The worldwide acceptance and use of polyurethane foams for insulation applications focuses competition and innovation toward making the best product at the best price. As research organizations mobilize to improve the use of insulation materials, it is natural to focus on those aspects that can exhibit the most improvement. Analyzing the performance of an insulation foam means quantifying the physical contributions of each structural and chemical feature for its influence on thermal conduction. Thermal conductivity or the "^"-factor" is the proportionality constant between the heat flow per unit area (q) and the thermal gradient through the substance in question (Eq. 8.3) .
The lower the thermal conductivity of a substance (in our case a polyurethane foam), the lower the heat flow rate per unit area across the sample for a given temperature gradient, and the better its performance as an insulation material. Commercial insulation materials are normally identified by their "R" value which is product specific based on its thickness L, and is related to q by Equation 8.4 :
The units of A" in the SI system are (watts/(meterx Kelvin)) and also often expressed in terms of imperial units ((BTUxin.)/(hxft2x °F)) where a British thermal unit (BTU) is the energy to raise a pound of water by 1 °F or about 1055 J.
When a polyurethane foam is applied as an insulation, it displaces air and substitutes the gas entrapped in the foam (95% of the substitution by volume) and the polyurethane material. The foam walls and windows serve to scatter and impede radiative conduction of heat. At the same time, the polyurethane acts as a conduction path for heat transport through the material. Not counting the influence of one path on another (an interaction term) the ^-factor for a polyurethane foam is the sum of three conduction terms (Eq. 8.5):
Where K ,, is the solid conduction component of the foam ^"-factor, K is the gas solid 1 7 gas ° conduction component of the foam ^"-factor, and K , , is the radiative component 1 7 radiative 1 of the foam ^"-factor [53-55].
Given the very low densities of rigid foams (typically >95% pore volume), the solid component is usually ignored in a first approximation but can account for about 10% of the total heat transfer across a foam . The radiative component is that associated with temperature difference and not gradient. It is commonly associated with black-body emission and is thus a function of foam thickness and density, but not the material itself. This factor can account for about 15% of the heat transfer across a foam . The remaining heat conduction is associated with that due to transfer through the gas phase. Consequently, both the solid and radiative components are minor contributors to the overall thermal foam conduction and even significant improvements in their ^"-factor contribution would be minor in overall effect. Accordingly, the gas conduction component has become the focus of almost all attempts to improve the insulation properties of polyurethane rigid foams [58-60].
A simple first-order expression for the thermal conductivity of gases is obtained from kinetic molecular theory (Eq. 8.6) :
Where T is absolute temperature, mo is the molecular weight of the gas molecules, b is the molecular diameter, / is the number of degrees of freedom per molecule, iVo is Avogadro's number (6.023 xlO23 mol-1) and kB is Boltzmann's constant (1.38xlO_23J/K). This equation predicts that the thermal conductivity of gases is independent of pressure, proportional to the square root of the system temperature, and has inverse proportionality to molecular mass. This equation is confirmed by experiment at ambient temperatures and pressures . Refinements of Equation 8.6 have been developed by taking into account the potential for energy transfer between molecular vibration or rotational modes into translational energy . Equation 8.6 can be expressed in terms of gas viscosity v, density p, pressure P, molecular speed u, mean free path lo, specific heat co, and the gas constant R (8.314J/(molK) as Equation 8.7.
Terms in Equation 8.7 can be calculated using the relationships in Equation 8.8.
For air the mean free path (in micrometers) is 2.24xl0^t7TX)/P(atm) or about 0.07 |jm at room temperature.
Rigid foam designers have employed gas mixtures as a means to improve overall insulation cost and performance. The thermal conductivity (^-factor) of a mixture of gases at normal temperatures and pressures has been described by the Riblett equation (8.9) . Variations and refinements have been proposed to take account of observed deviations from Equation 8.9.
While mixing gases has been one route to improve the gas conductivity of polyurethane insulation foam, another route has been to take advantage of the Knudsen effect [64-67]. The Knudsen effect takes into account the fact that thermal transport by the gas phase assumes that that the mean free path between molecular collisions is negligible compared with a foam cell size. The calculated mean free path for air calculated above is about 70 nm. Compared to the typical foam cell size (on the scale of a millimeter), it is safe to assume that the temperature of the insulation gas and the neighboring foam wall are the same. The Knudsen number (Eq. 8.10) is a dimension-less number that characterizes the pressure dependence of the thermal conductivity of a gas where L is the length over which energy and momentum are transported—in this case the foam cell size.
The typical case as outlined above is the case where the Knudsen number is very small. However, in the case of very low pressure or very small L (cell size) the pressure and corresponding temperature of a cell gas at any point will be more nearly the same since direct (or specular) and diffuse reflection of gas molecules off of the cell wall surface will be effectively isotropic (or more so) and no surface will be significantly hotter than any other. Thus, at very small foam cell size, less than 100 nm, the foam is effectively acting as if the gas has a lower ratio of heat transport to temperature gradient than would be expected. In the limit of this phenomenon, the insulation foam with a very large Knudsen number would appear to be a foam with evacuated cells and the K of Equation 8.7 would tend toward zero.
As will be discussed Section 8.5, this phenomenon has been a research path numerous academic and industrial laboratories have pursued to most influence insulation technology. Some exotic technologies such as aerogels and xerogels [68-70] have provided very small pore sizes, but generally have relatively poor structural integrity and high cost. Obtaining very small pore sizes in conventional insulation materials such as polyurethanes have proven difficult without sacrificing low foam density that then causes an undesirable increase in the ^"-factor contribution from solid conduction.