Journal of Mathematics
Volume 2019 (2019), Article ID 8649308, 16 pages
https://doi.org/10.1155/2019/8649308

Research Article

A Leap-Frog Finite Difference Method for Strongly Coupled System from Sweat Transport in Porous Textile Media

Qian Zhang1,2 and Chao Huang1,3

¹College of Mathematics and Statistics, Shenzhen University, China
²Shenzhen Key Laboratory of Advanced Machine Learning and Applications, China
³Guangdong Key Laboratory of Intelligent Information Processing, China

Correspondence should be addressed to Qian Zhang; mazhangq@szu.edu.cn

Received 25 March 2019; Accepted 15 May 2019; Published 12 June 2019

Academic Editor: Tepper L. Gill

Copyright © 2019 Qian Zhang and Chao Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we present an uncoupled leap-frog finite difference method for the system of equations arising from sweat transport through porous textile media. Based on physical mechanisms, the sweat transport can be viewed as the multicomponent flow that coupled the heat and moisture transfer, such that the system is nonlinear and strongly coupled. The leap-frog method is proposed to solve this system, with the second order accuracy in both spatial and temporal directions. We prove the existence and uniqueness of the solution to the system with optimal error estimates in the discrete norm. Numerical simulations are presented and analyzed, respectively.

1. Introduction

Single/multicomponent flow in porous textile media attracted considerable attention in the last several decades. See [1–4] for the single-component models and [5–9] for the multicomponent models. In this paper, we study the multicomponent sweat transport coupled with vapor and heat in porous textile media. In [10], Ye et al. proposed a quasi-steady-state single-component model which consists of a steady-state air equation and dynamic state equations for other components. Under certain conditions, the multicomponent model reduces to a new single-component model, and the physical process can be viewed as sweat transport (vapor and heat flow) governed by the conservation of mass and energy:

where is the porosity of the media, is the vapor concentration, is the temperature, is the thermal conductivity, is the latent heat of evaporation/condensation, and is the molecular weight of water. The effective volumetric heat capacity is defined by

where is the molar heat capacity and is the volumetric heat capacity of fiber.

By Darcy’s law, the gas velocity is defined as

where is the permeability and is the dynamic viscosity, which usually is density-dependent for the compressible flow. Here we choose a linear form of , where is a certain constant.

By the Hertz-Knudsen equation [11], the phase change rate is defined as

where is a positive constant, the saturation pressure is determined from experimental measurements [12], and the pressure is given by , where is the universal gas constant.

With nondimensionalization, the sweat transport process (1)-(2) can be described by the following system:

where , , , and is a smooth and increasing function satisfying .

Since the right boundary is exposed to environment and the left boundary is connected to the body, we consider commonly used Robin type boundary conditions

and the initial conditions

Physically, parameters , , , , , and are nonnegative constants [1, 2, 6]. We define initial condition parameters with and being positive constants.

Due to the strong nonlinearity and the coupling of the system, both theoretical and numerical analyses of the system are difficult. Numerical analysis for some related systems of parabolic/elliptic equations can be found in [13–20]. Existence and uniqueness of a classical solution for a steady-state model was given in [10]. Existence of a weak solution for the corresponding dynamic models was given in [21, 22]. Positivity of temperature and nonnegativity of vapor density were also proved here. Recently, a finite difference method second-order in space and first-order in time for the system (6)-(12) was presented in [23], where the backward semi-implicit Euler scheme is applied in the temporal direction and central finite difference approximations are used in the spatial direction. In [23], authors presented optimal error estimates under the assumption that the step size and are smaller than a positive constant.

In this paper, we propose an uncoupled leap-frog finite difference method for the system (6)-(12) with second-order accuracy in both spatial and temporal directions. We prove the existence and uniqueness of a solution to the finite difference system with optimal error estimates in the discrete norm, under the condition that the mesh size and are smaller than a positive constant which depends solely upon the physical parameters involved in the equations. Due to the strong nonlinearity and the coupling of equations, the method presented in [23] does not apply to the leap-frog scheme directly. One of the difficulties is to show convergence of the numerical solution without restriction on the grid ratio. In this paper, we assume that the solution to the system (6)-(12) satisfies that

for some positive constants , , , and .

The manuscript is organized as follows: in Section 2, we present an uncoupled leap-frog finite difference method for the nonlinear sweat transport system. In Section 3, we prove the existence and uniqueness of the solution to the sweat transport system with the optimal error estimate in the discrete norm. Numerical results will be presented in Section 4 to support our theoretical results.

2. The Leap-Frog Finite Difference Scheme

For convenience of calculations, we add the equation (6) times into the equation (7); thus the governing system (6)-(7) can be modified as

Due to the practical interest in a long time period, say 8–24 hours, we present an uncoupled leap-frog finite difference scheme in the temporal direction and the central finite difference (volume) scheme in the spatial direction for the above system with the initial/boundary conditions (8)-(12).

Let be a positive number, let be a uniform partition in , and let be a uniform partition in , where and are the step size in the spatial and temporal directions, respectively. Denote in the spatial cell and let be a mesh function defined on , where . Some notations are introduced below:

from which

The discrete system is defined by

and the discrete initial conditions

where

The computational procedure of the uncoupled leap-frog scheme at each time step is listed below:

Step 1. The vapor concentration can be calculated by solving the tridiagonal linear systems defined in (18)-(20).

Step 2. With the updated vapor concentration , we can get and correspondingly.

Step 3. Finally, the temperature can be obtained by solving the tridiagonal linear system (21)-(23).

3. The Leap-Frog Scheme and the Optimal Error Estimate

In this section, we will show the existence and uniqueness of the solution to the system (18)-(26) with optimal error estimates in the discrete norm. Let and be two mesh functions on . We define the inner product and norms by

Let be the solution of the system (6)-(12) and , . The error functions are defined by

We state our main result in the theorem below:

Theorem 1. Suppose that the solution of the system (6)-(12) is in , satisfying (13). Then there exist positive constants and , independent of and , such that, when , the finite difference scheme (18)-(26) is uniquely solvable and

To prove the theorem, we make a stronger assumption that there exists , independent of , such that the inequality,

holds for . We prove the assumption and the theorem by induction method. By the initial condition (26), this is true for . In the next subsection we will show that this is also true for . In this part, we let be a generic positive constant, which is associated with the physical parameters , , , , , , , the parameters involved in initial and boundary conditions and the solution of the system (6)-(12). is independent of time step , mesh size , , and constant .

3.1. The Leap-Frog Scheme and Preliminaries

For convenience of calculations, we further introduce some notations. Let ; thus the sweat transport system (6)-(7) can be reduced to

with the initial and boundary conditions

The discrete leap-frog system (18)-(23) is modified as

Let and . We denote by and the corresponding finite difference solution and

We get

and the initial conditions

where

Subtracting the system (36) from the system (38), we get the error equations

and

and by (40), we can directly derive the inequality

To prove our main theorem, the following formula will be often used:

In the following lemma, we present discrete Sobolev interpolation formulas and the proof can be found in [24].

Lemma 2. Let and be two mesh functions. Then for any positive constant ,

Lemma 3.

Proof. From (30) for , we have

When , with the inverse inequality we have

When , by taking in Lemma 2,

The first part of (54) is obtained and the second part and the inequality (55) can be proved similarly.

In addition, by Lemma 3, there exist constants and such that, when ,

and

3.2. The Existence and Uniqueness

Since the coefficient matrix in the system (18)-(20) is strictly diagonally dominant, thus the system (18)-(20) has a unique solution . Here we will discuss the boundedness of .

Multiplying (41)-(43) by , , and , respectively, we get

With (44), (51), (59), (60), and (40), we see that

and by using (60) again, we have

and with (52),

Substituting the last three equations into (61) results in

where we have noted . Moreover, by the assumption of the induction,

Since we have the fact that , thus

When , we can get the inequality as

Since are independent of , by (13) when and are small enough,

Now we try to prove our main theorem. By noting (44), (60), (40), and Lemma 3,

We can see that, when , the assumption of induction and (68) show that

and when , by (68),

which means there exists an independent of , such that

Multiplying the error equation (41) by leads to

that is,

We can see that, when ,

and when , with Lemma 2,

where is independent of . Then there exists , when ,

With a time step condition , we can see that the coefficient matrix of the system (21)-(23) is strictly diagonally dominant. Thus this system has a unique solution .

3.3. The Optimal Error Estimate

We have proved the existence and uniqueness of the solution to the system and have derived the estimate (65) for . In this part, we try to derive an estimate for .

Multiplying (45)-(47) by , , and , respectively, we try to estimate each term below:

According to Lemma 2, (40), (51), and (60), three terms on the left can be bounded by

and

By (70), for those terms in the right hand side, we obtain

Taking the last six equations into (79), we obtain

Besides, we introduce a notation as , and adding the first three equations into (36) by , , and , respectively, we have

We now estimate the terms in (84). By (51), we denote

Using (60) again, we get

and with Lemma 2, we have

Moreover, by noting the fact that

adding (87) in (84) and using Lemma 2 again, we further get

Multiplying the last equation with and adding it into equation (65), we get

Finally we estimate . Multiplying the error equation (45) by and summing up the resulting equations for , we have

For the first term, we have

By (51), we have

For the second term, we have

where we noted the fact that . From (94) we can get

Then we estimate the term , and to , respectively. From (46), we have

A straightforward calculation with Lemma 2 leads to

where we noted the fact that and . Similarly we estimate . From (47), we have

such that

For those terms in the right hand side, with (70), we have

Plugging the last six equations into (95), we get

Multiplying the last equation with and adding it into (90), we have

Letting , we get

Letting

we have

from which we can find and , such that

When ,

and

Since is independent of , with

we find that (30) holds for . The induction and the proof of the theorem are completed.

4. Numerical Examples

We now numerically evaluate the performance of the proposed leap-frog scheme.

Example 1. First, we test the accuracy of our algorithm in an artificial example which is taken from [23]. The system is

with the boundary conditions (8)-(12), where , , , and are coefficients decided by the exact solution

with being constants.
We apply the uncoupled leap-frog finite difference method to solve the artificial example. We choose and . Since the proposed scheme is of the second order in both spatial and temporal directions, we take such that the error bound is proportional to . We present the -norm errors and the order of convergence in Table 1 with at different time level. We can see clearly from Table 1 that the -norm errors for both components are proportional to , which confirms our theoretical analysis.

Table 1: Numerical results of Example 1 with .

Order

2.914E-03
1.004E-03
0.254E-03
1.99

8.163E-03
2.178E-03
0.527E-03
2.03

1.397E-02
0.350E-02
0.084E-02
2.04

2.032E-02
0.496E-02
0.118E-02
2.05

Order

9.509E-03
3.795E-03
0.947E-03
2.00

3.860E-03
1.042E-03
0.253E-03
2.03

7.115E-02
1.799E-02
0.439E-02
2.02

1.069E-01
0.265E-01
0.065E-01
2.02

Example 2. In the second example, we discuss a typical clothing assembly in the textile industry [2, 4, 25]. The clothing assembly consists of three layers, in the middle is porous fibrous media, and the outside cover is exposed to a cold environment with fixed temperature and relative humidity while the inside cover is exposed to a mixture of air and vapor at higher temperature and relative humidity. In this paper, polyester porous media with laminated or nylon cover materials are tested. To compare with the experimental data in [12], a water equation is added to equations (1)-(2):

where is water content, is the density of water, is the porosity with liquid water content, and is the porosity without liquid water content. We have

and the effective heat conductivity is defined by

where is the thermal conductivity of gas and is the thermal conductivity of the fiber-water mixture [2, 6, 7], given by

The values of these physical parameters for polyester media are presented in Table 2. Other parameters values can be found in [2, 6, 7].
The initial conditions for the vapor, temperature, and water content are given by

We apply the uncoupled leap-frog finite difference method for solving the sweat transport system defined in (1)-(2) coupled with the water equation (115). Since only the right side of the water equation includes and , therefore, the water equation is calculated separately. Numerically, at each time step, we first find solution , by procedure (18)-(26), and then can be solved by following nonnormalized discrete formate:

Then we evaluate the parameters explicitly in (18)-(26) based on . Here all numerical results are obtained by taking the time step size and spatial mesh size . We present numerical results of vapor, temperature, and water content at 8 hours and 24 hours, respectively, for the porous polyester media assembly with laminated cover in Figure 1 and with nylon cover in Figure 2. The comparisons between numerical results of water content and experimental measurements [12] are given in last two subfigures, where the blue lines represent the numerical solution and the red line is given by experimental measurement.

Table 2: Physical parameters for batting materials.

parameter
polyester
unit

kgm⁻³

kgm⁻³

0.993

W m⁻¹K⁻¹

W m⁻¹K⁻¹

J m⁻¹K⁻¹

m

Figure 1: Numerical results for 10-pile polyester batting sandwiched by two layers of laminated fabric.

Figure 2: Numerical results for 10-pile polyester batting sandwiched by two layers of nylon fabric.

5. Conclusion

As a subsequent work of [23], we have presented an uncoupled leap-frog finite difference method for the sweat transport system in porous textile media, which is governed by a strongly coupled, nonlinear parabolic system. Optimal error estimates were presented, which imply that the numerical scheme is unconditionally stable. Both theoretical analysis and numerical example indicate that the current scheme is second order accurate in both the temporal and spatial directions. Since the scheme is decoupled for the system, the method can be applied efficiently for problems in higher-dimensional space. Under certain time-step restrictions, the analysis can also be extended to the multidimensional problems.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors wish to thank Professors Z. Sun and W. Sun for helpful discussions. This research was partially supported by National Natural Science Foundation of China (Nos. 11601346, 11501377), Shenzhen Key Laboratory of Advanced Machine Learning and Applications, Guangdong Key Laboratory of Intelligent Information Processing, and Interdisciplinary Innovation Team of Shenzhen University.

References

M. K. Choudhary, K. C. Karki, and S. V. Patankar, “Mathematical modeling of heat transfer, condensation, and capillary flow in porous insulation on a cold pipe,” International Journal of Heat and Mass Transfer, vol. 47, pp. 5629–5638, 2004.

J. Fan, X. Cheng, X. Wen, and W. Sun, “An improved model of heat and moisture transfer with phase change and mobile condensates in fibrous insulation and comparison with experimental results,” International Journal of Heat and Mass Transfer, vol. 47, no. 10-11, pp. 2343–2352, 2004.

C. V. le, N. G. ly, and R. Postle, “Heat and Moisture Transfer in Textile Assemblies: Part I: Steaming of Wool, Cotton, Nylon, and Polyester Fabric Beds,” Textile Research Journal, vol. 65, no. 4, pp. 203–212, 1995.

Y. li and Q. Zhu, “Simultaneous Heat and Moisture Transfer with Moisture Sorption, Condensation, and Capillary Liquid Diffusion in Porous Textiles,” Textile Research Journal, vol. 73, no. 6, pp. 515–524, 2003.

P. W. Gibson and M. Charmchi, “Modeling convection/diffusion processes in porous textiles with inclusion of humidity-dependent air permeability,” International Communications in Heat and Mass Transfer, vol. 24, no. 5, pp. 709–724, 1997.

H. Huang, C. Ye, and W. Sun, “Moisture transport in fibrous clothing assemblies,” Journal of Engineering Mathematics, vol. 61, no. 1, pp. 35–54, 2008.

C. Ye, H. Huang, J. Fan, and W. Sun, “Numerical study of heat and moisture transfer in textile materials by a finite volume method,” Communications in Computational Physics, vol. 4, no. 4, pp. 929–948, 2008.

Q. Zhang, B. Li, and W. Sun, “Heat and sweat transport through clothing assemblies with phase changes, condensation/evaporation and absorption,” Proceedings of the Royal Society A Mathematical, Physical and Engineering Sciences, vol. 467, no. 2136, pp. 3469–3489, 2011.

Q. Zhang, “Mathematical modeling and numerical study of carbonation in porous concrete materials,” Applied Mathematics and Computation, vol. 281, pp. 16–27, 2016.

C. Ye, B. Li, and W. Sun, “Quasi-steady-state and steady-state models for heat and moisture transport in textile assemblies,” Proceedings of the Royal Society A Mathematical, Physical and Engineering Sciences, vol. 466, no. 2122, pp. 2875–2896, 2010.

F. E. Jones, Evaporation of Water, Lewis Publishers Inc, Michigan, Mich, USA, 1992.

J. Fan, X. Cheng, and Y. S. Chen, “An experimental investigation of moisture absoption and condensation in fibrous insulations under low temperature,” Experimental Thermal and Fluid Science, vol. 27, pp. 723–729, 2002.

W. Dai and G. Li, “A finite difference scheme for solving parabolic two-step micro-heat transport equations in a double-layered micro-sphere heated by ultrashort-pulsed lasers,” Numerical Methods for Partial Differential Equations, vol. 22, no. 6, pp. 1396–1417, 2006.

R. Eymard, J. Fuhrmann, and K. Gartner, “A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problems,” Numerische Mathematik, vol. 102, pp. 463–495, 2006.

R. Eymard, T. Gallouët, R. Herbin, and A. Michel, “Convergence of a finite volume scheme for nonlinear degenerate parabolic equations,” Numerische Mathematik, vol. 92, pp. 41–82, 2002.

C. V. Pao, “Numerical analysis of coupled systems of nonlinear parabolic equations,” SIAM Journal on Numerical Analysis, vol. 36, no. 2, pp. 393–416, 1999.

H. J. Schroll, “Convergence of implicit finite difference methods applied to nonlinear mixed systems,” SIAM Journal on Numerical Analysis, vol. 33, no. 3, pp. 997–1013, 1996.

W. Sun and G. Yuan, “Stability condition for difference schemes for parabolic systems,” SIAM Journal on Numerical Analysis, vol. 38, no. 2, pp. 548–555, 2000.

H. Wang, “An optimal-order error esti mate for a family of ELLAM-MFEM approximations to porous medium flow,” SIAM Journal on Numerical Analysis, vol. 46, no. 4, pp. 2133–2152, 2008.

Y. Yuan, “The upwind finite difference fractional steps methods for two-phase compressible flow in porous media,” Numerical Methods for Partial Differential Equations, vol. 19, no. 1, pp. 67–88, 2003.

B. Li, W. Sun, and Y. Wang, “Global existence of weak solution to the heat and moisture transport system in fibrous porous media,” Journal of Differential Equations, vol. 249, no. 10, pp. 2618–2642, 2010.

B. Li and W. Sun, “Global existence of weak solution for nonisothermal multicomponent flow in porous textile media,” SIAM Journal on Mathematical Analysis, vol. 42, no. 6, pp. 3076–3102, 2010.

W. Sun and Z. Sun, “Finite difference methods for a nonlinear and strongly coupled heat and moisture transport system in textile materials,” Numerische Mathematik, 2011.

A. A. Samarskii and B. B. Andreev, Finite Difference Methods for Elliptic Equation, Nauka, Moscow, Russia, 1976.

J. Fan, Z. Luo, and Y. Li, “Heat and moisture transfer with sorption and condensation in porous clothing assemblies and numerical simulation,” International Journal of Heat and Mass Transfer, vol. 43, pp. 2989–3000, 2000.