Numerical methods to simulate moisture dynamics in fibrous sheet
- Hidekazu Yoshioka^{1}Email authorView ORCID ID profile,
- Kotaro Fukada^{1} and
- Ichiro Kita^{1}
https://doi.org/10.1186/s13362-017-0040-1
© The Author(s) 2017
Received: 17 January 2017
Accepted: 14 July 2017
Published: 28 July 2017
Abstract
Comprehension of moisture dynamics in fibrous sheets is indispensable in a wide variety of industrial areas. This paper proposes a practical mathematical model, which is referred to as the 2-D extended porous medium equation (PME), to physically describe moisture dynamics in fibrous sheets under evaporative environment. A numerical method, which is referred to as the 2-D dual-finite volume method (DFVM), to approximate its solutions in a stable manner is also presented so that the moisture dynamics is reasonably simulated. The 2-D DFVM, which can optionally be equipped with isotone numerical fluxes, is examined with test cases to show its satisfactory accuracy and versatility. The parameters and coefficients involved in the mathematical model for a non-woven fibrous sheet are identified with laboratory experiments. Numerical simulation of moisture dynamics in the horizontally or vertically placed sheet is performed as a demonstrative application example of the present model and the numerical method.
Keywords
1 Introduction
Comprehension of moisture dynamics in fibrous sheets are necessary in a wide variety of industrial areas, such as chemical engineering [1, 2], environmental engineering [3, 4], precise agriculture [5], and sanitary engineering [6]. Moisture dynamics in porous media such as fibrous sheets is macroscopic appearance of microscopic liquid water transport in multiply-connected pore network structures [7, 8]. In practical applications, moisture dynamics both in fibrous sheets and on their surface, the latter is due to evaporation driven by the humidity dynamics between the sheets and atmosphere [9, 10], is of importance for manufacturing better products in paper and textile industry [11–13]. Assessing moisture dynamics in non-woven fibrous sheets is therefore indispensable for development of products that are better fit-for-purpose.
Moisture dynamics occurring in the fibrous sheets have been described with nonlinear conservation laws based on the classical fluid dynamics [14, 15]. Numerically approximating solutions to nonlinear conservation laws requires applications of a conservative and stable numerical method, so that reasonable numerical solutions are obtained, such that positivity and/or monotonicity of certain physical quantities are realized. One common numerical method for nonlinear conservation laws is the finite volume method (FVM) based on local conservation principles [16]. The core of the FVM is evaluating numerical fluxes on cell interfaces, which determine accuracy and stability of numerical solutions [17, 18]. For solving conservation laws with dominant advection in particular, numerical fluxes should guarantee unconditional stability in space and should possess least numerical diffusion effects. This issue is more delicate for degenerate conservation laws than for those with non-degenerate counterparts because of possibly vanishing diffusion coefficients [19–21]. Mathematically, degeneration of the diffusion coefficients significantly affects qualitative behavior of the solutions to the conservation laws [22, 23]. Degeneration of diffusion coefficients would cause numerical instabilities even with sufficiently fine meshes when a standard numerical method is utilized [24]. Handling source terms in nonlinear partial differential equations like conservation laws also requires devised techniques for stable computation [25, 26]. Analyzing the moisture dynamics that we focus on would require sophisticated numerical techniques due to its complexity, which is our motivation of writing this paper with a particular emphasis on numerics.
This paper focuses on an FVM for porous medium equations (PMEs), which govern moisture dynamics in porous media, such as soils [27–29], fibrous sheets [4, 30, 31], and debris materials [32]. Similar differential equations are encountered in different physical problems, such as heat conduction problems in plasma [33, 34], combustion of liquids [35, 36], evaporation dynamics of volatile liquids [37], chemotaxis of cells and organisms [38, 39], and astrophysics [40]. Detailed, extensive mathematical and numerical analyses on the PMEs have been performed so far [41–43]. Zambra et al. [44] proposed an oscillation-free FVM with the weighted essentially non-oscillatory interpolation. The Discontinuous Galerkin methods are finite volume analogues of finite element methods, which also are conservative [44, 45]. Finite Volume Element Methods are conservative methods that concurrently use finite element and finite volume discretization schemes, which have also been effectively used for PMEs [46, 47].
Literature indicate that the concept of the isotonicity on numerical fluxes, originally stated in Ortega and Rheinboldt [48] and has later been discussed in Fuhrmann and Langmach [49], can possibly help develop stable and physically consistent FVMs in a simple manner. According to Fuhrmann and Langmach [49], numerical fluxes in the FVMs for nonlinear conservation laws such as PMEs should equip with the isotonicity for computing oscillation-free numerical solutions without extremely fine computational meshes. Their numerical fluxes for the conventional PMEs are isotone and can compute oscillation-free and non-negative numerical solutions; however, they would not be suitable for the PMEs of moisture dynamics in fibrous sheets as later demonstrated in this paper.
The purpose of this paper is to propose and examine a practical, stable FVM for 2-D PME-type equations that is referred to as the 2-D dual-finite volume method (DFVM), focusing in particular on application to moisture dynamics in fibrous sheets under evaporative environment. The governing equation of the moisture dynamics is an extended PME having nonlinear advection and evaporation terms [6, 50]. Yoshioka and Unami [51] originally proposed a prototype DFVM for simulating solute dispersion phenomena in surface water networks (connected graphs) consisting of 1-D reaches (lines or curves) and 0-D junctions (vertices). Yoshioka et al. [6] and Yoshioka and Triadis [50] later extended the DFVM for nonlinear conservation laws in 1-D domains and connected graphs. The original DFVM is unconditionally stable in space for linear parabolic equations and is unconditionally stable in both space and time when equipped with a fully-implicit temporal discretization method [51]. Numerical fluxes are evaluated with analytical solutions to linearized local two-point boundary value problems in the DFVM. This numerical technique is referred to as the fitting technique. Fitting technique have successfully been used for solving differential equations encountered in a variety of problems, such as option pricing [52, 53], optimal controls [54], and reactive transport phenomena [55, 56]. Numerical fluxes with the fitting technique guarantee first-order spatial convergence for linear problems [57, 58]. This paper provides an isotone numerical flux specialized for the 2-D extended PMEs and its application to test cases and a realistic problem. Stability of the present DFVM for the extended PMEs is achieved with a new isotone numerical flux. Nonlinear source terms involved in the extended PMEs are dealt with using an operator-splitting technique [59] to compute stable and physically consistent numerical solutions.
The rest of this paper is organized as follows. Section 2 presents the 1-D and 2-D PMEs. Their regularized counterparts for well-posing the problems are also presented in this section, which are used in numerical computation in this paper. Section 3 presents the discretization procedure of the 2-D DFVM. Section 4 presents applications of the 2-D DFVM and 2-D extended PME to test cases and a realistic problem. Section 5 concludes this paper.
2 Mathematical model
The extended PMEs for moisture dynamics in fibrous sheets subject to evaporation are presented. The 1-D extended PME [50], although it is not explicitly used in this paper, is firstly presented as a reduced counterpart of the present 2-D model for self-contentedness.
2.1 1-D extended porous medium equation
The 1-D extended PME was formulated for simulating 1-D longitudinal moisture dynamics occurring in porous media with a striped shape. It is physically reasonable to assume that moisture profiles in such a sheet are transversely homogenous, leading to a longitudinally 1-D model. The saturation at each longitudinally 1-D position x of a sheet at the time t is denoted by \(\theta = \theta ( t,x ) \), and is normalized in \([ 0,1 ] \) as \(0 \le u = ( \theta - \theta_{\mathrm{r}} ) ( \theta_{\mathrm{s}} - \theta_{\mathrm{r}} ) ^{ - 1} \le 1\) where \(\theta_{\mathrm{s}}\) and \(\theta_{\mathrm{r}}\) are the maximum (saturated) and minimum (residual) water contents of the sheet.
2.2 2-D extended porous medium equation
Finally, in this section, remarks on boundary conditions to be supplemented to the 2-D extended PME and its regularized counterpart are provided with particular emphasis on the wetting process. From a mathematical viewpoint, at least some boundary conditions should be supplemented to these PMEs in general, since they are partial differential equations of the parabolic-type, although with possible degeneration of the coefficients. From a physical view point, a reasonable boundary condition for simulating a wetting process where a part of the boundary of the domain (\(\partial \Omega_{\mathrm{W}}\)) touches liquid water is the Dirichlet boundary condition \(u = 1\) meaning that the boundary \(\partial \Omega_{\mathrm{W}}\) is fully-wet. Of course, it is theoretically possible to specify the net flux \(\mathbf{F} \cdot \mathbf{n}\) on \(\partial \Omega_{\mathrm{W}}\), but directly measuring the flux is, at least to our experience, technically not easy. On the other part of the boundary, a simple and appropriate boundary condition is the no-flux condition \(\mathbf{F} \cdot \mathbf{n} = 0\) where the inner product between the flux vector F and the outward boundary normal vector n vanishes. This boundary condition represents that no liquid water escapes from the boundary, which is a reasonable assumption for moisture dynamics with evaporation as focused on in this paper later.
3 2-D dual-finite volume method
The discretization procedure of the 2-D DFVM is explained in this section.
3.1 Spatial discretization
3.2 Operator-splitting algorithm
3.3 Evaporation sub-step
3.4 Advection-diffusion sub-step
3.4.1 Finite volume formulation
3.4.2 Numerical flux
The value of \(\overline{u_{\rho ( i,l ) }}\) is specified to determine \(F_{i,l}\). The three evaluation schemes for \(\overline{u _{\rho ( i,l ) }}\) are presented in this paper, which are the central (CE) scheme, fully-upwind (FU) scheme, and the isotone (IS) scheme. The CE and FU scheme have been used in Fuhrmann and Langmach [49], and the IS scheme is a new numerical method proposed in this paper. The approximation \(\varepsilon = 0\) is used in this and next sections because it is assumed to be a sufficiently small positive constant such that it does not significantly affect accuracy of numerical solutions.
3.4.3 Isotonicity on the CE scheme
The CE scheme possibly gives unphysical (negative) numerical solutions because it does not comply with (36) and (37) even with the absence of advection term [32].
3.4.4 Isotonicity on the FU scheme
3.4.5 Isotonicity on the IS scheme
3.4.6 Temporal integration
4 Application
4.1 Test cases
The 2-D DFVM is applied to a series of test cases for its verification of accuracy and stability. Note that the aim of these test cases is not comparing the 2-D DFVM with other methods but to confirm its satisfactory convergence to analytical solutions, since the purpose of this paper is establishment of a mathematical model and a simple numerical method that can reasonably simulate the 2-D moisture dynamics. In each test case, the independent variables are appropriately scaled without the loss of generality. The 2-D DFVM is developed in a standard C++ environment. Each unstructured triangular computational mesh in what follows has been generated with free software VORO Ver. 3.17 (available at http://www32.ocn.ne.jp/~yss/voro.html). The 2-D DFVM with the operator splitting technique is expected to perform first-order accuracy in time according to Li et al. [59]. Note that using the Crank-Nicolson scheme (\(\theta = 0.5\)) in the temporal integration of the advection-diffusion sub-step yields oscillatory numerical solutions in the test cases examined below. The exact solutions in some test cases are partially \(u > 1\), which is physically invalid. These test cases therefore solely concern computational accuracy of the 2-D DFVM and do not discuss physical meaning of the solutions.
4.1.1 Test 1: Barenblatt problem
Computed error norms and the order of convergence for Test 1
Mesh | Error norm | Order of convergence in space | ||||
---|---|---|---|---|---|---|
m = 2 | m = 3 | m = 4 | m = 2 | m = 3 | m = 4 | |
32 × 32 | 3.66E − 02 | 6.29E − 02 | 7.69E − 02 | |||
64 × 64 | 2.12E − 02 | 4.12E − 02 | 5.70E − 02 | 7.88E − 01 | 6.10E − 01 | 4.32E − 01 |
128 × 128 | 8.54E − 03 | 2.22E − 02 | 3.40E − 02 | 1.31E + 00 | 8.92E − 01 | 7.45E − 01 |
256 × 256 | 3.85E − 03 | 1.25E − 02 | 2.28E − 02 | 1.15E + 00 | 8.29E − 01 | 5.77E − 01 |
4.1.2 Test 2: steady problem with evaporation
Computed error norms and the order of convergence for Test 2
Time increment Δ t | Error norm | Order of convergence in time | ||||
---|---|---|---|---|---|---|
32 × 16 | 64 × 16 | 128 × 16 | 32 × 16 | 64 × 16 | 128 × 16 | |
0.01 | 5.04E − 02 | 5.77E − 02 | 6.05E − 02 | |||
0.001 | 6.35E − 03 | 5.01E − 03 | 5.51E − 03 | 9.00E − 01 | 1.06E + 00 | 1.04E + 00 |
0.0001 | 1.29E − 02 | 5.65E − 03 | 2.31E − 03 | −3.08E − 01 | −5.22E − 02 | 3.78E − 01 |
4.1.3 Test 3: travelling wave solution
Computed error norms and the order of convergence for Test 3
Mesh | Error norm | Order of convergence in space | ||
---|---|---|---|---|
FU scheme | IS scheme | FU scheme | IS scheme | |
348 cells | 6.95E − 03 | 5.11E − 03 | ||
1,296 cells | 4.29E − 03 | 3.39E − 03 | 6.98E − 01 | 5.92E − 01 |
4,949 cells | 1.67E − 03 | 9.80E − 04 | 1.36E + 00 | 1.79E + 00 |
19,565 cells | 7.61E − 04 | 5.96E − 04 | 1.14E + 00 | 7.17E − 01 |
4.2 Demonstrative application
A demonstrative application example of the 2-D (regularized) extended PME and the 2-D DFVM to simulating moisture dynamics in an existing non-woven fibrous sheet is presented in this sub-section where the advection, diffusion, and evaporation terms do not vanish.
4.2.1 Parameter identification
To be used in applications, the extended PME has three coefficients of the functions of u to be estimated: the pressure head \(\psi ( u ) \), the hydraulic permeability \(K ( u ) \), and the evaporation rate \(E_{\mathrm{s}}u^{q}\). This sub-section presents our experimental results to identify reasonable ranges of the parameters involved in these coefficients. In the experiments, a non-woven fibrous sheet for water absorption was used where the product number is not presented in this paper. Thickness of the sheet is 10^{−3} (m). The company manufactured this product for multiple purposes, such as materials for vehicles, materials used in civil engineering, water transportation and adsorption in agricultural fields, and sanitary purposes. The parameters \(\theta_{\mathrm{M}}\) and \(\theta_{ \mathrm{r}}\) for this sheet has been estimated as \(\theta_{\mathrm{M}} = 0.48\) and \(\theta_{\mathrm{r}} = 0.07\), respectively.
Pressure head
As in the case for the conventional hanging water column method to be applied to soil materials [73], matric potential was steadily controlled for each fixed pressure −n (kPa) with non-negative integers \(0 \le n \le 6\). The moisture weight percentage of the material, which is denoted by w, was measured at each value of the matric potential. This experiment was repeated twice to reduce statistical errors in the identification process of the matric potential. A standard nonlinear least square method is applied to identify the pressure head \(\psi ( u ) \).
Hydraulic permeability
A tensiometer (Stec Co., Ltd.) was utilized. At the beginning of the experiment, a sheet material that is fully wet was prepared, and it was drained from the wet state until it finalizes the drainage of water. During the drainage process, pressure, moisture flux, and moisture weight percentage of the material were measured at one-second interval. This experiment was carried out seven times to reduce statistical errors in the identification process of the hydraulic permeability: four times with the pressure head of −25 (cm) and the remaining three with −35 (cm). The length of the period for the drainage was 40 to 50 (min). Assuming the conventional Darcy’s flux [73], the hydraulic permeability was identified from the experimental results. A standard nonlinear least square method is applied to identify the hydraulic permeability \(K ( u ) \) as a function of u.
Evaporation rate
The evaporation term is identified with a fully wetted sheet placed in a nearly temporally homogenous environment (a closed laboratory room) where the relative of about 42% to 44% and the temperature of 25 to 26 degree Celsius. The experimental setting is simple: a fully wetted sheet is placed on an electric balance and its weight is measured at the interval of 30 seconds. The measured data is calibrated with analytical solutions to the ODE \(\frac{\mathrm{d}u}{\mathrm{d}t} = - E_{ \mathrm{s}}u^{q}\). A standard nonlinear least square method is applied to identify the parameters q and \(E_{\mathrm{s}}\).
4.2.2 Computational examples
An application of the presented mathematical model and the numerical method to moisture dynamics in the non-woven fibrous sheet is carried out where it is placed horizontally (\(\alpha = 0\)) or vertically (\(\alpha = 0.5\pi \)). The identified coefficients and parameters in the previous sub-sections are used in the numerical simulation here. The focus here is to track moisture dynamics in a sheet with the two contrasting placement patterns under an evaporative environment.
A sheet with a circular shape with the radius of 0.05 (m) is considered, which is considered as the domain \(\Omega = \{ ( x _{1},x_{2} ) \mid x_{1}^{2} + x_{2}^{2} < 0.05^{2} \} \) in the dimensional form. The finest computational mesh for the unit circular domain with 38,588 triangular cells and 19,565 dual cells is used where its spatial scale has been accordingly changed so that the above-mentioned domain is created. The sheet is initially assumed to be dry (\(u = 0\)), which is specified as the initial condition at the time \(t = 0\) (s). The part of the boundary of the domain \(( x_{1},x _{2} ) \) with \(0 \le x_{1} \le 0.05\) (m) in the dimensional form is assumed to be fully wet (\(u = 1\)) and the other part of the boundary is considered as the no-flux boundary with the conventional zero-flux condition. The time increment Δt is set as 0.001 in the non-dimensional form, which corresponds to 24.0 (s) in the dimensional form.
It should be finally noted that both the IS scheme has also been applied to the numerical simulation, but relative error between the numerical solutions with the FU and IS schemes was smaller than several percent. This is considered to be due to the relationship \(m \gg p\) for the examined non-woven fibrous sheet where p is sufficiently small to affect the isotonicity of the IS scheme established at \(p = 0\).
5 Conclusions
This paper presented the 2-D extended PME for moisture dynamics in fibrous sheets under evaporative environment. Each coefficient involved in the 2-D extended PME is possible to identify with physical experiments, and this statement was checked with laboratory experiments. The 2-D DFVM for approximating solutions to the equation was also presented in this paper with detailed spatial and temporal integration schemes. The concept of isotonicity was effectively incorporated into the evaluation of the numerical flux to compute physically reasonable numerical solutions. An operator-splitting technique based on the analytical solutions to the ODEs was applied to the temporal discretization procedure to handle the problems with nonlinear source terms. Application of the 2-D DFVM to test cases demonstrated its satisfactory computational accuracy and stability. A demonstrative application example of the 2-D extended PME and the DFVM showed their potentially high functionality to simulating essentially 2-D moisture dynamics in fibrous sheets under evaporative environment.
Although this paper focused solely on the moisture dynamics in fibrous sheets, the presented model can be applied to the dynamics in other materials, such as papers, and even to different phenomena if the coefficients involved in the PME are appropriately modified [74–76]. The present DFVM can be used for solving similar differential equations arising in the other research areas where the equations of the PME-type are of central use. Future research will focus on exploring more relevant functional forms of the coefficients of the extended PMEs based on physical experiments, which requires sampling larger number of measured data with higher accuracy. Although the 2-D extended PME assumed homogeneity and isotropy of sheets, real problems of moisture dynamics in non-woven fibrous sheets would involve spatially heterogeneous and/or anisotropic sheets. Heterogeneity can be partly resolved with the parameters or the coefficients, but appropriate internal boundary conditions will be necessary on the boundary of heterogeneity as in Kuraz et al. [77, 78]. Transport phenomena of chemical substances whose dynamics typically follows certain conservation laws can be simulated basted on the moisture dynamics with the present model. From a mathematical side, theoretical numerical analysis of the 2-D DFVM will be performed to deeper comprehend its properties, theoretical order of convergence in particular. Mathematical analysis on the regularization for the nonlinear degenerate parabolic PDEs like PMEs will also be addressed. Approaching from mathematical, experimental, and numerical sides to these un-resolved issues will be carried out in future research.
Declarations
Acknowledgements
The authors thank to the reviewers for providing valuable comments on this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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