Environ. Sci. Technol. 2009, 43, 2381–2387
Compartmental Models for Continuous Flow Reactors Derived from CFD Simulations ¨ GGER, MARKUS GRESCH, RAPHAEL BRU ALAIN MEYER, AND WILLI GUJER* Swiss Federal Institute of Aquatic Science and Technology, Eawag, 8600 Du ¨ bendorf, Switzerland, and Institute of Environmental Engineering, ETH Zurich, 8093 Zurich, Switzerland
Received June 16, 2008. Revised manuscript received January 6, 2009. Accepted January 7, 2009.
Reactor modeling is of major interest in environmental technology. In this context, new contaminants with higher degradation requirements increase the importance of reactor hydraulics. CFD (Computational Fluid Dynamics) may meet this challenge but is expensive for everyday use. In this paper, we provide research and practice with a methodology designed to automatically reduce the complexity of such a highdimensional flow model to a compartmental model. The derivation is based on the concentration field of a reacting species which is included in the steady state CFD simulation. While still capturing the most important flow features, the compartmental model is fast, easy to use, and open for process modeling with yet unknown compounds. The inherent overestimation of diffusion by compartmental models has been corrected by locally adjusting turbulent fluxes. We successfully applied the methodology to the ozonation process and experimentally verified it with tracer experiments. The loss of information was quantified as a deviation from CFD performance prediction for different reactions. With increasing discretisation of the compartmental model, these deviations diminish. General advice on the necessary discretisation is given.
Introduction Finding an appropriate mathematical description of the transport processes is a major step in building a model to predict the performance of a reactor. For continuous flow reactors, this task frequently relies on comparing measured and simulated residence time distributions (RTD). This approach has been shown to perform reasonably well in many cases, but it has distinct limitations which become increasingly relevant when reactions with high conversion rates are considered: (i) systematic errors caused by a model structure that is not flexible enough to capture all the details of a RTD; (ii) time of mixing effects that lead to erroneous predictions in reactor performance depending on the kinetics of the reactions involved (1); and (iii) experimental determination of the RTD with possibly large errors made due to nonhomogeneous tracer addition, nonrepresentative sampling, or insufficient temporal resolution (2). In any case, a reactor in operation is required, even though the information gained from such an experiment would be particularly helpful in the design stage. * Corresponding author phone: +41 44 823 5036; fax: +41 44 823 5389; e-mail:
[email protected]. 10.1021/es801651j CCC: $40.75
Published on Web 02/25/2009
2009 American Chemical Society
Some of these limitations can be overcome by using Computational Fluid Dynamics (CFD). It makes predictions about the flow field, including the transport of reactive species, mainly on the basis of fundamental physical principles. With the appearance of general purpose codes, CFD has become increasingly popular in environmental technology (3-6). This tool is of special interest in cases where experimental data are difficult to obtain, such as for predicting disinfection efficiency in water treatment (7, 8). However, CFD involves an excessive need for computational resources. This becomes particularly critical if a large number of compounds are involved. Second, the application of such a tool requires special know-how which is hardly available to most experts interested in the outcome of such an analysis. In contrast, compartmental models are computationally inexpensive and much easier to use. This is a precondition for many numerical techniques used for process description and optimization such as sensitivity, uncertainty, and scenario analyses (9, 10). So we cannot expect CFD to replace compartmental models in the short to medium term. In environmental technology, emerging contaminants have important implications for model building: compounds of interest change much faster than the life span of a treatment reactor. As a consequence, whereas reactor geometries and hydraulic behavior of reactors are stable, questions of reactor performance change with every new contaminant of which we become aware (11). All these reasons suggest the usefulness of combining CFD and compartmental models, i.e., use CFD to produce an accurate prediction of the flow field and compartmental models to predict the conversion. This requires a reasonable transition from the CFD model to the compartmental model. Some propositions have been made so far (12-18). The major issues identified are (i) the choice of good criteria to aggregate computational cells of the CFD model to compartmental volumes; (ii) the preservation of macroscopic flow features; and (iii) the incorporation of turbulent flow properties in compartmental volumes. Besides more empirical approaches based on the qualitative inspection of the flow field (15, 17), either an approach using geometric subdivision of the reactor (19) or similarity in flow-related properties (e.g., energy dissipation rate, viscosity, species concentration) are used as aggregation criteria (13, 14, 18). In applications such as polymerization or crystallization, whose rates of reaction are strongly dependent on flow properties, it proved decisive to use these properties as a merging criterion (20). In this paper, the focus lies on homogeneous reactions in continuous flow systems, where the time of reaction (macroscopic flow field) and earliness of mixing are of particular importance (1). A timerelated property will therefore be proposed. The preservation of macroscopic flow features requires the flow not to change direction along surface elements aggregated to one compartmental volume (17). Both purely geometric subdivision and aggregation based on flow-related criteria have the disadvantage that such flow features (e.g., recirculation loops) could be destroyed. While a higher resolution (more compartments) generally improves the situation, it is more useful to examine the flow field carefully with respect to changes in flow directions before aggregation. We suggest an algorithm that performs this examination automatically. Turbulence is most often included only implicitly in compartmental models: the single compartments are assumed to be completely mixed, which automatically adds diffusion to a system of compartments. As in tank-in-series VOL. 43, NO. 7, 2009 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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If we first think of a representation in a tank-in-series model, we obtain the following for the advective part of eq 1 at the right edge of the control volume (Figure 1): -
∫ bu · c · bn dS ≈ -Q · c
FIGURE 1. Layout and definitions for a turbulent plug-flow reactor. models, where the amount of added dispersion/diffusion is controlled by adjusting the number of tanks, the number of volumes used in the compartment model ultimately determines the diffusivity of the whole model. In contrast, Guha et al. (19) used a more rigorous approach. They extracted information on turbulent diffusion from the CFD simulation and included it as a bidirectional flow between adjacent volumes of the compartment model. As an accurate representation of turbulent mixing has great impact on the RTD and consequently on conversion rates, we follow this approach. But we upgrade this method further by tackling the problem of excess diffusion that is introduced by this more physical accounting of turbulence in compartmental models.
Methodology for Compartmental Model Generation We start with the general idea clarifying the basic concept of compartmental models and discuss the problem of excess diffusion relating to this concept. In a second step, we present an automatic cell-merging algorithm that generates the compartmental model from a steady state CFD simulation. General Idea. A compartmental model consists of a number of fully mixed volumes connected to each other by exchange fluxes. Two basic steps are required to successfully reduce the complexity of the CFD model to a compartment model: (1) Meaningful aggregation of computational CFD cells to the volumes of the compartment model. (2) Quantification of advective and turbulent fluxes at the interfaces of the aggregated cells. Due to the assumption of complete mixing within one volume, compartment models inherently include diffusive processes. Together with the true turbulent diffusion at the interfaces that is added as a separate flux component to the model, this leads to overestimating the total diffusion. This effect could be minimized by using a large number of volumes leading to a more complex model. Alternatively, turbulent fluxes can be adjusted to account for the diffusion related to the model assumption of complete mixing. We follow this approach and must therefore first quantify this effect. From a purely numerical point of view, a compartmental model is a coarse finite volume representation of the advection equation using an upwind scheme. As a first-order scheme in space, it is numerically diffusive. This property is used to model turbulent diffusion. As numerical diffusion can be estimated by using a higher-order numerical scheme (21), it can be accounted for later when turbulent fluxes are introduced in the compartmental model. We will exemplify this in the following sections using a turbulent plug-flow reactor. See Figure 1 for definitions. The results will then be generalized to a nonrestricted compartmental model. The starting point is the integral form of the advectiondiffusion equation for a control volume V with a surrounding surface S (eq 1):
∫ ∂c∂t dV ) -∫ bu · c · bn dS + ∫ D V
S
S
T,eff ·
∇c · b n dS
(1)
where c is the concentration of the transported species, u is the flow velocity, and DT,eff is the turbulent diffusion coefficient. 2382
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(2)
i
A
This corresponds to an upwind scheme that is of first order in space. If we think of a numerical solution to the advection diffusion equation using a central scheme that is of second order in space, the same part of eq 1 becomes: -
∫ bu · c · bn dS ≈ -(u · A
)
(
ci+1 + ci ci+1 + ci · A ) -Q · 2 2
)
(3)
Taking the difference between eqs 2 and 3, we obtain an estimation of the numerical diffusion introduced by the tankin-series model. -Q ·
(
)
ci + ci+1 Q - (-Q · ci) ) - · (ci+1 - ci) 2 2
(4)
This has the same structure as the numerical approximation of true diffusion which reads as follows for the right edge of the control volume:
∫D A
n T,eff · ∇c · b
dS ≈ DT,eff · A ·
(
)
DT,eff · A ci+1 - ci ) · (ci+1 - ci) ∆x ∆x
In fact, we can express eq 4 in terms of a diffusion coefficient DT,comp: -
DT,comp · A Q Q ∆x · (ci+1 - ci) ) · (ci+1 - ci) with DT,comp ) · 2 ∆x 2 A
If we subtract this from the case for true turbulent diffusion, we get a compartmental model with largely reduced numerical diffusion, or in other words which compensates excess in diffusion due to the model assumptions. Turbulence is now represented by the adjusted turbulent diffusion coefficient DT,adj DT,adj ) DT,eff - DT,comp
(5)
To keep the model numerically stable, turbulent fluxes have to exceed this compensation. This is consistent with demanding the grid Peclet number to be smaller than 2, which is a common criterion for grid-size determination (21, 22). Turbulence in a Compartmental Model. Turbulence has to be transformed from a turbulent diffusion coefficient into a flux in order to fit into the framework of compartmental models. This is done by introducing back-mixing between adjacent volumes. The relationship between the back-mixing flowrate Reff and the turbulent diffusion coefficient DT,eff is given by: DT,eff ·
ci+1 - ci · A ) Reff · ci+1 - Reff · ci ∆x
For the back-mixing flowrate Reff, we obtain: Reff )
DT,eff · A ∆x
Using the adjustment of the diffusion coefficient (eq 5), the adjusted back-mixing flowrate R becomes: R ) (DT,eff - DT,comp) ·
DT,eff · A Q A Q ) - ) Reff ∆x ∆x 2 2
(6)
Since R has to be a positive number, it also imposes a constraint on the minimal discretisation needed in the compartmental model (see Discussion). If variable grid
spacing and variable cross sections are allowed, we get a similar relationship to the derivation above for the backmixing flowrate: Ri ) Ri,eff -
Qi 2 · ∆xi 2 · Ai · DT,eff,i with Ri,eff ) 2 ∆xi+1 + ∆xi ∆xi+1 + ∆xi
(7)
which is not much different from eq 6 for smooth transitions in ∆x. Cell-Merging Algorithm: Merging Based on Similar Concentrations. Because we focus on homogeneous reactions in continuous flow systems, the aggregation strategy has to preserve the residence time distribution (RTD) and the time of mixing (1). The RTD is determined by the macroscopic flow field, including mean advection, dispersion over the cross section, and recirculation as well as dead zones. Time of mixing relates to the evolution of the RTD along the reactor and is determined by the local mixing intensity (turbulent diffusion). While time of mixing mainly demands a locally accurate representation of turbulent diffusion processes, the representation of major flow features is linked directly to the structure of the compartmental model. The compartmental model presented in this work has the characteristics of a one-dimensional model. Independent of the complexity of reactor geometries, most continuousflow systems have one dominant (spatial) dimension leading from the inlet to the outlet of the reactor. This dimension becomes obvious by taking the concentration of a reactive tracer that is either produced or degraded as a measure. However, one dimension is insufficient to capture flow features such as internal recirculations. For this reason, we introduce an additional dummy dimension that allows these structures to be resolved. The basic principle of model derivation showing the preservation of important flow structures is explained with the aid of Figure 2, which serves as an illustrative example. We start with a CFD simulation based on the classical approach of the Reynolds decomposition and statistical modeling of turbulence by a turbulence model (RANS simulation). The vector field in Figure 2 (left) is a result of such a simulation and shows the direction of the mean flow. The steady state solution of the flow field is used to calculate the reactive transport of a species. If a zero-order reaction (production) is used, a special physical interpretation may be made: merging will be based on the mean age of water parcels at a given location in the reactor. Shaded areas in Figure 2 show the scalar field of concentration obtained this way. For illustration only five levels are shown, the first of them at the inlet. Computational cells between two iso levels of concentration are merged to compartments (e.g., cells between levels 4 and 5 result in the yellow areas in Figure 2, left and right). This gives the basic structure of the compartmental model as a cascade of volumes with different sizes. Within one layer, a further distinction is made between areas where the concentration increases in the flow direction (dark shades) and those where it decreases in the flow direction (light shades). This is the case when water parcels mix with younger water, leading to a decrease in concentration, which is a typical feature of a recirculation zone. The cascade can therefore be split into two lines to resolve this structure. This way, a compartmental model with a total of 2n volumes is obtained, where n is the number of iso-levels: n volumes are attributed to forward flow and n volumes are attributed to backward flow. In the approach presented here, computational cells that build one volume of the compartmental model do not necessarily have to be connected. Due to flow separation, isolated regions with similar concentrations could evolve. These regions are merged to one volume.
FIGURE 2. Visualization of the merging concept. 2D-CFD simulation for a reactor with a distinct recirculation zone. Areas between iso-levels of concentration are colored (left). Compartmental model based on the CFD simulation (right). Colored areas correspond to each other. Only advective fluxes are shown. Because they are functionally identical, this procedure is feasible as it significantly reduces the number of volumes needed. Preserved Flow Features. The compartment model represents advection by a sequence of ideally mixed reactors. Here, only five levels are used to clarify the basic concept. They are certainly not enough to adequately represent advection. Recirculation zones are important flow features leading to tailing in the RTD. When recirculation occurs, there must not only be a zone with increasing concentration in the flow direction but another one with decreasing concentration in that direction. This property is represented by the condition ∇c · b v > 0 for forward flow and ∇c · b v < 0 for backward flow. The dividing line between the two flow regimes satisfies ∇c · b v ) 0. Gradients in mean velocity give rise to concentration gradients in the cross section of a reactor (dispersion). If the flow is not mixed at the outlet (e.g., by an overfall or by contraction in an outlet pipe), this effect is substantial and is accounted for by taking outflow from different volumes of the compartmental model. Delimitation of Volumes and Quantification of Fluxes. After determining the resolution (number of levels n), the computational cells are merged to compartmental volumes using the following rules: Cells that fulfill c ∈(ci-1,ci] and ∇c · b v g 0 are merged to Vi,for Cells that fulfill c ∈(ci-1,ci] and ∇c · b v < 0 are merged to Vi,back where c0 ) cmin and ci ) cmin + i · ∆c with i from 1 to n and ∆c ) (cmax - cmin)/n The advective links are quantified by integrating flow velocities over the iso-line (2D) or iso-surface (3D) that serves as an interface between adjacent volumes. v ·b n dAi for cell surfaces with b v ·b ng0 Qi,forward ) ∫Ai b v ·b n dAi for cell surfaces with b v ·b n
