An inverse problem of Bilayer textile thickness determination in dynamic heat and moisture transfer

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An inverse problem of Bilayer textilethickness determination in dynamicheat and moisture transferDinghua Xu a , Lei Wen a & Boxi Xu ba Department of Mathematics , Zhejiang Sci-Tech University ,310018 , Zhejiang , Chinab College of Mathematical Sciences , Fudan University , 200433 ,Shanghai , ChinaPublished online: 09 Sep 2013.

To cite this article: Dinghua Xu , Lei Wen & Boxi Xu , Applicable Analysis (2013): An inverseproblem of Bilayer textile thickness determination in dynamic heat and moisture transfer,Applicable Analysis: An International Journal, DOI: 10.1080/00036811.2013.835042

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Applicable Analysis, 2013http://dx.doi.org/10.1080/00036811.2013.835042

An inverse problem of Bilayer textile thickness determination indynamic heat and moisture transfer

Dinghua Xua∗, Lei Wena and Boxi Xub

aDepartment of Mathematics, Zhejiang Sci-Tech University, Zhejiang 310018, China; bCollege ofMathematical Sciences, Fudan University, Shanghai 200433, China

Communicated by Yongzhi Xu

(Received 18 June 2013; accepted 12 August 2013)

An inverse problem of bilayer textile thickness determination in dynamic heatand moisture transfer is presented satisfying the heat–moisture comfort levelof human body. Heat and mass transfer law in bilayer textiles is displayed byproving the existence and uniqueness of solution to the coupled partial differentialequations with initial-boundary value conditions. The finite difference method isemployed to derive the numerical solution to partial differential equations. Theregularized solution of the inverse problem is reformulated into solving functionminimum problem through the Tikhonov regularization method. The goldensection method is applied to solve the direct search problem and achieve theoptimal solution to the inverse problem. Numerical algorithm and its numer-ical results provide theoretical explanation for textile materials research anddevelopment.

Keywords: inverse problems of thickness determination; partial differentialequations of parabolic type; well-posedness; regularized solution; bilayer textilematerials; numerical algorithm

AMS Subject Classifications: 35R30; 35K51; 65M32; 80A20

1. Introduction

Heat-moisture transfer models based on body–clothing–environment system aim to studycharacteristics of the heat moisture transfer inside textile materials. It has been foundthat the physical processes such as heat conduction, heat diffusion, heat radiation andheat condensation have decisive effects on the heat–moisture comfort level of humanbody.[1–5] Heat–moisture transfer inside the textile materials has usually been formulatedas the systems of ordinary/partial differential equations.

Under the different climate conditions, people ask for different types of clothes. Themost important purpose of clothes is that clothes can meet the human body’s comfort,such as heat–moisture comfort, pressure comfort, friction comfort and aesthetic comfort.Besides the environmental conditions such as temperature, moisture and wind speed, thefactors influencing human body heat–moisture comfort level include thickness, textureand structure of textile materials. One of the main purposes of textile material design is to

∗Corresponding author. Email: [email protected]

© 2013 Taylor & Francis

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determine thickness, type and pore structure of textile materials on the condition that humanbody feels comfortable.

Given the time-varying environmental temperature and moisture, one tries to find outthe optimal thickness and type of the textile material when one takes human body comfortas the design objective. Traditional methods employed to determine the textile thicknesswere empirical or experimental methods. In order to theoretically determine the thicknessof textile materials, we adopt the regularization idea to solve inverse problems of textilematerial design. This research can provide theoretical basis and scientific explanation forthe functional textile designs.[6]

The definite problems of solving ordinary/partial differential equations based on theheat–moisture transfer model are usually called direct problems. Meanwhile taking humanbody comfort as design objective, the problems of determining textile material thicknessand type are called the inverse problem of textile material design.

Some researchers [7–9] put forward steady heat–moisture transfer models, and studythe determination of textile material thickness and type when the thermal condensationhappens. The results show that is feasible and effective to employ scientific theories andmethods to study the property of the textile material.

Based on dynamic heat–moisture transfer models or the definite problems of partialdifferential equations, the researchers [10–12] derived coupled nonlinear partial differentialequations when heat conduction, heat diffusion and heat radiation are taken into consid-eration. And the corresponding research results [10,11] on inverse problems of the textilematerial thickness design serve as theoretical guidance to the textile material design.

The well-posedness of direct problems is explained by existence, uniqueness and sta-bility of the solutions. Hence is imperative to verify the applicability of the heat–moisturetransfer model,[13] which can provide a scientific explanation for the physical processes inheat–moisture transfer.

In the functional textiles design, heat-moisture transfer models can proclaim character-istics of heat and mass transfer in multilayered textiles.[14] For example, under the hightemperature conditions one ask for high waterproof and adiabatic textile materials so asto prevent human body from being fried. Meanwhile under intense radiation environmentconditions, one asks for textiles of radiation protection.

This paper intends to discuss inverse problems of bilayer textile thickness determinationbased on the dynamic heat–moisture transfer model at low temperature. We show that thereexists a unique solution to the direct problem of heat–moisture transfer model inside thebilayer textiles. And then, we continue on the inverse problem of determining optimalthickness of inner and outer porous batting. By means of the Tikhonov regularizationmethod, we formulate the inverse problem of determining textile thickness into a functionoptimization problem with penalty terms. The golden section method is employed to directsearch in alternating directions, we have achieved the regularized solution, namely thethickness of the bilayer textiles. The optimal regularized solution shows that the bilayertextile thickness may be simultaneously determined to meet the actual requirement of humanbody heat–moisture comfort.

2. Dynamic heat and moisture transfer model in bilayer textiles

2.1. NomenclatureCa(x, t) water vapour concentration in the inter-fibre void space (kg m−3)

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Applicable Analysis 3

C∗a (x, t) saturated water vapour concentration in the inter-fibre void space (kg m−3)CE (t) water vapour concentration between outer covering fabrics and surrounding (kg m−3)C0(t) water vapour concentration in inner covering fabrics (kg m−3)C1(t) water vapour concentration in outer covering fabrics (kg m−3)CI (t) initial water vapour concentration in batting (kg m−3)

Cv effective volumetric heat capacity of the fibrous batting (kJ m−3 K−1)Da diffusion coefficient of water vapour in the air (m2 s−1)

FL (x, t) total heat radiation incident travelling to the left (kJ m−2 s−1)FR(x, t) total heat radiation incident travelling to the right (kJ m−2 s−1)

RH(x, t) relative humidity of the surroundings (%)RH0(t) relative humidity in the microclimate area (%)RH1(t) relative humidity in the outer environment (%)T (x, t) temperature in fabrics (K or ◦C)

T0(t) temperature in the microclimate area (K or ◦C)T1(t) temperature in the outer environment (K or ◦C)TI (t) initial temperature in batting (K or ◦C)

�(x, t) the total rate of (de)sorption,condensation,freezing and/or evapouration (kg m−3s−1)ξi surface emissivity of the inner and outer covering fabrics (i=1:inner fabric; i=2:outer

fabric)κi effective heat conductivity of the fibrous batting (i=1:inner batting, i=2:outer

batting) (kJ m−1 K−1 s−1)λ latent heat of (de)sorption of fibres or condensation of water vapour (kJ kg−1)β radiative sorption constant of the fibres (m−1)σ Boltzmann constant (kJ m−2 K−4 s−1)εi porosity of the fibrous batting (i=1:inner fabric; i=2:outer fabric)τ effective tortuosity of the fibrous batting

ω1 water vapour resistance of inner and outer fabricshc convective vapour transfer coefficient (m s−1)

2.2. Mathematical formulation of dynamic heat and moisture transfer model in bilayertextiles

We make some assumptions for the dynamic heat and moisture transfer in the body–clothing–environment system (Figure 1) at low temperature [14]:

Figure 1. Body-clothing-environment system.

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(A1) The inner and outer fibrous batting are isotropic in fiber arrangement and materialproperties.

(A2) Volume changes of the fibres due to changing moisture and water content areneglected and effective tortuosity is a constant.

(A3) Sorption and condensation are considered only.(A4) Radiation absorption coefficient of different materials are regarded as constant.(A5) Sweat is neglected when human body is motionless.

Under the above assumptions, we conclude the heat and moisture transfer equationstogether with the corresponding initial–boundary value conditions. Let �1 = (0, L1) ×(t1, t2), �2 = (L1, L1 + L2) × (t1, t2).

(I) In inner and outer porous batting, the temperature T (x, t), heat radiation FL(x, t)and FR(x, t) satisfy the following partial differential equations⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

Cv∂T∂t = ∂

∂x

(κ1

∂T∂x

)+ ∂ FL∂x − ∂ FR

∂x + λ�, (x, t) ∈ �1

Cv∂T∂t = ∂

∂x

(κ2

∂T∂x

)+ ∂ FL∂x − ∂ FR

∂x + λ�, (x, t) ∈ �2∂ FL∂x = βFL − βσ T 4, (x, t) ∈ �1 ∪ �2

∂ FR∂x = −βFR + βσ T 4, (x, t) ∈ �1 ∪ �2

(1)

together with the initial condition

T (x, t1) = TI (x), x ∈ [0, L1 + L2] (2)

left boundary value vconditions{T (0, t) = T0(t), t ∈ [t1, t2](1 − ξ1)FL(0, t) + ξ1σ T 4(0, t) = FR(0, t), t ∈ [t1, t2] (3)

intermediate contact surface conditions⎧⎪⎨⎪⎩

limx→L−

1

T (x, t) = limx→L+

1

T (x, t), t ∈ [t1, t2]lim

x→L−1

κ1∂T (x,t)

∂x = limx→L+

1

κ2∂T (x,t)

∂x , t ∈ [t1, t2] (4)

and right boundary value conditions{T (L1 + L2, t) = T1(t), t ∈ [t1, t2](1 − ξ2)FR(L1 + L2, t) + ξ2σ T 4(L1 + L2, t) = FL(L1 + L2, t), t ∈ [t1, t2].

(5)The coupled partial differential equations together with the initial–boundary value

conditions are called Direct Problems (DP).

(II) In the inner and outer porous batting, water vapour concentration Ca(x, t) satisfiespartial differential equations⎧⎨

⎩ε1

∂Ca∂t = Daε1

τ∂2Ca∂x2 − �, (x, t) ∈ �1

ε2∂Ca∂t = Daε2

τ∂2Ca∂x2 − �, (x, t) ∈ �2

(6)

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together with the initial conditions

Ca(x, t1) = CI (x), x ∈ [0, L1 + L2] (7)

intermediate contact surface conditions⎧⎪⎨⎪⎩

limx→L−

1

Ca(x, t) = limx→L+

1

Ca(x, t), t ∈ [t1, t2]

limx→L−

1

Daε1τ

∂Ca(x,t)∂x = lim

x→L+1

Daε2τ

∂Ca(x,t)∂x , t ∈ [t1, t2] (8)

and right boundary value conditions⎧⎨⎩

Ca(L1 + L2, t) = C1(t), t ∈ [t1, t2]Daε2

τ∂Ca(x,t)

∂x

∣∣∣x=L1+L2

= CE (t)−Ca(L1+L2,t)ω1+h−1

c, t ∈ [t1, t2]. (9)

The problem of determining the left boundary C0(t) = Ca(0, t) by the above partialdifferential equations and the initial, right boundary value conditions is called inverseproblem (IP).

(III) About the total water accumulation rate�(x, t)([7]) and relative humidity RH(x, t).Total water accumulation rate �(x, t) is uniquely determined by temperature T (x, t),

namely �(x, t) = �(x, t; T ). The following relationships are valid:⎧⎨⎩ �(x, t) = Da

τ

∂2C∗a (x,t)

∂x2 − ∂C∗a (x,t)∂t , (x, t) ∈ �1 ∪ �2

C∗a (x, t) = 216.5 × 10−6 Vap(T (x,t))

T (x,t) , (x, t) ∈ �1 ∪ �2

(10)

where s = T − 273.16, and Vap(T ) is given by the following empirical formula

Vap(T ) ={

1013.25e13.3185s−1.976s2−0.6445s3−0.1299s4, T ≤ 273.16

1010.5380997− 2663.91T , T > 273.16

(11)

Relative humidity RH(x, t) is defined as

RH(x, t) = T (x, t)Ca(x, t)

216.5 × 10−6Vap(T (x, t))= Ca(x, t)

C∗a (x, t)

, (x, t) ∈ [0, L1 + L2] × [t1, t2].(12)

3. Mathematical formulation of An Inverse Problems of bilayer Textile ThicknessDetermination (IPTTD)

We formulate the IPTTD as follows.IPTTD: Given the human body heat–moisture comfort indexes in the microclimate

area,[15] namely temperature (32 ± 1)◦ C, relative humidity (50 ± 10)%, wind speed(25±15)cm s−1,[15] we suppose that the environmental temperature and relative humidity(T e(t), RHe(t)) are given. By means of solving (DP) and (IP), we should determineoptimum thickness L1 and L2 in the body–clothing–environment system to make surethat people feel heat–moisture comfortable. That is to say, the values of temperature andhumidity in the microclimate area lie stably in heat–moisture comfort indexes (intervals).

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Least squares method can be used to solve the above IPTTD. Through the numericalsolution to DP and IP, we can numerically obtain the temperature T0(t j ) = T (0, t j ) andhumidity C0(t j ) = Ca0, t j ) in inner covering fabric (x = 0) at any time t j . Relativehumidity RH0(t j ) = RH(0, t j ) is henceforth obtained according to the formula (12).RH0(t j ) is rewritten as RH(0, t j ; L1, L2) since it varies as the thickness L1 and L2 ofinner and outer porous batting. In general, we adopt relative humidity RHA = 50% inthe sense of the most comfortable expectation in the paper. The objective function of leastsquares is J (L1, L2) =∑n

j=1

∣∣RH(0, t j ; L1, L2) − RHA∣∣2.

In practical applications, there are some suitable requirements on textile material design.For example we hope the clothing be kept much thinner, so we to need make a limitationon the thickness of inner and outer porous batting, for example, the thickness satisfiesL1 + L2 ≤ Lmax. Similarly, we also hope the clothing be kept much lighter, that is to say,we have the limitation such as ρ1L1 + ρ2L2 ≤ Kmax. In this paper, we only considerthe constraint on the thickness, i.e. L1 + L2 ≤ Lmax. In order to solve the optimizationproblems of objective function with constraint conditions, we employ the regularizationmethod, where we add a penalty term to the above objective function. Denote Jα(L1, L2)

by

Jα(L1, L2) =n∑

j=1

∣∣∣RH(0, t j ; L1, L2) − RHA

∣∣∣2 + α |L1 + L2 − Lmax|2 . (13)

We call the minimizer (L†1, L†

2) of the objective function the regularized solution ofIPTTD:

Jα(L†1, L†

2) = min Jα(L1, L2),

where the parameters α is called the regularization parameter. The regularized solution ofthe above objective function is exactly the optimal solution which meet both heat-moisturecomfort indexes and thickness limitation condition.

4. Existence and uniqueness for the DP

4.1. Decoupling of the DP

The heat radiation function FL and FR are firstly derived from the last two equations in (1)and subsequently substituted in the first two equations in (1), therefore the DP is decoupledto the following parabolic equations with respect to the temperature T (x, t){

Cv∂T∂t = ∂

∂x

(κ1

∂T∂x

)+ �(T ) − 2βσ T 4 + λ�(T ), (x, t) ∈ �1

Cv∂T∂t = ∂

∂x

(κ2

∂T∂x

)+ �(T ) − 2βσ T 4 + λ�(T ), (x, t) ∈ �2(14)

together with the initial and boundary value conditions:⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

T (x, t1) = TI (x), x ∈ [0, L1 + L2]T (0, t) = T0(t), t ∈ [t1, t2]lim

x→L−1

T (x, t) = limx→L+

1

T (x, t), t ∈ [t1, t2]lim

x→L−1

κ1∂T (x,t)

∂x = limx→L+

1

κ2∂T (x,t)

∂x , t ∈ [t1, t2]T (L1 + L2, t) = T1(t), t ∈ [t1, t2]

(15)

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where �(T (x, t)) (x, t) ∈ �1 ∪ �2 is written as

�(T (x, t)) = β2σe−βx[∫ x

0e+βy T 4(y, t) dy + θ1(t; T )

]

− β2σe+βx[∫ x

0e−βy T 4(y, t) dy + θ2(t; T )

], (16)

and the two coefficients are

θ1(t; T ) = ξ1

βT 4(0, t) − (1 − ξ1)θ2(t; T )

θ2(t; T ) = 1

(1 − ξ1)(1 − ξ2)βe−β(L1+L2) − βeβ(L1+L2)

×[(1 − ξ2)βe−β(L1+L2)

∫ (L1+L2)

0eβy T 4(y, t) dy + βeβ(L1+L2)

×∫ (L1+L2)

0e−βy T 4(y, t) dy + ξ1(1 − ξ2)e−β(L1+L2)T 4(0, t)+ξ2T 4(L1+L2, t)

].

4.2. Existence and uniqueness of the solution to the DPWe define the operator B : H1((t1, t2), H2(0, L1 + L2)) → L1((t1, t2), L1(0, L1 + L2))by

B(T ) ≡⎧⎨⎩

B1(T ) = T + r[Cv

∂T∂t − ∂

∂x

(κ1

∂T∂x

)− �(T ) + 2βσ T 4 − λ�(T )

], (x, t)∈�1

B2(T ) = T + r[Cv

∂T∂t − ∂

∂x

(κ2

∂T∂x

)− �(T ) + 2βσ T 4 − λ�(T )

], (x, t)∈�2

(17)

where r ∈ R is a given constant.We make the following assumptions:

(H1) T (x, t) ∈ C([0, L1 + L2]×[t1, t2])∩ H1((t1, t2), H2(0, L1 + L2)), i.e. there existconstants Tmin > 0, Tmax > 0 and M > 0 such that Tmin ≤ T (x, t) ≤ Tmax and

‖T (x, t)‖H1((t1,t2),H2(0,L1+L2))≤ M;

(H2) T (x, t) ∈ W 1,1((t1, t2), L∞(0, L1 + L2)), i.e. there exists a constant G > 0 suchthat ∫ t2

t1

∥∥∥∥∂T

∂t(·, t)

∥∥∥∥L∞

dt < G.

Lemma 4.1 For any T1, T2 satisfying (H1), and let L ∈ [0, L1 + L2], we have∫ t2

t1

∫ L

0

∣∣∣∣β2σeβx∫ x

0e−βy

[T 4

1 (y, t) − T 42 (y, t)

]dy

∣∣∣∣ dx dt

� C1 ‖T1 − T2‖L2((t1,t2),L2(0,L1+L2)),

where C1 = 4L32 (t2 − t1)

12 β2σeβL T 3

max.

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Proof∫ t2

t1

∫ L

0


0e−βy

[T 4

1 (y, t) − T 42 (y, t)

]dy

∣∣∣∣ dx dt

≤∫ t2

t1Lβ2σeβL

∫ L

0

∣∣∣(T 21 + T 2

2 )(T1 + T2)

∣∣∣ |T1 − T2| dy dt

≤ Lβ2σeβL(∫ t2

t1

∫ L

0

∣∣∣(T 21 + T 2

2 )(T1 + T2)

∣∣∣2 dy dt

) 12(∫ t2

t1

∫ L

0|T1 − T2|2 dy dt

) 12

≤ 4L32 (t2 − t1)

12 β2σeβL T 3

max‖T1 − T2‖L2((t1,t2),L2(0,L1+L2)).

�

Lemma 4.2 For any T1,T2 satisfying (H1), and let L ∈ [0, L1 + L2],we have∫ t2

t1

∫ L

0|�(T1) − �(T2)| dx dt � C�‖T1 − T2‖L2((t1,t2),L2(0,L1+L2))

,

where

C� = 4L32 (t2 − t1)

12 β2σ

[2eβL +

∣∣∣∣∣ (eβL + 1 − ξ1)[(1 − ξ2)β + βeβ(L1+L2)

](1 − ξ1)(1 − ξ2)βe−β(L1+L2) − βeβ(L1+L2)

∣∣∣∣∣]

T 3max.

Proof ∫ t2

t1

∫ L

0|�(T1) − �(T2)| dx dt

≤∫ t2

t1

∫ L

0


0e−βy

[T 4

1 (y, t) − T 42 (y, t)

]dy

∣∣∣∣ dx dt

+∫ t2

t1

∫ L

0

∣∣∣∣β2σe−βx∫ x

0eβy[T 4

1 (y, t) − T 42 (y, t)

]dy

∣∣∣∣ dx dt

+∫ t2

t1

∫ L

0

∣∣∣β2σ [eβx + e−βx (1 − ξ1)][θ2(t; T1) − θ2(t; T2)]∣∣∣ dx dt

The above three integrals are calculated by the Lemma 4.1, so we obtain the Lemma 4.2. �

Lemma 4.3 When the environmental temperature T > 273.16, the saturation vapourpressure is

Vap(T ) = 1010.5380997− 2663.91T .

Moreover for any T1, T2 satisfying (H1), we have

|Vap(T1) − Vap(T2)| ≤ N |T1 − T2| ,where N = P 2263.91 ln 10

T 2min

, P = 1010.5380997− 2663.91Tmax .

Proof Because the function T (x, t) is continuous, we have |Vap(T )| ≤ P . By theLagrange’s mean value theorem, Lemma 4.3 is valid. �

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Lemma 4.4 For any T1,T2 satisfying (H1),and let L ∈ [0, L1 + L2], we have∫ t2

t1

∫ L

0

∣∣∣∣Vap(T1)∂T1

∂tT −2

1 − Vap(T2)∂T2

∂tT −2

2

∣∣∣∣ dx dt

� C2

∥∥∥∥∂T1

∂t− ∂T2

∂t

∥∥∥∥L2((t1,t2),L2(0,L1+L2))

+ C3‖T1 − T2‖L2((t1,t2),L2(0,L1+L2)),

where C2 = PT −2min L

12 (t2 − t1)

12 , C3 = (2PT −3

min + N T −2min)M.

Proof Mentioning that∣∣∣∣Vap(T1)∂T1

∂tT −2

1 − Vap(T2)∂T2

∂tT −2

2

∣∣∣∣=∣∣∣∣Vap(T1)

∂T1

∂tT −2

1 − Vap(T1)∂T2

∂tT −2

2 + Vap(T1)∂T2

∂tT −2

2 − Vap(T2)∂T2

∂tT −2

2

∣∣∣∣≤ |Vap(T1)|

∣∣∣∣∂T1

∂tT −2

1 − ∂T2

∂tT −2

1 + ∂T2

∂tT −2

1 − ∂T2

∂tT −2

2

∣∣∣∣+∣∣∣∣∂T2

∂tT −2

2

∣∣∣∣ |Vap(T1) − Vap(T2)|

≤ PT −2min

∣∣∣∣∂T1

∂t− ∂T2

∂t

∣∣∣∣+ (2PT −3min + N T −2

min)

∣∣∣∣∂T2

∂t

∣∣∣∣ |T1 − T2| ,

we have∫ t2

t1

∫ L

0

∣∣∣∣Vap(T1)∂T1

∂tT −2

1 − Vap(T2)∂T2

∂tT −2

2

∣∣∣∣ dx dt

≤∫ t2

t1

∫ L

0PT −2

min

∣∣∣∣∂T1

∂t− ∂T2

∂t

∣∣∣∣ dx dt +∫ t2

t1

∫ L

0(2PT −3

min + N T −2min)

∣∣∣∣∂T2

∂t

∣∣∣∣ |T1 − T2| dx dt

≤ C2

∥∥∥∥∂T1

∂t− ∂T2

∂t

∥∥∥∥L2((t1,t2),L2(0,L1+L2))

+ C3‖T1 − T2‖L2((t1,t2),L2(0,L1+L2)).

�

Lemma 4.5 For any T1, T2 satisfying (H1) and let L ∈ [0, L1 + L2], we have

∫ t2

t1

∫ L

0|λ�(T1) − λ�(T2)| dx dt � C�,1

∥∥∥∥∂2T1

∂x2− ∂2T2

∂x2

∥∥∥∥L2((t1,t2),L2(0,L1+L2))

+ C�,2

∥∥∥∥∂T1

∂x− ∂T2

∂x

∥∥∥∥L2((t1,t2),L2(0,L1+L2))

+ C�,3

∥∥∥∥∂T1

∂t− ∂T2

∂t

∥∥∥∥L2((t1,t2),L2(0,L1+L2))

+ C�,4‖T1 − T2‖L2((t1,t2),L2(0,L1+L2)),

where

C�,1 =(

abcPT −3min + acPT −2

min

)L

12 (t2 − t1)

12 ,

C�,2 =(

2ab2cPT −5min + 8abcPT −4

min + 4acPT −3min

)M,

C�,3 =(

abPT −3min + a PT −2

min

)L

12 (t2 − t1)

12 ,

C�,4 = ab2c(

5PT −6min+N T −5

min

)M2+ 4abc

(4PT −5

min+N T −4min

)M2+ 2ac

(3PT −4

min+N T −3min

)M2

+ abc(

3PT −4min + N T −3

min

)M + ac

(2PT −3

min + N T −2min

)M

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+ ab(

3PT −4min + N T −3

min

)M + a

(2PT −3

min + N T −2min

)M,

a = 216.5 × 10−6λ, b = 2263.91 ln 10, c = Da

τ.

Proof By means of the following relationship

λ�(T1) − λ�(T2)

= −ab

(Vap(T1)

∂T1

∂tT −3

1 − Vap(T2)∂T2

∂tT −3

2

)

+ a

(Vap(T1)

∂T1

∂tT −2

1 − Vap(T2)∂T2

∂tT −2

2

)

+ ab2c

(Vap(T1)

(∂T1

∂x

)2

T −51 − Vap(T2)

(∂T2

∂x

)2

T −52

)

− 4abc

(Vap(T1)

(∂T1

∂x

)2

T −41 − Vap(T2)

(∂T2

∂x

)2

T −42

)

+ 2ac

(Vap(T1)

(∂T1

∂x

)2

T −31 − Vap(T2)

(∂T2

∂x

)2

T −32

)

+ abc

(Vap(T1)

∂2T1

∂x2T −3

1 − Vap(T2)∂2T2

∂x2T −3

2

)

− ac

(Vap(T1)

∂2T1

∂x2T −2

1 − Vap(T2)∂2T2

∂x2T −2

2

),

that is Lemmas 4.3 and 4.4, we arrive at the Lemma 4.5. �

Theorem 4.6 Assume that (H1) and (H2) are satisfied. For any (L1+L2)12 (t2−t1)

12 < 1

5and the constant r satisfying

0>r >1

5max

⎧⎪⎪⎨⎪⎪⎩

−1

8βσ T 3max(L1 + L2)

12 (t2 − t1)

12 + C� + C�,4

,−1

Cv(L1 + L2)12 (t2 − t1)

12 + C�,3

,

−1

C�,2,

−1(κ1L

121 + κ2L

122

)(t2 − t1)

12 + C�,1

⎫⎪⎪⎬⎪⎪⎭.

there exists a unique solution T (x, t) ∈ H 1((t1, t2), H2(0, L1 + L2)) ∩ W 1,1((t1, t2),L∞(0, L1 + L2)), such that B(T (x, t)) = T (x, t) .

Proof For any T1, T2 satisfying (H1) and (H2), we have

‖B(T1) − B(T2)‖L1((t1,t2),L1(0,L1+L2))

=∫ t2

t1

∫ L1

0|B1(T1) − B1(T2)| dx dt +

∫ t2

t1

∫ L1+L2

L1

|B2(T1) − B2(T2)| dx dt

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≤∫ t2

t1

∫ L1+L2

0

[ ∣∣∣T1 − T2 + 2rβσ(

T14 − T2

4)∣∣∣+

∣∣∣∣rCv

(∂T1

∂t− ∂T2

∂t

)∣∣∣∣+ |r(�(T1) − �(T2))| + |r (λ�(T1) − λ�(T2))|

]dx dt

+∫ t2

t1

∫ L1

0

∣∣∣∣r(

∂

∂x

(κ1

∂T1

∂x

)− ∂

∂x

(κ1

∂T2

∂x

))∣∣∣∣ dx dt

+∫ t2

t1

∫ L1+L2

L1

∣∣∣∣r(

∂

∂x

(κ2

∂T1

∂x

)− ∂

∂x

(κ2

∂T2

∂x

))∣∣∣∣ dx dt

�∫ t2

t1

∫ L1+L2

0

∣∣∣1 + 2rβσ(T1 + T2)(

T 21 + T 2

2

)∣∣∣ |T1 − T2| dx dt

+ |r |Cv

∫ t2

t1

∫ L1+L2

0

∣∣∣∣∂T1

∂t− ∂T2

∂t

∣∣∣∣ dx dt

+|r |∫ t2

t1

∫ L1+L2

0|�(T1)−�(T2)| dx dt+|r |

∫ t2

t1

∫ L1+L2

0|λ�(T1)−λ�(T2)| dx dt

+ |r |κ1

∫ t2

t1

∫ L1

0

∣∣∣∣∂2T1

∂x2− ∂2T2

∂x2

∣∣∣∣ dx dt + |r |κ2

∫ t2

t1

∫ L1+L2

L1

∣∣∣∣∂2T1

∂x2− ∂2T2

∂x2

∣∣∣∣ dx dt .

From the above five Lemmata and the choice of the constant r , we have

‖B(T1) − B(T2)‖L1((t1,t2),L1(0,L1+L2))

� (L1 + L2)12 (t2 − t1)

12 ‖T1 − T2‖L2((t1,t2),L2(0,L1+L2))

+ |r |(

8βσ T 3max(L1 + L2)

12 (t2 − t1)

12 + C� + C�,4

)‖T1 − T2‖L2((t1,t2),L2(0,L1+L2))

+ |r |(

Cv(L1 + L2)12 (t2 − t1)

12 + C�,3

) ∥∥∥∥∂T1

∂t− ∂T2

∂t

∥∥∥∥L2((t1,t2),L2(0,L1+L2))

+ |r | C�,2

∥∥∥∥∂T1

∂x− ∂T2

∂x

∥∥∥∥L2((t1,t2),L2(0,L1+L2))

+ |r |(

(κ1L121 + κ2L

122 )(t2 − t1)

12 + C�,1

)∥∥∥∥∂2T1

∂x2− ∂2T2

∂x2

∥∥∥∥L2((t1,t2),L2(0,L1+L2))

< ‖T1 − T2‖H1((t1,t2),H2(0,L1+L2)).

By the Fixed Point Theorem,[16,17] there exists a unique solution T (x, t), such thatB(T (x, t)) = T (x, t). This completes the proof. �

5. Numerical algorithm and simulation of the DP and IP

The DP and IP are numerically solved by the finite difference method.First, we discretize the independent variables x ans t as follows.

in time:

t j = t1 + jτ0, j = 0, 1, . . . , n, τ0 = t2 − t1n

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in space:{xi = ih1, i = 0, 1, . . . , m1, h1 = L1

m1

xi = L1 + (i − m1)h2, i = m1 + 1, m1 + 2, . . . , m1 + m2, h2 = L2m2

5.1. Numerical solution to the DP

By decoupling the DP, the resulting Equation (14) is a non-linear partial differential equa-tions. Finite difference method is employed to obtain the computational scheme for thetemperature T j

i ≈ T (xi , t j ).

(I) In the inner porous batting: i = 1, 2, . . . , m1 − 1

Cv

T j+1i − T j

i

τ0= κ1

T j+1i+1 − 2T j+1

i + T j+1i−1

h21

+ �ji − 2βσ

(T j

i

)4 + λ�ji . (18)

(II) In the intermediate contact surface: i = m1

κ2T j

m1+1 − T jm1

h2= κ1

T jm1 − T j

m1−1

h1. (19)

(III) In the outer porous batting: i = m1 + 1, m1 + 2, . . . , m1 + m2 − 1

Cv

T j+1i − T j

i

τ0= κ2

T j+1i+1 − 2T j+1

i + T j+1i−1

h22

+ �ji − 2βσ

(T j

i

)4 + λ�ji . (20)

where �(xi , t j ) is approximated by

�ji ≡

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

Daτ

(C∗a )

ji+1−2(C∗

a )ji +(C∗

a )ji−1

h21

− (C∗a )

ji −(C∗

a )j−1i

τ0, i = 1, 2, . . . , m1 − 1

Daτ

((C∗

a )jm1+1−(C∗

a )jm1

h2− (C∗

a )jm1−(C∗

a )jm1−1

h1

)2

(h1+h2)− (C∗

a )jm1−(C∗

a )j−1m1

τ0, i = m1

Daτ

(C∗a )

ji+1−2(C∗

a )ji +(C∗

a )ji−1

h22

− (C∗a )

ji −(C∗

a )j−1i

τ0, i = m1 + 1, m1 + 2, . . . , m1 + m2−1

(21)�

ji can be computed by Simpson’s numerical integration formula

�ji = β2σe−βxi

(U j

i + (θ1)j)

− β2σe+βxi(

V ji + (θ2)

j)

(22)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

U ji = h1

3

⎛⎝ f j

0 + 4

i2∑

k=1f j2k−1 + 2

i2 −1∑k=1

f j2k + f j

i

⎞⎠ , i = 1, 2, . . . , m1

U jm1+i = U j

m1 + h23

⎛⎝ f j

m1 + 4

i2∑

k=1f jm1+2k−1 + 2

i2 −1∑k=1

f jm1+2k + f j

m1+i

⎞⎠ , i = 1, 2, . . . , m2

V ji = h1

3

⎛⎝g j

0 + 4

i2∑

k=1g j

2k−1 + 2

i2 −1∑k=1

g j2k + g j

i

⎞⎠ , i = 1, 2, . . . , m1

V jm1+i = V j

m1 + h23

⎛⎝g j

m1 + 4

i2∑

k=1g j

m1+2k−1 + 2

i2 −1∑k=1

g jm1+2k + g j

m1+i

⎞⎠ , i = 1, 2, . . . , m2

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and

f jk = e+βxk (T j

k )4, g jk = e−βxk (T j

k )4, (θ1)j = ξ1

β(T j

0 )4 − (1 − ξ1)(θ2)j ,

(θ2)j = 1

(1 − ξ1)(1 − ξ2)βe−β(L1+L2) − βeβ(L1+L2)

[(1 − ξ2)βe−β(L1+L2)U j

m1+m2

+βeβ(L1+L2)V jm1+m2

+ ξ1(1 − ξ2)e−β(L1+L2)(T j0 )4 + ξ2(T

jm1+m2

)4].

Meanwhile the initial and boundary value conditions are discretized as follows⎧⎪⎨⎪⎩

T 0i = TI (xi ), i = 0, 1, . . . , m1 + m2

T j0 = T0(t j ), j = 0, 1, . . . , n

T jm1+m2

= T1(t j ), j = 0, 1, . . . , n

(23)

By the above scheme, we can derive the approximate solution T ji of the temperature T (x, t).

5.2. Numerical solution to the IP

It is evident that there exists a unique solution for the IP. In fact, if CI (x) ∈ C[0, L1 +L2], and for any T (x, t) satisfying (H1) and (H2), that is to say �(x, t) is uniquelydetermined by temperature T (x, t), then there exists a unique solution Ca(x, t) whichbelongs to H1((t1, t2), H2(0, L1 + L2)). But the IP is unstable, so we need use stabilizationalgorithm.[6,18]

The Equation (6) together with the initial and boundary value conditions (7)–(9) arerewritten as two parabolic equations, hence the numerical solution arrives.

The water vapour concentration (Ca)ji ≈ Ca(xi , t j ) in the outer porous batting can be

computed by finite difference scheme.⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

ε2(Ca)

j+1i −(Ca)

ji

τ0= Daε2

τ

(Ca)j+1i+1 −2(Ca)

j+1i +(Ca)

j+1i−1

h22

− �ji , i =m1 + 1, m1+2, . . . , m1+m2−1

(Ca)0i = CI (xi ), i = m1, m1 + 1, . . . , m1 + m2

(Ca)jm1+m2

= C1(t j ), j = 0, 1, . . . , n

Daε2τ

(Ca)jm1+m2

−(Ca)jm1+m2−1

h2= CE (t j )−(Ca)

jm1+m2

ω1+h−1c

, j = 0, 1, . . . , n

(24)

Because temperature T is unique, � is uniquely determined. Therefore by the Equation (24),the intermediate boundary value (Ca)

jm1 ≈ Ca(xm1 , t j ) is determined by the right boundary

value.Likewise, the water vapour concentration (Ca)

ji of inner porous batting is computed by

the similar scheme.⎧⎪⎪⎪⎨⎪⎪⎪⎩

ε1(Ca)

j+1i −(Ca)

ji

τ0= Daε1

τ

(Ca)j+1i+1 −2(Ca)

j+1i +(Ca)

j+1i−1

h21

− �ji , i = 1, 2, . . . , m1

(Ca)0i = CI (xi ), i = 0, 1, . . . , m1

Daε2τ

(Ca)jm1+1−(Ca)

jm1

h2= Daε1

τ

(Ca)jm1−(Ca)

jm1−1

h1, j = 0, 1, . . . , n

(25)

Through the intermediate boundary value (Ca)jm1 derived from the scheme (24), we derive

the left boundary value (Ca)j0 ≈ Ca(0, t j ) by the scheme (25).

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Table 1. Porosity and heat conductivity of three textile materials.

Material Wool Polyester Polypropylene

Porosity 0.915 0.88 0.87heat conductivity 0.052 0.084 0.071

Table 2. Physical parameters in the DP and IP [11]

Cv = 1715.0 kJ m−3 K−1 Da = 2.5 × 10−5 m2 s−1 β = 8 m−1

σ = 5.672 × 10−8 kJ m−2 K−4 s−1 ω1 = 64.99 τ = 1.2hc = 2.294 × 10−3 m s−1 ξ1 = ξ2 = 0.9 λ = 2593 kJ kg−1

0 1 2 3 4 5510152025

5101520253035

t (h)

x (mm)

T

0 1 2 3 4 5510152025

0

0.5

1

1.5

2

2.5x 10−5

t (h)

x (mm)

Ca

Ca (k

g/m

3 )

8 10 12 14 16 18 20 2247.5

4848.5

4949.5

5050.5

5151.5

52

t (h)

RH

0 (%)

RH0

(a) (b)

Figure 2. case 1 (Wool and Polyester): (a) distribution of temperature T (x, t), vapour waterconcentration Ca(x, t); (b) relative humidity RH0(t) in the microclimate area.

For given thickness of inner and outer porous batting, L1 and L2 respectively, the relativehumidity RH j

0 in the microclimate area is approximated by the following relationship

RH j0 ≡ RH(0, t j ; L1, L2) = T j

0 (Ca)j0

216.5 × 10−6Vap(T j0 )

= (Ca)j0

(C∗a )

j0

. (26)

5.3. Numerical Simulation

In the numerical simulation, we select three representative fabrics such as wool, polyesterand polypropylene fibre, see Table 1. Through computation, we derive the temperature,vapour water concentration within the textiles and relative humidity in microclimate area.Other parameters are given in Table 2.

For example, we set thickness L1 = L2 = 2.5 × 10−3 m, time interval is t1 = 8h,t2 = 20h. The initial temperature is TI (x) = 1◦C and the vapour water concentra-tion CI (x) = 1.0 × 10−3 kg m−3. The boundary temperature is T0(t) = 32 ◦C, T1(t) ∈[1◦C, 11◦C]. The relative humidity is RH1(t) ∈ [30%, 60%]. The environmental vapourwater concentration CE (t) is 1% higher than C1(t).

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0 1 2 3 4 5510152025

5101520253035

t (h)

x (mm)

T

0 1 2 3 4 5510152025

00.5

11.5

22.5

3x 10−5

t (h)

x (mm)

Ca

Ca (k

g/m

3 )

8 10 12 14 16 18 20 2253

53.554

54.555

55.556

56.557

57.5

t (h)

RH

0 (%)

RH0

(a) (b)

Figure 3. case 2 (Polypropylene and Polyester): (a) distribution of temperature T (x, t), vapour waterconcentration Ca(x, t); (b) relative humidity RH0(t) in the microclimate area.

0 1 2 3 4 5510152025

5101520253035

t (h)

x (mm)

T

0 1 2 3 4 5510152025

00.5

11.5

22.5

3x 10−5

t (h)

x (mm)

Ca

Ca (k

g/m

3 )

8 10 12 14 16 18 20 2259

60

61

62

63

64

65

t (h)

RH

0 (%)

RH0

(a) (b)


0 1 2 3 4 5510152025

5101520253035

t (h)

x (mm)

T

0 1 2 3 4 5510152025

00.5

11.5

22.5

33.5

x 10−5

t (h)

x (mm)

Ca

Ca (k

g/m

3 )

8 10 12 14 16 18 20 2267

68

69

70

71

72

73

74

t (h)

RH

0 (%)

RH0

(a) (b)


Figure 2–5 show the distribution of the temperature T (x, t) and the vapour water con-centration Ca(x, t) and relative humidity RH0(t) for the four kinds of textile combination,see Table 3.

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Table 3. Various combination of inner and outer porous fabrics.

inner porous batting outer porous batting feature of ε and κ relative humidityRH0(t)

case 1 wool polyester ε1 > ε2, κ1 < κ2 [47%, 52%]case 2 polypropylene polyester ε1 < ε2, κ1 < κ2 [53%, 58%]case 3 polyester polypropylene ε1 > ε2 ,κ1 > κ2 [59%, 65%]case 4 polyester wool ε1 < ε2, κ1 > κ2 [67%, 74%]

From the numerical simulation results of the above four cases, it can be concluded thatunder the same environmental conditions, the combination of different textile materialsplays a critical role in determining the relative humidity in the microclimate area.

In case 1, the combination of wool and polyester keeps the relative humidity RH0(t)in the interval [47%, 52%], which meets the standard of heat–moisture comfort of humanbody. Meanwhile, the inner and outer fabrics are exchanged in order (see case 4), the RH0(t)will be kept in [67%, 74%] without reaching the standard of the heat–moisture comfort.The comparison between the two cases suggests that the order of the double layer fabricsdetermines the heat–moisture comfort level.

Compared with case 1, case 2 selects a combination with different inner fabric and sameouter fabric. In these two combinations, heat conductivity of inner fabric is always smallerthan the outer fabric, but the porosity of inner fabric and outer fabric is kept different, i.e.inner fabric porosity is bigger than the outer in case 1 and just the reverse in case 2. TheRH0(t) is kept in [53%, 58%] for case 2. The case 2 suggests that the textile porosity andfabric combination affect the comfort index. Similar conclusions can also be drawn fromthe comparison between case 2 and case 4.

Compared with case 1, case 3 suggests that the textile heat conductivity has greatinfluence on the comfort index. The same conclusion can also be drawn from the comparisonbetween case 2 and case 4.

According to the cross-comparison between case 1, 2 and 3, we find that the textile heatconductivity has much more influence on the comfort index than the textile porosity. Thesame conclusion can be made from the cross-comparison of case 2, case 3 and case 4.

Therefore, the results of numerical simulation suggest that different combinations oftextile materials affect the comfort index. The result also proves that the theoretical analysisand numerical simulation can provide proper guidance and scientific explanation for textilematerials development.

6. Numerical algorithm and simulation of the IPTTD

6.1. Numerical algorithm of the IPTTD

When the thickness of inner porous batting (or outer porous batting) is given, the objectivefunction of two variables (13) is simplified to objective function of one variable. We employdirect search method, for example, Golden Section Method, to make one-dimensional searchfor the above objective function.

For example, fix the thickness L2 = 2.0 × 10−3m. The objective function Jα(L1, L2)

is the function of thickness L1. The objective function exists one minimizer from thegraph (Figure 6).

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0 1 2 3 4 50.05

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Figure 6. The minimizer of objective function Jα(L1, L2) about L1 for given thickness of outerporous batting L2 = 2.0 × 10−3 m.

For minimization problem of the objective function of two variables, we can employdirect search method, for example, alternating direction method or random search method,to obtain optimal solution. Here we adopt the alternating direction search method. Thereforethe algorithm of finding the minimizer of the objective function in the given domain can bedescribed as following steps:

(1) Set initial thickness (L1, L2), the maximum thickness Lmax, stop parameter δ andregularized parameter α.

(2) Compute the objective function J = Jα(L1, L2)

(a) compute numerical solution T ji of temperature T (x, t) by solving the

DP (18)–(23) and hence �ji of �(x, t);

(b) compute numerical solution (Ca)jm1 on the intermediate contact boundary

by solving the IP (24), similarly go on to compute (Ca)j0 in the microclimate

area by the IP (25);(c) compute relative humidity RH j

0 by the formula (26), hence we get the valueof the objective function J = Jα(L1, L2).

(3) Fix L1, repeat Step 2 and obtain the minimum of objective function J ∗ =Jα(L1, L∗

2) about thickness L1 by direct search method. If |J ∗ − J | < δ, stopthe computation; otherwise let L2 = L∗

2 and J = J ∗, go to Step 4;(4) Fix L2, repeat Step 2 and obtain the minimum of the objective function J ∗ =

Jα(L∗1, L2) about thickness L2 by direct search method. If |J ∗ − J | < δ, stop the

computation; otherwise let L1 = L∗1 and J = J ∗, go to Step 3.

When |J ∗ − J | < δ, (L1, L2) is the regularized solution which satisfies the human bodyheat-moisture comfort.

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(a) (b)Figure 7. Example 1: the minimizer of the objective function Jα(L1, L2) about case 1 and case 2.

6.2. Example 1: simultaneous determination of both thickness L1 of inner porous battingand thickness L2 of outer porous batting

In example 1, the physical parameters, the initial and boundary value conditions of tem-perature, water vapour concentration and relative moisture are same as the DP and the IP.In numerical simulation, we set Lmax = 4.0 × 10−3 m, δ = 10−4 and α = 104. Herewe mention that the choice of regularized parameter α is bigger than usual choice sincethe two terms in the objective function are not treated as dimensionless. The dimensionlesstreatment will reduce the choice of α by the L-curved method.

Firstly, we consider the same combination of textile material as case 1: wool andpolyester batting. The optimal thickness of (L1, L2) is determined by the alternatingdirection search method. The numerical results are shown in the left graph of Figure 7,where we take L1-ordinate and L2-abscissa . The more deeper blue the area is, the smallerthe value of Jα(L1, L2) is. The red points are minimum points in each step of alternatingdirection search process and the arrow represents its search direction. The dashed linesatisfies the constraint condition L1 + L2 = Lmax. Above numeral results show thatthe minimizer of Jα(L1, L2) is (L†

1, L†2) = (1.2 × 10−3m, 2.8 × 10−3m), that is to say,

we obtain the regularized solution which satisfies the constraint condition and makes theobjective function minimal.

Secondly, the same methods are employed for the combination of textile material:polypropylene fibre and polyester batting in case 2. The right graph of Figure 7 shows theminimizer of the objective function Jα(L1, L2) is (L†

1, L†2) = (1.4×10−3m, 2.6×10−3m).

Obviously, due to the same requirements of textile material design, different combinationsof textile material have different results of thickness. As shown in Figure 7, since the heatconductivity of polypropylene fibres (0.071 kJ m−1 K−1 s−1) is higher than that of wool(0.052 kJ m−1 K−1 s−1), the thickness needs to be increased to achieve thermal effects.

Finally, the numerical results are shown in Figure 8 for the case 3 (polyester andpolypropylene) and case 4 (polyester and wool). As two combinations of textile materialin case 3 and case 4 both make RH0(t) outside of heat-moisture comfort interval, theminimizer for the two cases is (L†

1, L†2) = (3.8 × 10−3m, 0.2 × 10−3m), which indicates

that single-layer batting is more suitable, meanwhile combinations of bilayer textile material

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(a) (b)Figure 8. Example 1: the minimizer of the objective function Jα(L1, L2) about case 3 and case 4.

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(a) (b)Figure 9. Example 2: the minimizer of the objective function Jα(L1, L2) about case 1 and case 2with noise.

are inappropriate. The numerical simulation results in the two cases prove that cloths madeof single-layer textile material make people feel more comfortable.

The numerical result of Example 1 illustrates that alternating direction search methodcan find out the minimizer of the objective function Jα(L1, L2) effectively. Moreover, theminimizer can meet the actual requirements of textile materials design and realistic heat–moisture transfer rules.

6.3. Example 2: simultaneous determination of both thickness L1 of inner porous battingand thickness L2 of outer porous batting with noisy measurments

Now we consider how much influence the noisy environmental measurements of temper-ature and vapour water concentration have on the objective function and the minimizer.Here, we keep the parameters in Example 1 unchanged, but 1% random noise is added tothe initial conditions and boundary value conditions.

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Figure 10. Example 2: the minimizer of the objective function Jα(L1, L2) about case 3 and case 4with noise.

The numerical results of case 1 and case 2 with noisy measurements are shown inFigure 9 and their minimizers of the objective function Jα(L1, L2) are (L†

1, L†2) = (1.2 ×

10−3m, 2.8 × 10−3m) and (L†1, L†

2) = (1.4 × 10−3m, 2.6 × 10−3m), respectively, whichare the same as that of Example 1. For case 3 and case 4 with noisy measurements (seeFigure 10), their numerical results are also the same as that of Example 1.

The results of numerical simulation prove that the model of IPTTD is well proposed andnumerical algorithms have nice stability. Besides, numerical algorithms and its numericalresults provide theoretical and scientific explanation for textile materials design.

7. Concluding remarks

The well-posedness of the DP verifies the existence and uniqueness of solution to theheat–moisture transfer model, which not only provides theoretical foundation to numericalsolution of the DP, but also determine the uniqueness of �(x, t) in the IP. Hence, the DP alsoprovides theoretical foundation to research of the IP. On the basis of different combinationsof textile materials and given thickness of inner and outer porous batting, we make numericalsimulation to achieve the distribution of temperature and water vapour concentration inporous batting for the case 1–4. Further, we go on judging whether the body heat–moisturecomfort in microclimate area arrives, and verify feasibility of numerical simulation.

In the IPTTD, the optimal thickness of the inner and outer porous batting material hasbeen determined through numerical simulation. Both Example 1 with exact measurementsand Example 2 with noisy measurements show the numerical results of thickness of innerand outer porous batting when the material textures are given. The numerical results provethe feasibility of the theory and algorithm of IPTTD. Moreover, the numeral results confirmrationality of the objective function and stability of the algorithm of the IPTTD, whichprovides theoretical basis to design bilayer textile materials.

For the coupled non-linear partial differential equations, which are based on the dynamicheat-moisture transfer process, the finite difference scheme and its grid ratio are essentialfor the accuracy of numerical solution. Meanwhile, as the golden section method is locallyconvergent in alternating direction search procedure, the numerical results are dependent

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dramatically on the choices of initial iteration guess. So in our further research, we will focuson the stability of the DP, the well-posedness of the IPTTD and the multiple parametersdetermination of multilayered textile materials in the forthcoming papers.

AcknowledgementsThis work is supported by the National Natural Science Foundation of China (No. 11071221 and10561001), Natural Science Foundation of Zhejiang Province (No. LQ12A01024) and Science foun-dation of Zhejiang Sci-Tech University (No. KY2012015).

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[2] Li Y, Li FZ, Liu YX, Luo ZX. An integrated model for simulating interactive thermal processesin human-clothing system. J. Therm. Biol. 2002;29:567–575.

[3] Fan JT, Wei XH. Heat and moisture transfer through fibrous insulation with phase change andmobile condensates. Int. J. Heat Mass Trans. 2002;19:4045–4055.

[4] Fan JT, Cheng XY, Sun WW. An improved model of heat and moisture transfer with phasechange and mobile condensates in fibrous insulation and comparison with experimental results.Int. J. Heat Mass Trans. 2004;47:2343–2352.

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[10] Xu DH, Ge MB. Thickness determination in textile material design: dynamic modeling andnumerical algorithms. Inverse Problem. 2012;2:035011.

[11] Xu DH, Ge MB, Cheng RL. nverse problems of textile material design based on comfort ofclothing. Commun. Appl. Comput. Math. 2012;3:332–341.

[12] Ye C, Huang HX, Fan JT, Sun WW. Numerical study of heat and moisture transfer in textilematerials by a finite volume method. Commun. Comput. Phys. 2008;4:929–948.

[13] Xu YH, Xu DH, Zhang LP, Zhou XH. A new inverse problem for the determination of textilefabrics. Inverse Problem in Science and Engineering. in press.

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[15] Huang J. Cloting Comfort. Beijing: Scienc Press; 2008.[16] Yosida K. Functional analysis(sixth version). Beijing: Springer-Verlag, Beijing World Publishing

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An inverse problem of Bilayer textile thickness determination in dynamic heat and moisture transfer