A multiscale hybrid simulation method for simulating biofouling in membrane channels

Abstract

The application discloses a multiscale hybrid simulation method for simulating biological contamination in a membrane channel, and through an innovative quasi-steady coupling strategy, a finite element model describing continuous multi-physical fields is efficiently integrated with a cellular automaton model describing discrete biological behaviors. In the method, by introducing a Darcy penalty term in the fluid dynamics equation, the macroscopic morphology of the biological membrane is dynamically equivalent to a porous medium, and the blocking effect of the biological membrane on the flow field is accurately captured. Meanwhile, in the cellular automaton model, a unique spatial expansion mechanism containing chain pushing and macroscopic desorption is adopted, and the spatial spread and peeling of the biological membrane due to internal growth are simulated with high fidelity; the application can realize high-resolution dynamic simulation of the long-period evolution process of biological contamination of the membrane system at a low calculation cost, and provides a scientific and reliable simulation tool for the research and development of anti-pollution membrane elements and the optimization of operation strategies.

Description

Technical Field

[0001] This invention relates to the field of water treatment technology, specifically to a multi-scale hybrid simulation method for simulating biological contamination within membrane channels. Background Technology

[0002] Membrane fouling remains a major technical bottleneck restricting performance and increasing operation and maintenance costs. Membrane fouling takes many forms, among which biofouling, due to its complexity and destructive nature, is widely recognized as one of the most challenging. Biofouling refers to the attachment, reproduction, and secretion of extracellular polymers by microorganisms on the membrane surface and feed spacer, ultimately forming a biofilm with a complex three-dimensional structure. The presence of this biofilm directly leads to a series of serious problems, including: significantly increased operating pressure drop, continuously decreasing permeate flux, increased frequency of chemical cleaning, and even irreversible damage to membrane elements, severely impacting the economics and stability of such membrane systems.

[0003] To mitigate biofouling, existing technologies primarily focus on optimizing the physical structure of pressure-bearing membrane elements (such as improving the geometry of the feed gasket) and optimizing operating conditions. For example, traditional mesh-like diamond gaskets aim to enhance mass transfer and reduce contaminant deposition by altering the hydrodynamic conditions near the membrane surface. However, these structures have limited capacity to improve mass transfer and often come at the cost of significantly increased channel pressure drop and higher energy consumption. Therefore, developing novel and highly efficient antifouling membrane element structures and formulating more scientific operating strategies have become research hotspots in this field.

[0004] In the development of the aforementioned anti-fouling structures and strategies, traditional research methods mainly rely on experiments. However, experiment-driven research models, especially pilot-scale or field-scale experiments, are not only costly and time-consuming, but more importantly, they are difficult to capture and observe in real time and at high resolution the complex hydrodynamics, mass transport, and key dynamic information of biofilm evolution within the membrane channels (such as around the mesh fibers).

[0005] Numerical simulation has become a powerful tool as an alternative. However, existing simulation models and methods still have significant limitations. On the one hand, many models, in order to simplify calculations, often ignore the dynamic and strong coupling feedback between the three-dimensional spatial morphology of biofilms and hydrodynamic and mass transfer processes. On the other hand, models capable of handling such complex couplings are usually computationally very expensive, making them difficult to apply to simulating the long-term evolution of biofouling processes that can last for days or even weeks, and failing to achieve an effective balance between physical fidelity and computational efficiency. In addition, existing methods generally lack high-fidelity and computationally efficient descriptions of key physical behaviors within biofilms, such as spatial pushing, spreading, and macroscopic desorption caused by growth. Summary of the Invention

[0006] Purpose of the invention: The purpose of this invention is to provide a multi-scale hybrid simulation method for simulating biocontamination within membrane channels, thereby solving the problems of high cost, low efficiency, and low fidelity in existing technologies.

[0007] Technical solution: The multi-scale hybrid simulation method for simulating biofouling within membrane channels according to the present invention includes the following steps:

[0008] (1) Establish a computational domain containing the geometry of the membrane and the influent grid, and discretize the computational domain into a cellular automata grid, and set the initial biomass spatial distribution;

[0009] (2) Advance by time step, and perform the following operations sequentially within each time step:

[0010] (21) Transfer the current biomass spatial distribution to a physical field solver based on continuum mechanics. In the solver, the biofilm region is equivalent to a porous medium, and the steady-state fluid velocity field and steady-state substrate concentration field are obtained by solving.

[0011] (22) The steady-state velocity field and substrate concentration field are interpolated and mapped onto the cellular automata grid;

[0012] (23) In a cellular automaton model, update biomass growth based on the mapped local substrate concentration and handle the attachment events of new biomass;

[0013] (24) In the cellular automata model, cells with biomass exceeding a preset threshold are identified, and spatial expansion and desorption mechanisms are triggered to handle excess biomass.

[0014] (25) Repeat steps (21)-(24) for coupled iteration until the simulation reaches the preset total duration.

[0015] Furthermore, in step (21), the biofilm region is equivalent to a porous medium by introducing a Darcy penalty term into the fluid dynamics equation. The magnitude of the Darcy penalty term is related to the permeability of the biofilm region.

[0016] Furthermore, the local permeability related to the Darcy penalty term is dynamically determined by a binary permeability model. In each time step, the model defines the cell region in the computational domain where the local biomass concentration exceeds a preset biomass threshold as a biofilm region with a preset extremely low permeability, while other regions are defined as fluid regions with extremely high permeability, thereby transforming the macroscopic morphology of the biofilm into a spatially varying fluid resistance field.

[0017] Furthermore, in step (21), the steady-state fluid velocity field is obtained by solving the Navier-Stokes equations, which include at least inertial terms, pressure gradient terms, and viscous forces; the steady-state substrate concentration field is obtained by solving the mass transport equations, which balance the convection terms, diffusion terms, and bioreaction consumption terms related to local biomass concentration and substrate concentration, which describe changes in substrate concentration.

[0018] Furthermore, the bioreaction consumption term in the material transport equation is determined based on a pre-defined microbial growth kinetic model.

[0019] Furthermore, the microbial growth kinetic model is the Monod kinetic model, which couples the specific growth rate of biomass with the local substrate concentration through a nonlinear functional relationship.

[0020] Furthermore, in step (24), the spatial expansion mechanism is as follows: when all the neighboring positions of a cell are non-empty cells, a chain pushing mechanism is activated to transfer excess biomass along a selected direction.

[0021] Furthermore, when the iterative transfer along all selectable directions is blocked due to encountering solid boundaries or reaching a preset number of transfer steps, the chain-pushing mechanism identifies the excess biomass that cannot be accommodated as a macroscopic desorption event and removes it from the computational system of the cellular automata model to achieve adaptive regulation of biofilm volume.

[0022] Furthermore, in step (25), the quasi-steady-state coupling strategy is used. The strategy takes advantage of the physical property that the time scale of biofilm growth is much larger than the time scale of hydrodynamic relaxation. In each coupling time step, the dynamically evolving biomass distribution is used as the static boundary condition for solving the steady-state physical field, thereby significantly reducing the computational cost of long-period evolution while ensuring the high fidelity of the physical process.

[0023] Furthermore, in step (1), the initial biomass spatial distribution is set on a predetermined surface based on a preset pollution scenario. The predetermined surface is selected from at least one of the following: the surface of the reverse osmosis membrane; the surface of the feed grid.

[0024] Beneficial Effects: Compared with existing technologies, this invention has the following significant advantages: This invention efficiently integrates a finite element model describing continuous multiphysics fields with a cellular automata model describing discrete biological behavior through an innovative quasi-steady-state coupling strategy. In this method, by introducing a Darcy penalty term into the fluid dynamics equations, the macroscopic morphology of the biofilm is dynamically equated to a porous medium, accurately capturing its blocking effect on the flow field. Simultaneously, the cellular automata model employs a unique spatial expansion mechanism including chain-like pushing and macroscopic desorption, faithfully simulating the spatial spread and peeling of the biofilm due to internal growth. Compared with existing technologies, this invention can achieve high-resolution dynamic simulation of the long-term evolution of biofouling in this type of membrane system at a lower computational cost, providing a scientific and reliable simulation tool for the research and development of antifouling membrane elements and the optimization of operational strategies. Attached Figure Description

[0025] Figure 1 This is a flowchart of the coupling iteration process of the present invention;

[0026] Figure 2 This is a schematic diagram of the geometric structure of the simulation computation domain of the present invention;

[0027] Figure 3 This is a schematic diagram of the Darcy penalty term coupling mechanism of the present invention;

[0028] Figure 4 This is a schematic diagram of the "chain-like pushing" space expansion mechanism of the present invention;

[0029] Figure 5 This is a schematic diagram of the biomass desorption mechanism of the present invention. Detailed Implementation

[0030] The technical solution of the present invention will be further described below with reference to the accompanying drawings.

[0031] like Figure 1 As shown, this embodiment of the invention provides a multi-scale hybrid simulation method for simulating biofouling within membrane channels. Figure 1The coupled iterative process of this method is illustrated in detail. The entire simulation process progresses in discrete time steps. At the beginning of each time step, the cellular automata model (implemented in MATLAB in this example) provides the three-dimensional spatial distribution information of the biofilm at the current moment. This information is passed to the finite element model (implemented in COMSOL Multiphysics in this example) to update the physical properties (such as fluid resistance field and bioreaction rate field) in the computational domain. Subsequently, the finite element model solves for the steady-state fluid velocity field and substrate concentration field at that moment. These continuous physical field results are interpolated and then passed back to the cellular automata model as environmental conditions driving discrete behaviors such as biomass growth, attachment, and spatial expansion within that time step. This iterative cycle repeats continuously until the preset total simulation time is reached, thereby achieving dynamic simulation of the long-term evolution of biofouling. The process includes the following steps:

[0032] Step 1: Constructing a Continuous Physical Field Model (Microscopic Scale) At the microscopic scale, this embodiment employs a continuous medium model based on the finite element method (FEM) to solve for the key physical fields controlling the evolution of macroscopic biofilms—namely, the fluid velocity field and the nutrient substrate concentration field. The computational domain of this model accurately includes the complex geometry of the influent mesh, describing the hydrodynamics and mass transport processes within the channel with high fidelity. For example... Figure 2 As shown in the figure, this is a schematic diagram of the geometric structure of the simulation computational domain used in this invention. This diagram visually illustrates the physical space of the simulation, which mainly consists of two parallel membrane surfaces and feed spacer filaments placed between them. This geometric structure accurately replicates a typical repeating unit in a reverse osmosis membrane element. The complex configuration of the feed spacer is a key factor affecting the hydrodynamic conditions within the channel, mass transfer efficiency, and the spatial distribution of biofilm attachment and growth. Therefore… Figure 2 The computational domain shown is the foundation of the entire multi-scale simulation method. All subsequent calculations regarding fluid flow, substrate concentration distribution, and biofilm evolution are performed within this precisely defined geometric space.

[0033] Step 11: To ensure the physical realism and computational feasibility of the model, the following boundary conditions were applied in this embodiment: Inlet boundary: A fully developed laminar velocity profile was given, with an average velocity of 0.163 m / s; the inlet substrate concentration was set to a constant value. Outlet boundary: Zero-pressure reference and stress-free conditions were applied to allow free outflow; substrate transport was handled using outflow flux conditions. Solid wall: All membrane surfaces and mesh surfaces were set to no-slip boundary conditions; for solute transport, an impermeable zero-flux boundary was used. Lateral boundary: Periodic boundary conditions were used to simulate an infinitely extending mesh structure and eliminate boundary effects.

[0034] Step 12: Introducing the fluid dynamics model with the Darcy penalty term: In this embodiment, it is assumed that the fluid is a Newtonian fluid and the flow is laminar, with the flow field within the channel controlled by the steady-state incompressible Navier-Stokes equations. To accurately simulate the drag effect of biofilm dynamic growth on the flow field, this embodiment introduces the Darcy penalty term, thus treating the biofilm region as an equivalent porous medium. The modified momentum equation is specifically expressed as follows:

[0035]

[0036] in, It is the fluid density. It is a velocity vector. It's pressure. It is dynamic viscosity. It is the Darcy penalty source term that acts on the fluid as an additional volume force.

[0037] Darcy's Punishment Source Item The mathematical expression is:

[0038]

[0039] In the formula, It is the permeability of the medium, characterizing the ability of a porous medium to allow fluid to pass through.

[0040] This embodiment employs an efficient binary permeability model to achieve coupling between the discrete biomass field and the continuous flow field. Figure 3 The principle of this coupling mechanism is illustrated schematically. It demonstrates the information transformation process: the discrete biomass distribution in the cellular automata (CA) model is transferred and mapped to the continuous computational domain of the finite element model. In this process, the region occupied by the biofilm is assigned an extremely low permeability ( This results in significant fluid resistance; while the fluid region is endowed with extremely high permeability (…). To ensure it remains unaffected, a biomass threshold is first defined. For any location within the computational domain, its permeability... The value depends on the local biomass concentration at that location calculated by the cellular automata model. :

[0041]

[0042] In this expression, It is a preset effective permeability constant representing a mature biofilm, and its value is set to be extremely small to generate strong flow resistance. In the fluid region, This causes the Darcy penalty term to disappear naturally. In this way, the biomass field updated by the cellular automaton at each time step can be directly transformed into a spatially "on-off" Darcy penalty field, efficiently capturing the "blocking" effect of the biomembrane on the mainstream.

[0043] Step 13: Substrate Transport and Bioreaction Model: This embodiment assumes that biomass growth is limited by the concentration of a soluble substrate (e.g., dissolved oxygen). The steady-state substrate distribution in the liquid and biofilm phases is controlled by equations incorporating convection, diffusion, and reaction terms:

[0044]

[0045] in, It is the substrate concentration. It is the effective diffusion coefficient. The velocity field is obtained by solving a fluid dynamics model. This is the volumetric reaction rate (i.e., the substrate consumption rate). In this embodiment, to simplify the model parameters, it is assumed that the effective diffusion coefficient of the matrix inside the biofilm is the same as that in the liquid phase.

[0046] Biomass specific growth rate The Monod dynamic model is used for description:

[0047]

[0048] in It is the maximum specific growth rate. It is the half-saturation constant. This refers to the substrate concentration and the substrate consumption rate. Proportional to the rate of biomass growth, as indicated by the yield coefficient. (Convert the biomass produced per unit mass of substrate consumed) to:

[0049]

[0050] in, The volumetric reaction rate, or substrate consumption rate, describes the amount of nutrients consumed per unit volume due to metabolic activity within the biomembrane. The yield coefficient is defined as the biomass produced per unit mass of substrate consumed. The volume growth rate of biomass. The maximum specific growth rate is the theoretical maximum growth rate of microorganisms when the substrate supply is sufficient and unrestricted. This refers to the local substrate concentration, which is the concentration of nutrients that microorganisms depend on for growth in a specific local area. The half-saturation constant is equal to the substrate concentration required for the specific growth rate to reach half of its maximum value, reflecting the affinity of microorganisms for the substrate. Local biomass concentration, representing the density or quantity of biofilm at that location.

[0051] Step 2: Constructing a Discrete Dynamic Growth Model of the Biofilm (Macroscale): At the macroscale, this embodiment uses a cellular automata (CA) model to simulate key discrete behaviors of the biofilm, such as attachment, growth, spatial expansion, and detachment. The computational domain is uniformly discretized into... Each rectangular cell.

[0052] Step 21: Basic Model Setup: Define two state variables for each cell that evolve over time: one is biomass concentration. , representing time Time located in cell The biomass within. Secondly, the occupied state variables. At a specific moment The value represents the cell Attributes.

[0053] The occupied state variable The specific meanings are as follows: This represents a liquid phase empty cell. This state is dynamic, and its value can become 1 during subsequent attachment or expansion steps. This represents a cell occupied by a biofilm. This state is also dynamic, and may change from 0 to 1 due to an attachment event. This represents a permanent solid structure (such as a mesh). Once a cell is defined as a solid structure at the initial moment, its state will remain constant at 2 throughout the simulation and will not participate in any state transitions. Based on the above definition, the state matrix... As a whole, it can dynamically reflect the spatial distribution changes of biofilm and fluid phase over time. The interactions between cells are defined by the von Neumann neighborhood, that is, each cell interacts directly only with its six neighbors above, below, left, right, front, and back.

[0054] Step 22: Adhesion Mechanism: This embodiment defines three adhesion mechanisms: membrane surface adhesion only, spacer surface adhesion only, and simultaneous adhesion of the membrane and spacer surfaces. At each time step... Inside, according to the preset adhesion rate Determine the number of new cells to be attached. From all current potential attachment sites (i.e., empty liquid cells adjacent to the membrane or septum), random samples are drawn without replacement. Each cell updates its state to "occupied". and assign a small initial biomass. .

[0055] Step 23: Biomass growth: For each cell occupied by a biofilm Its biomass is determined by the specific growth rate based on the local substrate concentration transferred from the microscopic model. Update with Monod dynamics:

[0056]

[0057] in, It is the concentration of biomass. It is a comparison of growth rate. It is the set time step.

[0058] Step 24: Spatial Expansion and Desorption Mechanism: When the biomass concentration of any cell exceeds a preset expansion threshold... At this time, spatial expansion and pushing events are triggered to simulate spatial sprawl. The detailed logic of this mechanism is as follows:

[0059] (a) Traversal and selection: Traverse all occupied cells and mark cells that exceed the threshold as cells to be expanded.

[0060] (b) Priority Expansion: For each cell to be expanded, its excess biomass is used as the payload to be transferred. Its six neighbors are searched in a random order; if a liquid-phase empty cell is found... If so, the biomass payload is directly transferred to the empty cell, and its state is updated to "occupied". .

[0061] (c) Chain pushing: such as Figure 4 As shown, a "chain push" mechanism is used to address the problem of no expandable space around cells with excess biomass. If all neighbors are not empty cells (i.e., all are occupied by biofilm or solids), a neighbor already occupied by a biofilm is randomly selected, and the chain push process is initiated. The biomass load is exchanged with the biomass of that neighboring cell, and this neighbor is used as a new starting point to continue searching for empty cells in the same direction.

[0062] (d) Desorption simulation: such as Figure 5 As shown, in order to simulate the macroscopic desorption mechanism of biomass, the failure of the "chain pushing" process is handled. If the pushing process encounters solid cells... If no empty cells are found, either by calculating the domain boundary or by reaching the preset maximum number of push steps, the expansion is considered a failure. The biomass load that cannot be accommodated in this area is considered to have detached due to excessive internal compressive stress and is removed from the system. This mechanism provides a crucial self-regulating function for the macroscopic morphology of biofilms.

[0063] Step 3: Quasi-steady-state coupling mechanism of multi-scale model: One of the core innovations of this embodiment lies in the adoption of an efficient quasi-steady-state coupling strategy, which is based on the physical fact that there is a huge timescale separation between hydrodynamic relaxation (seconds) and biofilm growth (hours to days). Figure 1 (Diagram of the coupled iterative process) As shown, the long-term evolution of biofilms is decomposed into a series of discrete time steps. At the beginning of each coupled iterative step, the spatial distribution of biomass at the current time, calculated in a cellular automata model (e.g., implemented in MATLAB), is used. This data is passed to a finite element model (e.g., implemented in COMSOL Multiphysics software). The finite element model maps this biomass distribution to a spatially varying permeability field (i.e., the Darcy penalty term) and a biological reaction rate field, and then solves for the steady-state velocity field at that moment. and substrate concentration field .

[0064] Subsequently, these steady-state continuous physical fields calculated at high resolution are transmitted back to the macroscopic cellular automata grid via an interpolation algorithm, serving as constant environmental conditions driving biomass attachment, growth, and spatial expansion within that time step. Through this iterative data exchange between the two models, this invention efficiently captures the dynamic feedback between biofilm development and the fluid environment.

Claims

1. A multiscale hybrid simulation method for simulating biofouling in a membrane channel, characterized in that, The method comprises the following steps: (1) establishing a calculation domain containing a membrane and an inlet grid geometry, discretizing the calculation domain into a cellular automaton grid, and setting an initial biomass spatial distribution; (2) advancing in time steps, and in each time step, sequentially performing the following operations: (21) transferring the biomass spatial distribution at the current time to a physical field solver based on continuous medium mechanics, equivalently treating the biofilm region as a porous medium in the solver, and solving to obtain a steady-state fluid velocity field and a steady-state substrate concentration field; wherein the biofilm region is equivalently treated as a porous medium by introducing a Darcy penalty term in the fluid dynamics equation, and the size of the Darcy penalty term is related to the permeability of the biofilm region; wherein the local permeability related to the Darcy penalty term is dynamically determined by a binary permeability model, which defines the cell region in the calculation domain where the local biomass concentration exceeds a preset biomass threshold as a biofilm region with a preset extremely low permeability, and defines other regions as fluid regions with extremely high permeability, thereby converting the macroscopic morphology of the biofilm into a spatially varying fluid resistance field; (22) interpolating and mapping the steady-state velocity field and the substrate concentration field onto the cellular automaton grid; (23) in a cellular automaton model, updating biomass growth according to the mapped local substrate concentration, and processing the attachment event of new biomass; (24) in the cellular automaton model, identifying cells with biomass exceeding a preset threshold, and triggering spatial expansion and detachment mechanisms to process excess biomass; wherein the spatial expansion mechanism is as follows: when all adjacent positions of a cell are non-empty cells, a chain pushing mechanism is activated to transfer excess biomass in a selected direction; wherein when the chain pushing mechanism is blocked in iterative transfer in all available directions due to encountering a solid boundary or reaching a preset transfer step number, the excess biomass that cannot be accommodated is identified as a macro-detachment event, and is removed from the calculation system of the cellular automaton model to realize adaptive adjustment of the biofilm volume; (25) repeating steps (21)-(24) for coupled iteration until the simulation reaches a preset total time; wherein a quasi-steady-state coupling strategy is used; the strategy takes advantage of the physical characteristics that the time scale of biofilm growth is much larger than the relaxation time scale of fluid dynamics, and in each coupling time step, the dynamically evolving biomass distribution is used as a static boundary condition for solving the steady-state physical field, thereby significantly reducing the calculation cost of long-period evolution while ensuring high fidelity of the physical process.

2. The multi-scale hybrid simulation method of simulating biofouling in a membrane channel according to claim 1, wherein, In step (21), the steady-state fluid velocity field is obtained by solving the Navier-Stokes equation containing at least an inertial term, a pressure gradient term, and a viscous force; the steady-state substrate concentration field is obtained by solving the material transport equation, which balances the convection term, the diffusion term, and the biological reaction consumption term related to the local biomass concentration and the substrate concentration.

3. The multi-scale hybrid simulation method of simulating biofouling in a membrane channel according to claim 2, wherein, The biological reaction consumption term in the material transport equation is determined according to a preset microbial growth kinetics model.

4. The method of claim 3, wherein, The microbial growth kinetics model is a Monod kinetics model, which couples the specific growth rate of biomass and the local substrate concentration by a non-linear function.

5. The method of claim 1, wherein, In step (1), setting the initial biomass spatial distribution is performed on a predetermined surface according to a predetermined pollution scenario, the predetermined surface being selected from at least one of: a surface of a reverse osmosis membrane; a surface of an intake grid.

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