A multi-dimensional concentration correction method for solving multi-component dust gas model

Abstract

The present application belongs to the technical field of numerical simulation of porous media, and particularly relates to a concentration correction method applicable to multi-dimensions for solving a multi-component dust-gas model, which comprises the following steps: Step 1: constructing a complete form of DGM equation set according to actual gas components, chemical reactions and porous medium parameters; Step 2: constructing a linear correction term for the DGM equation set established in Step 1; Step 3: iteratively solving on the basis of the linear correction term obtained in Step 2, and gradually correcting until the material flux and molar concentration meet preset constraint conditions, and then the concentration correction is completed. The method does not ignore the pressure gradient term, and can calculate the influence of pressure difference on gas transmission in the porous medium; the number of gas components is not limited, and the method is applicable to the solution of 1D, 2D and 3D dust-gas models; the method applies strong coupling between equations to solve the correction amount, so the solution speed is fast; practice shows that the constructed algorithm is not sensitive to initial values and is stable in calculation.

Description

Technical Field

[0001] This invention belongs to the field of porous media numerical simulation technology, specifically relating to a multidimensional applicable concentration correction method for solving multi-component dust gas models. Background Technology

[0002] Multicomponent transport in porous media is common in oil shale mining, fuel cells, and chemical reactors. Numerical simulations of mass transfer in porous media with nano- and micro-pore sizes typically require consideration of Knudsen diffusion. Commonly used mass transfer models in the literature include the Fick model (FM), the Stephen-Maxwell model (SMM), and the dust-gas model (DGM). Theoretically, FM is only applicable to two components, and the SMM model is only suitable for multicomponent transport in non-porous media and cannot be used for porous media. DGM, however, is a mass transfer model based on gas dynamics theory, treating the solid as a non-movable component, and is therefore applicable to multicomponent transport in porous media. Numerous studies have shown that DGM has higher accuracy in describing mass transfer phenomena in porous media under both reactive and non-reactive conditions.

[0003] However, due to the implicit relationship between mass flux, molar concentration, and pressure in the DGM, it cannot be expressed using a general equation, and therefore, this model is not currently available in commercial software. Secondly, current methods for solving the DGM have limitations to varying degrees, including limitations in dimension (less than 3), number of components (less than or equal to 3), and equation simplification (primarily ignoring the pressure gradient term). One-dimensional and two-dimensional models are not realistic, a number of components (less than or equal to 3) limits the model's applicability, and equation simplification only holds true under certain conditions. Therefore, developing a general algorithm for solving the DGM model is highly valuable. Summary of the Invention

[0004] (a) Technical problems to be solved

[0005] The technical problem to be solved by this invention is: how to provide a multidimensional applicable concentration correction algorithm based on the finite volume method for solving multi-component dust gas models.

[0006] (II) Technical Solution

[0007] To address the aforementioned technical problems, this invention provides a multidimensional applicable concentration correction method for solving multi-component dust gas models. The concentration correction method includes the following steps:

[0008] Step 1: Construct a complete set of DGM equations based on the actual gas composition, chemical reaction structure, and porous media parameters;

[0009] Step 2: Construct linear correction terms for the DGM equation system established in Step 1;

[0010] Step 3: Based on the linear correction term obtained in Step 2, perform iterative solution and gradually correct until the mass flux and molar concentration meet the preset constraints, then the concentration correction is completed.

[0011] In step 1, the transport of the k-th gaseous component in a porous medium where a chemical reaction occurs can be represented by the complete DGM equations as follows:

[0012]

[0013]

[0014]

[0015] in: For gradient operators,

[0016] The k-th gaseous component mentioned above will be referred to as component k hereafter.

[0017] J k =[J k,x J k,y J k,z ] T J is the vector representing the flux of component k along the x, y, and z directions in a rectangular coordinate system, i.e., the mass flux; l S is a vector representing the flux of component l, which is different from component k, in the x, y, z directions of a rectangular coordinate system; k X is the mass source for component k; k X l The molar concentrations of components k and l are respectively, where mass source S k It is the molar concentration of the component X k The function of X; T Total molar concentration; The effective Knudsen diffusion coefficient of component k; B represents the effective double diffusion coefficients of components k and l; g Permeability; μ is dynamic viscosity; p is pressure; R is molar gas constant; T is temperature; N is the number of gas components.

[0018] The ultimate goal is to solve for variable J. k X k The distribution of p, i.e., the mass flux J of component k. k molar concentration X k And the distribution of pressure p.

[0019] In step 1, the DGM equations already implicitly contain the momentum equation, so no additional momentum equation is needed.

[0020] In step 2, since formula (2) in step 1 is an implicit equation, i.e., the mass flux J k Cannot be expressed as molar concentration X k Since the gradient or divergence is an explicit function of pressure p, it cannot be directly solved in conjunction with formula (1); therefore, the concentration correction method is used to solve the formula set in step 1 above.

[0021] In step 2, linear correction terms are constructed for the DGM equation system established in step 1; the specific process is as follows:

[0022] First, rearrange the terms in formula (2):

[0023]

[0024] The first term on the right-hand side makes formula (4) an implicit equation and cannot be used as a correction term; the second term X k and This is a coupling relationship, a nonlinear term, and not suitable as a correction term; removing the first two terms on the right side of the equation yields a linear correction term:

[0025]

[0026] Wherein, the superscript “′” indicates the correction amount; J' k J represents k The correction amount, X' k X represents k The correction amount; the corrected amount is equal to the original amount plus the correction amount:

[0027]

[0028]

[0029] Where n is the number of solution steps; The purpose is to introduce formula (2) in the iteration, and the quantity before correction is obtained by interpolation using formula (8) to accelerate convergence; formula (8) is derived from formula (2);

[0030]

[0031] Substituting formula (6) into formula (1) yields the following: Poisson's equation:

[0032]

[0033] Based on the problem to be solved, Boundary conditions; given boundary concentration X k In the case of the boundary, That is, the concentration correction is 0;

[0034] Solving formula (9) yields Then, by using formulas (6) and (7), we can obtain... and

[0035] In step 3, based on the linear correction term obtained in step 2, an iterative solution is performed to gradually correct the concentration until the mass flux and molar concentration meet the preset constraints. The specific process is as follows:

[0036] Step 31: Given the component flux J k and molar concentration X k initial value and

[0037] Step 32: Initialize the value and Substituting into the right side of formula (4), we get...

[0038] Step 33: [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require the full context.] Substituting into formula (6) will yield The expression;

[0039] Step 34: Substituting the expression into formula (1), we get

[0040] Step 35: [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require the full context.] Substituting into formula (5), we get

[0041] Step 36: [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require the full context.] and Substituting into formulas (6) and (7) respectively, we obtain the results. and

[0042] Step 37: Repeat steps 31 to 36 until the preset constraints are met, that is, the solution ends and the concentration correction is completed.

[0043] The preset constraint conditions are as follows:

[0044]

[0045]

[0046] As can be seen from the solution process in step 3, the concentration correction method is not limited by the number of components and can be applied to one-dimensional, two-dimensional and three-dimensional numerical simulations.

[0047] Wherein, N = 5.

[0048] The multi-component dust gas includes CH4, CO, CO2, H2, and H2O.

[0049] (III) Beneficial Effects

[0050] Compared with existing technologies, the present invention proposes a multidimensional applicable concentration correction method for solving multi-component dust gas models, the advantages of which are as follows:

[0051] (1) The method provides a solution algorithm for a complete dust gas model, which does not ignore the pressure gradient term and can calculate the influence of pressure difference on gas transport in porous media.

[0052] (2) The method provides a solution algorithm for a multi-component dust gas model, where the number of gas components is not limited, for example, greater than or equal to 5. Chemical reactions between different gas components are also taken into account.

[0053] (3) The method provides a multidimensional applicable dust and gas model solution algorithm, which can be applied to the solution of 1D, 2D and 3D dust and gas models.

[0054] (4) The algorithm constructed by the method described above uses strong coupling between equations to solve for corrections, thus the solution speed is fast.

[0055] (5) Practice shows that the constructed algorithm is not sensitive to initial values ​​and has good computational stability. Attached Figure Description

[0056] Figure 1 This is a schematic diagram of the iterative solution process.

[0057] Figure 2 A schematic diagram of CH4 concentration calculated for a 2D model.

[0058] Figure 3 A schematic diagram of the total pressure distribution calculated for a 3D model. Detailed Implementation

[0059] To make the objectives, contents, and advantages of the present invention clearer, the specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and examples.

[0060] To address the aforementioned technical problems, this invention provides a multidimensional applicable concentration correction method for solving multi-component dust gas models. The concentration correction method includes the following steps:

[0061] Step 1: Construct a complete set of DGM equations based on the actual gas composition, chemical reaction structure, and porous media parameters;

[0062] Step 2: Construct linear correction terms for the DGM equation system established in Step 1;

[0063] Step 3: Based on the linear correction term obtained in Step 2, perform iterative solution and gradually correct until the mass flux and molar concentration meet the preset constraints, then the concentration correction is completed.

[0064] In step 1, the transport of the k-th gaseous component in a porous medium where a chemical reaction occurs can be represented by the complete DGM equations as follows:

[0065]

[0066]

[0067]

[0068] in: For gradient operators,

[0069] The k-th gaseous component mentioned above will be referred to as component k hereafter.

[0070] J k =[J k,x J k,y J k,z ] T J is the vector representing the flux of component k along the x, y, and z directions in a rectangular coordinate system, i.e., the mass flux; l S is a vector representing the flux of component l, which is different from component k, in the x, y, z directions of a rectangular coordinate system; k X is the mass source for component k; k X l The molar concentrations of components k and l are respectively, where mass source S k It is the molar concentration of the component X k The function of X; T Total molar concentration; The effective Knudsen diffusion coefficient of component k; B represents the effective double diffusion coefficients of components k and l; g Permeability; μ is dynamic viscosity; p is pressure; R is molar gas constant; T is temperature; N is the number of gas components.

[0071] The ultimate goal is to solve for variable J. k X k The distribution of p, i.e., the mass flux J of component k. k molar concentration X k And the distribution of pressure p.

[0072] In step 1, the DGM equations already implicitly contain the momentum equation, so no additional momentum equation is needed.

[0073] In step 2, since formula (2) in step 1 is an implicit equation, i.e., the mass flux J k Cannot be expressed as molar concentration X k Since the gradient or divergence is an explicit function of pressure p, it cannot be directly solved in conjunction with formula (1); therefore, the concentration correction method is used to solve the formula set in step 1 above.

[0074] In step 2, linear correction terms are constructed for the DGM equation system established in step 1; the specific process is as follows:

[0075] First, rearrange the terms in formula (2):

[0076]

[0077] The first term on the right-hand side makes formula (4) an implicit equation and cannot be used as a correction term; the second term X k and This is a coupling relationship, a nonlinear term, and not suitable as a correction term; removing the first two terms on the right side of the equation yields a linear correction term:

[0078]

[0079] Wherein, the superscript “′” indicates the correction amount; J' k J represents k The correction amount, X' k X represents k The correction amount; the corrected amount is equal to the original amount plus the correction amount:

[0080]

[0081]

[0082] Where n is the number of solution steps; The purpose is to introduce formula (2) in the iteration, and the quantity before correction is obtained by interpolation using formula (8) to accelerate convergence; formula (8) is derived from formula (2);

[0083]

[0084] Substituting formula (6) into formula (1) yields the following: Poisson's equation:

[0085]

[0086] Based on the problem to be solved, Boundary conditions; given boundary concentration X k In the case of the boundary, That is, the concentration correction is 0;

[0087] Solving formula (9) yields Then, by using formulas (6) and (7), we can obtain... and

[0088] In step 3, based on the linear correction term obtained in step 2, an iterative solution is performed, gradually correcting the concentration until the mass flux and molar concentration meet the preset constraints. The iterative solution process is illustrated below. Figure 1 The specific process is as follows:

[0089] Step 31: Given the component flux J k and molar concentration X k initial value and

[0090] Step 32: Initialize the value and Substituting into the right side of formula (4), we get...

[0091] Step 33: [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require the full context.] Substituting into formula (6) will yield The expression;

[0092] Step 34: Substituting the expression into formula (1), we get

[0093] Step 35: [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require the full context.] Substituting into formula (5), we get

[0094] Step 36: [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require the full context.] and Substituting into formulas (6) and (7) respectively, we obtain the results. and

[0095] Step 37: Repeat steps 31 to 36 until the preset constraints are met, that is, the solution ends and the concentration correction is completed.

[0096] The preset constraint conditions are as follows:

[0097]

[0098]

[0099] As can be seen from the solution process in step 3, the concentration correction method is not limited by the number of components and can be applied to one-dimensional, two-dimensional and three-dimensional numerical simulations.

[0100] Wherein, N = 5.

[0101] The multi-component dust gas includes CH4, CO, CO2, H2, and H2O.

[0102] Example 1

[0103] This embodiment uses the steam reforming of methane in a nickel-based catalyst as a case study, involving five gaseous components: CH4, CO, CO2, H2, and H2O. Each gaseous component reacts within a porous catalyst. The geometric model consists of a free channel and a porous medium, with the porous medium being the nickel-based catalyst. The gas mixture flows into the channel from the inlet, is transferred to the porous medium region, and undergoes flow and diffusion, resulting in the following three reactions:

[0104] 1. Steam reforming

[0105]

[0106] 2. Water vapor conversion

[0107]

[0108] 3. Methane reversion

[0109]

[0110] The equations are discretized using the finite volume method with a co-located grid, and a concentration correction algorithm is implemented through self-programming. The material source S(X) is defined based on the reaction rate equation. k The double diffusion coefficient and Knudsen diffusion coefficient of the gas were calculated. The flow inside the channel is laminar, with given velocity and concentration boundary conditions at the inlet, zero pressure at the outlet, and zero normal concentration gradient. The remaining boundary conditions are no-slip boundary conditions and zero mass flux.

[0111] Simulations were performed for both 2D and 3D scenarios. Figure 2 and Figure 3 The CH4 concentration and total pressure distributions in the flow channel and porous media regions are presented respectively.

[0112] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the technical principles of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A multidimensional and applicable concentration correction method for solving multi-component dust gas models, characterized in that, The concentration correction method includes the following steps: Step 1: Construct a complete set of DGM equations based on the actual gas composition, chemical reaction structure, and porous media parameters; Step 2: Construct linear correction terms for the DGM equation system established in Step 1; Step 3: Based on the linear correction term obtained in Step 2, perform iterative solution and gradually correct until the mass flux and molar concentration meet the preset constraints, then the concentration correction is completed. In step 1, for porous media where chemical reactions occur... k The transport of various gas components can be represented in the complete form of the DGM equations as follows: （1） （2） （3） in: For gradient operators, ; The above-mentioned k The gaseous components, hereinafter referred to as components. k ; , as components k In a rectangular coordinate system x, y, z A vector composed of fluxes in a direction, i.e., mass flux; Different from the components k Components l In a rectangular coordinate system x, y, z A vector composed of flux in a direction; Components k The source of quality; , Components k and components l The molar concentration of the mass source It is the molar concentration of the component. The function; Total molar concentration; Components k The effective Knudsen diffusion coefficient; Components k and components l The effective double diffusion coefficient; For penetration rate; Dynamic viscosity; For pressure; The molar gas constant; For temperature; The quantity of gas components; The ultimate goal is to solve for the variables. , and The distribution of components k mass flux molar concentration and pressure Distribution; In step 2, since formula (2) in step 1 is an implicit equation, i.e., mass flux Cannot be expressed as molar concentration and pressure The explicit function cannot be directly solved in conjunction with formula (1) to solve for the gradient or divergence; therefore, the concentration correction method is used to solve the formula set in step 1 above. In step 2, linear correction terms are constructed for the DGM equation system established in step 1; the specific process is as follows: First, rearrange the terms in formula (2): （4） The first term on the right-hand side makes formula (4) an implicit equation and cannot be used as a correction term; the second term and This is a coupling relationship, a nonlinear term, and not suitable as a correction term; removing the first two terms on the right side of the equation yields a linear correction term: （5） The superscript "′" indicates the correction amount; express The correction amount, express The correction amount; the corrected amount equals the original amount plus the correction amount: （6） （7） in, To determine the number of steps; This is to introduce formula (2) in the iteration, and the quantity before correction is obtained by interpolation using formula (8) to accelerate convergence; formula (8) is derived from formula (2); （8） Substituting formula (6) into formula (1) yields the following: Poisson's equation: （9） Based on the problem to be solved, Boundary conditions; at a given boundary concentration In the case of the boundary, That is, the concentration correction is 0; Solving formula (9) yields Then, through formulas (6) and (7), we can obtain... and .

2. The multidimensional applicable concentration correction method for solving multi-component dust gas models as described in claim 1, characterized in that, In step 1, the DGM equations already implicitly contain the momentum equation, so no additional momentum equation is needed.

3. The multidimensional applicable concentration correction method for solving multi-component dust gas models as described in claim 1, characterized in that, In step 3, based on the linear correction term obtained in step 2, an iterative solution is performed to gradually correct the concentration until the mass flux and molar concentration meet the preset constraints. The specific process is as follows: Step 31: Given the component flux and molar concentration initial value and ; Step 32: Initialize the value and Substituting into the right side of formula (4), we get ; Step 33: [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require the full context.] Substituting into formula (6) will yield The expression; Step 34: Substituting the expression into formula (1), we obtain ; Step 35: [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require the Substituting into formula (5), we get ; Step 36: [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require the , , and Substituting into formulas (6) and (7) respectively, we obtain the results. and ; Step 37: Repeat steps 31 to 36 until the preset constraints are met, that is, the solution ends and the concentration correction is completed.

4. The multidimensional applicable concentration correction method for solving multi-component dust gas models as described in claim 3, characterized in that, The preset constraint is as follows: Official (10) Official (11).

5. The multidimensional applicable concentration correction method for solving multi-component dust gas models as described in claim 4, characterized in that, As can be seen from the solution process in step 3, the concentration correction method is not limited by the number of components and can be applied to one-dimensional, two-dimensional and three-dimensional numerical simulations.

6. The multidimensional applicable concentration correction method for solving multi-component dust gas models as described in claim 5, characterized in that, The value of N is 5.

7. The multidimensional applicable concentration correction method for solving multi-component dust gas models as described in claim 5, characterized in that, The multi-component dust gas includes CH4, CO, CO2, H2, and H2O.

Images