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*Heating Element*

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**Dryer Heating Element Test**

## Kinetic finite element model to optimize sulfur vulcanization: application to extruded EPDM weather-strips.

INTRODUCTION

A weather-strip is typically an extruded elastomers bulb with

complex geometry and curved shape. Typically, weather-strips (1) are

installed in automotive industry and civil engineering, in order to

prevent water leakage, block exterior noises, minimize body and window

vibration, and provide some shock absorbing capacity. Initial use of

weather-strip seals in automotive applications was aimed at

accommodating for manufacturing variations. Lately, the isolation issue

became important, and the key design was to better isolate the passenger

compartment from dust, air, and water leakage. Nowadays, the general

trend in seal design is the isolation of the passenger compartment from

noise and vibration. Isolation is again the main role played by

weather-strips installed in windows and doors. Here, stiffness,

strength, isolation capacity, and aging resistance are key issues to be

considered, since the average life of a building is typically longer

than that of a car.

Weather-strips are realized by elastomers, which typically are

viscoelastic materials. Their mechanical properties are strain,

frequency, and temperature dependent. In addition, being the geometry of

weather-strip often rather complex, this generally implies that a

numerical model has to be performed with a high level of detail. Unlike

plastic and other materials, elastomers have characteristics such as

high flexibility, high elasticity, and high elongation. Their major

drawback is that they require vulcanization during the manufacturing

process, and (2) they cannot be reshaped after curing.

Ethylene-propylene-diene monomers (EPDM) are elastomers utilized in a

wide range of applications, including weather-strips. The advantage of

EPDM is its outstanding resistance to heat, ozone, and weather. Usually,

EPDM vulcanizates are composed of rubber, filler (carbon black and

calcium carbonate), curatives (fatty acids, zinc oxide, accelerators,

and sulfur), antidegradants, and processing aids.

As a matter of fact, weather-strip producers are interested in the

improvement of cured EPDM final mechanical performance, at the same time

limiting the production costs. For instance, sulfur is normally

preferred to peroxides merely for economic reasons, despite the fact

that the performance of rubber cured with peroxides--in terms of final

mechanical properties--may be sensibly higher than that of rubber

vulcanized with sulfur. Indeed, from a chemical point of view, sulfur

vulcanization determines transversal chains constituted by more than one

sulfur atom (link energy 270-272 kJ/mol) whereas for peroxides the link

is created between two back-bone carbons belonging to contiguous chains

(energy 346 kJ/mol). In addition, it has to be emphasized that, when

peroxides cure method is used, the rubber base could exhibit a peculiar

smell, which is obviously inacceptable for the production of

weather-strips.

Despite the wide diffusion of sulfur vulcanization and the fact

that its discovery and utilization go back to Goodyear (3-6), the

chemistry of vulcanization remains an open issue. In this field, among

the others, historical contributions by Ding and Leonov (7) and Ding et

al. (8) are worth noting. Globally, it can be stated that the majority

of the available approaches are models enforced to resemble to peroxidic

laws.

Another important issue to consider is reversion, which occurs

quite frequently in practice. From a macroscopic point of view, it

consists of remarkable decrease of rubber vulcanized properties at the

end of the curing process. Chen et al. (9) have shown that this

phenomenon seems to appear when two reactions are competing during

vulcanization. Reversion is often associated with high-temperature

curing. For instance, Loo (10) demonstrated that, as the cure

temperature rises, the crosslink density drops, thus increasing the

degree of reversion. Morrison and Porter (11) confirmed that the

observed reduction in vulcani-zate properties is caused by two reactions

proceeding in parallel, i.e., desulfuration and decomposition, see Table

1. Generally speaking, this could give rise to the items with

considerable thickness and undergoing different temperatures gradients

during curing a strongly inhomogeneous final level of vulcanization.

TABLE 1. Products and schematic reaction mechanisms of accelerated sulfur vulcanization of polydiene and EPDM elastomers. Reaction Compounds Process/reaction Kinetic Model ID constant constants NA [S.sub.8] + Mechanical mixing NA NA accelerators + by open roll mill, ZnO + stearic internal mixers, acid [right and/or extruders arrow] soluble ai T < sulphurate zinc 100[degrees] C complex (A) + polydiene elastomers (P) (a) [MATHEMATICAL Allylic [K.sub.1] [K.sub.1] EXPRESSION NOT substitution REPRODUCIBLE IN ASCII] (b) [MATHEMATICAL Disproportionate [K.sub.2] [K.sub.2] EXPRESSION NOT REPRODUCIBLE IN ASCII] (c) [MATHEMATICAL Oxidation [K.sub.3] [~.K] EXPRESSION NOT REPRODUCIBLE IN ASCII] (d) [MATHEMATICAL Desulfuration [K.sub.4] EXPRESSION NOT REPRODUCIBLE IN ASCII] (e) [MATHEMATICAL De-vulcanization [K.sub.5] EXPRESSION NOT REPRODUCIBLE IN ASCII]

Very recently Milani and Milani (12) have proposed a simple kinetic

numerical model to predict EPDM reticulation level, which may also take

into account reversion. The model is a simplified one and relies into

the derivation of a single second order nonhomogeneous differential

equation, representing the degree of reticulation (or conversely the

torque resistance) of rubber in dependence of curing time. Kinetic

parameters to set in the kinetic model are only three and they can be

evaluated by at least two torque curves performed on the same blend at

two different vulcanization temperatures. Cure tests have to be

maintained at fixed vulcanization temperature and may be performed by

means of both traditional oscillating disc (ODR) and rotor-less (13),

(14) (RPA2000) cure-meters.

In the present article a RPA2000 cure-meter is utilized to perform

the experimentation. Such device has a test chamber with controlled

stable vulcanization temperature allowing the storage of 5 gm of

product, with diameter 20 mm and height 12.5 mm (total neat volume 8

[cm.sup.3]). Basically, there are no perceivable geometric effects

related to the torque measure because (1) the test is fully standardized

and (2) in case of RPA2000 devices the ODR is missing. Quite small

secondary torques may be present as a consequence of material viscosity

and plates friction, which are obviously negligible in practice. Also in

case of experimentations conducted with ODRs, it is worth remembering

that the dimension of the disc is relatively small, allowing to

disregard inertia forces, always present when mechanical elements move

and responsible of secondary geometric effects.

After experimental data reduction, the aforementioned kinetic model

is adopted to predict rubber degree of vulcanization during the

industrial curing process of a thick weather-strip used in civil

engineering applications. Once evaluated the kinetic constants involved

in the reticulation process, the second phase relies in implementing

kinetic model parameters within a nonstandard finite element (FEM)

software for a thermal analysis of the item. Such approach follows a

relatively long tradition regarding FEMs applied to then-no--mechanical

problems of weather-strips installed within devices subjected, after

thermal curing, for static and dynamic loads (15-19). FEM is, indeed,

recognized as the most suitable technique to interpret the

thermo--mechanical behavior of vulcanized rubber items with complex

geometries, giving the possibility to quickly study combined nonlinear

3D problems, with error estimates and error reduction upon mesh

refinements. The software developed allows obtaining, element by

element, temperature profiles at increasing curing times. In addition,

it is possible to evaluate output mechanical properties (tensile

strength, tear resistance, and elongation) increase as a function of

curing time. In this case, the numerical database collected in the first

phase (reticulation kinetic model) is used, allowing a point by point

estimation of any output mechanical property.

The blend studied to realize the weather-strip is a mix of two

different EPDMs (Dutral TER 4049 and 9046) with a medium amount of

propylene content (ca. 31% or 40% in weight) and 9% or 4.5% in weight on

ENB (5-ethylidene-2-norbornene), vulcanized through accelerated sulfur,

as described in detail next. Once evaluated the final mechanical

properties of the item point by point, a compression test is numerically

simulated, assuming that the rubber behaves as a Mooney-Rivlin material

under large deformations and using contact elements between the

compression device and the item. From an industrial point of view, the

numerical approach may be useful to optimize (especially in economic

terms) (i) vulcanization time, (ii) energy utilization, (iii)

temperature of vulcanization, and (iv) accelerators quantities. The

procedure is quite general and can be used in presence of any rubber

blend, provided that suitable experimental data are at disposal to

characterize crosslinking reactions at different temperatures.

THE KINETIC NUMERICAL MODEL: A REVIEW

The recently presented kinetic model (20) is utilized to evaluate

the degree of vulcanization reached by a rubber specimen subject to

thermal predefined conditions (constant temperature) and vulcanized with

sulfur. The model relies into a second order differential equation with

solution evaluable in closed form, having only three kinetic constants

to be determined. Constants are usually evaluated by means of rheometer

experimental tests realized following the ASTM D 2084 and D 5289 methods

(13).

Focusing exclusively on EPDM rubber, the commonly accepted basic

reactions involved-see also Refs. [5. 2124 and Table 1, are:

(a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(C.) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(d) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(e) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where P and A are the polymer (EPDM) and soluble sulfureted zinc

complex ([S.sub.8] + accelerators + ZnO + stearic acid) respectively,

[P*.sub.1] is the pendent sulfur (crosslink precursor), [P.sub.v] is the

reticulated EPDM. [K.sub.1,...,5] are kinetic reaction constants, which

depend only on reaction temperature, [MATHEMATICAL EXPRESSION NOT

REPRODUCIBLE IN ASCII] [Q.sub.x]and [D.sub.e] are the matured crosslink,

the oxidation product and diaryl-disultide, respectively. Reaction (a)

in Eq. / represents the allylic substitution in Table 1, reaction (b) is

the disproportionation, whereas reactions (c-e) occurring in parallel

are respectively the oxidation, the de-sulfuration and the

devulcanization.

Chemical reactions occurring during sulfur vulcanization reported

in Eq. I obey the following rate equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

The set of differential Eq. 2 may be obviously solved numerically

by means of standard Runge--Kutta procedure (25), (26). However, this

approach may become tedious both from a numerical and practical point of

view. Closed form solutions are obviously preferable.

In (20) it is shown how the concentration of vulcanized polymer

[P.sub.v](t) within the material during the vulcanization time range is

ruled by the following single second order non-homogeneous differential

equation with constant coefficients:

[d.sup.2][P.sub.v]/d[t.sup.2] + [K.sub.2]d[P.sub.v]/dt +

[[~.K].sup.2][P.sub.v] =

[K.sub.1][K.sub.2][P.sub.0.sup.2]/[([P.sub.0][K.sub.1]t + 1).sup.2] (3)

Having indicated with [[~.K].sub.2] the following constant:

[[~.K].sub.2] = [K.sub.2]([K.sub.3] + [K.sub.4] + [K.sub.5]) +

[K.sub.3.sup.2] + [K.sub.4.sup.2] + [K.sub.5.sup.2] (4)

It can be shown that Eq. 3, after some reasonable simplifications

on the nonhomogeneous term fully explained in Ref. (20), may be solved

in closed form and the solution is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

As it is possible to notice, in Eq. 5 kinetic constants to be

determined are only three, namely [K.sub.1], [K.sub.2], and

[[bar.K].sub.2]. The most straightforward method to provide a numerical

estimation of kinetic constants is to fit Eq. 5 on experimental

cure-curves, normalized scaling the peak value to Po (for instance equal

to 1) and translating the initial torque to zero, as suggested by Ding

and Leonov (7). As a rule, variables [K.sub.1], [K.sub.2], and

[[~.K].sup.2] are estimated through a standard nonlinear least square

routine. The initial part of the curve, before scorch point, which is

typically linked to viscosity exhibited by a fluid, cannot enter into

the optimization process, because it is obviously ruled by other

physical mechanisms.

EPDM Blends Under Consideration

Two different EPDMs (Dutral TER 4049 and Dutral TER 9046)

experimentally tested in [20.1 are reanalyzed in this work. The

characteristics of such EPDM blends, in terms of Mooney viscosity and

compositions are summarized in Table 2. In the same table, a hypothetic

product derived ad hoc mixing the two experimentally tested blends is

also indicated. Generally speaking, from a practical point of view, an

elastomeric blend is an interesting method commonly used in the rubber

industry in order to increase final properties of rubber items. It has

been shown in many applications that, while two polymers may be

virtually mutually insoluble, blends may be industrially produced, which

are macroscopically homogeneous and have improved properties, provided

that mechanical mixing is suitable and viscosities after mixing are

sufficiently high to prevent gross phase separation. In this situation a

master-batch process would be desirable, i.e., the component polymers

should be precompounded with vulcanizing agents and additives, and then

individual stocks should be blended in desired proportions. In our

specific case, the two rubber types are quite similar in terms of

molecular weight, molecular weight distribution, chemical properties,

and structure. They exhibit different cure rates because different

amounts of ENB are present, namely 1% and 2% in terms of % moles in the

polymers. The mix under consideration deserves to be studied for the

following reasons: (1) it may be produced by means of the same catalyst

system and the same process used for Dutra! TER 4049 and 9046; (2) when

dealing with Mooney viscosity number, differences do not exceed 30

points between the two products, meaning that the mix between them do

not modify the MWD extensively; (3) the composition in terms of ethylene

of both principal polymers is about the same and in any case in the

range in which the polymers are perfectly amorphous.

TABLE 2. Composition of Dutral 9046, Dulral 4049 and Duiral 9046-4049 blend. Mix 70% Product type Dutral 9046 Dutral 4049 4049-30% 9046 Propylene (wt%) 31 40 37.3 ENB (wt%) 9.0 4.5 5.85 Ethylene (wt%) 60 55.5 56.85 ML(1 + 4) [Degrees]C 67 93 85.2 ML(1 + 4) 125[Degrees]C 49 76 67.9

[FIGURE 1 OMITTED]

For all types of EPDM and the hypothetical mix considered, the same

formulation was used, with the aim of comparing the experimental data

with the data derived numerically from the solution of the ordinary

differential equation system Eq. 2--hereafter labeled as ODEs system

model for the sake of clearness--and the single second order

differential Eq. 5--hereafter labeled as single EQ-DIFF model. Rheometer

curves at 160 and 180[degrees]C are shown in Fig. 1. It is interesting

to notice that Dutral TER 9046 exhibits more visible reversion at

180[degrees]C, whereas the behavior of Dutral TER 4049 is less critical.

Being the mix composed by 70% Dutral TER 4049 and only 30% Dutral TER

9046, its reversion is expected to be quite little and in any case

limited to the final vulcanization range at 180[degrees]C. Reversion is

totally absent at 160[degrees]C. In Table 3, the compounds formulation

adopted (in parts per hundred resins) is summarized. The experimental

compounds were generally prepared in an internal mixer and by adding

both curing and accelerating agents on the roll mixer, maintaining the

temperature of mixing lower than 100[degrees]C.

TABLE 3. Compounds formulation adopted (in phr). Ingredients Description phr Polymer Dutral 4049, 9046 100 Zinc oxide Activator 5 Stearic acid Coagent 1 HAFN 330 Carbon black 80 Paraffin oil Wax 50 Sulfur Vulcanization agent 1.5 TMTD Tetramethylthiuram disulfide 1.0 MBT Mercaptobenzothiazole 0.5

A comparison among experimental data, ODEs system and single

EQ-DIFF at temperatures equal to 180 and 160[degrees]C is represented in

Fig. 2 for Dutral TER 4049, in Fig. 3 for Dutral TER 9046 and in Fig. 4

for the mix between 4049 and 9046, respectively. Convergence curves are

also represented for both models. Each curve represents the difference

between experimental and numerical normalized torque at a given

iteration number. The mix of products here presented will be adopted for

studying the production phase of whether-strips analyzed in the

following sections. Here it is worth noting that, for these particular

items, it is crucial to have at disposal a rubber compound with a

rheometer test curve, which reaches asymptotically the maximum Mooney

viscosity, without exhibiting reversion. As it has been demonstrated in

Ref. (20) in a different technical problem, the overall vulcanization

level of an item may be numerically predicted by means of the

determination of temperature profiles and the knowledge of rheometer

curves at different temperatures. In general, vulcanization density

depends on curing temperature and time as well as on thickness of the

items, obviously at given recipe used. Compounds exhibiting reversion at

temperatures lower than 220[degrees]C should be avoided, since may

result into items over-vulcanized near the skin. Conversely, from an

economical point of view, it is important to reduce vulcanization time

and hence to use high temperatures, blends that vulcanize more quickly

(as Dutral TER 9046), but without reversion (as Dutral TER 4049): for

these reasons, the polymer mix here proposed appears to be a good

compromise between quality of final reticulation and reduction of

production time needed.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Cured Rubber Mechanical Properties

To evaluate numerically the degree of vulcanization of a given

rubber item, it is necessary to link degree of vulcanization with output

mechanical properties. There are many important parameters that

producers are interested to maximize at the end of the vulcanization

process, as for instance tear resistance, tensile strength, tension set,

etc. (27), (28). Unfortunately, in almost the totality of the cases,

such parameters cannot be maximized at the same time. In this article,

we will focus, for the sake of simplicity, only on a single output

parameter (tensile strength).

As experimental evidences show, there is a precise relationship

between torque values obtained in the vulcanization curve and output

mechanical properties of a real cured item (see for instance Refs. (27),

(29-40). In Fig. 5, several experimental data available in the

literature concerning torque values M and tensile strength exhibited by

vulcanized commercial products are represented, along with a possible

numerical experimental data fitting. The numerical relation between

torque and tensile strength found for Dutral blends and represented in

Fig. 5 will be used in the following section to perform some simulations

on a real weather-strip. An interesting aspect of the experimental

relation existing from torque and tensile strength is that it is not

necessarily mono-tonic, i.e., tensile strength may slightly decrease

increasing torque resistance. In the same figure, the relation between

hardness and percentage vulcanization degree is also depicted. The curve

is taken from Coran (21) and depends on the blend used. Having at

disposal such relationship, from the thermal model proposed hereafter it

is possible to evaluate point by point the vulcanization degree at the

end of the thermal process, and hence the local hardness of the item.

From well-known numerical relationships (usually based on empirical

formulas or equations based on theoretical formulations, e.g., Gent, BS

903, ASTM 1415, Qi et al. (41-43) it is finally possible to estimate

resultant Young Modulus E of the rubber after vulcanization, to be used

within the mechanical compression simulations reported at the end of the

article.

Another important issue to underline is that, obviously, the

representation of Fig. 5 is able to give correct information on output

parameters only at constant temperatures (note that T [not equal to]

[T.sub.n] being T rubber temperature and [T.sub.n] curing agent

temperature), i.e., results cannot be applied directly to a rubber

infinitesimal element subjected to curing. Indeed, the actual

temperature of a point inside an item subjected to heating is rather

variable and such variability can be estimated only resorting to

numerical methods. In general, each point P of the item has its own

temperature profile T = T(P,t). At a fixed time during the curing

process, each point has also its own torque value, which can be

calculated from the database collected previously. For each value of the

torque reached, a corresponding value of the output parameter can be

identically evaluated. In this way, torque-exposure time and output

property-exposure time (or alternatively output property-temperature)

profiles can be numerically estimated. Nonetheless. Fig. 5 gives a

precise (although approximate) idea of the complex behavior of rubber

during vulcanization, addressing that a strong variability of output

mechanical properties is possible when changing curing time and

vulcanization temperature.

Vulcanization of Thick Weather-Strips with Accelerated Sulfur

In a weather-strip subjected to any vulcanization process, an

inhomogeneous distribution of temperatures between internal (cool) zone

and (hot) skin occurs. With the aim of optimizing a production line,

many parameters have to be chosen carefully but, among the others, the

following variables play a crucial role: exposure time and temperature

of the heating phase. The software presented here--fully developed in

Matlab (44) language--is basically an assemblage of two elemental

blocks, further subdivided into sub-blocks.

BLOCK 1: for each node of the item (already discretized by means of

FEMs) at fixed values of the input parameters [T.sup.n] (curing agent

temperature) and [t.sup.c] (curing time), temperature profiles are

evaluated by solving numerically a Fourier's heat transmission

problem in two-dimensions (45-51). Since, closed form solutions are

rarely available and usually refer to simple geometries, a FEM (52)

discretization is needed in the most general case. For each node of the

item and for each value of exposure time and curing temperature

inspected numerically, local final mechanical properties are evaluated.

The average uniaxial tensile strength is considered as objective

function. To maximize this quantity, lour sub-block instructions are

repeated node by node.

SUB-BLOCK 1.1: at increasing times, the temperature of each node

varies (increasing until the end of the vulcanization phase and then

quickly decreasing during cooling). Each point is thus characterized by

a couple of values representing temperature and time.

SUB-BLOCK 1.2: for each temperature value collected in the previous

sub-block, the corresponding cure curve (evaluated numerically with the

model described previously and having at disposal kinetic constants of

the partial reactions) is uploaded. Here, it is worth noting that each

node of the item to vulcanize undergoes continuously different

temperatures, thus passing from a cure curve to a contiguous one (more

precisely to a contiguous curve associated to a higher temperature in

the heating phase).

SUB-BLOCK 1.3: for each point of the item, the torque-time diagram

is evaluated making use of all the numerical cure curves provided by the

single differential equation model described previously.

SUB-BLOCK 1.4: tensile strength and hardness (or elastic modulus)

to be used in mechanical simulations may be finally evaluated (e.g.,

tensile strength--time and tensile strength--temperature profiles) for

each node or point as a function of the torque reached by the point at

successive time steps. This is possible only having at disposal a

relation between final torque and tensile strength (or hardness) as

those provided in Fig, 5. Usually experimental data may be useful in

this step.

[FIGURE 5 OMITTED]

BLOCK 2: the procedure summarized in BLOCK 1 should be repeated

changing input parameters (i.e., total curing time and vulcanization

agent temperature). A very direct way to evaluate the best input couple

([t.sup.c] [T.sup.n]) is probably to subdivide the input domain in a

regular grid of points, but for practical purposes random attempts and

the experience of the user should be sufficient. Very advanced

algorithms based on Genetic Algorithms or bisection has been already

presented in Refs. (51), (53), (54). They are not used here, because the

knowledge of the blend behavior drives almost directly to the choice of

the most indicated production parameters to adopt.

The Modified Heat Transmission Problem

A modified heat transmission problem has to be solved to evaluate

item temperature profiles, accounting for the chemical reactions

occurring during vulcanization. We schematically subdivide the curing

process into two phases: heating and cooling. In the first phase,

elastomers are exposed to high temperatures in order to activate

crosslinking and thus vulcanization, whereas in the second phase rubber

is kept to ambient temperature through air and/or water. In the most

general case of 2D items, temperature profiles for each point of the

element are obtained solving numerically Fourier's heat equation

law (45-51):

[[rho].sub.p][c.sub.p.sup.p]([partial derivative]T/[partial

derivative]t) - [[lambda].sub.p][[gradient].sup.2]T -

[r.sub.p][DELTA][H.sub.t] = 0 (5)

where [[rho].sub.p], [c.sub.p.sup.p] and [[lambda].sub.p] are EPDM

density, specific heat capacity, and heat conductivity, respectively;

[DELTA][H.sub.r](KJ/mol mol) is rubber specific heat (enthalpy) of

reaction and [r.sub.p] [mol/([m.sup.3] sec)] is the rate of

crosslinking.

When heat transmission at the external boundary is due to

convection and radiation (extrusion process), the following boundary

conditions must be applied in combination with the field problem:

[[lambda].sub.p][partial derivative]T(P, t)/[partial

derivative]n(P) + h(T(P, t) - [T.sub.n]) + [q.sub.rad] = 0 (7)

where h is the heat transfer coefficient between EPDM and

vulcanizing agent at fixed temperature T, [T.sub.n] is vulcanizing agent

(e.g., nitrogen) temperature, P is a point on the object surface, n is

the outward unitary vector on P. and [q.sub.rad] is the heat flux

transferred by radiation. Radiation contribution for the vulcanization

of complex 2D geometries may not be determined precisely, but the

following formula may be utilized without excessive inaccuracies:

[q.sub.rad] = [sigma]([T.sub.n.sup.4] - T[([R.sub.p],

t).sup.4])/[1/[[epsilon].sub.p] +

[A.sub.p]/[A.sub.n](1/[[epsilon].sub.n] - 1)]

where [sigma] = 5.67 x [10.sup.8] W/([m.sup.2][K.sup.4]) is the

Stefan--Boltz-mann constant, s are emissivity coefficients, [A.sub.p,n]

are the areas of heat exchange (p: rubber item; n: curing agent).

The same considerations hold for the cooling phase, with the only

differences that (a) the actual cooling agent temperature has to be used

and (b) that heat exchange for radiation is negligible, i.e.:

[[lambda].sub.p][partial derivative]T(P, t)/[partial

derivative]n(P) + [h.sub.w](T(P, t) - [T.sub.w]) = 0 (9)

where [h.sub.w], is the water (air) heat transfer coefficient,

[T.sub.w] is the water (air) cooling temperature, and all the other

symbols have been already introduced.

Initial conditions on temperatures at each point at the beginning

of the curing process are identically equal to the room temperature.

Initial conditions at the beginning of the cooling phase are obtained

from the temperature profiles evaluated at the last step of the cooling

zone, i.e., at T(P,[alpha][t.sub.c]), where ate is the heating interval,

and P is a generic point belonging to f2. To solve partial differential

equations system (6)-(9), standard elements (46), (47), (52) are used.

The procedure has been completely implemented in Matlab (44) language

and interfaced with the first phase of the procedure. In this way,

resultant FEM temperature profiles at each time step are directly

collected from the numerical analysis and utilized for the evaluation of

output rubber mechanical properties by means of an integrated tool.

For the discretization of the weather-strip, which is schematically

represented by its cross-section, 4-and 3-noded 2D elements are

utilized. Temperature field interpolation is assumed linear inside each

element, i.e., = [N.sup.e][T.sup.e], where, for 4-noded elements,

[T.sup.e] = [[[T.sup.1], [T.sup.2], ... [T.sup.4]].sup.T] is the vector

of nodal temperatures, [N.sup.e] = [[N.sup.1], [N.sup.2], ... [N.sup.4]]

is the vector of so-called shape functions [N.sup.i]( i = 1, ... 4) and

P is a point of coordinates [x.sub.p], [y.sub.p], and [z.sub.p] inside

the element. Following consolidated literature in this field, e.g., see

Ref. (55), in the numerical simulations reported, the following

parameters have been used: EPM/EPDM density [[rho].sub.p] = 922

kg/[m.sup.3], rubber specific heat capacity [c.sub.p.sup.p], = 2700

J/(kg [degrees]C), [[lambda].sub.p] = 0.335 W/(m [degrees]C),

[DELTA][H.sub.r] = 180 kJ/mol, water heat transfer coefficient [h.sub.w]

= 1490.70 W/([m.sup.2] [degrees]C), curing agent heat transfer

coefficient h = 900 W/([m.sup.2] [degrees]C) (for air heat transfer

coefficient we assume h = 5 W/([m.sup.2] [degrees]C), [[epsilon].sub.p]

= 0.60, [[epsilon].sub.c] = 0.70, water cooling temperature [T.sub.w] =

25[degrees]C.

The FEM discretization and the geometry of weatherstrip considered

are sketched in Fig. 6 (top subfigure). In the same Fig. 6 (bottom

subfigure), temperature contours obtained during a FEM simulation at

increasing instants are represented (nitrogen temperature [T.sub.n] =

160[degrees]C, curing interval [alpha][t.sub.c], = 300 sec). As it is

possible to notice, internal points heating is sensibly slower with

respect to the skin, as also schematically represented in Fig. 7a, where

temperature profiles of three different points of the item are depicted.

This may influence the final quality of the vulcanized rubber. Points

are labeled as A, B, and C. B and C are near the external boundary of

the item. whereas A is an internal point. It is therefore expected that

B and C thermal behavior is similar and at the same time quite different

to point A behavior. Data adopted in the thermal simulations are in

agreement with literature indications (56), (57). For the sake of

simplicity, specific heat capacity [c.sub.p.sup.p] and conductivity

[[lambda].sub.p] are assumed constants in the simulations. In reality,

such parameters reasonably exhibit certain variability in the

temperature range inspected. In particular heat capacity [56] ranges

from 1700 J/(kg [degrees]C) at 80[degrees]C to 2900 J/(kg [degrees]C) at

200[degrees]C, with an almost linear behavior in such temperature range,

whereas conductivity decreases asymptotically from 0.355 to 0.335 W/(m

[degrees]C). In Fig. 7b and c, part of the huge amount of results

obtained from a sensitivity analysis conducted varying material heat

capacity and conductivity is summarized. In particular, temperature

profiles for points A (b) and B (c) are represented, assuming three

different constant values for the heat capacity ([c.sub.p.sup.p], equal

to 1700, 2200, and 2700 J/ (kg [degrees]C), respectively) at fixed

conductivity [0.335 W/(m [degrees]C)], a linear behavior of

[c.sub.p.sup.p] with either [[lambda].sub.p] constant or 'P

asymptotically variable following (58). As it is possible to notice, the

variability of the temperature profiles is quite small and in any case

the error introduced assuming [c.sub.p.sup.p] equal to 2700 J/(kg

[degrees]C) is in the most unfavorable case within 10%, fully acceptable

from an engineering standpoint. The greatest variability is obviously

experienced for internal points, whereas for points near the core,

temperature profiles almost coincide. Finally, authors experienced very

little variability with [[lambda].sub.p].

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

The thermal behavior during vulcanization of the core and the skin,

i.e., points A and B, subjected to the vulcanization conditions

previously discussed and also with a different [T.sub.n], =

200[degrees]C, are represented from Figs. 8-11. The relationship used to

evaluate the final tensile strength of the point is depicted in Fig. 5a,

which is a reasonable hypothesis of the blend behavior under mechanical

tests. As already pointed out, there is very little or absent reversion

even at high temperatures when a Dutral TER 4049 70% and Dutral TER 9046

30% mix is utilized. In particular, the temperature profile of point A

for both [T.sub.n] = 160[degrees]C and [T.sub.n] = 200[degrees]C, Fig. 8

(2nd and 4th row, 2nd column sub figures), shows a gradual initial

temperature increase, meaning that Point A passes very slowly from

rheometric curves at low temperatures to rheometric curves at high

temperatures. The vulcanization phase stops at 300 sec and in both cases

([T.sub.n] = 160[degrees]C and 200[degrees]C) the point is not able to

reach the maximum possible temperature, i.e., external nitrogen

temperature. This was expected, because A is positioned in the core of

the item. Such curves are numerically interpolated, as discussed

previously, from experimental data using the kinetic model proposed, see

Fig. 9, once that constants [K.sub.1], [K.sub.2], and [[~.K].sup.2] are

at disposal. When the maximum temperature is reached, point A follows

the rheometric curve associated to such temperature, stabilizing its

behavior at increasing times on that rheometer curve. In other words,

point A follows almost completely the rheometer curve at around

150[degrees]C and 190[degrees]C, respectively. When curing finishes,

temperature decreases quickly. Here, two paths should be followed

numerically for point A, represented in Fig. 9 with circles and crosses.

The numerical behavior represented by crosses is not real, because it

would be referred to a material without any memory of the vulcanization.

Therefore, it relies on a material that ideally jumps from a rheometer

curve at high temperature to the successive at low temperature

(reversible process). On the contrary, the real behavior of the material

under consideration is represented by the circles, indicating that the

level of vulcanization reached at the end of the curing process cannot

be inverted reducing temperature, but can only be stopped.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

Once that the curing time-torque diagram of the rheo-metric curve

is known numerically (2nd and 4th row, 1st column diagrams) for the

point under consideration, tensile strength reached increasing exposure

times can be easily determined through the relation sketched in Fig. 5.

For point A, the exact vulcanization history in terms of curing

time-[[sigma].sub.t], and temperature-[[sigma].sub.t] diagrams is

represented in Fig. 8 (1st and 3rd row, 1st sub-figures). The same

considerations can be repeated for points B (or similarly C) of the

item, thus giving the possibility to evaluate the average final tensile

strength. In particular, in Figs. 10 and 11 the curing behavior of point

B is represented, whereas point C data are omitted for the sake of

brevity. From a comparative analysis, it is rather evident that points B

and C reach maximum vulcanization temperature more quickly with respect

to point A. since the cure curves followed by such points are very near

with each other. Also in this case, the reversion range is very little

even at 160[degrees] and, as expected, it occurs after that the peak

strength has been reached.

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

From a detailed analysis of points A, B, and C behavior, it can be

argued that at 160[degrees] the item is slightly undervulcanized,

because the thickness of the core would require either a higher

vulcanization external temperature or a longer exposure time. The

technical usefulness of the approach proposed seems rather clear, since

manufacturers could calibrate (with a few numerical attempts) both

curing time and vulcanization temperature of the production phase, to

avoid reversion and under-vulcanization, thus minimizing the industrial

process cost.

Mechanical Compression Test on the Weather-Strip

After a detailed numerical analysis of the industrial vulcanization

process, a FE mechanical analysis of weather-strip installed in the body

of a door (civil engineering application) is finally conducted using

Strand7 commercial software package. The mechanical simulations

reproduce a typical experimental compression test, where a part of the

boundary (shown in Fig. 6) is supposed clamped and a rigid punch

compresses the opposed boundary. Displacements in the x- and

y-directions are restricted only in such region, to simulate the actual

constraints acting where the weather-strip is installed in a door.

Contact elements between the punch (Fig. 6) and the rubber material are

utilized to simulate mono-lateral contact. Zero gap elements available

in Strand 7 are used to model contact.

A two constants Mooney--Rivlin model under large deformation

hypothesis is used for rubber, with parameters [C.sub.10] and

[C.sub.20], respectively equal to 0.270 and 0.160 MPa. Such constants

are suitably chosen in order to fit as close as possible, the

experimental uniaxial stretch--strain curve available for the blend

used. Here it is worth noting that, from the vulcanization model

proposed in the previous section, only the initial Young Modulus is

indirectly known, once that the average vulcanization level is evaluated

and hence the final hardness S is at disposal. For a vulcanization

executed at [T.sub.n], = 185[degrees]C and [t.sub.e] = 600 sec a very

good vulcanization level with average hardness equal to 56 is obtained.

The corresponding Young modulus is evaluated by means of BS 903 as 2.58

MPa. The elastic shear modulus is always E/3 and is linked with

Mooney--Rivlin constants by means of the following formula: G =

2([C.sub.10] + [C.sub.20]). A further condition is necessary to set

univocally [C.sub.10] and [C.sub.20], meaning that the uniaxial

stretch--stress curve test is needed. In absence of specific

experimental data available, from authors experience, the typical

uniaxial behavior of the compound is well approximated, at least for

stretches lower than 3, assuming [C.sub.20] roughly equal to 0.6

[C.sub.10].

In such a situation, defined the stretch as the ration between the

length in the deformed configuration divided by the length in the

undeformed state, let [[lambda].sub.1] = [lambda] be the stretch ratio

in the direction of elongation and [[sigma].sub.l] = [sigma] the

corresponding stress. In uniaxial stretching, the other two principal

stresses are zero, since no lateral forces are applied ([[sigma].sub.2]

= [[sigma].sub.3] = 0). For constancy of volume, the incompressibility

condition [[lambda].sub.1] [[lambda].sub.2] [[lambda].sub.3] = 1 gives:

[[lambda].sub.2] = [[lambda].sub.3] = 1/[square root of [lambda]].

(10)

In the most general case, the strain energy function for a

two-constant Mooney--Riv lin model is given by:

W = [C.sub.1]([I.sub.1] - 3) + [C.sub.2]([I.sub.2] - 3) (6)

where [I.sub.1.sup.2] + [[[lambda].sub.2.sup.2] +

[[lambda].sub.2.sup.3] and [t.sub.2] = [[lambda].sub.1.sup.-2] +

[[[lambda]sub.2.sup.-2] + [[[lambda].sub.3.sup.-2].

The FE model, as mentioned earlier, is constituted by

two-dimensional elements, which are supposed here to undergo a plane

strain mechanical condition under large displacement and contact

elements. Quadrilateral 4-noded elements with bubble-shaped

extra-functions and triangular linear elements are utilized to mesh the

rubber weather-strip. To speed up computations, only half of the device

is meshed for symmetry, applying suitable boundary conditions on

displacements on the symmetry axis. The compression device is assumed to

be a rigid body, since it is supposed to exhibit very little deformation

when compared to rubber. The compression device is loaded by means of an

increasing horizontal displacement up to 4 mm from its unloaded

position. Displacement sub-steps equal to 0.2 mm are imposed on the FE

model to facilitate convergence.

Contact forces distributions at successive time-steps are

represented in Fig. 12a. It is particularly evident that the compressed

contact zone, as expected, spreads at successively increased

displacements of the compression device. A direct integration of contact

forces allows the evaluation of the force to apply to the compression

device at successive displacements, as represented in Fig. 12b. The

loading condition obviously justifies the increase of first derivative

of the load at increased displacements, since the contact surface

becomes larger. Figure 12b shows the overall behavior is very important

for producers, because it indicates the overall expected stiffness of

the device under design loads. Such stiffness should fall within some

lower and upper bounds to be acceptable. The global behavior of the

device depends not only on the rubber compound used but also on the

vulcanization process used to cure it, since elastic properties are

Shore A dependent. It is therefore crucial to have at disposal of an

integrated thermo--chemo--me-chanical software as that used here to

predict such behavior.

For the sake of completeness in Fig. 12b compression curves

obtained assuming a hardness S equal to 65 and 45, respectively are

represented. For the blend and the specific item under consideration,

such hardness values are obtained respectively when the following

vulcanization conditions are assumed: [T.sub.n] = 195[degrees]C with

[t.sub.e] = 500 and [T.sub.n] = 175[degrees]C with [t.sub.e] = 650. The

same hypotheses assumed previously are adopted for Mooney--Rivlin

constants [C.sub.10] and [C.sub.20]. From the sensitivity analysis

conducted, it is very straightforward to conclude that the evaluation of

the overall stiffness of the item in presence of different vulcanization

conditions is crucial, especially when the item is installed as

weather-strip in a real door. As a matter of fact, from an engineering

point of view, the knowledge of such stiffness may be very useful,

especially when a maximum displacement threshold must be respected, as

for instance for large windows subjected to wind.

[FIGURE 12 OMITTED]

CONCLUSIONS

In the present article, the possibility to utilize, in

weather-strip for industrial production processes, two different

ter-polymers has been investigated by means of a comprehensive numerical

model. The first ter-polymer has a low amount of ENB, whereas the second

exhibits the maximum amount of ENB nowadays possible in EPDM production

plants. For both polymers. experimental rheometer curves were at

disposal at two different temperatures. The behavior of a mix of both

polymers, 70-30% in weight, has also been investigated numerically. In

particular, rheometer curves of the mix at two different temperatures

have been numerically extrapolated from components experimental data

available, taking into account their presence in the mix in terms of

relative percentage in weight. The vulcanization behavior of the mix has

been then numerically studied within the production line of a real

weather-strip. Generally, weather-strips are produced from ter-polymers

with an ENB content equal to around 6. but this custom is almost always

empirical. In this framework, this article gives a numerical insight

into the most suitable blend to be used in the production of such kind

of items.

From a production point of view, it is well known that a huge

amount of different EPM--EPDM commercial products are available.

Differences are mainly due to the ENB content, the distribution of

molecular weights, and the Mooney viscosity. The choices to mix two

different products with low and high ENB content, respectively, appears

interesting, since blends with an intermediate behavior could

potentially increase the final mechanical quality of the vulcanized

item, as well as reduce production time and hence costs. From the

producer point of view, costs reduction is derived not only from the

reduction of materials to stock, but also from the possibility to

modulate time and temperature used to obtain an optimal vulcanization of

the final item. It has been shown, indeed, that a ter-polymer with low

ENB content exhibits a rheo-metric curve monotonically increasing, but

with low rate of vulcanization. Conversely, a ter-polymer with high ENB

content shows a very good vulcanization rate but exhibiting also

reversion, especially at high temperatures. The present article shows

how an intermediate blend, obtained ad hoc mixing the aforementioned

products, may have both good vulcanization rate (intermediate between

the products with low and high ENB content) and no or very little

reversion. Furthermore, it has to be emphasized that the possibility to

mix ter-polymers with different molecular weights and Mooney viscosity

higher than 60, should modify the molecular weight distribution of the

mix, with the important advantage to obtain an ENB uniform distribution

in the back bone, which should allow a higher crosslinking homogeneity.

From the numerical model proposed, it seems that, at least

theoretically, it is possible to obtain mixtures whose vulcanization can

be tuned as a function for industrial production requirements, with an

obvious increase of the flexibility. A comparison with experimental data

on the production line, unavailable at this stage, will be obviously

rather useful to have an insight into the actual predictive capabilities

of the comprehensive numerical approach proposed.

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G. Milani, (1) F. Milani (2)

(1.) Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133

Milano, Italy

(2.) CHEM.00 Consultant, Via J.F. Kennedy 2, 45030 Occhiobello,

Rovigo, Italy

Correspondence to: G. Milani; e-mail: gabriele.milani@polimi.it

Published online in Wiley Online Library .(wileyonlinelibrary.com).

[c] 2012 Society of Plastics Engineers

DOI 10.1002/pen .23270