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 aswhere 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 aswhere 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 conditionsand the initial conditionsPhysically, 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 byand the discrete initial conditionswhere
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 byWe 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 towith the initial and boundary conditionsThe discrete leap-frog system (18)-(23) is modified as
Let and . We denote by and the corresponding finite difference solution andWe getand the initial conditionswhere Subtracting the system (36) from the system (38), we get the error equations
andand 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 haveWhen , with the inverse inequality we haveWhen , 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 getWith (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 asSince 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 thatMultiplying the error equation (41) by leads tothat 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 byand By (70), for those terms in the right hand side, we obtainTaking 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 haveWe now estimate the terms in (84). By (51), we denote Using (60) again, we get and with Lemma 2, we haveMoreover, by noting the fact that adding (87) in (84) and using Lemma 2 again, we further getMultiplying 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 haveFor the first term, we have
By (51), we have For the second term, we havewhere we noted the fact that . From (94) we can getThen we estimate the term , and to , respectively. From (46), we have
A straightforward calculation with Lemma 2 leads towhere 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 getMultiplying the last equation with and adding it into (90), we have
Letting , we getLetting 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 solutionwith 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.
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 haveand the effective heat conductivity is defined bywhere 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 byWe 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.
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.