type static mixers. Their study considered low Reynolds numbers only (200 < Re < 5,000) as well as two different geometries of screen mixers, a small and a very wide mesh opening ((M  b) = 0.5 and 3 mm, b = 0.3 and 1 mm, respectively). They analyzed their data by fitting the measured concentration profiles using the well-known single-parameter axial dispersion model without attempting to find the RTD function, and concluded that this method can only be employed over a very short range of Re and therefore proposed a two-parameter model to fit the concentration profiles instead. This proposed model accounted for the velocity profile in the stream cross-section as well as a modified axial dispersion coefficient. The pitfall of this work is that the data is not easily interpretable because the reported axial dispersion coefficient does not correspond to the well-known coefficient in the axial dispersion model; rather, it is one factor of a two-parameter model. Consequently, this investigation deals with characterizing the macro-mixing in tubular reactors/contactors equipped with screen-type static mixers. This will be achieved by measuring the

residence time distribution (RTD) function in the reactor under various operating conditions and geometrical/design configurations in order to quantify the axial dispersion coefficient. In addition, the effect of varying the various operational parameters on the pressure drop of the reactor/contactor will also be reported and discussed. To the authors’ knowledge, very few studies [18] measured the RTD in turbulent flows in static mixers. Further, this will be the first study that attempts to measure the RTD function at large Re values in screen-type static mixers, and the reported results are therefore expected to validate the assumption of plug flow at high flow velocities. This investigation will also highlight the importance of agreeing on the right choice of the characteristic parameters to calculate dimensionless numbers (i.e. Reynolds number) and will clearly show the pitfalls of wrongly choosing them. 2. Experimental work The residence time distribution is determined experimentally by injecting an inert chemical, molecules, or atoms into the reactor and measuring its concentration in the effluent stream as a function of time using a suitable measurement tool. For this purpose, dyes, salts, and radioactive elements amongst others, have commonly been used to measure the residence time in a variety of reactors/contactors by means of measuring light intensity, conductivity and radioactivity, respectively. In addition, there exist several techniques of injection that can be used to measure the RTD, namely, pulse, step, and imperfect pulse, each of which has its advantages, disadvantages and a different method of analysis to extract the desired information from measurements. In the current study, a pulse injection of a highly concentrated aqueous salt solution was used to measure the RTD in the reactor/contactor by means of measuring the changes in electrical conductivity in reverse osmosis (RO) water of low initial conductivity. However, to mitigate the problems that might arise from entrance effects to the mixing section and the assumption of a perfect pulse input, the analysis was performed assuming an imperfect pulse technique. This was accomplished by measuring the electrical conductivity at both entrance and exit to the mixing section. The following section therefore describes the experimental setup used in this study in addition to the various experimental techniques employed for measuring RTD. Moreover, the method of analysis will also be detailed. 2.1. Experimental setup The continuous flow experimental setup used in the present investigation is schematically illustrated in Fig. 1. Reverse osmosis (RO) water was collected in a 500 L tank from which it was fed to the static mixer loop using a constant speed centrifugal pump (Pedrollo, model PKm80). The manually regulated flow rate of water entering the mixing section was measured by means of a digital flow meter (Omega model: FP7002A) and was varied between 4 and 20 L/min for the RTD experiments (which corresponds to pipe Reynolds numbers ranging from 2300 to 11,500). The reactor section consists of a 855 mm vertical transparent Plexiglas pipe (50.1 mm OD and 44.1 mm ID) in which the mixing section is 600 mm long. The reactor section was flanged to the setup and placed 700 mm downstream from the liquid entrance to ensure radial flow uniformity. Commercially available stainless steel (type 304) screen-type static mixing elements (Universal Wire Cloth Co., Morrisville, PA, USA), the characteristics of which are given in Table 1 (and their geometry represented in Fig. 2), were held apart using a set of transparent Plexiglas cylindrical spacers (44 mm OD and 37 mm

951

K. Abou Hweij, F. Azizi / Chemical Engineering Journal 279 (2015) 948–963

Fig. 1. Schematic representation of the experimental setup (1) supply/drain tank, (2) centrifugal pump, (3) gate valve, (4) ball valve, (5) flow meter, (6) pressure sensors, (7) conductivity sensors, (8) injection port, (9) check valve.

Table 1 Characteristics of the investigated stainless steel plain-weave wire cloth. Mesh number

M  103 (m)

b  103 (m)

(M  b)  103 (m)

a (%)

20 30 50 80

1.27 0.8382 0.508 0.3175

0.4064 0.3048 0.2286 0.1397

0.8636 0.5334 0.2794 0.1778

46.2 40.8 30.3 31.4

ID). Forty (40) spacers of 20 mm length were used to ensure that the screens remained perpendicular across the flow direction and helped maintain the inter-screen spacing at the desired value by retaining them inside a larger outer pipe whose ID is 1 mm larger than the outer diameter of the spacers (Fig. 3). The effective diameter of the mixing section which contained a varying number of static elements is thus 37 mm with a total length of 600 mm. It should be noted that the numerical value of the mesh number (cf. Table 1) indicates the number of meshes per linear inch (i.e. mesh number = 25.4 (mm)/M (mm)). The setup was provided with an injection valve through which the tracer pulse was injected, and two pressure transducers

675 mm

855 mm

Injection Valve 75 mm Fig. 3. Internal configuration of the reactor section.

Fig. 2. Geometry of the woven screen.

(Omega model PX302-015G) that were fixed at the top and bottom of the reactor section, 855 mm apart. Furthermore, conductivity measurements were taken at the inlet and outlet to the mixing section by means of two identical conductivity sensors (Omega model CDCE 90-X) placed at 75 and 675 mm from the injection point, respectively. The measurement cells of these conductivity sensors were aligned with the centerline of the pipe. Data from all sensors was collected at a frequency of 10 Hz by means of a National Instruments data acquisition board (NI USB-6218) and a specially developed LabView program. In addition to studying the effect of varying screen geometries (as highlighted in Table 1) on the RTD, experiments were also conducted while varying the number of screen elements. Measurements were acquired in the presence of 1, 2, 4, and 8 screens inside the mixing section. Consequently, and based on the number of screens in the system, Ns, the inter-screen spacing, L, was changed to ensure a proper distribution of the screens over the entire mixing section volume while keeping this distance independent of the screen geometry. Whereas an inter-screen spacing of 300 mm was used in the presence of two screens, the distance was reduced to 120 mm for 4 screens and further decreased to

952

K. Abou Hweij, F. Azizi / Chemical Engineering Journal 279 (2015) 948–963

3.1. RTD analysis

Table 2 Operating conditions for RTD experiments. Q (L/min)

U (m/s)

Re (–)

4 6 8 10 12 15 20

0.062 0.093 0.124 0.155 0.186 0.232 0.310

2300 3450 4600 5750 6900 8600 11,500

60 mm in the presence of 8 screens. Moreover, the first screen element was always held at 60 mm above the inlet conductivity sensor. The various flow rates and the corresponding flow velocity and Reynolds number at which the RTD experiments were performed are detailed in Table 2. It should be noted, that in the case of pressure drop experiments, data was collected with 8 screen-type static mixing elements in the system and the flow rate in the reactor unit was varied over a larger span ranging from 4 to 40 L/min, which corresponds to Re values between 2300 and 23,000. Such discrepancy between the flow rates depending on the type of experiment (RTD vs. pressure drop) was dictated by the inability of the conductivity sensors to detect the variation in the conductivity due to a pulse injection at higher velocities. 2.2. Experimental method To measure the residence time distribution inside the reactor/contactor, electrical conductivity was measured at both the inlet and outlet of the mixing section after imposing a pulse injection of a highly conductive solution in the low conductivity RO water (200 lS) used in all experiments. To ensure that no accumulation of salt occurs, which would subsequently impose an increase in the initial conductivity of the water, RO water was drawn from one clean supply tank and the outlet of the mixing section was directly drained or collected in a separate tank. Because of the linear relationship between its concentration and its electrical conductivity, an aqueous solution of potassium chloride, KCl, was used as the tracer in all experiments. A stock solution of 100 g/L corresponding to 1.34 mol/L was prepared using deionized water, from which 5.8 mL were repetitively injected in the system as a pulse disturbance. This volume was kept constant throughout the experiments by using the same injection syringe, and all injections were performed in less than 300 ms. Depending on the operating conditions, this meant the maximum conductivity would range between 2000 and 3000 lS, which is substantially larger than the base conductivity of the water. 3. Methodology In the current work, the effect of varying the number of screens (1, 2, 4, and 8 screens) and screen geometry (four different geometries related together by a geometric progression, mesh size varying by a factor of 1.6) is investigated. In addition, conducting the RTD experiments in a Re range (2300 6 Re 6 11,500) that is rarely explored was undertaken in this study. Furthermore, in lieu of fitting the experimentally measured concentration profiles with a single parameter model and/or assuming a perfect pulse injection, the current work will attempt to extract the E-function from the inlet and outlet concentration profiles by assuming an imperfect pulse experiment. Such approach helps in better accounting for the flow non-idealities and offering a better alternative to the assumption of a perfect Dirac delta pulse injection, and/or not accounting for the RTD function in the analysis.

Residence time theory is a well-established, mature technology that deals with measuring the time fluid particles spend inside the reactor vessel boundaries. Except for the special case of plug flow, velocity profiles and molecular diffusion will cause a distribution of residence times [15]. Typically, measuring the residence time distribution from a pulse injection, requires the monitoring of the outlet concentration of the tracer in order to obtain the exit-age distribution function, E(t), which is defined in Eq. (1).

CðtÞ EðtÞ ¼ R 1 CðtÞdt 0

ð1Þ

In other terms, E(t) is a quantitative measure of how much time different fluid elements spend in the reactor. Therefore mass continuity dictates that the E-function should obey the following constraint depicted in Eq. (2).

Z

1

EðtÞdt ¼ 1

ð2Þ

0

Furthermore, it is common to compare and analyze RTDs by means of their moments instead of their entire distributions, and typically the first two moments of the distribution are used. The first moment is the mean residence time and can be used as an indicator on whether there exist dead volumes or short-circuiting in the reactor. This is obtained by comparing the first moment of the distribution, tm, defined in Eq. (3), with the theoretical residence time, s, that is calculated as the ratio of the reactor volume to the volumetric flow rate ðs ¼ V=Q Þ.

tm ¼

Z

1

tEðtÞdt

ð3Þ

0

The second moment of the distribution is another important parameter that indicates the ‘‘spread’’ of the distribution and is calculated using Eq. (4).

r2 ¼

Z

1

ðt  tm Þ2 EðtÞdt ¼

0

Z

1

t2 EðtÞdt  t2m

0

ð4Þ

Another important parameter is the coefficient of variation (COV) which reflects a combination of the first and second moments of the distribution [3,25]. The coefficient of variation, defined in Eq. (5), is the ratio of the standard deviation to the mean and provides a measure of the relative spread of a distribution.

COV ¼

r

ð5Þ

tm

A complete distributive mixing is indicated by a value of COV = 0, while a COV = 1 is a sign of total segregation [3]. Even though care was always taken while performing the pulse experiments, in order to eliminate all sources of error associated with the assumption of a perfect pulse, or flow disturbances while entering the reactor/mixing section, it was decided to use the imperfect pulse technique to analyze the residence time distribution. In this approach, the injected pulse is not a perfect Dirac distribution rather it has a certain spread. Therefore, by measuring the inlet and outlet conductivities, and consequently the inlet and outlet concentrations of the tracer, the residence time distribution function can be obtained by the deconvolution of the inlet and outlet concentrations which are related by the convolution integral shown in Eq. (6).

C out ðtÞ ¼

Z

t

C in ðt0 Þ  Eðt  t0 Þdt

0

ð6Þ

0

3.2. Deconvolution method Several authors studied the numerical solution of the deconvolution integral [26–38] and several techniques are available in the

K. Abou Hweij, F. Azizi / Chemical Engineering Journal 279 (2015) 948–963

open literature for solving Eq. (6) and extracting the impulse response or residence time distribution function, E(t). A majority of these techniques rely on converting time domain measurements to the frequency or Laplace domains in order to extract E(t). For example, Mills and Dudukovic [26] presented two methods for the evaluation of model parameters from nonideal tracer responses based on matching the experimental and model predicted responses in the time domain. One of the methods (described in details, below) requires the time domain measurements to be converted to the frequency domain, where a filter to reduce the noise in the data is applied before reconverting the frequency domain E-function to its time domain equivalent. Their second method performs the calculation directly in the time domain however it requires a linear flow model to obtain the impulse response. Mao et al. [27] also presented an approach for the deconvolution by Fast Fourier Transform (FFT). In a fashion similar to the work of Mills and Dudukovic [26], their method requires a smoothing filter in the frequency and time domains. Nahman and Guillaume [28] studied both frequency and time domain methods for the deconvolution and concluded that the frequency domain method implemented via FFT is by far superior to the time domain methods. Furthermore, they proposed a simple regularization filter for offsetting the effects of noise which uses a low-pass filter. Bennia and Nahman [29], later modified the automatic deconvolution method proposed by Nahman and Guillaume [28] to ensure that causality (real and imaginary parts of a function to be related) is maintained in the deconvolution result. Recently, Hutter et al. [30] utilized the deconvolution filter for smoothing data as proposed by Nahman and Guillaume [28], however they modified it by adding a phase component as proposed by Bennia and Nahman [29]. The investigators were able to successfully solve the deconvolution problem and obtained results that corresponded to values available in the literature. Parruck and Riad [31] also studied the deconvolution by evaluating the performance of two filters, namely, the Nahman-Guillaume and the optimum compensation iterative frequency-domain technique over a wide range of signals, and concluded that each method is suitable for a specific case. The former technique was found to be more accurate, however, more sensitive to noise whereas the latter can tolerate higher levels of noise. Pananakis and Abel [32] solved the deconvolution problem for isothermal reaction rate using several methods, namely, time domain, frequency domain, optimized frequency domain, and iterative methods, and they concluded that the constrained iterative method in conjunction with a low pass noise filter, proved to be the most suitable. Other proposed methods to solve for E(t) in Eq. (6) are performed in the time domain directly, and rely on employing a flow model to describe the RTD function and then search for its parameters by fitting predictions to experimental measurements. For example, Boskovic and Loebbecke [33] proposed an empirical model for laminar flow regime in micromixers and the task of deconvolution is reduced to time-domain fitting to find the best parameter that matches predicted to experimental results. Similarly, Hafeli et al. [37] performed time domain fitting to find the E-function for flow through porous structures by using the well-known axial dispersion model as representative of the flow. Essadki et al. [34] solved the deconvolution problem in the frequency domain, and after converting the E-function to the time domain, it was fitted to known reactor models to find the axial dispersion coefficient or the Péclet number. However, no sufficient information on which solution method was employed, is reported in the work. Gutierrez et al. [35] solved Eq. (6) by performing a time domain discretization of the integral and by assuming that E(t) follows a known-model. Furthermore, Heibel et al. [36] employed the time domain deconvolution using a two-variable partial differential equation model coupled with a parameter

953

optimization along the lines proposed by Mills and Dudukovic [26] for their time-domain deconvolution technique. As discussed above, most of the commonly accepted and mature solution techniques call for solving the deconvolution problem by using Fourier transforms after normalizing the concentration data, because convolution in the time domain corresponds to multiplication in the frequency or Laplace domain [26,27]. Typically, fast Fourier transforms can be used in lieu of discrete Fourier transforms. Therefore, by switching from time- to frequency-domain, the convolution integral using normalized concentrations becomes,

e out ðxÞ FFTðC out ðtÞÞ ¼ FFTðC in ðtÞÞ  FFTðEðtÞÞ () C e in ðxÞ  e EðxÞ ¼C

ð7Þ

and E(t) can be obtained by calculating the inverse transform (IFFT) of the ratio of the Fourier transforms of the normalized outlet and inlet concentrations, as shown in Eq. (8).

!   e out ðxÞ FFTðC out ðtÞÞ C ¼ IFFT EðtÞ ¼ IFFT e in ðxÞ FFTðC in ðtÞÞ C

ð8Þ

Due to measurement noise, many high order harmonics will be created and consequently unrealistic oscillations of high frequency, and negative numerical values at many instants, will appear in the calculated E(t). These two conditions, however, bear no physical meaning [27]. To mitigate these problems, several investigators have therefore implemented noise filtering techniques [26,28,29,31,39]. In the current work, the filtering method in the frequency domain as proposed by Mills and Dudukovic [26], and which relies on the method of regularization that was refined by Hunt [39,40] [27], will be employed. For details concerning the mathematical derivation, the reader is referred to the works of Hunt [39,40] and Mills and Dudukovic [26]. In this approach a second-order smoothing filter was employed to remove noise from measured data according to Eq. (9)

e out ðxÞ= C e in ðxÞ C e EðxÞ ¼ eSðxÞeS  ðxÞ 1þc eC in ðxÞeC  ðxÞ

ð9Þ

in

where, c is the filtering parameter, and the term e in ðxÞ  C e  ðxÞ plays the role of a digital filter. ½c  e SðxÞ  e S  ðxÞ= C in e SðxÞ is the FFT of the smoothing formula coefficient series [1,2,1,0,. . .,0]; which is a matrix of length k, the number of sampling points. The quantities marked with an asterisk, Cin* and S* denote the complex conjugate of the inlet concentration and the smoothing formula series, respectively [26,27,39,40]. The length of the matrix, k, is equivalent to the number of sampled points in each experimental run. While the filtering parameter is unbounded and can assume any positive real value ðc 2 Rþ Þ the choice of c should be such that it only removes the frequency components of the errors and prevent over-smoothing by deleting essential frequency components of the solution [26]. The authors [26] proposed using the additivity property for the moments of the normalized tracer impulse responses [41], in order to identify c. This method relies on calculating the nth absolute moment using two methods. The first calculates it as the difference between the nth moment for the sampling and injection responses, whereas the second computes the nth moment directly from the estimated impulse response distribution. The appropriate value of the filtering parameter, c, is therefore the one that minimizes the error between the two to below a prescribed error. However, this requires a preliminary estimate of c, which can be obtained by solving a non-linear equation that relates

954

K. Abou Hweij, F. Azizi / Chemical Engineering Journal 279 (2015) 948–963

8

25

(a)

30-Mesh, 8 screens 30-Mesh, 4 screens 30-Mesh, 2 screens

7

20

6 5

ΔP (kPa)

ΔP (kPa)

80-Mesh, 8 screens 50-Mesh, 8 screens 30-Mesh, 8 screens 20-Mesh, 8 screens No screens

4 3 2

(b)

15

10

5

1 0

0 0

5000

10000

15000

20000

25000

0

5000

10000

Re (-)

15000

20000

25000

Re (-)

Fig. 4. Pressure drop across the mixer length, (a) effect of number of screen elements; (b) effect of screen geometry.

3.3. Models for non-ideal reactors

3

To further characterize the performance of tubular reactors/contactors equipped with screen-type static mixers by means of the residence time theory, the commonly used one parameter model, namely the axial dispersion model was employed in this investigation. This model is used when small deviations from ideal plug flow exist, and under such conditions, it should render comparable residence time distribution function, E(t) to the other commonly used tanks-in-series model [41,42]. The theory and mathematics behind this model is mature and thoroughly explained in reactor design textbooks. Consequently it will only be briefly presented here.

ΔP, predicted (kPa)

2.5

2

1.5

1

0.5

0

0

0.5

1

1.5

2

2.5

3

ΔP, experimental (kPa) Fig. 5. Parity plot of the correlated versus the experimental pressure drop for all screen geometries.

the mean squared error (MSE) between the approximate and exact solutions for the impulse response as given by Hunt [39] and shown in Eq. (10).

MSE ffi k 

k1 X i¼0



e out ðiÞ  C e  ðiÞ C out 1þc

1

eC in ðiÞeC  ðiÞ in eSðiÞeS  ðiÞ

2

ð10Þ

If an estimate of the MSE can be made, then a 1-D root finding method will permit a preliminary estimate of the filtering parameter c. In the current work, MatLab was used to perform these calculations after implementing a user defined function that would solve the equations to first obtain the value of the fitting parameter, c, and consequently the impulse response, E(t). Once the latter function is known, numerical integration was used to calculate the various moments of the distribution and perform further characterization of the reactor.

3.3.1. Axial dispersion model This model considers the flow inside the reactor to be plug flow on top of which is superimposed a certain degree of backmixing, the magnitude of which is independent of position in the vessel [41]. Because the material flowing inside the reactor/contactor is continuously redistributed by means of eddies, these disturbances can be calculated by analogy to molecular diffusion and can be written as shown in Eq. (11),

@C @ 2 C @ðUCÞ ¼ Dax  2  @t @z @z

ð11Þ

where C is the concentration of the tracer and Dax, is the longitudinal or axial dispersion coefficient, which characterizes the degree of backmixing in the reactor [41,42]. The solution of this model relies on the boundary conditions and involves the dimensionless Peclet number, Pe; which for tube flow is defined as Pe ¼ L  U=Dax , where L is the mixing chamber length, and U is the mean flow velocity. Because the tracer injection will be treated as an imperfect pulse with a certain distribution when entering the mixing section, the uniformly open system boundary conditions will be assumed in the current study. Under such conditions, the solution to Eq. (11) assumes the following form for the residence time distribution function [15,42],

1 EðHÞ ¼ sEðtÞ ¼  2

! rffiffiffiffiffiffiffiffiffiffiffi Pe Pe  ð1  HÞ2  exp  pH 4H

ð12Þ

where H is the dimensionless time, H = t/s. The mean residence time of this distribution is given as tm ¼ s  ð1 þ 2=PeÞ. However, the value of Pe can be calculated by

955

K. Abou Hweij, F. Azizi / Chemical Engineering Journal 279 (2015) 948–963

10

80-Mesh, 8 screens 50-Mesh, 8 screens 30-Mesh, 8 screens 20-Mesh, 8 screens No screens

10

80-Mesh, 8 screens 50-Mesh, 8 screens 30-Mesh, 8 screens 20-Mesh, 8 screens No screens

fi /2 (-)

f/2 (-)

1

1

0.1

(a)

(b)

0.1 1000.00

0.01 1000

10000.00

10000

100000

Rejet (-)

Re (-)

Fig. 6. Effect of (a) Reynolds number on the friction factor, and (b) Rejet on the interstitial friction factor, fi/2.

6

4.1. Pressure drop experiments 80-Mesh, 8 screens 50-Mesh, 8 screens 30-Mesh, 8 screens 20-Mesh, 8 screens

5.5 5

An important characteristic of static mixers is the generation of high pressure drop when compared to flow in empty pipe. For the purpose of comparison between several types of static mixers, pressure drop has also been reported by means of the Z-factor, the Newton number, Ne, or the Fanning friction factor, f =2 [43–46]. While the Newton number and the friction factor are directly related to each other following Eq. (14), the Z factor is the ratio of the pressure drop across the mixer to that across the same pipe without static mixing elements, as shown in Eq. (15).

4.5

Z (-)

4 3.5 3 2.5

Ne ¼ 2  f ¼

2

Z¼

1.5 1

0

5000

10000

15000

20000

25000

Re (-) Fig. 7. Effect of Re on the value of the Z factor.

solving the relationship that relates the mean residence time and variance of the impulse response function, E(t), with Pe, according to Eq. (13).

r2 t2m

¼

2Pe þ 8 Pe2 þ 4Pe þ 4

ð13Þ

4. Results and discussion In this investigation, various operational parameters, as well as reactor/contactor design geometries, were varied with the objective of identifying their effect on the residence time distribution, specifically on the axial dispersion coefficient. Hydrodynamic studies related to pressure drop requirements to drive the flow inside tubular contactors equipped with screen-type static mixers were also performed and the results are reported in the following sections.

DP  D

ð14Þ

L  q  U2

DPwith static mixers DPempty pipe

ð15Þ

Fig. 4a shows the variation of the pressure drop with the flow velocity (represented by the pipe Reynolds number) in the tubular reactor equipped with screen elements. It clearly shows that pressure drop increases with an increase in the Reynolds number. In addition, the effect of varying the number of elements on the pressure drop in the system for the case of 30-mesh screens can also be discerned in the figure. It is evident that a larger number of screen elements in the pipe produce a larger DP, an expected result. Furthermore, the effect of varying the screen geometry on the pressure drop in the system is depicted in Fig. 4b, where the pressure drop also increases with an increase in Re for all screen elements, with the increase being a function of the mesh number. An increase in the mesh number was found to increase DP. This could not be explained only by relating the pressure drop to the mesh open area since the 80-mesh and 50-mesh screens have comparable screen open areas (31.4% and 30.3%, respectively), however, it appears that the pressure drop is also a direct function of the mesh opening (M – b), which decreases with an increase in the mesh number (Table 1). This is evident from the correlation of DP (for eight screens only but of varying geometries) depicted in Eq. (16) with its parity plot shown in Fig. 5.

DPðkPaÞ ¼ 0:004  Re0:805  a0:397  ðM  bÞ

0:612

ðR2 ¼ 0:905Þ ð16Þ

956

K. Abou Hweij, F. Azizi / Chemical Engineering Journal 279 (2015) 948–963

Fig. 8. Raw conductivity data under different operating and design conditions, (a) Q = 4 L/min, N = 0; (b) Q = 12 L/min, 50-mesh, N = 2.

0.5

0.5

0.4 0.35 0.3 0.25 0.2 0.15 0.1

(b)

0.45

Normalized Concentration (−)

0.45

Normalized Concentration (−)

Inlet Outlet

(a)

Inlet Outlet

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.05 0 0

10

20

30

Time (s)

40

0 0

50

10

20

30

Time (s)

40

50

Fig. 9. Normalized concentration under different operating and design conditions, (a) Q = 4 L/min, N = 0; (b) Q = 12 L/min, 50-mesh, N = 2.

0.4

0.4 raw E(t) fitted E(t)

0.35 0.3

0.3

(a)

0.2 0.15

0.2 0.15

0.1

0.1

0.05

0.05

0

0 5

10

Time (s)

15

(b)

0.25 −1 E(t) (s )

E(t) (s−1)

0.25

−0.05 0

raw E(t) fitted E(t)

0.35

20

−0.05 0

5

10

Time (s)

15

20

Fig. 10. Plot of the ‘‘raw’’ impulse response, E(t), and its Gaussian fit; (a) N = 1, 20-mesh, Re = 3450; (b) N = 4, 50-mesh, Re = 4600.

957

K. Abou Hweij, F. Azizi / Chemical Engineering Journal 279 (2015) 948–963

0.5 Experimental Convolution

0.45 0.4

(a)

(b)

0.3 0.25

C

out

(−)

0.35

0.2 0.15 0.1 0.05 0 0

10

20

30

40

Time (s) 0.5 Experimental Convolution

0.45 0.4

(c)

(d)

Cout (−)

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0

10

20

30

40

Time (s) Fig. 11. Normalized outlet concentration, experimental vs. convolution of inlet concentration and impulse response; (a) N = 0, Re = 2,300; (b) N = 2, 20-mesh, Re = 6900; (c) N = 4, 50-mesh, Re = 3,450; (d) N = 8, 30-mesh, Re = 11,500.

It should be noted that a better regression line can be obtained using a different correlation type (cf. Appendix), however this work was presented to keep conformity with most studies where the pressure drop is correlated in the form DP / Rep . Moreover, the effect of varying the flow velocity on the friction factor, f/2, is reported in Fig. 6a. It is evident that the friction factor decreases almost linearly with increasing the flow rate through the mixer while it starts plateauing at higher Re (> 20,000) similarly to the behavior in an empty pipe. In order to account for the varying porosities of static mixers, many investigators [43–46] suggest the use of interstitial velocities while calculating the friction factor and Re. In such cases, the effective velocity term is that through the openings of the static mixer (which can be looked at as a combination of orifices) rather than the interfacial velocity in the open pipe. Under such conditions, the interstitial friction factor becomes equal to,

fi DP  D  a2 ¼ 2 4  L  q  U2

ð17Þ

where, a is the fraction open area of the screen. Similarly, the Reynolds number will also be calculated based on the interstitial velocity, where in this case, it is commonly known as the macroscopic jet Reynolds number, Rejet.

Rejet ¼

DUq la

ð18Þ

Fig. 6b shows the effect of accounting for the interstitial velocity through the screen mixers on the friction factor. It is very clear that regardless of the screen geometry, fi/2 values overlap until a critical Rejet is reached, the value of which appears to be dependent on the 1:4 (with R2 = 0.967, for screen geometry, where f i =2 / Rejet Rejet < 38,000 which is highlighted on Fig. 6b). Finally, the values of the Z factor in the current investigation were found to be much smaller than those reported in the literature for other types of static mixers. Fig. 7 shows the variation of the Z factor with the pipe Reynolds number, similarly to the pressure drop effect, the value of Z was found to increase with increasing Re. It is also very clear from this figure that the value of Z is a direct function of the screen geometry, with higher values of Z for higher mesh numbers. While Z plateaus at close to unity for large mesh openings, this value reaches a maximum of approximately 5 for the smallest mesh opening at relatively high Re (23,000). These values compare very favorably with those reported for other types of static mixers (e.g. Kenics, SMX, Lightnin, Komax, etc. . .), where the value of Z varies between 5.5 and 300 for the laminar flow regime only, with its value expected to increase rapidly to several hundred at higher Re [3,47]. Comprehensive summaries

958

K. Abou Hweij, F. Azizi / Chemical Engineering Journal 279 (2015) 948–963

12

Experimental Theorecal

Residence Time (s)

10

8

6

4

2

0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Superficial Velocity, U (m/s) Fig. 12. Comparison between mean (from RTD), tm, and theoretical, s, residence times in the mixing zone for all 119 experimental runs.

of Z values for mostly laminar flows are tabulated in the works of Kumar et al. [43] and Theron and Le Sauze [46] and results in the turbulent regime for certain mixer types are reported in [47]. This clearly highlights the fact that screen-type static mixers are more energy efficient when compared to other static mixers. 4.2. RTD experiments As previously stated, KCl was used to perform RTD experiments because the linear relationship between its conductivity and concentration in the range of investigated conditions. While the injected tracer was sent into the system in a very short period of time, the analysis adopted in the current work assumes an imperfect pulse condition and therefore recordings of both the inlet and outlet conductivities were performed at all times. A sample of the raw data obtained in a given experiment is shown in Fig. 8, where the ‘‘background’’ conductivity of RO water appears in the dataset. In addition, the effect of flow velocity on the

inlet and outlet conductivity distributions can be clearly discerned in Fig. 8b. At higher velocities the liquid volume containing the tracer is expected to spend a shorter time around the sensor, which translates into a narrower distribution of electrical conductivity at either the inlet or outlet of the mixing zone. Moreover, because the residence time in the system is shorter, the outlet conductivity distribution is closer to the inlet distribution. All these phenomena are clearly illustrated in Fig. 8b when compared to Fig. 8a. The raw conductivity data is then converted into normalized concentrations (shown in Fig. 9) in order to perform the necessary RTD calculations. The normalized concentrations would have an adjusted baseline at C = 0 which removes the ‘‘background noise’’ (conductivity of RO water). It should be noted that by normalizing the concentrations, the shape of the distributions was not affected. All experimental runs were conducted a minimum of three times to ensure repeatability of the results. In addition, a mass balance was conducted on each of these runs to check if the two sensors are measuring a comparable amount of salt (mass conservation) in the system and it was found that the difference between the inlet and outlet recorded amounts of KCl varied by 11.7% on average. 4.2.1. Selection of the fitting parameter An in-house computational subroutine to find the best fitting parameter was executed in MatLabÒ on each run in order to find the residence time distribution function, E(t), which would be used in analyzing the system. In addition to the aforementioned conditions of the conservation of the first and second moments of the distribution while searching for the best fitting parameter, c, an additional criterion was introduced in this work. This relied on calculating c that returns the distribution with the minimum fluctuations in the time domain. Mao et al. [27] discussed this issue and highlighted the fact that the RTD obtained using the Mills and Dudukovic [26] algorithm requires additional smoothing in the time-domain where negative peaks of no physical significance ~ xÞ. can appear after applying the inverse Fourier transform to Eð In order to eliminate such peaks of no significance, the computed E(t) function was fitted with a Gaussian distribution in the time domain. Such undertaking is justified by the fact that almost all impulse responses showed a Gaussian behavior as shown in Fig. 10.

0.6

2.5 Re = 3,450 Re = 5,750 Re = 8,600

0.5

2

(a)

(b) E(θ) (−)

0.4

E(t) (s-1)

Re = 3,450 Re = 5,750 Re = 8,600

0.3

1.5

1 0.2

0.5

0.1

0

0

5

10

15

0 0

0.5

Time (s) Fig. 13. Effect of flow velocity on E(t) and E(h); (N = 2, 30-mesh).

1

θ (−)

1.5

2

2.5

959

K. Abou Hweij, F. Azizi / Chemical Engineering Journal 279 (2015) 948–963

120

0.45

N = 2, 30-mesh No screens

(a) 100

(b)

0.4 0.35 0.3

COV (-)

Pe (-)

80

60

40

0.25 0.2 0.15 No screens

0.1

20

N = 2, 20-Mesh N = 4, 80-Mesh

0.05 0

0

5000

10000

0

15000

N = 8, 50-Mesh 0

5000

Re (-)

10000

15000

Re (-)

Fig. 14. Effect of increasing the flow velocity on Pe and COV.

(a)

(b) 100

Dax (cm2/s)

Dax (cm2/s)

100

10

10

N=8, 50-Mesh N=4, 50-Mesh N=2, 50-Mesh N=1, 50-Mesh

N=1, 20-Mesh N=1, 30-Mesh N=1, 50-Mesh N=1, 80-Mesh 1

1 0

5000

10000

15000

Re (-)

0

5000

10000

15000

Re (-)

Fig. 15. Effect of varying the design conditions on the axial dispersion coefficient: (a) effect of screen geometry; (b) effect of number of screens.

Further validation of this approach was concluded by checking the convolution of the inlet concentration measurements and the computed impulse response, according to Eq. (6), which should render the measured outlet concentration. Fig. 11(a–d) show the result of the convolution integral for various reactor/contactor designs and operating conditions, plotted against the normalized outlet concentrations. It is very clear that the computed RTD renders an exact description of the macromixing behavior of the system since all distributions obtained by the convolution integral fit very well the measured data. 4.2.2. Mean residence time A major use of the RTD analysis is to determine whether or not there exist dead volumes or bypassing in the reactor/contactor under investigation. For this purpose, the mean residence time, tm, computed from the RTD experiments using Eq. (3) is compared

with the theoretical residence time, ðs ¼ V=Q Þ in Fig. 12. As expected, the mean residence time in the mixer decreases with an increasing superficial velocity, however, if a dead (or stagnant) volume exists in the mixing zone, tm is expected to be smaller than s because the flow would have a smaller volume to sweep and would therefore exit faster. The situation is inverted in the case of bypassing or channeling, where a larger tm would be expected since part of the flow would immediately exit the system leaving a smaller effective flow rate to sweep the system [41,42]. It is obvious from Fig. 12, where all 119 experimental points are plotted, that the data is scattered around the theoretical value, with an average relative error of 8.4%, which indicates the absence of dead volumes or channeling in the system. The discrepancy between the data around the lowest superficial velocity can be attributed to a lower accuracy of the flow meter at those flow rates; nonetheless, the data will be presented as if it were exactly at that flow rate.

960

K. Abou Hweij, F. Azizi / Chemical Engineering Journal 279 (2015) 948–963

100

Dax (cm2/s)

Dax (cm2/s)

100

10

N=1, 20-Mesh N=1, 30-Mesh N=1, 50-Mesh N=1, 80-Mesh

(a) 1

10

(b) 1

0

100

200

300

N=8, 20-Mesh N=8, 30-Mesh N=8, 50-Mesh N=8, 80-Mesh

400

0

100

200

300

400

ReM (-)

ReM (-)

Fig. 16. Effect of screen geometry on the axial dispersion coefficient with respect to the mesh Reynolds number, ReM: (a) N = 1, (b) N = 8.

100

Dax (cm2/s)

Dax (cm2/s)

100

10

N=1, 20-Mesh N=1, 30-Mesh N=1, 50-Mesh N=1, 80-Mesh

(a) 1 1,000

10

10,000

100,000

Rejet (-)

N=8, 20-Mesh N=8, 30-Mesh N=8, 50-Mesh N=8, 80-Mesh

(b) 1 1,000

10,000

100,000

Rejet (-)

Fig. 17. Effect of screen geometry on the axial dispersion coefficient with respect to the macroscopic jet Reynolds number, Rejet: (a) N = 1, (b) N = 8.

4.2.3. Effect of flow velocity on RTD At higher flow rates, the residence time inside the mixing zone becomes smaller rendering a narrower E(t) distribution that is shifted toward shorter time values. This is clearly observed in Fig. 13a where E(t) is plotted for various flow rates in the same reactor configuration. However, the situation becomes different when dimensionless time, H, is used in order to normalize all distributions. In such cases, it becomes very clear that the dimensionless distribution E(H) becomes broader with an increase in the flow velocity (Fig. 13b). This reflects an increase in the axial dispersion in the system with an increase in the flow velocity. Using the one parameter model, presented in Eq. (12), the Péclet number, Pe, can be computed by fitting the equation to the computed E(H). The value of Pe was found to decrease with an increase in the flow rate, with its value being a direct function of the reactor geometry (Fig. 14a).

This is attributed to an increase in the axial dispersion coefficient as the velocity increases. Further, the flow in an empty pipe showed a lower value of Pe when compared to the reactor equipped with screen-type static mixers. Similar behavior is also observed when comparing the coefficient of variation along the length of the reactor. The COV was found to increase with an increase in the velocity with the values computed for pipes equipped with screen mixers being constantly smaller than those of an empty pipe. This finding highlights the fact that the presence of screen-type static mixers better homogenize the flow under the same operating conditions when compared to an empty pipe configuration. 4.2.4. Effect of screen geometry and number of screens on RTD To investigate the effect of varying the number of static mixer elements as well as their geometrical design on the macromixing

961

K. Abou Hweij, F. Azizi / Chemical Engineering Journal 279 (2015) 948–963

100

Dax (cm2/s)

Dax (cm2/s)

100

10

N=8, 80-Mesh N=4, 80-Mesh N=2, 80-Mesh N=1, 80-Mesh

(a) 1

10

0

50

100

N=8, 80-Mesh N=4, 80-Mesh N=2, 80-Mesh N=1, 80-Mesh

(b) 150

1

0

10,000

20,000

30,000

40,000

Rejet (-)

ReM (-)

Fig. 18. Effect of number of screens of the axial dispersion coefficient with respect to (a) the mesh Reynolds number, ReM; and (b) macroscopic jet Reynolds number, Rejet.

150

Dax, predicted (cm2/s)

Axial dispersion, Dax (cm2/s)

100

10

100

50

0 0

No screens With screens 1 0

5000

10000

15000

50

100

150

Dax, experimental (cm2/s) Fig. 20. Parity plot for the predicted vs. experimental axial dispersion coefficient values.

Re (-) Fig. 19. Comparison between the axial dispersion in the presence of screens with that for an empty pipe when plotted against the empty pipe Reynolds number, Re.

behavior of the system, the axial dispersion coefficient was calculated from the Pe values. The effect of varying the screen geometry is shown in Fig. 15a where the axial dispersion coefficient is plotted against the Reynolds number, and no clear effect could be discerned. Similarly, the effect of varying the number of screen elements inserted in the mixing zone is shown in Fig. 15b. It was interesting to note that while the axial dispersion coefficient continuously increases with an increase in the flow velocity, no significant effect for the screen geometry and number of screens could be highlighted in such a comparison. The empty pipe Reynolds number, Re, measures the ratio of inertial to viscous forces in the system with the pipe diameter being the characteristic length. However, for flow through woven

screens or grids, there exist several other characteristic lengths that could be employed when computing the Reynolds number, e.g. wire diameter, mesh opening, individual jet diameter, etc. . . While no agreement on which characteristic length to be used is available in the literature, the wire diameter remains a commonly acceptable length when investigating the turbulence of the flow through screens but the mesh size remains the most commonly employed length. In the latter case, the corresponding Re is called the mesh Reynolds number, ReM, and defined according to Eq. (19).

ReM ¼

MUq

l

ð19Þ

It is therefore interesting to note that when the axial dispersion coefficient is plotted against ReM, a clear effect of the screen geometry is observed where screens with the smallest mesh size, M, i.e. 80-mesh, exhibiting the largest overall axial dispersion (Fig. 16). Based on this dimensionless group, the larger the mesh opening,

962

K. Abou Hweij, F. Azizi / Chemical Engineering Journal 279 (2015) 948–963

3,000

Dax ðcm2 =sÞ ¼ expð0:12NÞ  Re0:43 M  ðM  bÞ

3:61

pffiffiffi N 2:61

ð0:018Nþ4:543Þ

 Rejet

ð20Þ

ΔP, predicted (Pa)

2,500 5. Conclusions

2,000 1,500 1,000 500 0 0

500

1,000

1,500

2,000

2,500

3,000

ΔP, experimental (Pa) Fig. 21. Parity plot for the pressure drop using various Reynolds numbers.

the smaller is the axial dispersion coefficient. However, such conclusion can be overturned by changing the dimensionless group where the effect against the macroscopic jet Reynolds number (defined in Eq. (18)) shows that the smaller the screen open area, a, the lower is Dax (Fig. 17) with screens of close open areas (i.e. 50- and 80-mesh) rendering almost an identical performance with larger number of screen elements inserted in the pipe. While such conclusions might appear contradictory, it is not, and the choice of the characteristic length to compute the Reynolds number remains an important criterion to judge the performance of the mixer and design it accordingly. Furthermore, no matter what is the choice of the Reynolds number, whether it is based on the macroscopic jet velocity or the mesh size or the empty pipe diameter, the effect of number of screen elements in the mixing zone remains negligible. Fig. 18(a) and (b) show the axial dispersion coefficient plotted against ReM and Rejet for different number of mixing elements of the same geometry. While it is very clear that a scatter exists, and as will be shown in the next section, this parameter was found to have a minor effect on the exponent of the respective Re numbers with no major statistical significance. 4.2.5. Comparison with flow in an empty pipe and regression analysis The advantages of utilizing screen-type static mixers are highlighted in Fig. 19, which shows the calculated axial dispersion coefficient for all experimental runs plotted against those for the runs where no static mixers were inserted in the pipe. When all the parameters are lumped together (i.e. number of screen elements, screen geometry, and flow velocity) it is very evident that screen-type static mixers allow for a smaller axial dispersion when compared to an empty pipe and therefore a better mixing performance. To the authors’ knowledge, no such data or information is reported for other types of motionless mixers at such high Re values so it can be used for comparison between the mixers. The data obtained in the current work were then correlated in order to be able to predict the performance of the mixer. The most significant parameters that played a role were found to be Rejet, ReM, and (M–b). This is clearly highlighted in Eq. (20) which predicts the value of the axial dispersion coefficient in the presence of screen elements only. The regression line (R2 = 0.991) was found to predict quite well the experimental results as shown in the parity plot presented in Fig. 20.

This study investigates the hydrodynamics and residence time distribution of one phase flow in tubular reactors/contactors equipped with screen-type static mixers. The measured pressure drop through various screen geometries and under different operating conditions was found to be a direct function of their mesh opening and screen fraction open area. From an energy efficiency perspective, screen mixers compared favorably with other types of static mixers, where the Z factor was found about two-orders of magnitude lower than values reported in the open literature for other types of motionless mixers. Furthermore, it was found that the axial dispersion coefficient in reactors equipped with screen mixers increases with an increasing flow velocity. However, the data could be interpreted in terms of various Reynolds numbers, depending on the characteristic length used. While the axial dispersion decreases with an increasing mesh size when compared on a mesh Reynolds number basis, the same data show that the dispersion decreases with the fraction open area of the screen when compared on the basis of Rejet. Such a conclusion is therefore left to the user to decide on which characteristic length is more important for its application. In addition, while its effect is somewhat evident, no statistical significance was found for the effect of number of screen elements in the pipe on the axial dispersion coefficient, in this study. Acknowledgments The financial support of the Lebanese National Council for Scientific Research (LNCSR) and the American University of Beirut, Lebanon, through its University Research Board is acknowledged. Appendix . The pressure drop data was found to better correlate using another regression line of a different form which employs various Reynolds numbers that are based on different characteristic lengths. This regression line is presented in Eq. (21), and its corresponding parity plot is shown in Fig. 21.

pffiffiffiffiffiffiffi DP ¼ 34:91a  4909:7b þ 7:6  103 Rejet  0:399 Reb þ 0:095ReM ðR2 ¼ 0:999Þ

ð21Þ

where DP is in Pa, b in m, and Reb is the wire Reynolds number, Reb ¼ b  U  q=l. It should be noted that the values reported here are for 8 screen elements but varying screen geometries. References [1] A. Ghanem, T. Lemenand, D. Della Valle, H. Peerhossaini, Static mixers: mechanisms, applications, and characterization methods – a review, Chem. Eng. Res. Des. 92 (2014) 205–228. [2] R. Andersson, B. Andersson, F. Chopard, T. Norén, Development of a multi-scale simulation method for design of novel multiphase reactors, Chem. Eng. Sci. 59 (2004) 4911–4917, http://dx.doi.org/10.1016/j.ces.2004.07.084. [3] R.K. Thakur, C. Vial, K.D.P. Nigam, E.B. Nauman, G. Djelveh, Static mixers in the process industries – a review, Chem. Eng. Res. Des. 81 (2003) 787–826. [4] E.B. Nauman, Reactions and residence time distributions in motionless mixers, Can. J. Chem. Eng. 60 (1982) 136–140, http://dx.doi.org/10.1002/ cjce.5450600123. [5] A. Peschel, B. Hentschel, H. Freund, K. Sundmacher, Design of optimal multiphase reactors exemplified on the hydroformylation of long chain alkenes, Chem. Eng. J. 188 (2012) 126–141.

K. Abou Hweij, F. Azizi / Chemical Engineering Journal 279 (2015) 948–963 [6] A.M. Al Taweel, F. Azizi, G. Sirijeerachai, Static mixers: effective means for intensifying mass transfer limited reactions, Chem. Eng. Process. Process Intensif. 72 (2013) 51–62, http://dx.doi.org/10.1016/j.cep.2013.08.009. [7] F. Azizi, A.M. Al Taweel, Turbulently flowing liquid–liquid dispersions. Part I: Drop breakage and coalescence, Chem. Eng. J. 166 (2011) 715–725, http:// dx.doi.org/10.1016/j.cej.2010.11.050. [8] A.M. Al Taweel, C. Li, H.G. Gomaa, P. Yuet, Intensifying mass transfer between immiscible liquids: using screen-type static mixers, Chem. Eng. Res. Des. 85 (2007) 760–765. [9] A.M. Al Taweel, J. Yan, F. Azizi, D. Odedra, H.G.G. Gomaa, Using in-line static mixers to intensify gas–liquid mass transfer processes, Chem. Eng. Sci. 60 (2005) 6378–6390, http://dx.doi.org/10.1016/j.ces.2005.03.011. [10] C. Chen, Dispersion and Coalescence in Static Mixers, Technical University of Nova Scotia, 1996. [11] F. Azizi, Design and Operation of Intensified Multiphase Processes, 2009. [12] M. El-Ali, Performance Characteristics of a Novel Liquid–Liquid Contactor, 2001. [13] F. Azizi, A.M. Al Taweel, Inter-phase mass transfer in turbulent liquid–liquid dispersions: a comparative analysis of models, Chem. Eng. J. 179 (2012) 231– 241, http://dx.doi.org/10.1016/j.cej.2011.10.063. [14] J.R. Bourne, M. Lips, Micromixing in grid-generated turbulence. Theoretical analysis and experimental study, Chem. Eng. J. Biochem. Eng. J. 47 (1991) 155– 162. [15] E.B. Nauman, Residence time theory, Ind. Eng. Chem. Res. 47 (2008) 3752– 3766. [16] P.V. Danckwerts, Continuous flow systems, distribution of residence times, Chem. Eng. Sci. 2 (1953) 1–13, http://dx.doi.org/10.1016/00092509(53)80001-1. [17] T.R. Keshav, P. Somaraju, K. Kalyan, A.K. Saroha, K.D.P. Nigam, Liquid phase residence time distribution for gas–liquid flow in Kenics static mixer, Chem. Eng. Process. Process Intensif. 47 (2008) 2275–2280. [18] R. Wadley, M.K. Dawson, LIF measurements of blending in static mixers in the turbulent and transitional flow regimes, Chem. Eng. Sci. 60 (2005) 2469–2478, http://dx.doi.org/10.1016/j.ces.2004.11.021. [19] P. Pustelnik, Investigation of residence time distribution in Kenics static mixers, Chem. Eng. Process. 20 (1986) 147–154. [20] Z. Kemblowski, P. Pustelnik, Residence time distribution of a power-law fluid in Kenics static mixers, Chem. Eng. Sci. 43 (1988) 473–478. [21] E.B. Nauman, D. Kothari, K.D.P. Nigam, Static mixers to promote axial mixing, Chem. Eng. Res. Des. 80 (2002) 681–685, http://dx.doi.org/10.1205/ 026387602760312890. [22] C.M.R. Madhuranthakam, Q. Pan, G.L. Rempel, Hydrodynamics in Sulzer SMX static mixer with air/water system, Ind. Eng. Chem. Res. 48 (2009) 719–726. [23] M. Rafiee, M.J.H. Simmons, A. Ingram, E.H. Stitt, Development of positron emission particle tracking for studying laminar mixing in Kenics static mixer, in: 14th Eur. Conf. Mix., Warszawa, Poland, 2013, pp. 389–394. [24] D. Ziolkowski, J. Morawski, The flow characteristic of the liquid streams inside a tubular apparatus equipped with static mixing elements of a new type, Chem. Eng. Process. 21 (1987) 131–139. [25] C.K. Harris, D. Roekaerts, F.J.J. Rosendal, F.G.J. Buitendijk, P. Daskopoulos, A.J.N. Vreenegoor, et al., Computational fluid dynamics for chemical reactor engineering, Chem. Eng. Sci. 51 (1996) 1569–1594. [26] P.L. Mills, M.P. Dudukovic´, Convolution and deconvolution of nonideal tracer response data with application three-phase packed-beds, Comput. Chem. Eng. 13 (1989) 881–898.

963

[27] Z.-S. Mao, T. Xiong, J. Chen, Residence time distribution of liquid flow in a trickle bed evaluated using FFT deconvolution, Chem. Eng. Commun. 169 (1998) 223–244, http://dx.doi.org/10.1080/00986449808912730. [28] N.S. Nahman, M.E. Guillaume, Deconvolution of time domain waveforms in the presence of noise, Boulder, CO, 1981. [29] A. Bennia, N.S. Nahman, Deconvolution of system impulse responses and time domain waveforms, IEEE Trans. Instrum. Meas. 79 (1990) 933–939. [30] C. Hutter, A. Zenklusen, R. Lang, P. Rudolf von Rohr, Axial dispersion in metal foams and streamwise-periodic porous media, Chem. Eng. Sci. 66 (2011) 1132–1141, http://dx.doi.org/10.1016/j.ces.2010.12.016. [31] B. Parruck, S.M. Riad, Study and performance evaluation of two iterative frequency-domain deconvolution techniques, IEEE Trans. Instrum. Meas. 33 (1984) 281–287. [32] D. Pananakis, E.W. Abel, A comparison of methods for the deconvolution of isothermal DSC data, Thermochim. Acta 315 (1998) 107–119. [33] D. Boskovic, S. Loebbecke, Modelling of the residence time distribution in micromixers, Chem. Eng. J. 135 (2008) S138–S146, http://dx.doi.org/10.1016/ j.cej.2007.07.058. [34] A.H. Essadki, B. Gourich, C. Vial, H. Delmas, Residence time distribution measurements in an external-loop airlift reactor: study of the hydrodynamics of the liquid circulation induced by the hydrogen bubbles, Chem. Eng. Sci. 66 (2011) 3125–3132, http://dx.doi.org/10.1016/j.ces.2011.02.063. [35] C.G.C.C. Gutierrez, E.F.T.S. Dias, J.A.W. Gut, Residence time distribution in holding tubes using generalized convection model and numerical convolution for non-ideal tracer detection, J. Food Eng. 98 (2010) 248–256, http:// dx.doi.org/10.1016/j.jfoodeng.2010.01.004. [36] A.K. Heibel, P.J.M. Lebens, J.W. Middelhoff, F. Kapteijn, J. Moulijn, Liquid residence time distribution in the film flow monolith reactor, AIChE J. 51 (2005) 122–133, http://dx.doi.org/10.1002/aic.10288. [37] R. Häfeli, C. Hutter, M. Damsohn, H. Prasser, P. Rudolf, V. Rohr, Dispersion in fully developed flow through regular porous structures: experiments with wire-mesh sensors, Chem. Eng. Process. Process Intensif. 69 (2013) 104–111. [38] M. Saber, C. Pham-Huu, D. Edouard, Axial dispersion based on the residence time distribution curves in a millireactor filled with b-SiC foam catalyst, Ind. Eng. Chem. Res. 51 (2012) 15011–15017, http://dx.doi.org/10.1021/ ie3017829. [39] B.R. Hunt, Biased estimation for nonparametric identification of linear systems, Math. Biosci. 10 (1971) 215–237, http://dx.doi.org/10.1016/00255564(71)90061-7. [40] B.R. Hunt, The inverse problem of radiography, Math. Biosci. 8 (1970) 161– 179, http://dx.doi.org/10.1016/0025-5564(70)90148-3. [41] O. Levenspiel, Chemical Reaction Engineering, Wiley, New York, 1972. [42] H.S. Fogler, Elements of Chemical Reaction Engineering, 1999. [43] V. Kumar, V. Shirke, K.D.P. Nigam, Performance of Kenics static mixer over a wide range of Reynolds number, Chem. Eng. J. 139 (2008) 284–295, http:// dx.doi.org/10.1016/j.cej.2007.07.101. [44] H.Z. Li, C. Fasol, L. Choplin, Pressure drop of Newtonian and non-Newtonian fluids across a Sulzer SMX static mixer, Chem. Eng. Res. Des. 75 (1997) 792– 796. [45] F. Theron, N. Le Sauze, A. Ricard, Turbulent liquid–liquid dispersion in Sulzer SMX mixer, Ind. Eng. Chem. Res. 49 (2010) 623–632. [46] F. Theron, N. Le Sauze, Comparison between three static mixers for emulsification in turbulent flow, Int. J. Multiphase Flow 37 (2011) 488–500, http://dx.doi.org/10.1016/j.ijmultiphaseflow.2011.01.004. [47] A.W. Etchells III, C.F. Meyer, Mixing in pipelines, in: Handb. Ind. Mix. Sci. Pract., 2004, pp. 391–477.