Thermal diffusivity.

Thermal diffusivity Heating and cooling are essential processes in the forming of most rubber articles. in the curing of rubber there are two main requirements: for efficiency the heating and cooling cycle must be kept as short as possible, but for uniformity in the finished products the temperature gradients in the article must be minimized. These are conflicting requirements. A compromise is often obtained on a trail and error basis leading to crude rule of thumb methods, or by inspired guesswork. The result of this may be waste of energy, inefficient use of machine time and the optimum properties of the compound may not be achieved.

Conflict between economic and quality considerations in the curing of relatively thick products is not a recent development. A paper (ref. 1) written 60 years ago referred to the modern tendency towards short cures with highly accelerated stocks often being limited in application by the slowness with which the heat penetrates the rubber. This is readily understandable in the light of published data on the temperature-time relationships at various distances in from the surface of a block of rubber during heating in a press. An example is given in figure 1: it shows, for instance, that whereas the middle of a 5 mm thick molding would reach approximately the press temperature of 140[degrees]C in rather less than 10 minutes, this period of heating would leave the center of a 5 cm thick molding only slightly warm, over three hours being needed to raise it to approximately 140[degrees]C.

In practice the position is not quite so bad as appears from such curves as those in figure 1. The thicker the cured article, the slower will its inside cool after the article has been removed from the curing environment. Thus if the center of a thick article has achieved sufficient cure to avoid porosity when the pressure is released it can be removed from the press and will continue to cure during the cooling part of the cycle.

The temperature lag effect has long been recognized as significant in the curing of thick articles, but with higher temperature cures and shorter cure times it becomes increasingly important in the curing of thin articles. Temperature measurement by thermocouples in 2 mm sheets heated between press platens at 260[degrees]C indicated that the centers of the sheets did not approach the platen temperature until about 30 seconds for a NR carcass compound or about 60 seconds for an IIR tread compound (ref. 2). The effect of this on cure distribution has been shown by Atkin and Nye (ref. 3); they pointed out that with curing temperatures in the region of 200[degrees]C even a filament only 0.8 mm in diameter can have a difference in the degree of cure between the inner and outer layers of a factor of 2.

So we have the situation that in any products more than about 5 mm thick, and even in much thinner ones cured at high temperatures, the temperature lag effect during the cure may result in large variations in the level of cure-sensitive properties between outer and inner layers. If the surface is close to its optimum cure state the center may well be badly undercured, causing the article to be, for example, too prone to heat up under rapid cyclic deformation. If the center is adequately cured, the surface may be badly overcured, resulting, for example, in reduced aging resistance.

Rule of thumb procedures, such as increasing cure time five minutes for every 1/4" thickness (beyond the first) to the center of a molding may well give an acceptable compromise in some instances, but if applied generally, they necessarily tend to be rather crude generalization, making no allowance for differences due to shape, composition or level of curing temperature. To calculate the best compromise the temperature distribution in the article during heating and cooling must be calculated, and this depends on a knowledge of the thermal diffusivity at the relevant temperatures.

General Theory

Equation of conduction of heat

For the flow of heat in one direction the heat flux J[sub.u] is related to the temperature gradient *[theta]/*x by Fourier's law

J[sub.u] = -K *[theta]/*x (1)

where K is the thermal conductivity. the minus sign indicates that the heat flows in the opposite direction to the temperature gradient. The form of equation (1) implies that heat conduction is a random process. If the energy were propagated without scattering then the heat flow would depend on the temperature difference between the end faces of the specimen instead of the temperature gradient (ref. 4).

The general equation from which the time dependent temperature distribution may be calculated is obtained from equation (1) and the equation of continuity

*J/*x[sub.u] = [rho]c *[theta]/*t (2)

where [rho] is the density, c is the specific heat, i.e. the heat capacity per unit mass, and t is the time. The equation of continuity is an expression of the conservation of energy.

The heat flux can be eliminated between equations (1) and (2) to give

[Mathematical Expression Omitted]

where [alpha] = K/[rho]c is the thermal diffusivity. Equation (3) is the equation of conduction of heat, in the absence of heat generation and convention, for heat flow in one direction. If the conductivity is independent of temperature it reduces to

*[sup.2][theta]/*x[sup.2] + 1 *[theta]/[alpha]*t (4)

which is the equation usually referred to. It has been shown that for rubbers, and hence plastics, except at melting transitions, the conductivity term in (3) is very small and equation (4) is adequate for most heat flow calculations (ref.5).

The parameter [alpha] was called the thermal diffusivity by Kelvin and the thermometric conductivity by Maxwell, but Kelvin's expression has been generally adopted. It measures the change in temperature which would be produced in unit volume of the substance by the quantity of heat which flows in unit time across unit area of a layer of the substance of unit thickness with unit temperature difference between its faces (re. 6). It is the parameter which determines the non-steady-state temperature distribution in the absence of heat generation and convection and is therefore essential for transient heat transfer calculations.

Fundamental significance of diffusivity 

Since [alpha] = K/[rho]c, diffusivity is often regarded as just a mathematical parameter rather than a material property. However, diffusivity is seen to be a fundamental material property if we think wholly in terms of energy.

The heat capacity per unit volume is given by c = *u/*[theta][sub.v[sup.1]] where u is the internal energy per unit volume. Obtaining from this an expression for the temperature gradient and substituting it into equation (1) gives

J[sub.u] = -[alpha] *u/*x (5)

Thus thermal diffusivity is the parameter relating energy flux to energy gradient, whereas conductivity relates the energy flux to the temperature gradient. The origin of the units mm[sup.2]/s, which are perhaps meaningless in themselves, now becomes apparent.

Thermal diffusivity measurement

To measure diffusivity it is only necessary to know the change in temperature with time at three co-linear points in the direction of the heat flow. For conductivity, although the temperature gradient can easily be obtained, the heat flux cannot be measured at a point; the total power to the heat source is known but it is difficult to control or calculate the movement of all the energy. Although diffusivity is easier to measure, and although it is the essential parameter for transient heat flow calculations, it has received less attention than conductivity. The scatter in the diffusivity data in the literature is not as great as for conductivity (possibly because not as many people have tried to measure it).

Basic principles

For certain boundary and initial conditions analytical solutions to equation (4) can be obtained. The experimental conditions are matched to these mathematical conditions as closely as possible and the appropriate solution is used to give a value for the diffusivity. The experiment can be repeated at different temperatures in order to obtain the temperature dependence of the diffusivity.

Diffusivity measurement methods based on analytical solutions to (4) have all had the same initial condition that the whole sample is at a constant uniform temperature. But three different types of boundary conditions have been employed: first the sample surface is subjected to a step change in temperature; second the surface is subjected to a linear rate of temperature rise; and third the surface is subjected to a periodic temperature fluctuation.

Sources of error

In the experimental techniques, which will be described below, there are a number of recognized sources of error. These have not always been avoided and although the measurement is basically straightforward the results in the literature are not always reliable. The significance of the errors will be discussed in connection with the methods of measurement and only a summary is given here:

* assuming an infinite surface heat transfer coefficient between the sample and the heating or cooling medium;

* ignoring lateral heat flow when it is significant;

* conduction along thermocouple leads;

* assuming that diffusivity is temperature independent;

* ignoring the thermal expansion of the samples.

Quenching methods

Thermal diffusivity has usually been measured using a quenching method, i.e. the solid sample at a uniform temperature is immersed in a temperature controlled bath at a different temperature. The rate of change of temperature at the center is then monitored with an embedded thermocouple. The sample dimensions are usually chosen so that lateral heat flow can be ignored and regular sample geometries, i.e. 'infinite' flat slabs, 'infinite' cylinders, or spheres, are used.

Quenching methods are based on the assumption that the thermal diffusivity is constant over the experimental temperature range. To obtain the temperature dependence of the diffusivity, the temperature range of interest is covered in a series of steps which are small enough for this assumption to be valid. The size of the step temperature change is a compromise between this requirement and the need to make accurate temperature difference measurements. The value which is usually chosen is about 5[degrees]C. In some experiments this point has been ignored and the measurement made with a much larger step. For example, the sample may be conditioned at room temperature and then immersed in boiling water. An average value for the diffusivity is obtained and the error depends on the temperature dependence of the diffusivity and hence changes from material to material.

Two methods have been used to obtain a value for the diffusivity from the experimental results. As was shown above, for the geometries in question the analytical solutions to the heat equation, which are in the form of infinite series, can be conveniently represented graphically. The first such collection of graphs was published by Williamson and Adams (ref. 7). By reference to such graphs a value for [alpha] can be obtained.

The second method is based on the rapid convergence of the series solutions. For example, after about 10% of the time for the center temperature to reach 99% of the surface temperature has elapsed, the second term in the series is about 1/2% of the first term. Thus after a certain time, only the first term is relevant. Now temperrature is always an exponential function of time; for example consider an infinite slab of thickness 2a, the term of the series for the dimensionless temperature at the center is given by

Y = 4/[pi] exp -[pi][sup.2]/4 [Phi] (6)

For long times this is the only significant term and the diffusivity can be obtained from the slope of a graph of log[sub.e] Y against t since [Phi] = [alpha]t/a[sup.2].

Linear heating method

The second type of boundary condition to be considered is that in which the sample surface is subjected to a linear rate of temperature rise. A method based on this has been developed by Shoulberg (ref. 8) for diffusivity measurements on polymer melts. He used two discs of his material with a thermocouple sandwiched between them; the diameter to thickness ratio was such that his sample could not be regarded as an infinite flat slab. The sample completely filled a cavity in an aluminium block and was melted in the apparatus. The aluminium block was heated electrically and the power was adjusted to give an approximately linear rate of temperature rise. Under his experimental conditions this lasted for about 30[degrees]C.

For this boundary condition the solution to the heat equation for the temperature at the center is (ref. 9)

[Mathematical Expression Omitted]

where k is the linear rater of temperature rise. After a long time (in Shoulberg's case about 12 minutes) the summation term is negligible and the temperature difference between the surface and the center becomes constant. The diffusivity is then given by

[alpha] = ka[sup.2]/2[delta][theta] (8)

where [delta][theta] is the temperature difference between the surface and the center of the sample, and a is the semi-thickness of the sample, i.e. the thickness of one of the discs. The temperature range of interest was covered in a series of 30[degrees]C steps, and using equation (8) he obtained an average value for the diffusivity for each step.

Periodic heating method

A method for measuring the diffusivity of solid polymers based on this type of boundary condition has been developed by Berlot (refs. 10 and 11). A disc sample of thickness 2a is held at uniform temperature and then a sinusoidal temperature fluctuation of angular frequency [omega] is imposed on the outer surfaces.

The amplitude ratio and phase of the temperature at the center are monitored with a thermocouple. Under these conditions the amplitude ratio, A, and phase, [Phi], are given by (ref. 12)

[Mathematical Expression Omitted]

k can be obtained from (9) or (10) and hence [alpha] from (11).

Continuous heating method

If the temperature dependence of diffusivity is taken into account then the temperature range from ambient up to, say, 250[degree]C can be covered in one experiment. With a quenching method a true step change in temperature is difficult to achieve because liquids capable of withstanding high temperatures tend to have high viscosities and this results in large temperature gradients close to the sample surface. Another objection to a quenching method is that the predominance of the quenching temperature throughout the experiment coupled with the extremely large initial temperature difference could give rise to computational difficulties. These problems, inherent in the quenching method, do not occur in a continuous heating method.

A method based on this principle was developed by Hands and Horsfall (ref. 13) and has been further developed by Smith (ref. 14). The apparatus is shown schematically in figure 2. To measure thermal diffusivity two disc shaped samples, 48 mm diameter and 4 mm thick, are placed together and a thermocouple is sandwiched between them for monitoring the center temperature. The samples are contained in the sample holder which consists of a central ring, two o-ring seals and two end plates, each one of which contains a heater element. The sample holder is spring loaded to allow for thermal expansion of the samples. Two other thermocouples are sealed into grooves in the end plates; the beads are flush with the surfaces and thus in direct contact with the surfaces of the sample sandwich. The thermocouples are arranged so that the three junctions lie as closely as possible to the axis of symmetry.

Enclosing the samples allowed measurements to be made on molten samples and eliminated errors caused by ignoring thermal expansion. In quenching methods, for example, the appropriate sampel dimension is measured at room temperature and this is the value which is used in the calculations. Since diffusivity depends on the square of the thickness any error caused by ignoring expansion is automatically doubled. This errror be a few percent depending on the temperature range covered.

In use the sample holder is supported inside a cylindrical stainless steel radiation shield; the general arrangement can be seen in the diagram. Power is supplied to the heaters, and the temperature rises smoothly to a pre-selected value. The three thermocouple outputs are scanned at 20 second intervals and the data is recorded. At the end of the run diffusivity is calculated as a function of temperature from equation (4). Typical examples are shown in figure 3 (polypropylene) and figure 4 (EPDM).

Pulse Heathing Method

If a disc shaped sample is irradiated on one surface for a short time using a flash tube or a pulsed laser, then the temperature time curve for the back face depends on the thermal diffusivity of the sample and the heat losses. If the pulse time [tau] is very much shorter than the time for the pulse to pass through the sample, then the heat losses can be ignored. Cape and Lehman (ref. 15) have given a criterion for this condition [a.sup.2 / [pi].sup.2.[alpha] > 10[tau]] where a is the thickness of the disc. Under these conditions the ratio of the temperature to the maximum temperature for the back face is given by (ref.16)

[Mathematical Expression Omitted]

the temperature as a function of time on the back face is usually measured with a radiation pyrometer (ref. 17). Analysis of the results had been discussed by Stuckes (ref. 18).

Cure calculations

In order to be able to calculate the cure level at any point in an object and hence the cure profile through the thickness, three types of information are required. We need to measure or calculate the temperature history at the points of interest. We need to know the relationship between cure and time for the compound under isothermal conditions; and we need an accurate mathematical model for the curing process.

Thermal diffusivity is needed to calculate the temperature history. Simple methods of calculation are available for regular geometries, but with the ready availability of computers a wider range of geometries and boundary conditions can be handled.

In current industrial practice the basic cure parameters for rubber are determined from measurements made on cure meters. These instruments measure a property, let us call it stiffness, which is approximately proportional to the hot shear modulus. The sample has a stiffness before any crosslinking has occurred, and stiffness increases from this minimum value to a maximum, during the course of the curing reaction, giving the familiar S shaped curve. The cure time at a given temperature is the time to reach, say, 90% of this stiffness change. Some instruments approximate more closely to the isothermal assumption than others.

The cure rate at any instant in the curing reaction is a function of temperature and the instantaneous cure level. The usual approach is to assume separation of variables and to work in terms of an equivalent cure time at an arbitrary reference temperature. On these assumptions the cure level dependence can then be eliminated. This leaves the temperature dependence and two methods have been discussed in the literature for this; temperature coefficient and activation energy.

The temperature coefficient of cure is defined as the ratio of the cure times for a 10[degrees]C change in cure temperature, thus [t=toQ.sup.([theta]-[theta]o)/10] where Q is the temperature coefficient, t is the cure time at a temperature [theta], and to is the cure time at some other temperature [theta]o. Optimum cure for the compound is chosen with reference to the cure time curves from a cure meter. The precise definition of optimum cure depends on the type of compound and on the end use, but usually 90% of the maximum cure level is used. The temperature coefficient can be obtained from the slope of a graph of log cure time against temperature.

An alternative approach is to assume that the temperature dependence can be represented by an Arrhenius equation from which an activation energy can be calculated, t = A exp (E/RT) where A is a constant, E is the activation energy, R is the gas constant, and T is the cure temperature in degrees kelvin. The activation energy can be obtained from the slope of a graph of log t against the reciprocal of the absolute temperature.

Cure meters which are not isothermal show an apparent decrease in both temperature coefficient and activation energy with increasing temperature. Isothermal cure meters, on the other hand, show that for most compounds the temperature coefficient is a constant and activation energy increases with temperature. During the curing of rubber a number of reactions occur simultaneously and this cannot be described by a single activation energy. The Arrhenius equation is, incidentally, sometimes used to show that the temperature coefficient approach is inaccurate.

The main problem with the traditional methods of cure integration is that they ignore the dependence of the cure rate on the cure level. This has a large effect on the cure distribution, especially in big articles, and can vary considerably between different rubber compounds. For example, in the case where two materials have different scorch times but the same 90% crosslink cure time, it is obvious that the cure reaction is proceeding faster in the latter stages for the material with the longer scorch time. A calculation based on a single point from the cure-time curve will give the same answer for both rubbers, although in practice they will have different cure distributions.

We have recently developed a new mathematical model. We do not assume separation of variables and we start with an empirical representation for the whole of the S-shaped cure-time curves. Some of the parameters are then made temperature dependent. Hence a function of the form c = f([theta],t) is obtained. This can be differentiated to give dc / dt = f ([theta], t)

The time is eliminated between these two equations to give dc / dt = q ([theta],c) which can be used to calculate the cure level for any temperature history.

This new model includes the cure level dependence as well as the temperature dependence because it is based on the whole of the isothermal cure-time curve instead of just a single point. This model was initially checked with laboratory cured slabs 25 mm thick and has been successfully applied to a number of industrial problems (ref. 19 and 20).

References

[1] Sherwood, T.K., Ind. Eng. Chem. 20, 1181 (1928).

[2] Smith, F.B., Rubber World 142, 89 (1960).

[3] Atkin, W. and Nye, F.H., J. Inst. Rubb. Ind. 3, 16 (1969).

[4] Kittel, C., Introduction to solid state physics, 3rd Edn., Wiley (1966), p. 168.

[5] Hands, D. and Horsfall, F., Rubb. Chem. tech. 50, 253 (1977).

[6] Carslaw, H.S. and Jaeger, J.C., conduction of heat in solids, 2nd Edn. Oxford, 1959, p. 9.

[7] Williamson, E.D. and Adams, L.H., Physical Review 14, 99 (1919).

[8] Shoulberg, R.H., J. Appl. Poly. Sci. 7, 1597 (1963).

[9] Carslaw, H.S. and Jaeger, J.C., op. cit. p. 104.

[10] Berlot, R., Plast. Mod. Elast. 18, 231 (1966).

[11] Berlot, R., Plast. Mod. Elast. 19, 117 (1967).

[12] Carslaw, H.S. and Jaeger, J.C., op. cit. p. 105.

[13] Hands, D. and Horsfall, F., Rubb. Chem. Tech. 50, 253 (1977).

[14] Smith, D., Ph.D. thesis, University of Bradford (1987).

[15] Cape, J.A. and Lahman, G.W., J. Appl. Phys. 34, 1909 (1963).

[16] Parker, W.J., Jenkins, R.J., Butler, C.P. and Abbott, G.L., J. Appl. Phys. 45, 2321 (1961).

[17] Taylor, R., British J. Appl. Phys. 16, 509 (1965).

[18] Parrott, J.E. and Stuckes, A.D., Thermal conductivity of solids, Pion, (1975) p. 37.

[19] Hands, D. and Horsfall, F., Kaut u. Gummi Kunst. 33, 440 (1980).

[20] Hands, D. and Horsfall, F., Paper 11, 124th ACS Rubber Div., Oct (1983).