Method for simulating fluid flow, method for designing mixing reactor, mixing reactor and method for controlling mixing reactor, and data processing device, computer program and computer readable medium

Abstract

The invention relates to a computer-implemented method for simulating a fluid flow, in which a computational grid (20) having grid points (30) is provided for calculating a distribution function (f, 2) of the fluid flow and / or a distribution function (f, 2) of a mixture of at least two substances at the grid points (30), comprising a lattice Boltzmann method, the lattice Boltzmann method has: a collision step in which the difference between the distribution function (f) and the equilibrium solution (feq) is calculated at each grid point (30); and a streaming step wherein the distribution function (f) at each grid point (30) is updated according to the streaming step; and / or the method comprises solving a lattice Boltzmann equation of the distribution function (f, 2), characterized in that the distribution function (f, 2) is represented in a tensor chain format and / or all operations performed for calculating the distribution function (f, 2) are performed using a distribution function (f, 2) in a tensor chain format, the distribution function (f, 2) and / or the tensor chain format comprising at least one tensor chain (3), the tensor chain (3) has at least one tensor chain core (4). The invention further relates to a method for designing a mixing reactor, to a mixing reactor, to a method for controlling a mixing reactor, and to a corresponding data processing device, to a corresponding computer program and to a corresponding computer-readable medium.

Description

Field of the Invention

[0001] The present invention relates to a computer-implemented method for simulating fluid flow, a computer-implemented method for designing a mixing reactor, a mixing reactor, and a computer-implemented method for controlling a mixing reactor. The present invention further relates to a data processing device configured to execute these computer-implemented methods, a corresponding computer program, and a corresponding computer-readable medium. Background Art

[0002] Known methods for simulating fluid flow include the Lattice Boltzmann method (LBM) and / or solving the Lattice Boltzmann equation. It is well known that the Lattice Boltzmann method is a discretization of the fundamental kinetic Boltzmann equation on a uniform lattice. Both the Lattice Boltzmann method and the Lattice Boltzmann equation are often used in computational fluid dynamics.

[0003] It is well known that the Navier-Stokes equation can be obtained from the Lattice Boltzmann method and / or the Navier-Stokes equation can be reproduced by the Lattice Boltzmann method, see, for example, Chen and Doolen, “Lattice Boltzmann method for fluid flows”, Annual review of fluid mechanics, 30(1):329–364, 1998 or Benzi, Succi and Vergassola, “The Lattice Boltzmann equation: theory and applications”, Physics reports (Review Section of Physics Letters) 222, No. 3, 145-197, 1992.

[0004] For complex flow geometries and / or high Reynolds number flows or mixtures, a fine grid (computational grid) is required, which in turn means a large number of grid points. At the same time, the way the velocity discretization occurs is such that in one time step, a particle can move exactly and / or only move to adjacent vertices of the lattice or remain in place.

[0005] Both of these effects mean that a large amount of memory is required. Usually, workstations do not have enough memory for this purpose, so a computer cluster or a distributed computing system with many distributed nodes needs to be adopted. This means that such simulations are mostly limited to academic purposes and / or situations where such computer systems are accessible.

[0006] In addition, due to the limited memory of each node, a large amount of communication may be required between nodes, so the computing speed may be reduced. That is to say, a large amount of network traffic may occur, and / or a large amount of data may have to be shared or transferred between nodes. Specifically, a node may need to wait for one or more other nodes before it can continue its own calculation. In some scenarios, the total amount of memory may be insufficient, so some data will be dumped to a storage device, and / or memory swapping will occur, thus greatly reducing the speed. Summary of the Invention

[0007] The present invention overcomes these obstacles. Compared with the prior art methods, the method according to the present invention requires much less memory. Therefore, compared with the methods known in the prior art, the method according to the present invention can be faster, more accurate and / or can achieve more complex geometries and / or higher Reynolds numbers.

[0008] Therefore, an object of the present invention is to provide a method for simulating fluid flow, a method for designing a mixing reactor, and a method for controlling a mixing reactor, which methods require less memory, can be faster and / or can be used on more conventional computing devices.

[0009] The method for designing a mixing reactor can allow the design of more complex geometries of the reactor with higher accuracy and within a reasonable time range. Further, in some embodiments, the method for designing a mixing reactor may be suitable for being executed by a workstation or a relatively small distributed system, and compared with the known methods, this method may be cheaper, more easily available and / or more feasible. Since the computing speed can be increased, it is easier to compare different designs.

[0010] The method for controlling a mixing reactor can achieve better control and / or more accurate mixing. Since the computing speed can be increased, in some embodiments, when mixing occurs in the controlled reactor, the results of the method can be used and / or control commands can be generated or updated, so that control commands can be generated and / or updated "in real time" and / or during the operation of the mixing reactor. In some embodiments, control commands can be generated and / or adjusted, such as reacting and / or responding to the actual mixing and / or flow in the mixing reactor. The method can achieve good mixing quality and / or mixing product quality, and / or reach or at least approach a given or desired quality.

[0011] Another object of the present invention is to provide a mixing reactor designed according to a method for designing a mixing reactor. A further object of the present invention is to provide a data processing device for performing one, several or all of the methods of the present invention, a computer program comprising instructions for performing one, several or all of the methods of the present invention, and a computer-readable medium comprising instructions for performing one, several or all of the methods of the present invention.

[0012] This object can be achieved by the method for simulating fluid flow according to claim 1, the method for designing a mixing reactor according to claim 5, the mixing reactor according to claim 7, the method for controlling a mixing reactor according to claim 8, the data processing device according to claim 13, the computer program according to claim 14, and the computer-readable medium according to claim 15. Particularly preferred embodiments are the subject matter of the dependent claims.

[0013] A first aspect of the present invention relates to a computer-implemented method for simulating fluid flow, wherein a computational grid having grid points is provided to calculate a distribution function of the fluid flow and / or a distribution function of a mixture of at least two substances at the grid points, the method comprising a lattice Boltzmann method, the lattice Boltzmann method having

[0014] - a collision step, wherein the difference between the distribution function and the equilibrium solution is calculated at each grid point; and

[0015] - a streaming step, wherein the distribution function at each grid point is updated according to the streaming step;

[0016] and / or the method comprises solving the lattice Boltzmann equation for the distribution function, wherein the distribution function is represented in a tensor chain format, and / or all operations performed for calculating the distribution function are performed in a tensor chain format.

[0017] The distribution function and / or the tensor chain format comprises at least one tensor chain, the tensor chain having at least one tensor chain core. The distribution function may be or comprise a tensor. The tensor chain format may comprise at least one tensor chain.

[0018] The distribution function may be a probability density function. The distribution function may be the probability density function of finding a fluid particle (e.g., finding a fluid particle at a grid point). The distribution function may be the fluid particle density.

[0019] In some embodiments, the distribution function may be the probability density function of finding particles of a substance in the mixture (e.g., finding particles of a substance at a grid point). The distribution function may be the substance density.

[0020] The equilibrium solution can be given by the following formula

[0021]

[0022] where e k can be the lattice velocity in the direction k, w k can be the weighting coefficient, ρ α can be the fluid density of substance α, u can be the macroscopic velocity, and / or c can be the lattice velocity. The superscript α can indicate the substance. It can be stipulated that the fluid density can be determined by the following formula

[0023]

[0024] The macroscopic velocity can be determined by the following formula

[0025]

[0026] In some embodiments, the flow can include a mixture of at least two substances α. A certain distribution function or the distribution function can be calculated for each substance α. In some embodiments, the flow can include a single substance, for example, when simulating the distribution function of a fluid flow without mixing. In some embodiments and / or in some of these cases, the index α can be removed from the equation and / or be implicit.

[0027] When simulating a mixture of at least two substances, it can be stipulated that the macroscopic velocity can be determined by the following formula

[0028]

[0029] This may be equivalent to density-weighting the corresponding macroscopic velocities of the corresponding substances.

[0030] It can be stipulated that in the collision step, the intermediate distribution function can be given and / or calculated by the following formula

[0031]

[0032] In the streaming step, the distribution function at the grid points can be updated by the following formula

[0033]

[0034] In some embodiments, it can be stipulated that the collision step can correspond to the right-hand side (r.h.s) in the following formula and the streaming step corresponds to the left-hand side (l.h.s.)

[0035]

[0036] In some embodiments, the lattice Boltzmann equation can be given by the following formula

[0037]

[0038] Here, the index α can be omitted and / or be implicit. The lattice Boltzmann equation can be different and / or include additional terms, depending on the application and / or the flow or mixture to be simulated. In some embodiments, as will be apparent and / or known to those skilled in the art, the lattice Boltzmann equation can include at least one term for external force and / or temperature effect.

[0039] Solving the lattice Boltzmann equation can include solving a system of linear equations

[0040]

[0041] In some embodiments, solving the lattice Boltzmann equation can include solving a system of linear equations

[0042]

[0043] The tensor chain can be or include a quantized tensor chain. The tensor chain core of the tensor chain can be shaped and / or refined into a quantized tensor chain core.

[0044] The tensor chain can have a tensor chain core corresponding to the grid coordinates respectively. Alternatively or additionally, the tensor chain can have a tensor chain core corresponding to the rate components of the flow, substance, and / or mixture.

[0045] In some embodiments, the tensor chain can have at least one tensor chain core corresponding to substance α. Alternatively or additionally, in some embodiments, a tensor chain is provided for each substance α.

[0046] In some embodiments, it can be stipulated that the tensor chain format includes a tensor chain having a substance index. In some embodiments, the tensor f(x, v, α) and / or f(x, v, t, α) can be provided. In some embodiments, a tensor can be provided for the distribution function, and the distribution function includes at least two, multiple, and / or all substances. In some embodiments, a single tensor can be provided for the distribution function, and the distribution function includes all substances.

[0047] Alternatively, it can be stipulated that at least two substances and / or each substance has a corresponding tensor chain and / or is given its own tensor chain. In some embodiments, a corresponding tensor f(x, v) and / or f(x, v, t) can be provided for each substance.

[0048] In some embodiments, in the case of two-dimensional flow and / or two-dimensional mixture, the symbol may correspond to or be equal to the symbol f(x1, x2, v x , v y , t, α). In some embodiments, in the case of three-dimensional flows and / or three-dimensional mixtures, the symbol may correspond to or be equal to the symbol f(x1, x2, x3, v x , v y , v z , t, α).

[0049] Another aspect of the present invention relates to a computer-implemented method for designing a mixing reactor for mixing at least two substances, wherein the method comprises providing an initial mixing reactor geometry having a computational grid with grid points, and the method comprises the following steps:

[0050] a) Simulating the mixing of at least two substances and / or the flow in and / or through the mixing reactor via any of the methods according to the present invention, and determining characteristic properties of the mixture and / or the fluid flow;

[0051] b) Modifying the mixing reactor geometry and / or the grid points, repeating the simulation using the modified mixing reactor geometry and / or grid points, and determining the characteristic properties of the mixture;

[0052] c) Comparing the characteristic properties with target values of the characteristic properties;

[0053] d) Iteratively repeating steps b) and c), and optimizing the mixing reactor geometry and / or the computational grid such that the mixing reactor geometry and / or the computational grid are optimized with respect to the characteristic properties and / or until a given accuracy of the characteristic properties relative to the target values is achieved.

[0054] The characteristic properties may be or may include at least one of a mixing fraction, a substance concentration, a characteristic mixing time, a characteristic residence time, and / or a mixing progress. In some embodiments, the characteristic properties may be or include properties derived from and / or related to at least one of a mixing fraction, a substance concentration, a characteristic mixing time, a characteristic residence time, and / or a mixing progress.

[0055] Modifying the mixing reactor geometry may include moving boundaries and / or changing the shape of the computational grid. Modifying the grid points may include adding grid points or removing grid points from the computational grid. Modifying the grid points may include coarsening or refining the computational grid.

[0056] Another aspect of the present invention relates to a mixing reactor for mixing at least two substances, the mixing reactor having a mixing reactor geometry determined by a method of designing a mixing reactor according to the present invention. In some embodiments, the geometry is or includes the internal geometry of the mixing reactor.

[0057] Another aspect of the present invention relates to a computer-implemented method for controlling a mixing reactor, the method comprising:

[0058] - Simulating the mixing of at least two substances and / or the flow in and / or through the mixing reactor via any of the methods according to the present invention, wherein the computational grid corresponds to the geometry of the mixing reactor;

[0059] - Generating and / or adjusting at least one control command for the mixing reactor based on the simulation.

[0060] The control command may include a command for controlling at least one flow into the mixing reactor and / or at least one flow out of the reactor, optionally the rate, volumetric flow rate, and / or mass flow rate. Alternatively or additionally, the control command may include a command for controlling a stirrer, optionally the angular rate or angular frequency. Alternatively or additionally, the control command may include a command for controlling a temperature control unit (e.g., a heater and / or a cooler). The control command may include a command for controlling or setting the cooling or heating temperature.

[0061] The method may include measuring fluid properties, optionally the rate and / or substance concentration, at at least one measurement point in the mixing reactor and comparing the measured fluid properties with the corresponding fluid properties obtained by the simulation. The control command may be generated, adjusted, and / or modified based on the comparison. The fluid properties may be measured using sensors. The sensors may be located at or near the measurement point. The measurement point may correspond to a grid point of the computational grid.

[0062] The control command may be generated and / or updated during or at the time of mixing at least two substances in the mixing reactor. The mixing reactor may be controlled by the control command.

[0063] It may be provided that after generating and / or updating the control command, the simulation of the mixing may be repeated. When the simulation is repeated, the calculation of the distribution function may take into account the generated and / or updated control command. In some embodiments, when the simulation is repeated, the calculation of the distribution function may take into account at least one of the forces, flow rates, and / or boundary conditions imposed and / or applied by the control command.

[0064] Another aspect of the present invention relates to a data processing device comprising a processor configured to perform the steps of any of the methods according to the present invention.

[0065] Another aspect of the present invention relates to a computer program comprising instructions which, when executed by a computer, cause the computer to perform the steps of any of the methods according to the present invention.

[0066] Another aspect of the present invention relates to a computer-readable medium comprising instructions which, when executed by a computer, cause the computer to perform the steps of any of the methods according to the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0067] The present invention will be described in further detail with reference to the following drawings:

[0068] Figure 1 : Exemplary grid;

[0069] Figure 2 : Exemplary tensor chain decomposition;

[0070] Figure 3 : Another exemplary tensor chain decomposition;

[0071] Figure 4 : Runtime comparison of an exemplary embodiment of the method according to the present invention with methods known in the prior art for a 2D test case (upper figure) and a 3D test case (lower figure);

[0072] Figure 5 : Exemplary mixing reactor;

[0073] Figure 6 : For Figure 5 Exemplary computational grid of the mixing reactor;

[0074] Figure 7 : Figure 6 Exemplary modified computational grid; and

[0075] Figure 8 : Figure 6 Another exemplary modified computational grid. DETAILED DESCRIPTION

[0076] The method according to the present invention may comprise and / or relate to the lattice Boltzmann method (LBM). The collision step and / or the streaming step may comprise, relate to the lattice Boltzmann method (LBM) and / or be part thereof. The method according to the present invention may comprise and / or relate to solving the lattice Boltzmann equation.

[0077] The lattice Boltzmann method (LBM) can be a discretization of the fundamental kinetic Boltzmann equation on a lattice and / or a computational grid. The lattice and / or the computational grid can be uniform. The lattice can include, be, and / or correspond to the computational grid.

[0078] By employing the lattice Boltzmann method and / or the lattice Boltzmann equation (LBE), the distribution function can be calculated at the grid points of the computational grid.

[0079] The distribution function can be the distribution function of the rate component of the flow in a certain direction (e.g., in the lattice direction) (and / or in the direction of the computational grid and / or along an axis). In some embodiments, the distribution function and / or the said distribution function can be the distribution function of a substance.

[0080] The distribution function can be a probability density function. The distribution function can be the probability density function having a flow rate (e.g., in the lattice direction, etc.) at the grid point and / or the probability of finding a fluid particle at the grid point. The distribution function can be the probability density function having a concentration of substance α at the grid point.

[0081] In some embodiments, it can be specified that more than one distribution function can be calculated and / or solved.

[0082] In Figure 1 , an exemplary computational grid (and / or lattice) is shown. The distribution function, flow properties, and / or flow rate, etc. can be discretized and / or calculated at the grid points of the computational grid. The discretization can be D2Q9 discretization. The computational grid can be 2D (two-dimensional). For example, the discretization of the rate space and / or the substance can be carried out in nine directions (see Figure 1 ).

[0083] Other discretizations are also possible. For example, the computational grid can be 3D (three-dimensional). The discretization can be D3Q15 discretization. The discretization can be D3Q27 discretization.

[0084] In the absence of external forces and at a constant temperature, the distribution function according to LBM can be given and / or calculated by the following formula

[0085]

[0086] where k is the index of the direction (e.g., in the case of D2Q9, k = 0, 1,..., 8, see for example Figure 1 ), τ is the dimensionless relaxation time, x is the position in space, e k is the lattice velocity, α is the substance, and Δt is the time step. In other words, it can be specified that the evolution of the distribution function with space and time can be given by the above equation. Here, f eq，αis the equilibrium distribution function of substance α, which can depend on position and time.

[0087] The above equation can be solved in two steps. In the first step (which can be, include, or correspond to the collision step), the intermediate distribution function can be calculated by the following formula

[0088]

[0089] The intermediate distribution function can be based on the difference between the distribution function and the equilibrium distribution function.

[0090] Then, in the second step (which can include or correspond to the streaming step), the distribution function at time step t + Δt and position x + e k at Δt can be determined as

[0091]

[0092] This is shown for illustrative purposes in Figure 1 (right), where the fluid particle 1 moves as determined in the streaming step. The fluid particle 1 can move from grid point 30 to an adjacent and / or neighboring grid point 30. Alternatively, the fluid particle 1 can remain at its current grid point 30.

[0093] In the case of D2Q9, the lattice velocity e k can be given, for example, by the following formula

[0094]

[0095] See Figure 1 . For different discretizations, such as D3Q15 or D3Q27, the choice of e k will be different. In other words, the lattice velocity can depend on the ratio of the distance Δx between grid points to the time step Δt, and / or can be such that within the time increment Δt, the distance Δx between adjacent grid points can be traversed.

[0096] Without loss of generality, in some embodiments, Δx = 1 and Δt = 1. In some embodiments, appropriate normalization and / or non - dimensionalization can be used.

[0097] For the two - dimensional case, the notation of the distribution function can be equivalent to f(x1, x2, v1, v2, α), while for the three - dimensional case, it is equivalent to f(x1, x2, x3, v1, v2, v3, α). It can be stipulated that the distribution function (or correspondingly, the corresponding tensor) has no specific index for time t. In some embodiments, it can be stipulated that the distribution function and / or the corresponding tensor are updated and / or overwritten at each time step.

[0098] Alternatively or additionally, for a two-dimensional case, the sign of the distribution function may be equivalent to f(x1, x2, v1, v2, t, α), and for a three-dimensional case, equivalent to f(x1, x2, x3, v1, v2, v3, α). It may be stipulated that the distribution function (or correspondingly, the corresponding tensor) has an index for time t. In some embodiments, it may be stipulated that the distribution function and / or the corresponding tensor contain a set of values for at least two and / or all time steps (e.g., distinct time steps of a simulation). The at least two time steps may be different and / or distinct time steps. In some embodiments, at least two, a plurality of, and / or all time steps of the tensor are consecutive and successively calculated time steps, and / or may be separated by Δt. Alternatively or additionally, at least two, a plurality of, and / or all time steps of the tensor may be separated by one or more intermediate time steps or time intervals (e.g., one or more intermediate time steps of a simulation), and / or separated by more than Δt.

[0099] In the case of a two-dimensional grid or lattice, the indices x1, x2 may correspond to the positions of grid points. In the case of a three-dimensional grid or lattice, the indices x1, x2, x3 may correspond to the positions of grid points. The index x1 may correspond to a direction and / or the index x and / or may also be referred to as x, the index x2 may correspond to a direction and / or the index y and / or may also be referred to as y, and / or the index x3 may correspond to a direction and / or the index z and / or may also be referred to as z.

[0100] The size of the index x1 may be equal to the number of grid points along the x1 axis (or x axis) of the computational grid. The size of the index x2 may be equal to the number of grid points along the x2 axis (or y axis) of the computational grid. The size of the index x3 may be equal to the number of grid points along the x3 axis (or z axis) of the computational grid.

[0101] The number of grid points along axis j may be n j . The index j may correspond to a direction in space and may be, for example, j = 1, 2 for a two-dimensional case and j = 1, 2, 3 for a three-dimensional case. In some embodiments, the number of grid points along at least two axes and / or along all axes may be equal, such that for all j, n j = n. In some embodiments, the computational grid and / or lattice may have n j = 2 m grid points for at least some axis j. In some embodiments, the computational grid and / or lattice may have n j = 3 mgrid points. For the two-dimensional case, the total number of grid points can be equal to n1n2, and / or for the two-dimensional case, the total number of grid points can be equal to n1n2n3. In some embodiments, the total number of grid points can be (2 m ) j . In some embodiments, the total number of grid points can be (3 m ) j .

[0102] In the case of a two-dimensional grid or lattice, the indices v1, v2 can correspond to lattice velocity components. In the case of a three-dimensional grid or lattice, the indices v1, v2, v3 can correspond to lattice velocity. The index v1 can correspond to lattice velocity and / or the index v x and / or can also be referred to as v x , the index v2 can correspond to direction and / or the index v y and / or can also be referred to as v y , and / or the index v3 can correspond to direction and / or the index v z and / or can also be referred to as v z .

[0103] The value of the lattice velocity component can be equal to -Δx / Δt, 0, or Δx / Δt (refer to the case of D2Q9 for lattice velocity e k given above). Here, Δx can be the distance between grid points, Δt is the time step, so c = Δx / Δt is the lattice velocity.

[0104] The dimensionless relaxation time τ can be related to the kinematic viscosity by the following formula

[0105]

[0106] The speed of sound c s can be given by , where c is the lattice velocity, c = Δx / Δt, where Δx is the distance between grid points, and Δt is the time step and / or the increment between adjacent time steps.

[0107] To solve the above equations, an equilibrium solution is needed. In some embodiments, the equilibrium solution can be or given and / or calculated by the following formula

[0108]

[0109] where ρ α is the fluid density of substance α, u is the macroscopic velocity and c is the lattice velocity. w k can be a weighting coefficient, which can depend on the direction k. For example, referring to Figure 1 (left), the weighting coefficient can be given by w o= 4 / 9, w1 = w2 = w3 = w4 - 1 / 9 and w5 = w6 = w7 = w8 = 1 / 3 6 are given. In some embodiments, the weighting coefficients can be different. The weighting coefficients can depend on the discretization scheme DxQy. The weighting coefficients can be chosen such that their sum is one, Σ k w k = 1.

[0110] The fluid density of substance α can be defined and / or calculated as

[0111]

[0112] The macroscopic rate can be given by the following formula

[0113]

[0114] That is, it can correspond to the density-weighted macroscopic rate of substance α. Herein,[[]]

[0115]

[0116] The present invention is not limited to the above equations. Depending on the purpose, external effects and / or external forces, temperature effects, etc., and / or other equations and / or models can be adopted.[[]]

[0117] Figure 2 and Figure 3 shows an exemplary embodiment of the tensor chain decomposition.[[]]

[0118] Figure 2 shows a schematic decomposition of the tensor A(x, y, z) into a tensor chain (TT), see Figure 2 the upper part of (where the tensor A is indicated by the reference numeral 2).[[]] Figure 2 Further shows the decomposition of the quantized tensor train format (QTT), see Figure 2 the lower part of. The distribution function can be or include the tensor 2.[[]]

[0119] A can be a function depending on x, y, and z. Alternatively or additionally, A can be a certain quantity, such as temperature, density, etc., which can vary, for example, in a three-dimensional space spanned by x, y, and z. The quantity can be given at the grid points at the positions x, y, and z. The tensor A can have n grid points in each direction x, y, and z, and / or the sizes of the indices x, y, and z can be n. The tensor A can be a three-dimensional tensor. This is indicated by Figure 2 the three branches x, y, and z at the upper left corner, where n indicates the number of entries for each index x, y, and z respectively.[[]]

[0120] In general, the tensor train format (TT format) can correspond to a decomposition of a (general) tensor A given by, for example, (see, e.g., Oseledets and Tyrtyshnikov, “Tt-cross approximation for multidimensional arrays”, Linear Algebra and its Applications, 432(1):70–88, 2010)

[0121]

[0122] where G j can be a tensor train core 5 (TT core 5), where G j and / or the tensor train core 5 is or can be a 3D tensor. The main characteristic of this representation is the rank r, which is equal to the maximum value among the indices β0, β1, …, β d and expresses the correlation and entanglement in the tensor. In the case of weak correlation (low rank r), this format can store and operate on tensors with logarithmic complexity in size.

[0123] In the present invention, a d-dimensional tensor A is mainly used. In some embodiments, each index of the tensor A can have n different values, so there are n d elements in the tensor A, which generally leads to the same complexity of operations. Importantly, any tensor can be represented as a tensor train of rank r via singular value decomposition (SVD) (see, e.g., Melven, Thies and Basermann, Performance of the low-rank tt-svd for large dense tensors on modern multicore cpus, SIAM Journal on Scientific Computing, 44(4):C287–C309, 2022) or cross approximation (see, e.g., Oseledets and Tyrtyshnikov, “Tt-cross approximation for multidimensional arrays”, Linear Algebra and its Applications, 432(1):70–88, 2010). This representation only has dnr 2elements. If the rank is restricted, as is indeed the case in PDE solutions (see, e.g., Dolgov, Khoromskij, and Oseledets, “Fast solution of parabolic problems in the tensor train / quantized tensor train format with initial application to the fokker–planck equation [Tensor train / quantized tensor train format for fast solution of parabolic problems and its initial application to the Fokker–Planck equation]”, SIAM Journal on Scientific Computing [SIAM Journal of Scientific Computing], 34(6): A3016–A3038, 2012, or Gourianov, Lubasch, Dolgov, van den Berg, Babaee, Givi, Kiffner, and Jaksch, “A quantum-inspired approach to exploit turbulence structures [A quantum-inspired approach to exploiting turbulence structures]”, Nature Computational Science [Nature Computational Science], 2(1): 30–37, 2022) and in the representation of certain smooth multidimensional functions (see, e.g., Rohrbach, Dolgov, Grasedyck, and Scheichl, “Rank bounds for approximating gaussian densities in the tensor-train format [Rank bounds for approximating Gaussian densities in the tensor train format]”, arXiv preprint arXiv:2001.08187, 2020), then exponential compression is obtained. Moreover, almost all algebraic operations in the TT format have a similar complexity (see Oseledets, “Tensor-train decomposition [Tensor train decomposition]”, SIAM J. Scientific Computing [SIAM Journal of Scientific Computing], 33: 2295–2317, 01 2011), as shown in the table below.

[0124]

[0125] It can be seen that in the case of a small rank value, all these operations are much more efficient when performed in the TT format. Another important TT algorithm (which can be used in some embodiments) is the cross-approximation technique (see Oseledets and Tyrtyshnikov, "TT-cross approximation of multi-dimensional arrays", Linear Algebra and its Applications [Linear Algebra and Its Applications], 432(1):70–88, 2010). This algorithm allows to recover from a given tensor a tensor chain of rank r that involves only O(dnr 2 ) of its elements. In the context of the present invention, this means that we can apply almost any function to the tensor chain for O(d 2 n 2 r 4 ) operations, since accessing the tensor chain elements also has complexity O(dnr 2 ), see operation 5 in the above table. If parallelization is used, the overall scaling can be boosted to O(dnr 2 ), where each iteration of the cross-approximation requires nr 2 calls to the tensor chain, and these calls can be executed in parallel; when the calls to the tensor chain can be parallelized, it also passes through d tt cores.

[0126] Since in physical problems the dimension of space is usually no higher than 3 (d = 3), in order to achieve more compression, the so-called quantized tensor train (QTT) format can be used. It is obtained by introducing artificial indices, i.e., by "shaping" the visible indices into two or three, as Figure 2 (lower part) and Figure 3 shown. For example, if the size of an index (e.g., index x1) is equal to 2 m , then the said index can be shaped into a tensor Similarly, if the size of an index (e.g., index x1) is equal to 3 m , then the said index can be shaped into a tensor It can be stipulated that the tensor train core 5 is further shaped and / or refined into a quantized tensor train core, as shown, for example, in Figure 2 (lower part).

[0127] For the QTT format, all of the above applies in the same way as for the standard tensor train format.

[0128] Figure 3 An exemplary distribution function f in the QTT format is shown, where the index α is omitted. In some embodiments, Figure 3 the distribution function f in Figure 3An exemplary distribution function for the upper part of the figure can be the distribution function in the D2Q9 case, where the grid size is equal to 3 for each axis d . Figure 3 An exemplary distribution function for the lower part of the figure can be the distribution function in the D2Q9 case, where the grid size is equal to 2 for each axis d . Further, the exemplary distribution function shown in the lower part of the figure can explicitly include an index t of size 2 s , and the index explicitly corresponds to different time steps. The distribution function f can be or include a tensor 2

[0129] For Figure 3 the exemplary distribution function shown, the velocities v x , v y each have only 3 values per axis (see the D2Q9 case in Figure 1 ), i.e., they can take the values -Δx / Δt, 0, and / or Δx / Δt. Without loss of generality, in some embodiments, the velocities can take the values -1, 0, and / or 1

[0130] All steps of the method according to the present invention can be performed using distribution functions in TT format and / or QTT format. All operations can be performed without departing from the TT format and / or QTT format. Thus, significant acceleration can be achieved compared to methods known in the art. Further, the memory required for the method according to the present invention may be much less than that of methods known in the art. For a given amount of memory, such as the RAM of a computer system and / or a given number of computer nodes in a distributed computing system, compared to the prior art, when using the method according to the present invention, a finer computational grid 20 can be used

[0131] For example, in some embodiments, for the collision step, the macroscopic velocity u is first calculated. The multiplication and summation with e k are both straightforward (operations 3 and 4 in the above table). To divide by the density ρ, cross approximation can be used. When calculating the velocities, all other operations only involve multiplication by constants, element-wise multiplication, and summation (operations 1 to 3 in the above table). Finally, the enlarged rank can be truncated using the rounding operation 6 in the above table

[0132] In some embodiments, for the streaming step consisting of applying a shift function on a tensor chain, a cross approximation algorithm can be used

[0133] Thus, all LBM steps can be implemented using a tensor chain. The algorithm based on the tensor chain provides O(d) runtime scaling, while the complexity of the classical implementation is O(3 2d )

[0134] InFigure 4 shows a runtime comparison of some methods according to the present invention with methods known in the prior art. The corresponding methods differ in that the method according to the present invention employs a QTT decomposition of the distribution function, while the methods known in the prior art do not use tensor train decomposition. Specifically, for the Taylor - Green Vortex in 2D and 3D, the LBM utilizing QTT decomposition according to an embodiment of the present invention is compared with the classical LBM known in the prior art. As is well known, the Taylor - Green Vortex has an analytical solution that can be used to compare the accuracy of different methods.

[0135] For the 2D case, the spatial domain of the exemplary test case is a square [0, 2π]×[0, 2π] with periodic boundary conditions. For the 3D case, the spatial domain of the exemplary test case corresponds to a cube with side length L = 2π and periodic boundary conditions (i.e., the cube [0, 2π] 3 ).

[0136] It is worth noting that for the 2D case, if 3 9*39 = 19683 * 19683 grid points are used to discretize the spatial domain, 12GB of memory is not sufficient to store the entire tensor f(x, y, v x , v y ). If double - precision is selected, approximately 19683 2 * 3 2 * 8 bytes ~ 25.97Gb of memory is required.

[0137] For the 2D test case, the flow of a single substance is simulated using QTT LBM and classical LBM respectively, where the density ρ = 1, the speed of sound c = 1, the initial velocity uo = 0.01 (so the Mach number Ma = 0.01), the initial distribution function f(t = o) = f eq and Re = 1. The resulting runtime as a function of the number of grid points is as shown in Figure 4 (the upper part), where QTT LBM corresponds to the solid line and classical LBM corresponds to the dashed line.

[0138] Obviously, when the computational grid is fine enough, the method according to the present invention has a significant acceleration compared to the prior art.

[0139] For the 3D test case, the following are selected: a small Re = 1, Mach number Ma = 0.01, speed of sound c = 1, area size L = 2π, and the number of points N = 81 along each axis (i.e., N = 3 4 ). In Figure 4In the following figure, it is plotted as a function of the time step Δt of a 3D test case during runtime, where the characteristic time corresponds to a suitably normalized dimensionless time step Δt. The error of QTT LBM with respect to the analytical solution is less than 7*10 -5 . Obviously, the method according to the present invention solves the problem faster than the methods known in the prior art.

[0140] Since the Courant-Friedrichs-Lewy (CFL) number of LBM must be 1 because the particle distribution function can only shift between adjacent lattice points within a single time step, when the computational grid is fine (i.e., Δx is small), the time step may be small, Δt = Δx. Therefore, a large number of iterations may be required.

[0141] However, the method according to the present invention also allows solving the lattice Boltzmann equation (LBE) in TT format and / or QTT format to overcome this problem. In some embodiments, the lattice Boltzmann equation can be given by

[0142]

[0143] Using an implicit scheme can produce

[0144]

[0145] where, for example, in 2D, fi = f(x, y, v x , v y ) and / or, for example, in 3D, f i = f(x, y, z, v x , v y , v z ) can be the distribution function at time step i. The explicit mention of species α is omitted hereinafter. However, the method is also applicable to calculating mixtures of at least two species α, where the equations can be adjusted as needed, which is obvious to those skilled in the art. Therefore, the linear system of equations

[0146]

[0147] can be solved, where b on the right side can correspond to the collision step, and A on the left side corresponds to the streaming step. This is equivalent to solving the linear system Ax = b, where both A and b can be represented in the TT format and / or the QTT format. Thus, an efficient linear system solving method can be used, such as the alternating minimal energy (AMEn) method (see Dolgov and Savostyanov, "Alternating minimal energy methods for linear systems in higher dimensions", SIAM Journal on Scientific Computing, 36(5): 1–24, September 2014), and the complexity of the method is O(dr 6 ).

[0148] In another way of using QTT according to the present invention to solve the LBE, the entire distribution function f = f(x, y, v x , v y , t) and / or f = f(x, y, z, v x , v y , v z , t) can be represented in the quantized tensor chain format. The explicit mention of substance α is omitted hereinafter. However, the method is also applicable to calculating mixtures of at least two substances α, where the equations can be adjusted as needed, which is obvious to those skilled in the art.

[0149] In other words, the distribution function f is in the quantized tensor chain format both in space and time, see for example Figure 3 the lower part of.

[0150] Then, the solution of the LBE can be reduced to the solution of the following non - linear equation

[0151]

[0152] where the corresponding matrix can be written down by the tensor product Here, I is the identity matrix, and e1 = (1, 0,..., 0) T is the first unit vector. Further, the rate matrix Δ1 is the matrix of the first - order derivative of the direct order

[0153]

[0154] In the case of periodic boundary conditions,

[0155]

[0156] Adjustments that are obvious to those skilled in the art can be made under different boundary conditions.

[0157] A t and A x can both be well represented in the QTT format of rank 3. To solve this non-linear equation, for example, the Picard iteration method can be used.

[0158]

[0159] To solve this equation at each step, an efficient tensor train method such as the AMEn algorithm mentioned above can be used.

[0160] Figure 5 An exemplary embodiment of the mixing reactor 10 is shown. In some embodiments, the mixing reactor 10 can be or include a T-shaped mixing reactor. The mixing reactor 10 and / or at least the mixing chamber of the mixing reactor can be T-shaped.

[0161] Alternatively or additionally, in some embodiments, the mixing reactor 10 can be or include a batch reactor and / or a flow reactor.

[0162] In Figure 5 the T-shaped reactor, fluids (e.g., fluids of different substances) can flow into the reactor at the arms of the T-shape ( Figure 5 the left side of Figure 5 ). The fluids can be mixed in the reactor, and the mixture can flow out at the long end of the T-shape (

[0163] the right side of

[0164] In some embodiments, the temperature of the first fluid and / or the second fluid may be the same. Alternatively, it may be provided that, for example when entering the mixing reactor 10, the temperature of the first fluid is different from the temperature of the second fluid.

[0165] The first fluid and / or the inflow 70 may enter the mixing reactor 10 at a specific inflow rate. It may be provided that the first fluid and / or the inflow 70 at a specific mass flow rate and / or volume flow rate enter the mixing reactor 10. The second fluid and / or the inflow 80 may enter the mixing reactor 10 at a specific inflow rate. It may be provided that the second fluid and / or the inflow 80 at a specific mass flow rate and / or volume flow rate enter the mixing reactor 10.

[0166] The inlet of the first fluid and / or the inflow 70 may have an inflow cross-section A I . The inlet of the second fluid and / or the inflow 80 may have an inflow cross-section A I , and the inflow cross-section may be the same as or different from the cross-section of the inlet of the first fluid and / or the inflow 70.

[0167] The first fluid and / or the inflow 70 may be or include a liquid, a gas, a multiphase flow, etc. The second fluid and / or the inflow 80 may be or include a liquid, a gas, a multiphase flow, etc.

[0168] In the mixing reactor 10, the first fluid and the second fluid may be mixed. In some embodiments, the first fluid and the second fluid may react to form at least one reaction product. In some embodiments, when mixed, at least one or more of density, temperature, composition, concentration, substance, pressure, etc. may change. In some embodiments, when mixed, the first fluid and the second fluid may react.

[0169] It may be provided that the density, temperature, composition, concentration, substance, pressure, etc. of the mixture may be different from those of the first fluid and / or the second fluid.

[0170] The mixture may leave the mixing reactor 10. The outflow 80 of the mixing reactor 10 may be or include the mixture. In some embodiments, the outflow 80 of the mixing reactor 10 may include a part of the first fluid and / or the second fluid.

[0171] The outflow 80 may leave the mixing reactor 10 via the outlet. The outlet may have an outlet cross-section A O . The outflow 80 may leave the mixing reactor 10 at a specific outflow rate. It may be provided that the outflow 80 at a specific mass flow rate and / or volume flow rate leaves the mixing reactor 10.

[0172] The mixing reactor 10 may have characteristic dimensions. In some embodiments, for example when the mixing reactor 10 is T-shaped, the characteristic dimension may be the mixing length L. For example, the mixing length L may be the length along which the inflows 70 and 80 may mix.

[0173] The mixing reactor 10 may have one or more sensors 100. The sensors 100 may be configured to measure or determine properties of the fluid, for example, at and / or near a measurement point. The measurement point may be located within the mixing reactor 10 and / or on the wall of the mixing reactor 10. The properties may be or include one or more of fluid rate, density, temperature, pressure, concentration, and / or composition. The properties may be or include the substance concentration, for example, the substance concentration of the fluid and / or mixture in the mixing reactor 10. The properties may be or include the composition of the fluid and / or mixture in the mixing reactor 10.

[0174] The mixing reactor 10 may have a temperature control unit 110. The temperature control unit 110 may be configured to heat and / or cool the wall of the mixing reactor 10 and / or impose a temperature profile and / or temperature distribution.

[0175] In some embodiments, the mixing reactor 10 may be or include a batch reactor (not shown in the figures). In some embodiments, the mixing reactor 10 may be or include a continuous reactor (not shown in the figures). It may be provided that the mixing reactor 10 has at least one stirrer (not shown in the figures). The stirrer may facilitate mixing in the mixing reactor. In some embodiments, the stirrer may be configured to stir or rotate the fluid in the mixing reactor 10.

[0176] A control unit (not shown in the figures) may be provided. The control unit may be part of or separate from the mixing reactor 10. The control unit may be configured to control the mixing reactor 10 and / or the mixing in the mixing reactor 10. In some embodiments, the control unit may be configured to perform one, multiple, or all of the methods of the present invention.

[0177] The mixing reactor 10 may have more than one and / or more than two inflows and / or outflows. The mixing reactor 10 may have a geometry different from a T-shape. The mixing reactor 10 may be configured to mix more than two inflows and / or segregates. The mixing reactor 10 may be configured to provide more than one mixture or product.

[0178] The method for simulating fluid flow is not limited to simulating the flow in the mixing reactor 10. In some embodiments, the method for simulating fluid flow may be used to simulate the fluid flow in, around, and / or through any shape, form, container, conduit, pipe, chamber, reactor, device, etc.

[0179] An embodiment of the method according to the invention is used to simulate fluid flow. The method can be used to determine the distribution function of fluid flow. Alternatively or additionally, the method can be used to determine the distribution function of a mixture of at least two substances. The method can be computer-implemented.

[0180] An embodiment of the method according to the invention is used to design the mixing reactor 10 and / or is used when designing the mixing reactor. The method can be computer-implemented.

[0181] The method for designing the mixing reactor 10 includes providing an initial mixing reactor geometry. The mixing reactor geometry can include or consist of a computational grid 20 having grid points 30. Alternatively or additionally, a computational grid and / or grid points can be created based on the initial mixing reactor geometry. Mixing reactor geometry. Figure 6 is shown in Figure 5 an exemplary initial mixing reactor geometry of the mixing reactor 10.

[0182] The number of grid points 30 and / or the distance Δx between adjacent grid points 30 can be exemplary. In some embodiments, a different number of grid points 30 and / or distance Δx can be selected or provided.

[0183] The computational grid 20 can be uniform and / or rectangular. In some embodiments, the computational grid 20 can be non-uniform and / or non-rectangular. It can be provided that the grid spacing can be different for different axes and / or vary at least locally.

[0184] The computational grid 30 and / or the simulation can use and / or have suitable boundary conditions. For example, wall boundary conditions (e.g., von Neumann and / or Dirichlet boundary conditions) can be employed to represent the (physical) walls of the mixing reactor 10. Inlets and / or outlets and / or inflows and / or outflows can be modeled and / or taken into account by suitable boundary conditions, such as determined or given curves and / or values of rate, pressure, density, temperature, etc. Inflows and / or outflows can be modeled and / or taken into account by suitable boundary conditions representing the (physical) inflows and / or outflows of the mixing reactor 10.

[0185] In some embodiments, the temperature control unit 110 can be modeled and / or taken into account by, for example, determined or given curves and / or values of the temperature at the boundary.

[0186] In some embodiments, when the mixing reactor to be designed has at least one stirrer, the stirrer, etc. can be considered.

[0187] A method for designing a mixing reactor 10 includes the step of simulating the mixing of at least two substances and / or the flow in and / or through the mixing reactor via a method for simulating fluid flow according to the present invention. The characteristic properties of the mixture and / or the fluid flow can be determined during or after performing the simulation. In some embodiments, the characteristic property can be or include at least one of a mixing fraction, a substance concentration, a characteristic mixing time, a characteristic residence time, and / or a mixing progress. The characteristic property can be or include one or more statistics, such as an average value, a variance, or a higher-order moment. It can be provided that the statistic can be an ensemble average, a time average, and / or a density-weighted average, and / or a statistic evaluated and / or calculated in space and / or time. In some embodiments, the characteristic property can be calculated at at least one, multiple, or all grid points or for at least one, multiple, or all grid points. The characteristic property can be or include a scalar. The characteristic property can be or include a directional quantity, such as a higher-order vector or tensor.

[0188] The method for designing a mixing reactor 10 further includes the step of modifying the mixing reactor geometry and / or the grid points 30. When modifying the mixing reactor geometry and / or the grid points 30, or after that, the simulation can be repeated using the modified mixing reactor geometry and / or the grid points 30. When repeating the simulation, or after that, the characteristic properties of the mixture can be determined.

[0189] The characteristic properties of successive repetitions of the simulation can be compared. In some embodiments, the characteristic properties of at least two, multiple, or all simulation runs with different reactor geometries can be compared. When comparing the characteristic properties, the characteristic properties can be compared with a target value of the characteristic property.

[0190] By repeating and / or iterating these steps, the mixing reactor geometry can be optimized. When the characteristic property is close enough to the target value, the iteration can be ended. When the accuracy of the characteristic property is close enough to the target value and / or the deviation from the target value is less than a given value, the iteration can be ended.

[0191] In some embodiments, when optimizing the mixing reactor geometry, more than one characteristic property can be considered. The mixing reactor geometry can be optimized in terms of more than one characteristic property.

[0192] Modifying the mixing reactor geometry can include moving boundaries and / or changing the shape of the computational grid 20. Alternatively or additionally, modifying the mixing reactor geometry can include adding grid points 30 or removing grid points from the computational grid 20, and / or coarsening and / or refining the computational grid 30.

[0193] For example, Figure 7 is shown Figure 6Exemplary embodiments in which the grid points 30 of the computational grid 20 have been modified. In some embodiments, the grid spacing (in the Figure 7 x-direction and / or y-direction) may have been locally adjusted and / or changed. Additionally or alternatively, the computational grid 20 has been coarsened. Grid points 30 have been removed from the computational grid 20. Figure 7 The embodiments shown in

[0194] Figure 8 are merely exemplary. Figure 6 Exemplary embodiments are shown in which the geometry of the mixing reactor has been modified. Specifically, the dimensions of the mixing reactor have been changed such that the characteristic properties are optimized, sufficiently close to the target values and / or within a given interval. For example, the characteristic dimensions may have been changed. In the Figure 8 exemplary embodiment, compared to the initial mixing reactor geometry, the mixing length L, the inlet (e.g., cross-section A I ) and / or the outlet (e.g., cross-section A O ) may have changed. The mixing length L may be shorter than the mixing length of the initial mixing reactor geometry (see Figure 6 ), one of the inlets may have a larger cross-section A I , the outlet may have a larger cross-section A O , and / or the length of one "arm" of the T-shaped structure (e.g., extending from the upper inlet) may have increased. Figure 8 The embodiments shown in

[0195] are merely exemplary.

[0196] After the mixing reactor geometry has been optimized, the resulting computational grid 30 can be used as the design of the (physical) mixing reactor 10. The design and / or technical drawings can be based on and / or interpreted from the resulting mixing reactor geometry.

[0196] Thus, a mixing reactor 10 for mixing at least two substances (e.g., as shown in Figure 5 ) can be designed based on and / or using the method for designing a mixing reactor according to the present invention.

[0197] Compared with the methods based on known technologies, the method for designing a mixing reactor according to the present invention can be faster and require less memory, especially for complex geometries. For example, complex geometries require a fine computational grid with many grid points, which in turn requires a large amount of memory for calculations. Thus, the method for designing a mixing reactor according to the present invention allows for a fast and accurate design of the mixing reactor.

[0198] The present invention further relates to a mixing reactor 10 (such as, for example, Figure 5As shown), the mixing reactor has a mixing reactor geometry obtained by a method for designing a mixing reactor according to the present invention. The geometry may be the internal geometry of the mixing reactor 10.

[0199] The present invention further relates to a method for controlling a mixing reactor 10 (e.g., Figure 5 the mixing reactor 10 shown therein).

[0200] The method for controlling the mixing reactor 10 includes the step of simulating the mixing of at least two substances and / or the flow in and / or through the mixing reactor via a method for simulating fluid flow according to the present invention. The computational grid corresponds to the geometry of the mixing reactor 10.

[0201] The method for controlling the mixing reactor 10 further includes the step of generating and / or updating at least one control command for the mixing reactor 10 based on the simulation.

[0202] The method for controlling the mixing reactor may be computer-implemented.

[0203] Updating the control command may include adjusting, modifying, and / or changing the control command, e.g., an existing and / or previously generated control command.

[0204] The control command may include one or more commands for controlling at least one flow into the mixing reactor and / or at least one flow out of the reactor. The control command may include a rate, a volumetric flow rate, and / or a mass flow rate. The control command may include a command for controlling a stirrer, e.g., an angular rate or an angular frequency. The control command may include a command for controlling a temperature control unit (e.g., a heater and / or a cooler) and / or a cooling or heating temperature.

[0205] Characteristic properties of the mixture and / or the fluid flow may be determined during or after performing the simulation. In some embodiments, the characteristic property may be or include at least one of a mixture fraction, a substance concentration, a characteristic mixing time, a characteristic residence time, and / or a mixing progress. The characteristic property may be or include a characteristic property for optimizing the mixing reactor geometry.

[0206] The characteristic property may be or include one or more statistics, such as an average value, a variance, or a higher-order moment. It may be provided that the statistic may be an ensemble average, a time average, and / or a density-weighted average, and / or a statistic evaluated and / or calculated in space and / or time. In some embodiments, the characteristic property may be calculated at at least one, multiple, or all grid points or for at least one, multiple, or all grid points. The characteristic property may be or include a scalar. The characteristic property may be or include a vector quantity, e.g., a higher-order vector or a tensor.

[0207] Control commands can be generated and / or updated such that the characteristic properties can approach the target value and / or deviate from the target value by less than a given value. Control commands can be generated and / or updated such that the flow and / or mixing in the (physical) mixing reactor 10 controlled by the control commands is equal to or at least sufficiently close to the simulated flow and / or mixing.

[0208] In some embodiments, the method for controlling the mixing reactor can include iterating at least one, several, or all steps, such as simulating and generating and / or adjusting control commands.

[0209] In some embodiments, the control commands can be at least partially or fully modeled by selecting appropriate boundary conditions for the simulation.

[0210] For example, the control commands can include the average flow rate, volumetric flow rate, or mass flow rate of the inflows (e.g., inflow 70 and / or inflow 80) such that a specific and / or desired mixing is achieved in the mixing reactor 10. Alternatively or additionally, the control commands can include heating the walls of the mixing reactor 10 with the temperature control unit 110 such that the walls can have a specific and / or desired temperature, which can facilitate mixing or is necessary for mixing. These control commands are merely exemplary.

[0211] The control commands can vary over time and / or be time-dependent. For example, the average rate and / or its mass flow rate or volumetric flow rate of the inflows (e.g., inflow 70 and / or inflow 80) can vary over time.

[0212] The method can include measuring fluid properties. For example, the fluid properties can be or include one or more of rate and / or substance concentration. The fluid properties can be measured at or near at least one measurement point in the mixing reactor 10. For example, the fluid properties can be measured by the sensor 100. The measured fluid properties can be compared with the corresponding fluid properties obtained by simulation. For example, the fluid properties obtained by simulation can be determined at one or several grid points 30 of the computational grid 20, which correspond to the (physical) measurement points of the mixing reactor 10 and / or the positions of the corresponding sensors 100. The control commands can be generated, adjusted, and / or modified based on the comparison. For example, the control commands can be generated, adjusted, and / or modified such that the measured physical quantity (e.g., the physical quantity measured by the sensor 100) is equal to or sufficiently close to the physical quantity from the simulation.

[0213] In some embodiments, control commands may be generated and / or updated during or at the time of mixing at least two substances in a mixing reactor. In some embodiments, it may be provided that the method for controlling the mixing reactor is performed "online", i.e., during or at the time of operating the mixing reactor. Alternatively or additionally, control commands may be generated and / or updated before operating the mixing reactor.

[0214] It may be provided that after generating and / or updating the control commands, the simulation of the mixing may be repeated and / or the method may be performed iteratively. When the simulation is repeated, the calculation of the distribution function may take into account the generated and / or updated control commands. For example, when the simulation is repeated, the calculation of the distribution function may take into account, for example, forces, flow rates, and / or boundary conditions imposed and / or applied by the control commands.

[0215] The mixing reactor may be controlled by control commands. The control unit may be configured to control the mixing reactor 10 according to the instructions via the control commands. In some embodiments, the control unit may be configured to perform the method for controlling the mixing reactor.

[0216] The invention further relates to a data processing device comprising a processor configured to perform the steps of a method for simulating fluid flow according to the invention. The invention further relates to a data processing device comprising a processor configured to perform the steps of a method for designing a mixing reactor according to the invention. The invention further relates to a data processing device comprising a processor configured to perform the steps of a method for controlling a mixing reactor according to the invention.

[0217] The data processing device may be or include a control unit, such as the control unit for controlling the mixing reactor. Alternatively or additionally, the data processing device may be or include a general-purpose computer and / or computer device, such as a distributed computing system, a cloud computing system, a computer cluster, a distributed workstation, a PC, a laptop, a tablet computer, a smartphone, a server, etc.

[0218] The present invention further relates to a computer program comprising instructions which, when the program is executed by a computer, cause the computer to perform the steps of the method for simulating fluid flow according to the present invention. The present invention further relates to a computer program comprising instructions which, when the program is executed by a computer, cause the computer to perform the steps of the method for designing a mixing reactor according to the present invention. The present invention further relates to a computer program comprising instructions which, when the program is executed by a computer, cause the computer to perform the steps of the method for controlling a mixing reactor according to the present invention.

[0219] The program can run in parallel and / or use multiple CPUs, computer cores, and / or nodes.

[0220] The present invention further relates to a computer-readable medium comprising instructions which, when executed by a computer, cause the computer to perform the steps of the method for simulating fluid flow according to the present invention. The present invention further relates to a computer-readable medium comprising instructions which, when executed by a computer, cause the computer to perform the steps of the method for designing a mixing reactor according to the present invention. The present invention further relates to a computer-readable medium comprising instructions which, when executed by a computer, cause the computer to perform the steps of the method for controlling a mixing reactor according to the present invention.

[0221] The computer-readable medium can be or include an optical disc (such as a floppy disc, compact disc (CD), digital versatile disc (DVD), Blu-ray disc, etc.), a memory (e.g., flash memory, EEPROM, RAM, etc.), a hard disk drive, an optical storage medium, or a magnetic storage medium, etc.

[0222] The features disclosed in the claims, the description, and the drawings can be related to the implementation of the present invention individually or in any combination.

[0223] List of Reference Signs

[0224] 1 Fluid particle

[0225] 2 Tensor

[0226] 3 Tensor chain

[0227] 4 Quantized tensor chain

[0228] 5 Tensor chain core

[0229] 6 Rate direction

[0230] 10 Mixing reactor

[0231] 20 Computational grid

[0232] 30 Grid point

[0233] 40 Inlet

[0234] 50 Outlet

[0235] 60 Sensor

[0236] 70 Inflow

[0237] 80 Inflow

[0238] 90 Outflow

[0239] 100 Sensor

[0240] 110 Temperature control unit

[0241] A I Inflow cross-section

[0242] A O Outflow cross-section

[0243] Δx Grid distance

[0244] L Mixing length

Claims

1. A computer-implemented method for simulating fluid flow, wherein: A computational grid (20) having grid points (30) is provided for computing the distribution function (f, 2) of the fluid flow and / or the distribution function (f, 2) of a mixture of at least two substances at said grid points (30), The method comprises a lattice Boltzmann method having: - a collision step, in which the distribution function (f) is calculated at each grid point (30) with the equilibrium solution (f eq ) - a streaming step, wherein the distribution function (f) at each grid point (30) is updated according to the streaming step; And / or the method comprises: solving the lattice Boltzmann equation of the distribution function (f, 2), characterized in that the distribution function (f, 2) is represented in a tensor chain format, and / or all operations performed to calculate the distribution function (f, 2) are performed using the distribution function (f, 2) in the tensor chain format, the distribution function (f, 2) and / or the tensor chain format comprises at least one tensor chain (3), and the tensor chain (3) has at least one tensor chain core (4).

2. The computer-implemented method of claim 1, wherein: The tensor chain (3) is or includes a quantized tensor chain.

3. A computer-implemented method as claimed in any one of the preceding claims, wherein: The tensor chain (3) has tensor chain cores (5) which correspond to grid coordinates and / or velocity components of the flow, respectively.

4. A computer-implemented method as claimed in any one of the preceding claims, wherein: The tensor chain (5) has at least one tensor chain core (5) corresponding to the substance and / or wherein a tensor chain is provided for each substance.

5. A computer-implemented method for designing a mixing reactor (10) for mixing at least two substances, wherein: The method comprises providing an initial mixing reactor geometry having a computational grid (20) with grid points (30), the method comprising the steps of: a) simulating the mixing of at least two substances and / or the flow in and / or through the mixing reactor (10) by means of a method according to any of the preceding claims and determining characteristic properties of the mixture and / or the fluid flow, optionally determining at least one of the mixing fraction, substance concentrations, characteristic mixing time, characteristic residence time and / or mixing progress; b) modifying the mixing reactor geometry and / or grid points (30), repeating the simulation using the modified mixing reactor geometry and / or grid points (30), and determining characteristic properties of the mixture; c) comparing the characteristic attribute with a target value of the characteristic attribute; d) iteratively repeating steps b) and c) and optimizing the mixing reactor geometry and / or the computing grid (20) such that the mixing reactor geometry and / or the computing grid (20) are optimized with respect to the characteristic property and / or until a given accuracy of the characteristic property relative to the target value is reached.

6. The method of claim 5, wherein: Modifying the hybrid reactor geometry comprises moving boundaries and / or changing the shape of the computational grid (20), and / or wherein, Modifying the grid points (30) includes adding grid points (30) or removing grid points from the computational grid (20).

7. A mixing reactor (10) for mixing at least two substances, the mixing reactor (10) having a mixing reactor geometry, optionally an internal geometry, determined via the method according to any one of the preceding claims 5 to 6.

8. A computer-implemented method for controlling a mixing reactor (10), the method comprising: - simulating the mixing of at least two substances and / or the flow in and / or through the mixing reactor (10) via a method according to any one of the preceding claims 1 to 4, wherein the computational grid (20) corresponds to the geometry of the mixing reactor (10); - generating and / or updating at least one control command for the mixing reactor (10) based on the simulation.

9. The method of claim 8, wherein: The control commands include one or more of the following commands: commands for controlling at least one stream flowing into (70, 80) the mixing reactor (10) and / or at least one stream flowing out of the reactor (90), optionally in the form of a rate, a volume flow rate and / or a mass flow rate; commands for controlling an agitator, optionally in the form of an angular velocity or an angular frequency; and / or commands for controlling a temperature control unit (110), optionally a heater and / or a cooler, optionally in the form of a cooling or heating temperature.

10. The method according to any one of claims 8 to 9, wherein: The method comprises measuring, optionally using a sensor (110), a fluid property, optionally a rate and / or a substance concentration, at at least one measuring point in the mixing reactor (10), and comparing the measured fluid property with a corresponding fluid property obtained from the simulation, wherein the control command is generated, adjusted and / or modified based on the comparison.

11. The method according to any one of claims 8 to 10, wherein: The control command is generated and / or updated during or when the at least two substances are mixed in the mixing reactor (10), wherein, optionally, the mixing reactor (10) is controlled by the control command.

12. The method according to any one of claims 8 to 11, wherein: After generating and / or updating the control commands, the simulation of the mixture is repeated, wherein, when repeating the simulation, the distribution function (f, 2) is calculated taking into account the generated and / or updated control commands, optionally forces imposed and / or applied by the control commands, flow rates and / or boundary conditions.

13. A data processing device comprising a processor configured to perform the steps of the method as claimed in any one of claims 1 to 6 and / or 8 to 12.

14. A computer program or a computer program product, comprising instructions which, when executed by a computer, cause the computer to perform the steps of the method as claimed in any one of claims 1 to 6 and / or 8 to 12.

15. A computer readable medium comprising instructions which, when executed by a computer, cause the computer to perform the steps of the method of any one of the preceding claims 1 to 6 and / or 8 to 12.

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