Linear Stability Analysis for a Reactive, Multiphase ... - ACS Publications

Article pubs.acs.org/IECR

Linear Stability Analysis for a Reactive, Multiphase Bubble Column Justin D. Smith, Daniel W. Crunkleton,* and Selen Cremaschi Russell School of Chemical Engineering, The University of Tulsa, 800 South Tucker Drive, Tulsa, Oklahoma 74104, United States S Supporting Information *

ABSTRACT: Reactive processes often depend on the dynamic states of the fluid in which the reaction takes place. In this study, we consider coupled momentum and solutal mass transport in a multiphase (gas/liquid) bubble column with a chemical reaction, in a configuration often found in columns used for microorganism growth. We model the reaction with a pseudoreaction term in the concentration balance equation, which represents the growth as a function of species and reactor conditions. We present a linear stability analysis on a two-dimensional bubble column with water as the continuous phase and carbon dioxide as the dispersed phase. It is concluded that in the range of variables examined the flow is always unstable. multiphase flows, however, has not been as thoroughly studied and is a further motivation of this work. The influence of flow phenomena on microorganism growth was investigated by Xiu and co-workers,23 who studied the stability of microbial growth in a continuously stirred tank reactor. Here, hydrodynamic phenomena were introduced into the equations of change as a dilution of the substrate and products within the reactor by an incoming substrate. A system of three simultaneous ordinary differential equations for the biomass, substrate, and product concentrations was derived and linearized. The eigenvalues of the system’s Jacobian were calculated for different values of the saturation constants, maintenance consumption coefficients, substrate uptake coefficients, maximum growth yields, and maximum product yields. The parameter space with stable eigenvalues was determined, and the authors concluded that at least two nonwashout, steadystate, stable solutions exist and are characterized by high biomass concentrations. Although this is a valid approach for a perfectly mixed system, this formulation neglects important hydrodynamic phenomena that would be present in many nonideal reactor configurations, such as the two-phase column reactor that is considered in this work. Later, Bees and Hill24 used a model based on the work of Pedley and Kessler25 to study microbial bioconvection and pattern formation within a finite-depth layer containing randomly swimming gyrotactic microbes. They determined cellular movement as a function of the cell concentration by analyzing the single-phase momentum and continuity equations, assuming that the density of the continuous phase varies linearly with concentration, similar to the Boussinesq approximation that is used to couple the momentum and energy conservation equations. An equilibrium solution for the finite depth suspension of cells was calculated while perturbing the mean cell swimming direction, cell diffusion tensor, suspension velocity, local cell concentration, and excess pressure. The analysis demonstrates how varying these variables manifests as

1. INTRODUCTION Reactive bubble columns have become an increasingly important area of research in several diverse engineering applications, including water purification,1 aromatic oxidation,2 and industrial solvent remediation.3 Numerous computational strategies have been proposed,4,5 including multiscaled approaches.6,7 Recently, the authors have begun investigating transport-assisted microorganism growth, specifically of algae, by increasing the mass transfer between bubbled carbon dioxide (CO2) in the gaseous phase and the microorganisms in an aqueous liquid phase.8,9 In this way, the efficiency of continuous systems can be combined with enhanced mass transfer in a bubble column. The parameter space needed to fully characterize the reactive flow in continuous-flow reactors is extremely large and involves several potentially important parameters, such as inertial states and reactant concentration and distribution. The growth of microorganisms, however, greatly exacerbates this situation by the inclusion of a host of organism-specific parameters. Experiments with physical reactors can determine these conditions; however, studying the entire parameter space would require an unreasonably large number of experiments and a high cost per experiment if the system is industrially sized or has high substrate costs. Furthermore, the presence of CO2 needed for microalgae growth (which is the main motivator for this study) complicates the hydrodynamics of the system and makes experimental characterization more complex. Linear stability analysis, on the other hand, minimizes the experimental expense while still providing practical guidance on the system performance. Linear stability analysis is an established technique for analyzing nonlinear flow problems and is the subject of several classic texts, such as those by Drazin and Reid10 and Chandresekhar.11 The technique’s ability to resolve regions of stable and unstable regimes has been applied in several diverse fields, such as crystal growth,12,13 extrusion,14 nanomaterial processing,15,16 and radiation heat transfer.17 The technique is often used to study interfacial problems,18−20 where effects such as Marangoni convection can play a critical role in determining the interfacial behavior.21,22 The application of stability analysis to the growth of microorganisms, specifically those growing in © 2015 American Chemical Society

Received: Revised: Accepted: Published: 12690

August 21, 2015 November 11, 2015 November 12, 2015 November 12, 2015 DOI: 10.1021/acs.iecr.5b03094 Ind. Eng. Chem. Res. 2015, 54, 12690−12698

Article

Industrial & Engineering Chemistry Research

the “Jackson roots” are always unstable, while the secondary roots are always stable. The current work extends the application of linear stability analysis of 2D reactive bubble columns, specifically tailored to those that could be applied to microorganism growth, using a normal-mode perturbation of the coupled Navier−Stokes and species balance equation, coupled through a Boussinesq term in the momentum equation. As with Rayleigh’s31 application of the Boussinesq approximation to convection currents, the variation in the density due to the difference in the concentration is neglected, except for where it changes the effects of gravity. Although our analysis intends to provide insight into the growth conditions that could be ideal for encouraging or preventing microbial growth, it also reveals how the addition of microbes and their growth changes the stability of an existing system. Additionally, this analysis is equally applicable for reactions in bubble columns that do not involve microbe growth. We conclude that introducing the reactive concentration equation always makes the flow unstable with the velocity, gas fraction, pressure, and algae concentration as the perturbation parameters. This result parallels an observation made by Nandapurkar et al.32 on an entirely different system, thermosolutal convection during dendritic solidification. In that study, the addition of gravity to the equations of change caused the system to move from a state with stable and unstable regimes to one with instabilities over the entire range of the parameter values considered. The paper is organized as follows: First, we integrate microbial growth into the equations of change for 2D bubble columns. Next, a solution of the dispersion relationship is obtained using a technique similar to that used by Jackson29 and Monahan and Fox.30 This solution is substituted into the equations of change. Thereafter, the method for perturbing and solving the perturbed equations is given. We also give the results of several cases for studies done using a sample microorganism, an alga of the species Chlorella.

changes in the overall microbial motion. They concluded that, for self-propelled microorganisms, it is important to treat the swimming speed as an independent random variable. Subramanian and co-workers26 studied a similar system, looking at the stability of chemotactic bacteria in a single-phase dilute suspension in the presence of a chemoattractant field, and concluded that the long-wavelength perturbations encountered by swimming bacteria are damped out by nonlinear effects. Saintillan and Shelley27 looked at the stability of the more broadly defined case of a suspension of self-propelled particles. They concluded that, for swimming particles, instabilities in the shear stress on the particles due to the fluid occur at long wavelengths. While these studies provide important frameworks for studying swimming microorganisms/particles within a liquid, the complexities encountered with microorganism growth in multiphase systems remain an open question. One of the simplest methods for studying the stability of a multiphase flow (whether with or without microorganism growth) is to consider a one-dimensional multiphase flow. Here, the momentum and continuity equations can be written for each phase, linearized, and reformulated into a second-order partial differential equation. This approach was used by LéonBecerril and Liné28 to study the stability of bubble-column reactors, with the goal of calculating the wave or wavepropagation speed, for different gas-phase volume fractions within a bubble column. They concluded that, for intermediate Reynolds numbers, defined by the authors as occurring when the superficial liquid velocity is 0.005 m/s and bubbles have a diameter of 0.8 mm, instability appears when the gas fraction nears 30%. For higher Reynolds numbers, instabilities occur at smaller gas fractions of around 10−15%. Jackson29 studied a system of fluidized particles within a continuous phase and investigated the stability analysis of the multiphase flow, taking into account the flow of the dispersed phase in multiple dimensions. This study determined a dispersion relationship for a fluidized bed and analyzed the effect of velocity perturbations of the continuous and dispersed phases, as well as perturbations of the pressure and dispersed-phase volume fraction. Jackson concluded that there were qualitative similarities between experimentally observed instabilities and those estimated by the linear stability analysis. This study also noted the similarities between a bubble rising in a liquid and a bubble rising in a fluidized bed. Monahan and Fox30 extended Jackson’s approach to gas− liquid flows and expanded it to study the multiphase stability of a nonreactive bubble column by using separate phase-specific momentum and continuity equations. They included several correlations for the drag, added-mass, lift, rotation, and strain forces for the bubbles, as well as expressions for the bubbleinduced turbulence modification to the turbulent viscosity. A base state in terms of the Reynolds number, drag force, drag coefficient, pressure, and dispersed-phase velocity was determined by setting the continuous-phase velocity to zero. The void fraction, pressure, and continuous- and dispersed-phase velocities were then perturbed, resulting in a system of six linearized equations from which a dispersion relationship was used to determine the stability of the system for different combinations of force terms. The authors concluded that the stability of the two-dimensional (2D) system is a function of the values of the force coefficients, specifically the bubble pressure coefficient. For their baseline homogeneous formulation, above a dispersed-phase volume-fraction-dependent minimum value of this bubble pressure coefficient, they found that what they term

2. METHODOLOGY When any sort of flow is modeled, continuity and momentum balance equations are the starting point, which for this study take the form ⇀ ∂ρ + ∇·(ρu ) = 0 ∂t

(1)

and ⎞ ⎛ ∂⇀ ⇀ ⇀ ⇀ ⇀ u ρ⎜⎜ + u ·∇u ⎟⎟ = −∇p + ∇·μeff [∇u + (∇u )T ] ⎠ ⎝ ∂t ⎯⇀ ⎯

+

∑ Ff f

⎯⇀ ⎯

+ ρg

(2)

in which ρ is the density, t is the time, u⇀ is the velocity, p is the pressure, μeff is the effective viscosity, F⇀ is the interphase force terms, and g⇀ is the gravitational vector. The linearized models used in this formulation (and the steps to transform eqs 1 and 2 to the form of eqs 4−7) are those used in a study from 2005 by Monahan and co-workers,30,33 as is the design of the column, shown in Figure 1. For multiphase flow models, separate equations for continuity and momentum must be written for each phase. Variables corresponding to the continuous phase are denoted by a subscript c, while those for the dispersed phase are denoted by a 12691

DOI: 10.1021/acs.iecr.5b03094 Ind. Eng. Chem. Res. 2015, 54, 12690−12698

Article

Industrial & Engineering Chemistry Research ⇀

∂ud1

0 = −αd0ρd +

∂t

⇀

+ αd0ρd ud0

⇀ C BTρc dbαd0 ud 0[∇2 uc1 2

∂ud1 ∂x

⇀

+ αd0ρc

∂uc1 − αd0μ0,c ∂t

⇀

+ ∇(∇· uc1)] − ρc C BP

⇀ ⇀ ⎛ α 2 ⎞⎛ α d 0 ⎞⎛ ∂uc1 ⎞ ∂ud1 d ⎟ ⎟⎟⎜ud 2⎜⎜ 0 ⎟⎟⎜⎜1 − − u d0 ∂x ⎟⎠ αdcp ⎠⎜⎝ 0 ∂x ⎝ αdcp ⎠⎝

⎛ 2α 3αd0 2 ⎞ d0 ⎟∇·αd − ρc C BPud0 2⎜⎜ − 1 αdcp2 ⎟⎠ ⎝ αdcp ⇀

⇀

+ (αd0μ0,d + C BTρc dbαd0 2ud0)[∇2 ud1 + ∇(∇· ud1)] ⇀ ⎛ ∂⇀ ∂ud1 ud1 ⎜ + u d0 + αd1(ρd − ρc ) g − αd0ρv C VM⎜ 0 ∂x ⎝ ∂t ⎯⇀ ⎯

⇀ ⇀ ⇀ ⎡ ∂ud1 ∂uc1 ∂uc1 ⎞ ⎢ ⎟ + u d0 + αd0ρv C ud0 − 0 ⎢ ∂x ∂x ∂t ⎟⎠ ⎣ ⎤ ⇀ ⇀ ⇀ T ⇀ T⎥ − ud0 i ·(∇uc1) − ud0 i ·(∇ud1) − αd0ρv 0 ⎥ ⎦

Figure 1. Diagram of the bubble column, showing continuous and dispersed phases as well as the directions within the column.

subscript d. The equations for the separate phases are coupled by the phase volume fractions αc and αd, where αc = 1 − αd (3)

⇀ ⇀ ⎡ ∂ud1 ∂uc1 ⇀ ⇀ ⎢ + ud0 i ·(∇uc1)T + u d0 Cs ud0 ⎢ ∂x ∂x ⎣

Accordingly, the continuity and momentum equations for the continuous phase are −

∂αd1 ∂t

⇀

+ (1 −

⇀ αd0)∇· uc1

=0

− u d0 i

(4)

and −ρc

⇀ ∂uc1

∂t

⇀

⇀

[∇2 uc1 + ∇(∇· uc1)] + αd1ρc β0ud0 i + αd0ρc ⇀ β1(ud1

−

⇀ uc1)

⇀ ⇀ ⎛ ∂⇀ ∂ud1 ∂uc1 ⎞ ud1 ⎜ ⎟ + u d0 − + αd0ρv C VM⎜ ⎟ 0 d ∂ x t ∂ t ⎝ ⎠

∂cr ⇀ + ∇·cruc1 = DAB∇2 cr + ra ∂t

− u d0 i

⎞

⇀ ⇀ ⎛ ∂ud1 ∂uc1 ⎜ + u d0 + αd0ρv Cs⎜ud0 0 dx ∂x ⎠ ⎝

⇀T ·∇ud1 ⎟⎟

⎞ ⇀ ⇀T ⇀ ⇀T − ud0 i ·∇uc1 + ud0 i ·∇ud1 ⎟⎟ ⎠ (5)

=0

∂t

+ u d0

∂αd1 dx

⇀

+ αd0∇· ud1 = 0

(8)

where μ is a growth rate. In this model, the growth rate ra, is a fraction of the existing concentration of the reacting species, cr. It is important to note that, by using this linear growth model, we assume that any substrate that may be necessary for growth (e.g., for microbe growth, light intensity, and nutrient concentration) is present at high enough concentrations such that it is not growth-rate-limiting. Because the case studied involves cellular reproduction, the reactants and products are the same. A specific type of kinetics for microorganisms, such as the Monod model,34 could be used to define the growth rate. However, a different

where CBT is the bubble-induced turbulence coefficient, db is the bubble diameter, β is the drag coefficient, CVM is the virtual mass force coefficient, C is the coefficient for lift and rotational forces, and CS is the strain coefficient. Those same equations for the dispersed phase are ∂αd1

(7)

where cr is the concentration of the reaction products in the column (i.e., the concentration of algal cells), DAB is the reactive species diffusivity in the continuous (liquid) phase, and ra is the rate of reaction of the reaction products (i.e., the produced cells). In this work’s case study of microorganism growth, we assume that the reactive species, microbes, move only because of the flow fields, and we neglect any form of taxis or motility. To ensure the applicability of this analysis to different types of kinetic models, a linear reaction rate of cellular generation is used, defined as ra = μcr (9)

⇀ ⇀ ⎛ ∂ud1 ∂uc1 ⇀ ⇀T ⎜ + u d0 − ud0 i ·∇uc1 − αd0ρv C ⎜ud0 0 dx ∂x ⎝ ⇀

⎥ ⎦

For 2D bubble columns with no reactions or microorganism growth, it is enough to analyze these equations to study the stability. We extend this analysis to incorporate microorganism growth, which is modeled as a chemical reaction, the effects of which are accounted for by adding a reactive species balance to the analysis, viz.

− ∇p1 + (μ0,c + C BTρc dbαd0ud0) ⇀

⎤

⇀ ·(∇ud1)T ⎥

(6)

and 12692


Article


in a manner similar to that of Monahan and Fox30 but including a concentration term, cr. We use this normal-mode expansion to solve the set of partial differential equations (4), (6)−(8), and (11) and reduce them to a set of algebraic equations. When the normal-mode equation (12) is substituted into the continuity equations (4) and (6), these become respectively

microorganism growth rate model would not lead to significantly different results within the region studied because most common microorganism growth kinetic models are approximately linear with a positive slope in the region of extremely low microorganism concentration. The small time scale of the perturbations in comparison with the time scales associated with cellular reproduction means that changes in the microbial growth rate due to these perturbations would be linear. Organism specific growth can still be accounted for by varying the microorganism growth rate, μ, used within the system of equations. To couple the species balance with the continuity and momentum equations, we use a mass-transfer analog to the Boussinesq approximation used in convection/conduction systems, where the concentration of the reactive species (i.e., the concentration of the microbe) is added to the continuousphase momentum equation. As with the thermal case, we assume in this study that the effect of mass transfer on the kinematics of the bubble-column continuous phase is only through this Boussinesq term in the momentum balance and takes the form of a linear change in the mass density of the continuous phase as a function of the reactant (i.e., microbe) concentration. This density increase takes the form of

−sαd1 + (1 − αd0)ik1uĉ 11 + (1 − αd0)ik 2uĉ 12 = 0

and (s + ud0ik)αd̂ 1 + αd0(ik1ud̂ 11 + ik 2ud̂ 12) = 0

(14)

where k1 is the vertical wavenumber perturbation and k2 is the horizontal wavenumber perturbation. Similarly, the continuousphase momentum equation (5) is transformed into 0 = −sρc uĉ 11 − p1̂ ik1 + ρc β0ud αd1 + αd0ρc β1(ud̂ 11 − uĉ 11) 1

0

2

+ (μ0,c + C BTρc dbαd0ud0)[−k uĉ 11 + (uĉ 11ik1 + uĉ 12ik 2) ι k̂ 1] + αd0ρv C VM(sud̂ 11 + ud0ik1ud̂ 11 − suĉ 11) 0

− αd0ρv Cud0(ik1uc11 + ik1ud̂ 11 − uĉ 11ik1 − ud̂ 11ik1) 0

⎯⇀

ρc g (cr1 − cr0)

(13)

+ αd0ρv Csud0(ik1uc11 + ik1ud̂ 11 + uĉ 11ik1 − ud̂ 11ik1)

(10)

0

⎯⇀ ⎯

+ ρc g cr1

where cr1 is the perturbed concentration and cr0 is the uniform state concentration. It should be noted that, in this work, for all variables except for k, a first numerical subscript of 0 indicates the base-state variable and a subscript of 1 indicates the perturbation variable. For the second numerical subscript of û and the first numerical subscript of k, a subscript of 1 indicates a vertical direction and 2 a horizontal direction. Substituting eq 10 into eq 5 yields the following coupled Navier−Stokes/species balance equations:

and 0 = −sρc uĉ 12 − p1̂ ik 2 + αd0ρc β1(ud̂ 12 − uĉ 12) 1

+ (μ0,c + C BTρc dbαd0ud0)[−k 2uĉ 12 + (uĉ 11ik1 + uĉ 12ik 2) ι k̂ 2] + αd0ρv C VM(sud̂ 12 + ud0ik1ud̂ 12 − suĉ 12) 0

− αd0ρv Cud0(ik1uc12 + ik1ud̂ 12 − uĉ 11ik 2 − ud̂ 11ik 2)

⇀

−ρc

∂uc1 ∂t

(15)

0

− ∇p1 + (μ0,c + C BTρc dbαd0ud0)

⇀ [∇2 uc1 ⇀ β1(ud1

+

⇀ ∇(∇· uc1)]

−

⇀ uc1)

+ αd0ρv Csud0(ik1uc12 + ik1ud̂ 12 + uĉ 11ik 2 − ud̂ 11ik 2) 0

(16)

⇀

+ αd1ρc β0ud0 i + αd0ρc

(k21

where k = + Upon substitution of eq 12 into the dispersed-phase momentum equation (7), we obtain

⇀ ⇀ ⎛ ∂⇀ ∂ud1 ∂uc1 ⎞ ud1 ⎜ ⎟ + u d0 − + αd0ρv C VM⎜ ⎟ 0 d ∂ x t ∂ t ⎝ ⎠

0 = −sρc uĉ 11 − p1̂ ik1 + ρc β0ud0αd̂ 1 + αd0ρc β1(ud̂ 11 − uĉ 11) 1

⎛ ⇀ ⇀T + u d0 − ud0 i ·∇uc1 − αd0ρv C ⎜⎜ud0 0 dx ∂x ⎝ ⇀ ∂uc1

⇀ ∂ud1

2

+ (μ0,c + C BTρc dbαd0ud0)[−k uĉ 11 + (uĉ 11ik1 + uĉ 12ik 2) ι k̂ 1] + αd0ρv C VM(sud̂ 11 + ud0ik1ud̂ 11 − suĉ 11) 0

⎞ ⎛ ⇀ ⇀T − ud0 i ·∇ud1 ⎟⎟ + αd0ρv Cs⎜⎜ud0 + u d0 0 dx ∂x ⎠ ⎝

⇀ ∂ud1

⇀ ∂uc1

⇀

⇀T

− ud0 i ·∇uc1

k22)1/2.

− αd0ρv Cud0(ik1uĉ 11 + ik1ud̂ 11 − uĉ 11ik1 − ud̂ 11ik1) 0

+ αd0ρv Csud0(ik1uĉ 11 + ik1ud̂ 11 + uĉ 11ik1 + ud̂ 11ik1) 0

⎞ ⇀ ⇀T ⎯⇀ ⎯ + ud0 i ·∇ud1 ⎟⎟ + ρc g (cr1 − cr0) ⎠

(17)

and 0 = −sρc uĉ 12 − p1̂ ik 2 + αd0ρc β1(ud̂ 12 − uĉ 12) 1

(11)

=0

+ (μ0,c + C BTρc dbαd0ud0)[−k 2uĉ 12 + (uĉ 11ik1 + uĉ 12ik 2)

As is typical of similar stability analyses, we next introduce a normal-mode expansion of the perturbation parameters, velocity, gas volume fraction, pressure, and reactant concentration, viz.

ik 2] + αd0ρv C VM(sud̂ 12 + ud0ik1ud̂ 12 − suĉ 12) 0

− αd0ρv Cud0(ik1uĉ 12 + ik1ud̂ 12 − uĉ 11ik 2 − ud̂ 11ik 2) 0

(uc⃗ 1 , ud⃗ 1 , αd1 , p1 , cr1)

+ αd0ρv Csud0(ik1uĉ 12 + ik1ud̂ 12 + uĉ 11ik 2 + ud̂ 11ik 2)

⎯⇀

= (uĉ 1 , ud̂ 1 , αd̂ 1 , p1̂ , cr̂ 1) exp(st ) exp(ik· x )

0

(12)

(18) 12693


Article

Industrial & Engineering Chemistry Research Finally, a similar process with the concentration equation (8) yields 0 = −(k 2DAB − s − μ)cr̂ 1 + (ik1uĉ 11 + ik 2uĉ 12)cr0 − μcr0 (19)

We now neglect the μcr0 term in eq 19 because Brune34 experimentally determined the product of the values of μ (h−1) and cr0 (g/cm3) to be on the order of 10−6, which is several orders of magnitude smaller than the other terms. Then, the concentration equation becomes 0 = −(k 2DAB − s − μ)cr̂ 1 + (ik1uĉ 11 + ik 2uĉ 12)cr0

(20)

The seven equations (13)−(18) and (20) are then simplified and terms collected for each of the seven variables ûc11, ûc12, ûd11, ûd12, α̂ d1, p̂1, and ĉr1. Equations 13−18 and 20 are put into matrix form, and the determinant of the coefficient matrix is taken to obtain a dispersion relationship. This procedure is used instead of an algebraic derivation similar to that of Monahan and Fox30 because the additional concentration equation drastically increases the complexity of the overall equation set, making it too large to manipulate in fully expanded form unless large parts of the equation have already been simplified. The resulting determinant can be further simplified and terms collected in terms of s, the disturbance growth rate. At this point, the equations are in a form that allows the stability analysis to be applied to the growth of microorganisms within the bubble column as a function of the disturbance wavenumber perturbation size, as well as other parameters. Mathematica, version 8.0, was used to simplify the equations and obtain the dispersion relationship, requiring 2 days of solution time on a computer with two 6-core Xeon processors and 32 GB RAM. The resulting dispersion relationship is too complex to reproduce in print form; therefore, the dispersion relationship is given as eq S1 in the Supporting Information, with values substituted for the physical properties of water and CO2.

Figure 2. Validation case: variation of the perturbation growth rate with the horizontal disturbance. k1 = 0. This result compares favorably with Figure 2c in Monahan and Fox.30

Fox,30 the maximum root when k2 = 0, shown in Figure 3, is positive over the range of 0 ≤ k1 ≤ 25.

Figure 3. Validation case: variation of the perturbation growth rate with the horizontal disturbance. k2 = 0. This result compares favorably with Figure 2a in Monahan and Fox.30

3. RESULTS Using eq S1, four cases are studied with microorganism growth as the reaction under study. First, we present a validation case, in which our formulation of the problem is verified with previous work by neglecting the growth rate and initial microbe concentration. Next, we investigate the effect of changes in the drag parameters on s and thus on the system stability. In the third and fourth cases, we investigate the effect of changes in {DAB, cr0, μr, αd0} and then {αd0, CBP} on s and the system stability. 3.1. Validation Case. For validation, we consider the “baseline model including all force terms” from Monahan and Fox,30 which allows for a direct comparison with this study by removing the terms containing microbial growth or microbe concentration perturbations from the dispersion relationship. To accomplish this, we create a submatrix by removing the last row and column from the multiphase coefficient matrix, which corresponds to the addition of the concentration balance, and then take its determinate. The specific case we consider is αd0 = 0.1, given in Figure 2, which compares favorably with Figure 2 of Monahan and Fox.30 In Figure 2, as in Monahan and Fox,30 the value of the maximum root when the vertical wavenumber k1 = 0 is everywhere negative indicates that the system is stable for all disturbance wavenumbers. Additionally, as in Monahan and

3.2. Microbial Concentration and Growth Rate Variation for Small Vertical or Horizontal Perturbations. 3.2.1. μ = 0 and Varying k1, k2, and Cr0. Having verified that the base system agrees with the results observed by Monahan and Fox,30 we now include the reaction parameters, particularly as they pertain to microorganism growth. This analysis shows not only the effects of perturbations in the velocity, pressure, and volume fraction variables but also the effect of perturbations in the concentration of microorganisms. We begin by detailing the behavior of the disturbance growth rate, s, as a function of the initial microorganism concentration and growth rate for small values of the vertical and horizontal disturbances, k1 and k2, without reaction. When the microbial growth rate is set to zero with the vertical wavenumber k1 = 0.01 and the concentration varying from zero to 5 × 10−5 g/cm3, the results, not reproduced here for brevity, show small values for s that are both constant and positive. The concentration gradient may induce motion in the vertical direction because of solute-induced buoyancy caused by an unstable concentration gradient with respect to the direction of gravity, however proving this would require additional study. Conversely, for a horizontal perturbation of the same magnitude where k2 = 0.01, the value of the maximum root is on the order of 12694


Article

Industrial & Engineering Chemistry Research 10−11. In this case, solutal convection may not be observed because of the absence of a destabilizing concentration gradient. 3.2.2. k1 = 0 and k2 = 0.001 and Varying μ. When the microbial growth rate is varied from 0−2.5 × 10−5 h−1 and the microbial concentration is held constant at 5 × 10−5 g/cm3, variation of s with the microbe growth rate can be seen in Figure 4. This figure reveals that the value of the disturbance growth rate

ensure a more homogeneous distribution of nutrients and less cell segregation with respect to light sources. 3.4. Base-Case Horizontal Perturbation. For studying horizontal disturbances, the same base case was used but with k2 as the disturbance variable and k1 = 0. Similar results, although different in magnitude, were observed with the presence of at least one positive root for all values of k2, as can be seen in Figures 6 At k2 ≈ 6.5, a transition is observed where two of the roots

Figure 4. Stability as a function of the microbial growth rate when cr0 = 0.0005, k1 = 0, and k2 = 0.001.

is positive and of the same order as the microbial growth rate, becoming approximately zero as the microbial growth rate approaches zero. The role of the increasing microbe concentration caused by the reaction term on the instability of the flow system is evident. This supports the main conclusion of this study, namely, that the presence of the microbes within the column leads to a corresponding increase in instability. 3.3. Base-Case Vertical Perturbation. To further quantify the effects of vertical disturbances observed in Figure 4, we take the baseline studied in the validation case with all force terms and add the microbe concentration and growth rate terms. Here, the gas volume fraction, αd0, was set to 0.1 and k2 = 0 for all calculations. The maximum root of s for this analysis is shown in Figure 5. It is important to note from this figure that the value of the maximum root is positive everywhere when k1 > 0, making the system unstable. When a stability analysis for other applications such as crystal growth is conducted, instability is undesirable;35 however, the instability may be advantageous for microbial growth because the convection it may cause can help to

Figure 6. Two positive roots seen in the base case with horizontal disturbance. k1 = 0, CVM = 0.5, c = 0.375, Cs = 0.125, CBT = 0.6, CBP = 0.2, and αd0 = 0.1.

which previously had the same value diverge from each other, as can be seen in Figure 7. This effect does not change the stability dynamics of the system, and its physical explanation is not immediately apparent. 3.5. Other Cases Considered for Horizontal and Vertical Disturbances. Several cases with both horizontal and vertical disturbances were considered. While all cases considered had different values for some of the roots, they all have similarities with the base cases described above. The biggest difference between these cases is that the bifurcation observed for the base case for horizontal disturbances, Figure 7, occurs at smaller values of k2 as αd0 increases, as can be seen in Figure 8, which has αd0 = 0.4 rather than 0.1, as is the case with Figure 7 [In this analysis, we are using the term “bifurcation” in the sense used by Drazin and Reed10 (“the branching of the curves of the equilibrium solution”).] This indicates that bifurcation happens at smaller wavenumbers for increasing gas volume fractions. The solutions of the determinant for all tested cases include at least one positive root, and therefore all of the systems described with these cases are unstable.

Figure 5. Base case with vertical disturbance. k2 = 0, CVM = 0.5, c = 0.375, Cs = 0.125, CBT = 0.6, CBP = 0.2, and αd0 = 0.1. 12695


Article


critical CBP in Figure 9, however, occurs at a slightly higher value than the one observed by Monahan and Fox.30 This shows that, as αd0 increases for these values of k1 and k2, the values of CBP at which stability can be achieved differ more and more from those that are physically realistic. Our solution differs, however, in the fifth root of s when the determinate is set equal to zero because of the concentration equation. This root is always unstable. Indeed, we can observe this in Figure 10, which shows contours of ℛ [s]

Figure 7. All roots for the base case with horizontal disturbance. αd0 = 0.1, k1 = 0, CVM = 0.5, c = 0.375, Cs = 0.125, CBT = 0.6, and cBP = 0.2. The inset is an enlargement of the square area indicated.

Figure 10. Contours of values of the maximum root for ℛ [s] (the real component of s) as a function of αd0 and CBP. Figure 8. All roots for the base case with horizontal disturbance. αd0 = 0.4, k1 = 0, CVM = 0.5, c = 0.375, Cs = 0.125, cBT = 0.6, and cBP = 0.2. The inset is an enlargement of the square area indicated.

(the real component of s) as a function of αd0 and CBP with k2 = 0 and k1 = 25. In Monahan and Fox,30 stability could be observed because this root is not present. We also investigate the stability of the system as a function of DAB and CBP, with all other parameters held constant. The value of CBP for which the fourth root becomes stable remains almost constant over the range of DAB. Similar results appear when we vary cr0 or μr and CBP while keeping other parameters constant. As with αd0, we see that the concentration root remains stable when any of the three quantities {DAB, cr0, or μr} are varied.

3.6. Variable CBP. Instead of varying the value of k, it is also possible to set the values of k1 and k2 as constants and vary other parameters in the dispersion relationship. One area of particular interest is the stability of a column at constant k for different values of αd0 and CBP. Monahan and Fox30 concluded that what they called the “Jackson roots” only achieve stability once CBP reaches a critical value and that this critical value is a function of αd0. We observe a similar behavior for our reactive, multiphase columns, as can be seen in Figure 9, where k2 = 0 and k1 = 25. The

4. CONCLUSIONS For a 2D reactive bubble column modeled by including a source term in the concentration balance equation, we conclude that the hydrodynamics are unstable within the parameter space explored. Although this investigation took the particular example of growing microorganisms in this column, the results presented are equally applicable for other, nonbiological, reactive systems. The observed instability stems from the presence of the concentration root and is due to the presence of the concentration equation and its associated reaction term, which was a new contribution of this study. This result is analogous to a study of a different system by Nandapurkar and co-workers,32 who found that neglecting gravitational forces caused a dendritic solidification system to become universally unstable. The overall stability of the other roots follows the pattern of those obtained by Monahan and Fox,30 with the main differences arising from the presence of the concentration equation and the

Figure 9. Critical value of CBP for “Jackson roots” where s = 0 as αd0 is varied. 12696


Article


(7) Van den Akker, H. Towards A Truly Multiscale Computational Strategy For Simulating Turbulent Two-Phase Flow Processes. Ind. Eng. Chem. Res. 2010, 49, 10780. (8) Smith, J.; Cremaschi, S.; Crunkleton, D. CFD Based Optimization of a Flooded Bed Bioreactor for Algae Production. Comput.-Aided Chem. Eng. 2012, 31, 910. (9) Smith, J.; Neto, A.; Cremaschi, S.; Crunkleton, D. CFD-Based Optimization of a Flooded Bed Algae Bioreactor. Ind. Eng. Chem. Res. 2013, 52, 7181. (10) Drazin, P.; Reid, W. Hydrodynamic stability; Cambridge University Press: Cambridge, U.K.2004. (11) Chandrasekhar, S. Hydrodynamic and hydromagnetic stability; Dover Publications, Inc.: New York, 1981. (12) Johnson, D.; Narayanan, R. Experimental Observation of Dynamic Mode Switching in Interfacial-Tension-Driven Convection Near a Codimension-Two Point. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1996, 54, 3102. (13) Müller, G. Convective Instabilities in Melt Growth Configurations. J. Cryst. Growth 1993, 128, 26. (14) Rothenberger, R.; McCoy, D.; Denn, M. Flow Instability in Polymer Melt Extrusion. J. Rheol. 1973, 17, 259. (15) Loscertales, I.; Barrero, A.; Márquez, M.; Spretz, R.; VelardeOrtiz, R.; Larsen, G. Electrically Forced Coaxial Nanojets for One-Step Hollow Nanofiber Design. J. Am. Chem. Soc. 2004, 126, 5376. (16) Li, F.; Yin, X. Y.; Yin, X. Z. Linear Instability of a Coflowing Jet Under an Axial Electric Field. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2006, 74, 036304. (17) Xu, B.; Ai, X.; Li, B. Stabilities of Combined Radiation and Rayleigh-Bénard-Marangoni Convection in an Open Vertical Cylinder. Int. J. Heat Mass Transfer 2007, 50, 3035. (18) Chaudhury, A.; Chakraborty, S. Dynamics of Mixing-Limited Pattern Formation in Nonisothermal Homogeneous Autocatalytic Reactors: a Low-Dimensional Computational Analysis. Ind. Eng. Chem. Res. 2011, 50, 4335. (19) Dey, M.; Bandyopadhyay, D.; Sharma, A.; Qian, S.; Joo, S. W. Charge Leakage Mediated Pattern Miniaturization in the Electric Field Induced Instabilities of an Elastic Membrane. Ind. Eng. Chem. Res. 2014, 53, 18840. (20) Khanna, R.; Jameel, A.; Sharma, A. Stability and Breakup of Thin Polar Films on Coated Substrates: Relationship to Macroscopic Parameters of Wetting. Ind. Eng. Chem. Res. 1996, 35, 3081. (21) Neogi, P.; Zahedi, G. Environmental Stress Cracking of Glassy Polymers. Ind. Eng. Chem. Res. 2014, 53, 672. (22) Kim, M.; Choi, C.; Yoon, D.; Chung, T. Onset of Marangoni Convection in a Horizontal Fluid Layer Experiencing Evaporative Cooling. Ind. Eng. Chem. Res. 2007, 46, 5775. (23) Xiu, Z.; Zeng, A.; Deckwer, W. Multiplicity and Stability Analysis of Microorganisms in Continuous Culture: Effects of Metabolic Overflow and Growth Inhibition. Biotechnol. Bioeng. 1998, 57, 251. (24) Bees, M.; Hill, N. Linear Bioconvection in a Suspension of Randomly Swimming, Gyrotactic Micro-Organisms. Phys. Fluids 1998, 10, 1864. (25) Pedley, T.; Kessler, J. A New Continuum Model For Suspensions of Gyrotactic Micro-Organisms. J. Fluid Mech. 1990, 212, 155. (26) Subramanian, G.; Koch, D.; Fitzgibbon, S. The Stability of a Homogeneous Suspension of Chemotactic Bacteria. Phys. Fluids 2011, 23, 041901. (27) Saintillan, D.; Shelley, M. Instabilities, Pattern Formation, And Mixing in Active Suspensions. Phys. Fluids 2008, 20, 123304. (28) León-Becerril, E.; Liné, A. Stability Analysis of a Bubble Column. Chem. Eng. Sci. 2001, 56, 6135. (29) Jackson, R. The Dynamics of Fluidized Particles; Cambridge University Press: Cambridge, U.K., 2000. (30) Monahan, S.; Fox, R. Linear Stability Analysis of a Two-Fluid Model For Air−Water Bubble Columns. Chem. Eng. Sci. 2007, 62, 3159. (31) Rayleigh, L. On Convection Currents in a Horizontal Layer of Fluid, When the Higher Temperature is on the Under Side. Philos. Mag. 1916, 32, 529.

Boussinesq term within the momentum balance. For cases with values of the concentration parameters, DAB, cr0, and μr, ranging from zero to conditions found in our test case of the growth of algae of the Chlorella species, it can be seen that there is a small effect on the amplitude of the positive roots from these variables. This can also be observed in the cases studied for variable CBP. The constant stability of the Jackson roots for a range of values of the concentration parameters, combined with the presence of the unstable concentration root, leads us to conclude that the stability of the Jackson root is mainly determined by the flow parameters rather than the concentration of the microbe. Moreover, we conclude that, although the dispersed bubble phase does cause large changes in the horizontal and vertical disturbance growth rate, the hydrodynamic stability of the system is determined principally by the concentration root. The fact that this root is always observed to be positive suggests amplified flow instability, which leads to greater intracolumn mixing and, therefore, enhanced growth of microorganisms in multiphasic bubble columns due to the increased convective mixing and interphasic mass transfer. This observed instability behaves as one would expect in a similarly constructed solutal convection problem. This concentration gradient, and its associated buoyancy, is expected when nonmotile microorganisms grow within a column reactor with multiphase flow conditions.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.5b03094. Equation S1 (PDF)

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This research was supported by Russell School of Chemical Engineering, University of Tulsa, and the Tulsa Institute of Alternative Energy.

■

REFERENCES

(1) Perdigoto, M.; Larachi, F.; Lopes, R. Multiphase Reacting Flow Studies of Bubble Column Ozonation Reactor For Water Remediation. Chem. Eng. Technol. 2013, 36, 137. (2) Lemoine, R.; Behkish, A.; Morsi, B. Hydrodynamic and Mass Transfer Characteristics in Organic Liquid Mixtures in a Large-Scale Bubble Column Reactor for the Toluene Oxidation Process. Ind. Eng. Chem. Res. 2004, 43, 6195. (3) Wu, C.; Kuo, C.; Chang, C. Homogeneous Catalytic Ozonation of C.I. Reactive Red 2 by Metallic Ions in a Bubble Column Reactor. J. Hazard. Mater. 2008, 154, 748. (4) Wang, Y.; Xiao, Q.; Yang, N.; Li, J. In-Depth Exploration of the Dual-Bubble-Size Model for Bubble Columns. Ind. Eng. Chem. Res. 2012, 51, 2077. (5) Bai, W.; Deen, N. G.; Kuipers, J. A. M. Numerical Investigation of Gas Holdup and Phase Mixing in Bubble Column Reactors. Ind. Eng. Chem. Res. 2012, 51, 1949. (6) Aboulhasanzadeh, B.; Thomas, S.; Taeibi-Rahni, M.; Tryggvason, G. Multiscale Computations of Mass Transfer From Buoyant Bubbles. Chem. Eng. Sci. 2012, 75, 456. 12697


Article

Industrial & Engineering Chemistry Research (32) Nandapurkar, P.; Poirier, D.; Heinrich, J.; Felicelli, S. Thermosolutal Convection during Dendritic Solidification of Alloys: Part I. Linear Stability Analysis. Metall. Trans. B 1989, 20, 711. (33) Monahan, M.; Vitankar, V.; Fox, R. CFD Predictions for FlowRegime Transitions in Bubble Columns. AIChE J. 2005, 51, 1897. (34) Brune, D. The Growth Kinetics of Freshwater Algae. Ph.D. Dissertation, University of Missouri, Columbia, MO, 1978. (35) Müller, G.; Neumann, G.; Weber, W. Natural Convection in Vertical Bridgman Configurations. J. Cryst. Growth 1984, 70, 78.

12698


Linear Stability Analysis for a Reactive, Multiphase ... - ACS Publications

Recommend Documents