14th European Conference on Mixing Warszawa, 10-13 September 2012

THE NETMIX® REACTOR: 3D CFD MODELLING AND PRESSURE DROP MEASUREMENTS

Carlos M. Fonte, M. Enis Leblebici, Madalena M. Dias, José Carlos B. Lopes

Laboratory of Separation and Reaction Engineering, Faculty of Engineering, University of

Porto, Rua Dr. Roberto Frias s/n, 4200-465 Porto, Portugal

lopes@fe.up.pt

Abstract. Three NETmix® prototypes with different geometry were used to obtain experimental data of pressure drop and a model for predicting pressure drop in NETmix® reactors was developed. This model incorporates a single adjustable parameter and it is only dependent on the geometric configuration of the network. The dynamic measurement of pressure drop was used to evaluate the mixing dynamics in the NETmix® chambers and, above the critical Reynolds number, the natural oscillation frequency was quantified. Furthermore, a three-dimensional Computational Fluid Dynamic (CFD) transport model was also developed and validated. The energy performance of the three NETmix® prototypes was quantified and shown to be competitive with the compared existing static mixers by evaluating the power number and Z factor. The new 3D CFD transport model allows the computation of transport properties and overcomes the need of obtaining experimental data each time a new NETmix® configuration is designed. Keywords: NETmix® reactor, pressure drop, power number, Z factor, CFD.

1. INTRODUCTION The NETmix® Reactor is a new technology [1] consisting of a network of mixing

chambers interconnected by transport channels (Figure 1a). Networks are generated by the repetition of unit cells where each unit cell consists of one chamber and two inlet and two outlet channels oriented at a 45º angle from the main flow direction and can either be constructed from cylindrical chambers and rectangular cross section area channels (2D unit cell, Figure 1b) or from spherical chambers and cylindrical channels (3D unit cell, Figure 1c).

(a) (b) (c)

Figure 1. a) Representation of the NETmix® network; b) 2D unit cell; c) 3D unit cell. Above a critical channel Reynolds number, the flow inside the mixing chambers evolves

to a self-sustained oscillatory laminar flow regime inducing local strong laminar mixing. This occurs due to the geometric characteristics of NETmix® network. A network model was developed to describe and predict the behaviour and performance of NETmix® [2]. From the point of view of modelling, it was shown that chambers can be assumed to behave as

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perfectly mixing zones and the channels as plug flow perfect segregation zones. The NETmix® Reactor can therefore be understood, along the main flow direction, as a plug flow reactor with local maximum mixing. The mixing degree was defined and quantified [2, 3] showing that mixing can be controlled effectively and efficiently making it particularly suited for complex and fast kinetics reactions. However, the performance of the NETmix® reactor, in terms of energy requirements, was yet to be studied and compared with others mixers.

The main goal of this work was the development of a model that describes the pressure drop along NETmix® reactors and its validation with experimental data and CFD simulations. From this pressure drop model, it is possible to evaluate the performance of any NETmix® reactor topology by defining two benchmarking properties, the power number and Z factor, for comparison with other type of mixers.

2. PRESSURE DROP MODELLING

2.1 Pressure drop model

The total pressure drop of each unit cell, ucpΔ , was modelled using an analogy with an equivalent pure resistive electric circuit [4], where the flow rate through the channels corresponds to the current across the resistance branches and the pressure drop between chambers corresponds to the voltage between nodes and is expressed by uc channelp RqΔ = where R is the total hydraulic resistance from the inlet to the outlet of the unit cell and channelq is the flow rate through the channel. This hydrodynamic model assumes the flow to be isothermal, incompressible and steady and the hydraulic resistance to be the sum of three terms, F LF NLFR R R R= + + where FR is the friction resistance in the channels, and LFR and

NLFR are hydraulic resistances related to the contractions and expansions in the chambers. The friction resistance in laminar regime is given by ( ) ( )2 21 2 32F o o

channel h channel h channelR f l q d A l d Aρ μ= = (1)

where f is the friction factor, ρ is the fluid density, ol is the channel length, hd is the channel hydraulic diameter, channelA is the channel cross sectional area. The linear flow resistance, LFR , in the chamber was defined by Koplik [5] as ( )* 232LF

h channelR l d Aμ= (2)

where * 2 hl d π= for this kind of geometry. This resistance exists whenever an unbounded jet flow is to be considered. When the fluid leaves the channel, the canonical friction term cannot model the energy transfer in the chamber as the flow is not bounded by the channel walls. Instead, the fluid’s solid channel is replaced with a viscous surrounding as long as the jet keeps its form. The nonlinear flow hydraulic resistance, NLFR , that takes into account the energy dissipation due to accelerations/decelerations in the mixing chambers is modelled as 21 2NLF

channel channelR Kq Aρ= (3) where K is a coefficient analogous to the coefficients for sudden contractions and expansions generally used in the calculation of pressure drop in pipe systems. The value of K only depends on the geometrical characteristics of NETmix® and can be determined from CFD simulations or experimental data.

The total pressure drop in the system, NETmixpΔ , can then be determined by the sum of the pressure drop for the total number of rows in the flow direction.

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From the dynamic measurement of pressure drop it is also possible to evaluate the mixing dynamics in the NETmix®. Above a certain Reynolds number in the channel, known as the critical Reynolds number, the flow inside the mixing chambers evolves to a self-sustained oscillatory flow regime inducing local strong laminar mixing. These oscillations are intrinsic to the nature of the flow and not determined by external factors [6]. The natural frequency of these oscillations, oscf , in NETmix® mixing chambers was proposed to be related to the residence time inside the chambers [6] and is expressed as ( )1 Reosc chamber h channelf V d Aτ μ ρ= = ⎡ ⎤⎣ ⎦ (4)

where τ is the residence time, chamberV is the volume of the chamber, and Re is the Reynolds number defined as Re h channeldρ υ μ= where channelυ is the average velocity at the channel.

2.2 Assessment of different mixers A common approach in the analysis of the power consumption in stirred tanks is to plot

the power curve, that is the relationship between the dimensionless power number and the Reynolds number. The dimensionless power number, pN , is defined as the ratio between the total amount of energy supplied to the system and the energy that is needed to cause the fluid motion necessary for mixing. For the NETmix® system operating in laminar regime, the power number can be expressed as ( ) ( ) ( ) ( )mixer mixer friction 1 2 64 Reo

P h hN p p p l d d Kπ⎡ ⎤= Δ Δ −Δ = + +⎣ ⎦ (5)

where mixerpΔ is the mixer’s pressure drop and frictionpΔ is the pressure drop due to friction.

Another common way of interpreting the pressure drop for static mixers is by the Z factor. It is defined as the ratio between the pressure drop through the static mixer, static mixerpΔ , and the pressure drop through an empty tube of equal length and diameter, empty tubepΔ . For the NETmix® geometry, the Z factor for laminar regime is expressed as ( ) ( )0 0

static mixer empty tube 2 64 Reoh hZ p p l d L d K Lπ ⎡ ⎤= Δ Δ = + + ⎣ ⎦ (6)

As can be seen in Equations 9 and 10, for the same Reynolds number, both the power number and the Z factor, are not dependent on the fluid properties, and are only dependent on the geometrical configuration of the NETmix® reactor.

2.3 Experimental setup

For these studies, three different NETmix® geometries, built for different purposes, were used: one geometry with spherical chambers and cylindrical channels and two geometries with cylindrical chambers and rectangular cross section area channels. The geometrical characteristics of the three prototypes are summarized in Table 1.

Water and glycerol solutions of 10%, 20% and 60% in mass were used in the measurements and the studies were carried out at room temperature. The differential pressure drop was measured with a pressure transducer at the inlet and outlet of each NETmix® prototype at the reactor’s inlet and outlet. Furthermore, the Lab-scale NETmix® 2D front cover was also drilled at 2/3 in height to measure the pressure difference along the time at two outlet channels from the same chamber. The pressure sensor was calibrated for each set of experiments so the voltage signal acquired could be converted to differential pressure. The pressure sensor sends voltage data to the computer via a data acquisition board.

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Table 1. NETmix® prototypes geometrical characteristics.

Geometry Lab-scale NETmix® 2D NETmix® 3D Multi-inlet

NETmix® 2D Number of rows, nx 29 49 65

Number of columns, ny 8 15 16 Chamber diameter, D (mm) 6.5 7 8.75 Channel diameter, d (mm) 1 1.5 1.5 Geometry depth, ω (mm) 3 ― 5.9 Channel length, lo (mm) 2 3 5.4 Total volume, V (mL) 23 140 1500

3. CFD MODEL Due to computer memory limitations, it is not possible to simulate, with the necessary

grid refinement, the whole NETmix® flow domain. Since the geometry and the expected pattern of the flow have a periodically repetitive nature, it is possible to simulate a larger system by modeling a small part that is far from its edge. The model’s 3D geometry consists of a portion of the lab-scale NETmix® 2D reactor prototype with 5 rows and 3 columns. The chambers of the first and third columns are half chambers and to their outer limits, a periodic boundary condition was applied as can be seen in Figure 2. At the chamber and channel walls a no-slip condition was assumed. A uniform velocity profile was applied to the network inlet channels and a constant and uniform pressure value was set at the network outlet channels. The fluid used in the simulation was water with constant physical properties at room temperature. The flow field simulation is achieved by numerical integration of the continuity and Navier-Stokes equations. The 3D simulations were performed in an Intel® Xeon® 3GHz 8-core machine running the finite-volume commercial CFD software ANSYS Fluent™ 13.0. A refined computational grid with 1.33 million elements was used. As an initial condition for the transient simulations, steady state simulations were performed. Non-friction walls were placed in the middle of the chamber to keep the flow segregated. The transient simulations were performed for a total flow time of 15τ .

Outlets

Inlets

Periodicsurfaces

Periodicsurfaces

Figure 2. Boundary conditions

4. RESULTS AND DISCUSSION

4.1 Pressure drop experimental results An averaged value of the dynamic measurement of the signal received from the pressure

transducer was converted to differential pressure. Figure 3 summarizes the pressure drop measurement results obtained for the three NETmix® prototypes and the respective model fitting. To find the value of K , the experimental data was fitted to the model by the least squares method and the model was considered to fit well the experimental data with maximum deviations of 15%.

The flow oscillations may be characterized by studying their frequency using power spectral analysis. The power spectral analysis was performed to the fluctuation of the signal

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with time obtained with the pressure transducer for the Lab-scale NETmix® 2D prototype. The signal was obtained from two measuring points at two outlets of a single chamber located at a middle region of the NETmix® reactor. For a given Reynolds number, the dominant peak indicates the natural frequency of oscillation inside the chambers and is shown in Figure 4. The frequency of oscillation is fitted by a universal model (Equation 4), with a maximum deviation of 10%, and is only dependent on inertial effects: the flow rate and reactor volume.

(a) (b) (c)

Figure 3. Pressure drop obtained experimentally and model fitting for a) Lab-scale NETmix® 2D, b) NETmix® 3D; c) Multi-Inlet NETmix® 2D.

4.2 Benchmarking Power number results for the three NETmix® prototypes are shown in Figure 5. These

results are compared with power number data of stirred tank mixers with different impeller designs obtained in the literature [7]. Results show that the power number for NETmix® decreases with increasing Reynolds number and becomes significantly lower than the corresponding values for stirred tanks.

Figure 4. Frequency of oscillation as a

function of the Reynolds number. Figure 5. Power Number plot for the NETmix® prototypes

and for several stirred tanks with different impellers [7]. Figure 6 shows the Z factor values for the NETmix® prototypes as a function of the

Reynolds number. In this range of Reynolds numbers, the three prototypes exhibit Z factor values one order of magnitude lower than the commercial static mixers data found in the literature [8] and therefore can be operated with smaller pressure drops and, consequently, with a reduced consumption of energy.

4.3 CFD Simulation Results The CFD simulation results were compared to the experimental data and to the model

fitting for the Lab-scale NETmix® 2D prototype (Figure 7). The experimental data and the CFD simulation results are in a good agreement with a maximum deviation of 15%. It can be concluded that CFD simulations can be used to evaluate the pressure drop through NETmix® reactor and overcome the need of experimental data for new configurations.

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Figure 6. Z factor for the NETmix® prototypes as a function of Re. The gray area corresponds to a typical

range of Z factors for some commercial statics mixers [8].

Figure 7. Pressure drop as a function of the Reynolds number obtained experimentally

and from CFD simulations.

5. CONCLUSIONS

Experimental data of pressure drop obtained from three different NETmix® prototypes was used to develop a model for prediction of the pressure drop with a single adjustable parameter that is only dependent on the geometric configuration of the network. The dynamic pressure measurements were also used to assess mixing dynamics in NETmix® reactor and to confirm and extend a previously proposed model for the frequency of oscillation in the chambers. NETmix®’s mixing efficiency was previously estimated. However from this work, its energy performance was quantified and shown to be competitive with existing mixers by evaluating PN and Z factor for the three NETmix® prototypes. The new 3D CFD transport model allows the computation of transport properties and overcomes the need of experiments each time a new NETmix® configuration is designed.

ACKNOWLEDGMENTS This work is partially financed by FEDER Funding through COMPETE - Programa

Operacional Factores de Competitividade, and by Portuguese National Funding through FCT - Fundação para a Ciência e a Tecnologia within the scope of project PEst-C/EQB/LA0020/2011. Carlos M. Fonte gratefully acknowledges the PhD grant from FCT (SFRH/BD/39040/2007).

6. REFERENCES [1] Lopes J.C.B., Laranjeira P.E., Dias M.M., Martins A.A., "Network mixer and related mixing process". PCT/IB2005/000647, February 2005. European Patent EP172643 B1, October 2008. [2] Laranjeira P.E., Martins A.A., Lopes J.C.B., Dias M.M., 2009. "NETmix®, a new type of static mixer: Modeling, simulation, macromixing, and micromixing characterization", AIChE Journal, 55, 2226-2243. [3] Gomes P.J., Fonte C.P., Santos R.J., Dias M.M., Lopes J.C.B., 2010. "Experimental and numerical characterization and quantification of mixing in a NETmix® reactor", in: ECCE 7 - 7th European Congress of Chemical Engineering & CHISA 2010- 19th internation Congress and Process Engineering, Process Engineerig Publisher, Prahe, Prague, Czech Republic. [4] Martins A.A., Laranjeira P.E., Lopes J.C.B., Dias M.M., 2007. "Network modeling of flow in a packed bed", AIChE Journal, 53, 91-107. [5] Koplik J., 1982. "Creeping flow in two-dimensional networks", Journal of Fluid Mechanics, 119, 219-247. [6] Laranjeira P.E., 2005. "NETmix® Static Mixer Modelling, CFD Simulation and Experimental Characterisation", in: Departamento de Engenharia Química, Universidade do Porto, Porto. [7] Bates R.L., Fondy P.L., Corpstein R.R., 1963. "Examination of Some Geometric Parameters of Impeller Power", Industrial & Engineering Chemistry Process Design and Development, 2, 310-314. [8] Thakur R.K., Vial C., Nigam K.D.P., Nauman E.B., Djelveh G., 2003. "Static Mixers in the Process Industries—A Review", Chemical Engineering Research and Design, 81, 787-826.

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