Dynamic Heat and Water Transfer Through Layered Fabrics
J. P. FOHR, D. COUTON, AND G. TREGLTIER
Laboratoired'utudes Ther-miques, Universite -de Poitiers, 86022 Poitiers, France
ABSTRACT
A mode] of heat and water transfer through layered fabrics, such as wearing clothing,
is developed in this paper. All particular properties of recently developed fabrics are
considered: hydrophobic or hydrophilic treatment, membranes glued onto a layer, and
surface modification of the textile (abrading). Physical phenomena taken into account are
sorption or desorption; free water condensation or evaporation; Jiquid, vapor, and ad-
sorbed water diffusion; and heat conduction and contact resistances between layers. The
model is dynamic for one-dimensional transfers. Considering this particular porous
medium, the energy and mass balance lead to a system of two differential equations
(moisture, temperature). The validity of the model is examrLined in light of basic experinments existing in the literature. The hygroscopic character of a fabric can be expressed by
a diffusion coefficient, which is a function of the water content. Two possible fonnulations
of this property are given.
Garmiients manufactured for specific activities are
growing worldwide. Many synthetic fibers produced by
chemical processes are offered on the market, and
clothes can be manutactured frorm numerous combinations of natural and synthetic fibers. Many different kinds
of clothing are adapted for particular uses: sports, industrial jobs, very hot or very cold weather. The comfort of
a gamwent is linked to several factors: lightness, heat and
vapor transport, sweat absorption and drying, wind and
impermeability. For example, winter sport clothes must
have good vapor transfer properties. A garmnent's comfort depends on the properties each of fabric layer and the
combination of all the layers worn, Each particular gariment-for example, a trenchcoat or a raincoat-has several different layers: an external waterproof layer, an
internal lining, and sometimes an internediate layer.
To create a high performance garment, a designer
considers fashion and other technical factors: fiber nature
and size (microfibers that have particular properties),
surface modification of fibers (hydrophobic or hydrophilic treatments), hydrophobic (Gortexg, for example)
or hydrophilic membranes fused to a textile layer,
weaving or knitting patterns, and abrasion of the fabric
surface.
To know how heat and water are transported through
each fabric, measurements of their physical properties
(heat conductivity, vapor diffusivity, air permeability ... .) must be made in laooratories. In this paper, we
Textile Res. J. 72(1),
1-12
propose mathematical models to simulate the response
of layered fabrics subjected to heat and sweating conditions.
Aims of the Model
Heat and water transfer in textile layers involves two
kinds of mathematical rnodels. The first, more frequently
referred in the literature, is either the Nordon-David
model or a derived form of it [2, 19]. Based on conservation equations, this global model consists of two differential coupled equations with variables for temperature and water concentration in air (C)) and in the fibers
of the textile (C9), which is generally the water adsorbed
by hygroscopic fibers. Cf is not in equilibrium with Ca,
but an empirical relation between the adjustable parameters is assumed: the rate, of sorption is a linear function
of the difference between the actual C, and the equilibrium value. The introduced coefficients are not directly
linked to the physical properties of the clothes. To simulate sorption experiments on wool samnples, Li and
Holcombe [13] improved the Nordon-David model considering a two-stage sorption process: "At the fiber surface the moisture content Is in equilibrium with the
moisture content of the adjacent air" (first stage), and
"moisture diffuses radially into a cylindrical medium
(fiber) with a constant diffusion coefficient" (second
stage). Some pararneters are adjusted from Watt's sorption experiments [29, 30] on wool fibers. This kind of
(2002)
I
0040-5 175/$ 15.00
TEXTILE RESEARCH JOURNAL
2
model does not take into account the occurrence of
liquid.
The seco-nd model type stems from an analysis of
transfers in porous media. The basic formulation is deduced L321 from a scale change: using volurme average
techniques, the macroscopic modei is deduced from the
microscopic state, and one mav speak of equations definied for a smal, averaging volume.
Many papers describe the nature of transfers in fibrous
insulation where the homogeneous character is obvious.
For example, Wijeysundera and Hawlader 133] studied
the effects of condensation and liquid transport on the
thermnal performance of fibrous insulation usinlg a semiempirical model and an experimental set-up, Vafai et atX.
[26, 27] analyzed the accumulation and migration of
moisture in a two-dimensional laver, and Mitchell et al.
[151 took into account forced air filtration and frosting.
Few papers exist in the literature on transfer in fabrics.
Le et al. [11, 12] studied steaming in a thick textile
assemnbly bed, taking into account the flow rate and
condensation or evaporation of water. Ghali e't al. [41
studied the wicking process from extremities of a vertical
fabric.
In this paper, we propose a model to determine the
occurrence of liquid in certain places in textile layers,
and thuLs liquid diffusion as well as vapor sorption. In a
small averaging volume, a local thermodynamic equilibrium between solid, liquid, and gas phases is assumed.
This hypothesis, which seemns satisfactory for a thick
layer of fibrous insulation such as rockwool, is qtuestionable when dealing with thin textile layers. Fabricss have
two scales of pores generated by the manufacturing process-interfiber and interyarn pores (Figure 1), and the
diffusion properties of heat and water in liquid and vapor
form are determined from this network of pores. We
focus our attention on vapor diffusion and adsorption on1
and in fibers. During a dynamric process, one might
expect that vapor would diffuse mainly through large
woof holes and adsorpion would occur mainly with
vapor diffusion through yarns and fibers. Thus, it is
interesting to determine if the thermodynamic equilibrium between the vapor and the s'lid is established for
the equivalert homogeneous porous medium of the textile layer along its thickness.
Gibson extends the macroscopic equations for a poro-us model of textile layers [5, 61. The sorption relation,
which represents the air/textile oisture equilibrium, is
associated with a mass flux equation similar to that in the
Nordon-David model, which expresses mass flux into or
out of fibers. The formulation of this "mass or energy
source term" is used in drying operations for granular
media where the grain behavior is expressed with a
global equation. When the exchanige parameter between
Fabri
Yar
Fiber
P.)
A
1
X
:
fiber
yom
porediamte
m
fir
I
I
Fwinar 1. A fabric structure and its model.
the grain and air is identified, the simulations are in
agreement with the experiments. We will discuss this
point later with the equations of our model and e
experimrental results of the literature.
Using the porous media method, we have developed in
this paper a model of heat and water transport through
layered fabrics. Our model takes into account all the
particular properties of actual fabrics, such as the hydrophobic or hydrophilic character of each fabric layer (or
mermbrane), which deternines water liquid displacement
[281, and the sorption-desorption effect, which is very
important for natural fibers, that is, the vapor/liquid
transformation o adsorbed water upon and into the fibers
[13, 141.
Moreover, we consider the occurrence of condensation
or evaporation in accordance with the environmental
conditions and their variations. Indeed, as with layered
walls in buildings, condensation can take place in certain
cdothing layers during winter. A garment can be subjected to changing environmental conditions-cold outside, temperate conditions inside-leading to varying
amiounts of stored water that have to be forseen by the
model. Thus, the model has to be dynam.ic.
We first consider that the elementary fabric layer of a
garment acts hormiogeneously (Figure 1). If some mechanical treatments of the surface of a given layer (abrad-
JANUARY
2002
3
ing, for example) create a heterogeneous character, it at each point of the layer. Thus, regain is only a function
should be possible to consider that fabric as comprising of the characteristics of air. Later we will examine the
two different layers. The complex geometric structure of possibility of improving this point. If free water is
one layer leads to physical parameters that are defined by present, vapor remains saturated. The dimensions do not
the averaging volume. We assume that all these param- change with water content.
eters can be determined with specific equipment for
individual textile layers. The mean character of a layer's
Symbols
physical parameters does not imply that they are constant
over the layer. Indeed, when clothes are worn, temperaspecific heat at constant pressure, J kg-' K- 1
ture and moisture gradients induce variations in the phys- C,
mass diffusivity
ical parameters of the fabric. For example, although the D
e
fabric
thickness, m
thermal conductivity values of fabrics are well known,
hb
specific
part of bound water enthalpy, J - kg-'
their dependence on moisture content has rarely been
differential
enthalpy of vapor liquid, J kg-'
studied. Using a transient method, Schneider et al. [22]
heat
exchange
coefficient, W m- K2 1
hi,
he
rneasured the effective thermal conductivity of textile
fabrics containing varying amounts of water. Jirsak et al. ki, ke mass exchange coefficient, m s-1
mass rate of condensation per volume, kg s- m 3
[10] compared the dynamic and static methods for dry m
pressure, Pa
textiles, and Azizi et al. [1] and Moyne et al. [18] p
time, s
showed the influence of moisture transfer on measure- t
T
temperature, K or °C
ments.
space variable, m
Fabrics are often layered to make up garments. The x
manufacturer assembles fabric layers (laminated, glued,
sewn), and the user puts on several layers of clothing. It Greek letters
follows that the contact conditions between these layers
water content of fabric, X = (Pb + Pd)P,
is variable, ranging from perfect to loose contact, related x
Ev
porosity
to the distribution of elementary surfaces in contact.
relative humidity of air
'p
Here, it is convenient to introduce contact resistance
A
conductivity, W m-1 K 1
parameters, as used by the thermal engineering sciences.
density, kg m- 3
The values of these parame-ters will be related to specific P
T
time period, s
measurements. To account for very fine membranes (micrometers), particular values of some parameters are Subscripts
expected. For example, a Gortex membrane acts as the
null value of a liquid diffusion coefficient and a contact a
dry air
resistance for vapor diffusion. A hydrophilic membrane b
bound water
collects sweat on certain areas of the skin and diffuses e
external
water into the fabric layer. To establish a one-dimen- eff
effective
sional model (through layered fabrics), we will treat this eq
equivalent
last phenomenon in a simplified form with uniform g
gas
boundary conditions for the skin side and the outer side.
internal
Thus, the main hypotheses of the model are as follows: in
initial
Each layer acts as a homogeneous one with physical I
free liquid
parameters that we will determine experimentally. The s
solid
dynamic model enables us to have variable boundary sat
saturation
conditions. Contact resistances should occur between v
vapor
different fabric layers. Transfers through layers are onedimensional, from the skin to the outer environment. Superscript
Heat and water transfers are associated with adsorbed
intrinsic
(regain) or free water condensation or evaporation. Ra- i
diation is taken into account outside the layers with an
equivalent exchange coefficient, Air filtration through
Physical Properties
fabric layers, as a consequence of wind or movement, is
not considered, nor is renewed air (pumping effect)
In this section we describe the physical properties of a
through openings. Fibers and air are in local equilibrium fabric layer and their mathematical expressions.
TEXTILE RESEARCH JOURNAL
4
HYGROSCOPIC CHARACTER
pb V,
The isothernm curves of equilibrium sorption/desorption are well known for almost all fibers. Natural fibers
('cotton, wool,...) are much mnore hygroscopic than
chemical fibers. The expression is [14, 16, ch. 7]
-DI, 9(pJ) a
-D
1 (p
)
(2a)
The bound water diffusion has an important influence
when a drying front occurs.
FREE WATER
x
=-7
feb, T), with
sb
(1)
Because of hysteresis, we have to distinguish betweeni
sorption and desorption curves. This effect is important
for artificial fibers, for example, fiberglass [23], where
regain is low, and minor for natural fibers where regain
is high. Furthermore, since data for textile fibers are
generally only given for sorptiorn (regain), we do not take
into account the hysteresis phenomenon in our mnodel.
Moreover, it is sufficient [16, ch. 7] to consider the loc:al
equilibrium between moist air and fibers as
Pb
X = -p- =~ffP)
(la)
.
The differential enthalpy of change in the liquid-vapor
phase, associated with sorption curves, is given by [16,
ch. 8]
Ah-,,, = Ah,, + hb(X)
'
(lb)
where hh is e specific part of bound water, tighter at the
substrate than free water.
We obtain data for bound water poinlt by point from
static conditions between moist air and fabric. When
clothing is worn, the temperature and moisture gradient
through the fabric layers induces a diffusion process [16,
ch. 9, 24, 25], which is generally writteni as
Pb V=
DhVpl, -
(2)
The coefficient D,, which can be a function of temperature and moisture, is difficult to measure and thus niot
well kniown. For example, Watt [29, 301 measured DIt
that corresponded to diffusion through the thickness of a
wool fiber.
At this point, thie heterogeneous character of the medium has to be considered. Indeed, bound water diffuses
along the surface of the fibers (all kinds of fibers), mainly
it the direction perpendicular to the fabric plane and into
the wall of the fibers (hygroscopic fibers such as cotton,
wool, etc.) in radial directions of fibers or yarns. Thus,
one expects that for the equivalent honmogeneous rnedium, bound water diffusion will be different in two
direction.s (Figure 1, x, normal to the fabric, and r, in the
radial direction of the equivalent fiber/larn, as given by
Free water is provided fromn a garment's external
boundary conditions (sweat, rain) or from internal condensation depending on moist air properties. It is easy to
observe sweat diffusion from particular places on the
skin (a 2D phenomuenon), and water diffusion through an
external layer (a ID phenomenon), as with rain on a
trenchcoat or leather shoes getting wet. This water moves
from one point to another, following the capillary properties of the fabric layers, a phenomenon that is well
illustrated when we observe the diffusion of a colored
liquid drop on a fabric surface [8, 36]. Wool and polyester have poor diffusion, but cotton has excellent diffusioni. Despite the heterogeneous structure of a fabric
layer, we assumne that a similar liquid diffusion takes
place in and across the fabric. Specific experiments are
necessary in order to determine the values of liquid
diffusion coefficients in relation to the transfer direction.
The liquid diffusion equation derives from extending
Darcy's law for partially saturated porous media where
the permeability factor depends on liquid density [32].
Few studies are available for heterogenous porous mnedia
[21]. The liquid pressure expressed in Darcy's law is
linked to the capillary pressure, which is the key function
for free xvater.
Capillary pressure expresses the hydrophilic or -phobic character of a textile Through the contact angle 0, [8]
and surface tension a- [16, ch. 10]. A porous network is
defined with a porous distribution r(p,l) [17]. For the
particular textile porou.s medium, because gas permeability is high and thickness layers are weak, it is convenient
to suppose pg is a constant, as does Vafai [26] for fibrous
insulation.
Thus, the combination of Darcy's equation and the
capillary pressure function leads to
p/'V1 = -D,Vp 1 - DI VT
(3)
Note the difference between density p (mass of the
constituent in the averaging volume around the point
divided by this volumne) and intrinsic density p' (same
mass divided by the part of the volume concerned wi-h
the phase) [321.
Capillary pre.sure induces a diffusive liquid nmovement. The coefficients D, and DT depend on liquid
density and temperature and are generally positive. The
JANUARY
2002
5
effect of the temperature gradient on diffusion is secondary to the liquid gradient.
HExr AND VAPOR DIFFJSION
Heat conduction through the porous medium constituted by the networkl of fibers, yarns, and air takes place
with an effective conductivity Aej [10, 16, ch. 23, 22].
The order of magnitude of this well known parameter is
close to that of air without movemiient. The conductive
flux is expressed as kX,fVT.
Vapor flux diffuses through the air in the textile medium following a gen.eralization of Fick's law:
p'V, = -DefPg'V (
)
(4)
The "hygroscopic factors" a 3 and a 4 are given in Appendix A.
(b) When air and fibers are not in equilibrium, the
condensation (evaporation) flux at the surface of the
fibers ti,, diffuses into (out ot) the fibers:
hb=
PT =
The coefficienits
f
(aj#I and
11.
a2 <
-Dff( 1 a 1 VP',i +
a-2VT)
the surface where the boundary
f(4)
Mass and Energy Balance
The balance equations are written according to Whitaker [321:
(8)
condition is PS
and only one node is fixed at the center of the
cylinder.
We niust point out that a choice has to be made as to
whether or not an equilibrium between air and fibers
exists. The volume averaging model is not compatible
with the two alternatives in the same set of equations [5,
6]. Furthernore, notice that the following equation is
probably not valid for textile media:
(5)
and a2 are given in Appendix A
2P)
(Sf)
We expect that this flux is greater than the diffusion flux
at the surface of the fibers.
Note that the form of this "mass source" term can be
obtained at the end of the numerical resolution of the
basic problem of moisture transfer into a cylinder from
=
The diffusion coefficient Deff has to account for the
tortuosity r of the network. The classical expression Deff
= DaE/Tr is not obvious in a fabric, and it is convenient
to consider Djef (X) where X is the water content, bound
or free water.
From the set of moist air equations (dry air and water
vapor are ideal gases), the vapor flux is expressed as
pJDbi
EV, 4
El, + El +E,+ 5
I
As a matter of fact, the swelling of solid and, above all,
of liquid under capillary forces is not compatible with
this relation for a nondeformable structure.
The sum of the three equations (6) gives the global
water conservation:
a
at (E,p]l + PI, + p,j) = -V(pJ'V., + PmVb + pj'V,)
e",
_t'+ i7(PiVv,) =-il-e
(9)
To this equation we have to add
aPt
at-1--+
ap,
(pb vb3
=
Pb
-- = f()
ilbe
(/_
6
at
=
or
a Pb
where m corresponds to the mass rate per unity volume
for free water condensation and mhb for bound water. For
the homogeneous equivalent medium (one textile layer),
the bound water conservation equation can be written as
follows:
(a) When air and fibers are in equilibrium, a ID
transfer is sufficient; the condensation (evaporation) flux
tm4, increases (decreases) the bound water, which diffuses
at the surface of the fibers:
pbV,
-Db(a 3 Vp,i - a 4 VI)
(model a)
PS
(
(7)
at
a
ax V
Pb
fix ,
- P,I)b_(
P
\ P8
-f(
)
(model b)
. (9a)
The energy conservation equation accounting for the
transformation of phases reads
pC,
at7
-
+ p§(,,V,VT
9
=
V(AejiVT)
+ Ahe,,h, + Ah,ih
,
(10)
6
TEXTILE RESEARCH JOURNAL
and pCp, the heat capacity of the medium, is as given in
Appendix A.
Equations 9 and 10, associated with 5, 6, 7, and 8 give
two coupled equations for the three variables, p,j, T, pl.
The system of these equations is completed with the state
relations for moist air:
p]i < P.,
4(T) X P,
0
(
P, > 0 <* P'12= PuVfa/MT
The heat and mass transfer problem. is expressed with
two independent variables, (pu', T) or (p,, T), following
the occurrence of free water. Note that thie variables p,j,
T are continuous throughout layered fabrics, which is not
the case for pi and pb.
Boundary conditions are as follows:
On the skin side, the conditions are
(p2VJ'4
A
I_T)
=-kJ(p,' -pi
h(T - Tn1)
=
(12)
where 1 is the node on the skiin side, and the x axis is
directed towards the outside. The heat transfer coefficient
includes the effects of high wave radiations. A liquid
sweating flux is given by
(pl/v1 ) I
7i 1 =
(12a)
The external side conditions are:
(p,J11J
-Aet(4x)
=
k,(pt,2 - p,4)
=
(T,TJ-
,
(13)
where n is the node on the external side.
Rain can be taken into account either with a liquid flux
if no dripping is expected, or with a quasi-instantaneous
wetting of the external layer if dripping is expected.
The kind of contact between textile layers can be
expressed by a condition of resistance in the flux. It is
necessary to choose a discretization node, or a grid point,
on each side of the layered sample, as we shall see later.
The Importance of Physical Parameters
The model introduces some physical parameters and
functions. The main ones are f( ), A f (plf Pl).
Deff(Pb), Db/Ph), Db(p), D,(p). Data on the hygro-
scopic furction f(4) are available for most fibers. Thermal conductivity AefC and vapor diffusivity Detf, which
depend on the yam diameter and the tightness of the
weave, are generally given for a standard value of air
moisture. In the global water conservation equation (9)
and for model a, it is convenient to associate vapor and
bound water diffusion in an equivalent coefficient D,,
eff al + Dla 3,
which takes into account two phe-
nomena that occur simultaneously during a dynamnic
transfer. De,(pb) will be obtained with dynamic experiments on a fabric sample, as done by Li and Holcombe
[ 13], for example, as we will see later. The liquid diffusion coefficient D1 is accessible for diffusion on the
textile layer surface, and it should be of the same order of
magnitude in the transverse direction. Experitnents need
to follow the front displacement along the surface from a
constant wet point. One can use, for example, a moisture
sensor derived from the electrical capacitance technique
[9, 16, ch. 20]. D, will come from the part of the model
that describes liquid diffusion.
Numerical Resolution
We use the finite volume method as described by
Patankar [20]. The control-volume defined around a grid
point, or node, concerns a homnogeneous medium, for
example, one fabric layer, where parameters and functions are continuous. Space integration over the volumre,
at a fixed time, leads to conservation of fluxes on the
interface between two control voltumes. It is possible to
split a fabric layer to create several control layers. Numerical tests show that it is rarely efficient to split a
textile layer more than once. Since the equation coefficients vary throughout the fabric layers, their values are
expressed with a linear variation between grid points
when they are the function of p, and T, and step-to-step
whern they are a function of pi. The particular treatment
of p, is justified because condensation or evaporation
takes place in a control-volume, and liquid diffusion has
a high dependence on contact resistance between layers.
It should also be possible to introduce a thermal contact
resistance between textile layers. Because temperature
and vapor density are continuous, this resistance is less
important.
These diffusion equations express various characteristic periods of time obtained with an appropriate ratio
similar to e 2ID or e2 /A, We obtain the following orders
of magnitude with e = 1 mm: moisture (vapor and
bound water) period T,i,, - 500 seconds (cotton) and
20 seconds (polyester), liquid water period -r,4 - 50
seconds (cotton) and .10,000)seconds (polyester), heat
period Theat 200 seconds (cotton) and 5 seconds (polyester). With these characteristic periods of diffusive
transfer time, an explicit resolution method can be sufficient when the time step is about one second. A semiimplicit path is possible. At each time step, the coefficients for grid point (i) are evaluated from actual values
in (i - 1) and previous values in (i + 1). The imnplicit
JANUARY
2002
17
solution is more accurate when boundary conditions
have sharp variations. Thus, the linear algebraic system
is solved in each grid point at each time with a test for the
condition of the existence of free water. A node is fixed
on each external face. the skin side and the external side.
0.040
0 035
0.030
y6 O 025
4,
X
0.020
4m
a 0.015
Model Validation Using Basic Experiments
The question of whether or not a model of homogeneous layers should be taken into account will arise when
we simulate well-chosen basic experiments. Fabrics are
porous media with pores of two sizes-interfiber and
interyam. We have to select experiments where vapor
and bound water diffusion are the main phenomena expected on and into the fibers rather than through the
yarns.
The excellent experiments of Wehner et al. 131] on
"dynamic water vapor transmission through fabric barriers" are suitable for a comparison with simulations. An
apparatus with a sorption cell gives the transient moisture content of the fabric sample and the moisture flux
across the fabric. The air temperature on both sides of the
fabric is kept at the same value, but a vapor pressure
difference between the two sides is fixed so that the
moisture concentration gradient is the major driving
force causing moisture flux across the fabric. For a
polyester fabric sample, the response delay is very short
(the water content is weak) and not very efficient for a
comparison with the simulation. For cellophane film it is
the expected response of a homogeneous medium. Such
an initially dry film is exposed to a 100%-0% relative
humidity gradient between the two sides of the film. The
weight increase following water adsorption starts from
the initial time of exposure, whereas the moisture flux
through the film appears after the delay necessary to
cross the thickness. The response for an initially dry
cotton fabric does not show such a delay before the
moisture flux crosses the tissue layer (Figure 2). The
parameter values needed for simulations of e, p5 DeffI
h, k ... are deduced from Wehner's data. Db estimated
from the data in the literature is not well known but has
a minor influence. The sorption equation is given elsewhere [13, 16, ch. 71. The parameters DIff, Db, and k
determine the established moisture flux across the fabric.
A response delay strongly depends on the exchange
coefficients h and k on each side. As expected, the
simulated outside rmoisture flux (Figure 2) obtained with
the air-fiber equilibrium model increases after a delay,
contrary to the experiment where the vapor flux using
"large holes" between the yarns increases as soon as the
experiment starts. However, the simulated moisture flux,
i
0.010
0.005
0
5
10
15
20
25
30
35
40
45
50
55
s0
Time (m;i)
F1GURE 2. Siniulation of one of Wehner's experiments [311. Response of moisture adsorption and moisture flux across an initially dry
cotton fabric exposed to a 100%-O% relattve humidity gratdient: (a)
inlet flux through one side of the fabric (x experiment, - simulation),
(b) outlet flux from the other side of the fabric (A exp., - sim.), (c)
adsorbed flux in the fabric (S exp.. - sim.). Simulation with the
air-fiber equ.ilibnum model: e
1.5 mm, p5- 330 kg/ln3, hi, h
2
9
= 14.7 W/m K, ki = k, = 0.0146 m/s, D10 -nm /s, D,fr
= 2.195 10-6 m2/s, T. - 20°C,jf(4d) =1(a
+ **et -- (!*02), a
= 1.985, * = 18.29, c = 15.93. F.ight nodes are distributed over the
fabric layer.
stocked and crossed, is close to the experimental one.
Note that the moisture adsorption rate is obtained from
a not very accurate derivation of the weight gain
function close to the starting time. At this point, it
appears that the air-fiber equilibrium model is not able
to describe perfectly such a Wehner-type experiment.
As a challenge, a rigorous macroscopic model could
be obtained from a microscopic one adapted to the
structure following Whitaker's method [32]. A simpler
and more classic model would be the parallel one: a
transfer flow through "large holes (l.h.)" (between
yarns) parallel to a transfer flow through "narrow
holes (n.h.)" (between fibers). Let us suppose, for
example, that such a model is defined only with a
spatial ratio wo,the volume percentage of large holes.
The density of the fabric is obtained by p, = (PstIah +
(I - °)Ps ni3 Likewise, D,,, and the moisture flux may
be calculated by taking into account -the boundary
layer resistance. Many "numerical experiments" are
possible by giving different values to the three parameters, Co, P,,I.hu.Ps, Df} lhIDeff. Figure 3 depicts a typical
result obtained with the set of values-0.1, 0.5, and
10. A systematic study could deliver a perfect adjustment of the experimental data. This example shows
that it is always possible to improve the air-fiber
equilibriumn model to make it more realistic. Although
these parameters should be measured, it was not possible to do this.
8
TEXTIL.E RESEARCH JOURNAL
would be perfect, which is obvious if the therrmal perturbation is negligible. This experiment can be used to
deduce the diffusion parameter Di, (Equation 9a) of the
nonequilibrium mnodel. Figure 6 shows the dependency
on water content.
0
6
1i
'5
20
25
30
36
40
45
50
60
0
-
0.12
Time (min)
~--------
FIGtURE 3. Simiiulation of Wehner s results with a model of large holes
1D,,
.
0,h
0 5, Def,jh/DJ
ps
parallel to narrow holes: w)=- 0.1, p
A second kind of basic experiment will allow us to
interpret the diffusion coefficient Deff in the air-fiber
equilibrium model. Li and Holcormbe [131 set up an
0.20
0
10
30
20
40
during a sorption phase when boundary conditions are
identical on both sides (20 0C, 99% relative air mnoistare,
given exchange coefficients). The wool fabric, initially
dlry, adsorbs water continuously, and the curve of water
content versus time can be deduced. Li and Holcombe
compared experimental results with three models: the
pure Fickian diffiusionr model, the Nordon-David model,
and a "two-stage mnodel," which we refer to as a "mass
source term" that expresses the diffusion of bound water
in the radial direction of fibers and yarns. The experiment-simulation comparison is excellent with this last
model. The sim-ulation of this sorption experiment with
our model (air-fiber equilibrium) is obtained with values
of the parameters of the paper [ 13]. e, p,5 f(q), Db. The
exchange coefficients (h, k) are deduced from tihe air
flow data around the sample and are not used as they are
given. Since the measured value of De is related to large
holes, we have to introduce another value related to an
equivalent homogeneous medium in order to obtain a
convenient response (Figure 4). At this point, we should
S0
60
70
S0
80
Time (min)
experimental apparatus for weighing a fabric sarnple
1-(tuRE 4. Simulation of Li and Holcombe experi-ments [13]. IJptake
of a< initially dry wool fabric in a sorption cell (99% relative humidity
2
and 20C.3: (a) simulation witb Dff = 1.91 10-5 m /s (given in the
with D, = 1.91
paper), (b) Li and Holcombe results, (c) simulationi
2
10 6 n2/s. e - 2.96 mnm, p, = 91.9 kg/rn3 , hi = ii- 13 Wfm K,
2
5.1 101.4 m /s. Eight nodes are
= 0.(13 oi,s, Di,
k, i,,
distributed over the fabric layer.
2.0E-06
if 1.BE-06tb;
o.
'
1.20-S8
|-
--
--
E
8.OE_07
0.24
E
-
0.20
t.2S
0.30
Water Content
0.32
0.34
0.36
x Pbp,
consider that this basic experiment can give the function
D,jj (p,) through De,qPh) using the air-fiber equilibrium
model. The method is detailed in Appendix B. The
deduced moisture diff:usion coefficient D, and thus D,.+
(the Db tenm appears negligible with respect to DejA3,
varies strongly with moisture (Figure 5). Thqis could
mean that water adsorption induces swelling, modifying
the tortuosity and volumetric porosity of the medium
DaoEjr7). The experimnental conditions do not
(D,etf
allow measurements of weak water content values. One
would expect that using a function D,ff(Pb) throughout
the range of variation, the agreement with the experiment
5. Moisture coefficient De_(JpJ,1p,)
FiGTTRu
Deq -- D,, Ui + Db6
3,
determined from Li and Holcombe's experiments with the air-fiber
equilibrium model.
Since the two models (air-fiber equilibrium or not) are
able to give an exact response for such a sorptioni experiment, the question is, which one is physically the best'?
Because the model with a inass source term expresses a
transfer normal to fiber/yarn, it seemns to be closer to
actual physical behavior. On the other hand, this model
JANUARY
2002
9
simulation and experiment are very similar (Figure 7).
Vapor sorption creates a heat source that expands over
time due to the conductive properties of the fabric. When
one layer of wool is added, the temnperature increases
after a delay. Crossing of the temperature curves during
the first part (0-5 seconds) is somewhat different whlen
comparing the simulation and the experimalent. The vapor
flux is distributed as bound water over all the layers, and
the competition between heat and vapor diffusion is
expressed with A&f, Deff, f(4), which are not explicitly
given but deduced from the data.
iOE-.04-
04
0i
21.0EtQ4
a Ie
.O
Water Content - ax
f
---
FIGURE 6. Moisture diffusion coefficient D, (PblPv) detemiined
from Li and Holc5ombe's experiments. 'The fiber is not in equilibriun
with air.
0045
-- II
0.C140
e7
0
W
p
j_+rv2____+
-
0135-
rv4
=r-
-
-,
_t_.e>
j70,030 ---_'E .
generally retains a constant value of Deq-7 which expresses the vapor diffusion mainly through the large
holes and does not take into account fiber swelling.
Therefore, the coefficient Db1 (Pb) has to account for the
vapor diffusion between the fibers and the bound water
diffusion into these fibers, together with the swelling.
Such a sorption experiment allows us to determine
Db [(Pb) if Deff is given. Whatever the chosen model,
sorption experiments are needed to determine a moisture
diffusion coefficient.
Simulation of Yasuda's Experiments
Using a diffusion column, Yasuda et al. [34, 35]
conducted transient heat and miioisture transfer expenments through several layers of fabrics. They tested
fibers of various compositions, the effect of surface modification, and the arrangement of some fabric layers.
Although these experiments are relevant and interesting,
physically speaking, the heat and vapor transfer in the
two spaces (chambers) surrounding the sides of the fabric samrple induce exchange coefficients that are not
precise (natural convection can occur), although they are
very important parameters. However, an evaluation is
possible. For this purpose, we have selected a particular
experiment for which data are available. The sample
consists of one or several wool fabric layers (up to five).
The fabric is conditioned beforehand in air at 1 80C and
8% relative humidity. The experiment starts when the air
condition on the bourdary of the space surrounding the
internal side is fixed at 37°C and 100% relative humidity.
The temperature and relative humidity of this space
(which we assume is in the center) are recorded and can
be compared to temperature and vapor density simulated
on the internal side of the fabric. The behaviors of the
- f
0.025
0.020 t
e
Pr--sur
2WlrVpor
0~~~~~~~~~~~~~2
g
8
f
(nmHg
-
0.01
0.005
_
-
<! EXpimento
-
Yssuda
105
0.000
0
-
-
5
10
-
15
20
46,
_
25
- --
30
'5
20
IO-n.e (mmi)
35
40
45
50
25
lO
|
55
60
Time (min)
_
44 ..i___ .__
|>40
50
g38 _ .1\fg:
~34
-T--T2
(m--
--T3'l.[
32 0
30
Eprmuorsd
- 1015
-T4
20
251---- 30
3-045
0
20
25
35
01Z 5------- -5Z5
6
30
t
0
S
10
1S
30
40
45
50
55
60
Time (min)
FiGURE 7. Simulation of one of the Yasuda sorption experiments [34,
35]. A variable number of wool fabrics (one to five) are tightly stacked
together. The air humidity in the space in front of the sample and the
temperature of the sample surface are measured. The sinulation gives
the preceding variables on the sample ssurface. Initial conditions of the
samples: equilibrium in air at 8% relative moisture and 18°C. Conditions of the chamber in front of the samisple: I00% relative humidity and
2
37°C. e (one layer)
0.32 mm, p0,
355 kghn3 , D,0 = 0- 6 m2/S,
D, = 10-ia m2 /s, h,
S W/m'K, he = I W/m2 K, k4 - 10 mts.
k, = 0. Three nodes are distributed into each layer and one node on
each of the bound surfaces.
Conclusions
The ID dynamic heat and moisture transfer model
we have proposed here is efficient at simulating almost
TEXTILE RESEARCH JOURNAL
10
all the sets of fabric layers actually worn for many
realistic activity conditions. It is obvious that the
choice of clothing depends on the weather and the
activity, but combinations of different clothes or different layers in a garment are not always adapted to
the situation. Our simulatiom of a set of fabrics during
wear shows that the behavior of sorne textile layers is
not the simple addition of individual properties. Each
set of textile layers determines an occurrence of condensation and a main characteristic period of time for
moisture transfer.
An efficient sim-ulation is the result of a well
adapted mnodel and accurately measured physical parameters. More work is needed to measure the diffusion coefficient for liquid and vapor/bound water and
contact resistances.
Appendix A
Vapor diffusion coefficients:
Cotton
Polyester
Wool
1.985
18.29
15.93
0.23
29.132
164.886
128.98
0.015
1.223
10,.347
8.777
0.358
a
b
c
tat
Differential bound water heat of sorption [16. ch. 81:
hb = c 0Ahjjexp(-cjX)-exp(-c
Xsa,)1
Cl
0.5135
0.5538
0.5086
26.976
10.04
121.449
Cotton
Wool
Polyester
Ce)
Coinductivity depends on the thickness and nature of the
weave. For example, from Schneider's measurements
[221:
A Wm ' K-)
< xt
a(aX
±
+ a2, x
= aO
A)t(Wm- K)--)=a 3 +a 4 x+ac5 2x2X>X,,I
T 2\ -1
'T
-0=
1)
. +
a0
Cotton
Wool
Polyester
p = 10 1300 Pa,
r = 287 mrs
2
0.048
0.039
0.04
Ahl(T) = L ± (C,,
-
i(T)
do = 23.58;
T+
d, = 40349.42;
d2 = 236
afL
_117a-
=
p=. im(lT)
aTa
sP p'aT
xf= ()
=a+-
0.076
0.0694
0.066
0.004
-0.0105
0.002
0.02
0.02
0.01
Cp,
polyester
Cp, wool
p kg m'9
Hygroscopic factors for bound vapor diffusion:
a3
0.47
0.427
7
+ (p2e,
*
p,Cpv3el,
Cps cotton = 1.380 kJ/kgK
_
Ps
a.
cl, = 4.180 k.J/kgK,
=____
7rC;
a4
Cpv= 1.860 kJ,kgK,
r;eC
(273 + T)r exp
a3
10W5
0
1.6
02
Cp, = 1.006 kJ/kgK,
Cp)T,
L = 2.5016 10" J kg
a1'
PC, = P,C, + (Pt, + pl,)Cz
K--,
r, = 461 m2 s2 K
p
I
ao'
-C 2
=
1.255 kJ/kgK,
1.360 kJ,/kgK
Appendix B
The equivalent vapor/bound water coefficient D e
takes into account two phenomena occurring simultaneously during dynamic transfer. Vapor diffuses iinside
the yarns through fibers and condenses in the form of
bound water around and into the fibers. Dey will be
measured with dynamnic experiments on a fabric sample
irn the manner of Li and Holcombe [13]. The model of
such an experiment, which keeps p, = 0 and T constant,
will be
JANUARY
2002
I].
ap 0
a
ax
dt
or
aPb
-,
at
aP,\
) D
----
3 ------
=
\
a '
q
a x /)
aPb\,
ID(Pb)
Px a /
d
a
8.
D = D-a3
9.
The boundary conditions are
10.
Pbs.rf= P,=AO-f)
-~D(pigu,f)(~'I)
=
k(p.., 1
-
p,)
The two sides of the fabric sample (thickness e) are
submitted to the same exchange conditions:
Pb(t) =
1
e/2
(e/2
J
pldx
is obtained by continuous weighing. The mass exchange
coefficient k is determined by the air flow rate along the
layer surfaces.
Some methods are available to solve this problem: the
equation pv - p, = f(x) * g(t) leads to a sernal solution. More directly, a parabolic profile for Pb also gives
a solution. One deduces the function D(pb,Surf). Initial
experimental conditions have to be accurately fixed if
one wants values at low moisture.
11.
12.
13.
14.
15.
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Manuscrip received Derember 22, 1998; ac(epted Marchi 22, 2000.
Parameters of Rotor Spun Slub Yarn
JUN WANG AND XItJBAO HUANG
Donig Hufa University, Shanghai, 200051, People's Republic of China
ABSTRACT
Using the mechanism of collecting and doubling in rotor spinning, the parameters of
rotor spun slub yar. are analyzed with special reference to slub length, factors affecting
slub length, and the inherent connection of slub length to sloib amplitude. The results show
that slub length increases with an increasing slub mnultiple. the slub length is always longer
than the rotor's circu-mferential length, and the factors affecting the slub length are rotor
diameter, normal yarn count, slub multiple, and servomotor performance.
The study o-f rotor spun fancy yarn. started in the late
1970s [1-6]. Rotor spun slub yam now commercially
produced by changing the feed roller speed, but theoretical research on its parameters, such as slub length, slub
amplitude, and slub space, etc, 'has not appeared in the
published literature. Thne special fancy effect of rotor spun
slub yarn is characterized by the longer slub length-i and is
Textile Res. .1.72(l), 12-_16 (2002)
quite different from ring spun slub yarn. In this paper, we
discuss in detail the parameters of rotor spun slub yarn
with special reference to slub length, the factors affecting
slub length, and the inherent connection of slub length to
slub amplitude in terms of the collecting and doubling
mechanism in rotor spinning, which is significant for the
practical production of rotor spun slub yarns.
0040-5175/$15.300
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TITLE: Dynamic heat and water transfer through layered fabrics
SOURCE: Textile Research Journal 72 no1 Ja 2002
WN: 0200100388001
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