Chemical and Physical Blowing Agents in Structural Polyurethane Foams: Simulation and Characterization.
Physical blowing agents such as n-pentane and methyl formate Methyl formate, also called methyl methanoate, is the methyl ester of formic acid. It is a clear liquid with an ethereal odor, high vapor pressure and low surface tension. and, for comparison, chemical blowing agents such as water were used to prepare structural polyurethane rigid foams of different densities by reaction injection molding Reaction injection molding OR RIM Molding is similar to injection molding except that a reaction occurs within the mold. The process uses thermoset polymers (commonly polyurethane) instead of thermoplastic polymers used in standard injection molding. . Experimental runs were carried out with formulations based on oligomeric isocyanate i·so·cy·a·nate
Any of a family of nitrogenous chemicals that are used in industry and can cause respiratory disorders, especially asthma, if inhaled. and a mixture of polyether pol·y·e·ther
A polymer in which the repeating unit contains two carbon atoms linked by an oxygen atom. polyols. The constitutive equations In structural analysis, constitutive relations connect applied stresses or forces to strains or deformations. The constitutive relations for linear materials are linear, and termed Hooke's law. for the vaporization vaporization, change of a liquid or solid substance to a gas or vapor. There is fundamentally no difference between the terms gas and vapor, but gas is used commonly to describe a substance that appears in the gaseous state under standard conditions of rate of the two blowing agents and the polymerization polymerization
Any process in which monomers combine chemically to produce a polymer. The monomer molecules—which in the polymer usually number from at least 100 to many thousands—may or may not all be the same. kinetics kinetics: see dynamics.
Kinetics (classical mechanics)
That part of classical mechanics which deals with the relation between the motions of material bodies and the forces acting upon them. data are reported. Experimental results were compared with the prediction of a simplified theoretical model, and they showed a satisfactory agreement in terms of temperatures and density profile. All the specimens were characterized by physical-mechanical properties such as hardness, impact strength, flexural strength Flexural strength is also known as modulus of rupture, bend strength, or fracture strength. Flexural strength is measured in terms of stress, and thus is expressed in pascals (Pa) in the SI system. and elastic modulus elastic modulus
or elastic constant
In materials science and physical metallurgy, any of various numbers that quantify the response of a material to elastic or springy deflection. and the results were reported in function of the densities. The best mechanical performance were obtained with the physical blowing agents, due to a better density distribution profile and a thicker skin layer.
In recent years, the Years, The
the seven decades of Eleanor Pargiter’s life. [Br. Lit.: Benét, 1109]
See : Time production of polyurethane foams has expanded greatly owing to owing to
Because of; on account of: I couldn't attend, owing to illness.
owing to prep → debido a, por causa de their enormous and increasing employment in several industrial fields, i.e., car manufacture, furniture, refrigeration refrigeration, process for drawing heat from substances to lower their temperature, often for purposes of preservation. Refrigeration in its modern, portable form also depends on insulating materials that are thin yet effective. , etc. This success is due to the intrinsic capability of polyurethane systems to change physical-mechanical properties as soon as reactants or processes are varied.
The present work deals with structural polyurethane foams, polymers with low-density foamed cores and high-density exterior skin to be used mainly as wood substitutes. Owing to their peculiar "sandwich" structure, these foams reveal good mechanical properties, such as flexural strength, tensile modulus, and elongation elongation, in astronomy, the angular distance between two points in the sky as measured from a third point. The elongation of a planet is usually measured as the angular distance from the sun to the planet as measured from the earth. , which depend primarily on skin thickness and density, while their lightness is related with the foamed inner layer. A proper choice of blowing agents is the main target of present research; new and alternative blowing agents must be discovered in order to replace traditional and well-known CFCs and HCFCs HCFCs: see chlorofluorocarbons. , which have been banned because of their environmental impact. The aim of this paper is to compare the physical-mechanical properties of specimens prepared with different blowing agents, for the purpose of replacing the above-mentioned dangerous compounds. In our experimental tests, n-pentane and methyl formate are used as physical blowing agents, while, for comparison, water is employed as ch emical blowing agent. Several different structural foam specimens are prepared with the three above mentioned blowing agents, with the employment of standard processing equipment for Reaction Injection Molding (RIM). This equipment makes it possible to prevent the capture of air during the mixing process, a typical drawback of manual mixing techniques. A method for obtaining kinetic data will be developed from the experimental temperature values in order to identify which parameters are most effective for foam structure control, Thereafter, a theoretical model based on previous works (1-7) is presented to simulate the temperature and density profile inside the mold. Simulation data will be correlated with experimental results. Finally, previous specimens will be tested to determine the effective mechanical properties that may be compared at the end.
* Glendion RM 601: a polyether polyol with average functionality equal to 4.3, hydroxyl hydroxyl /hy·drox·yl/ (hi-drok´sil) the univalent radical OH.
The univalent radical or group OH, a characteristic component of bases, certain acids, phenols, alcohols, carboxylic number = 395 mg KOH/g, viscosity = 2700 mPa[cdotp]s at 25[degrees]C (Enichem S.p.A.).
* Tercarol 1: polyether polyol with average molecular weight equal to 1000, hydroxyl number = 168 mg KOH/g, viscosity = 280 mPa*s at 25[degrees]C (Enichem S.p.A.).
* MDI (1) (Multiple Document Interface) A Windows function that allows an application to display and lets the user work with more than one document at the same time. VM 2085: oligomeric isocyanate of 4-4' biphenyl biphenyl /bi·phen·yl/ (-fen´il) diphenyl.
polychlorinated biphenyl (PCB) any of a group of chlorinated derivatives of biphenyl, used as heat-transfer agents and electrical insulators; they are methane diisocyanate of average functionality = 2.85 and %w NCO NCO
NCO noncommissioned officer
NCO n abbr (Mil) (= noncommissioned officer) → Uffz. = 31 (I.C.I.).
* DMCEA: dimethylcycloexylamina catalyst (Abbot).
* Tegostab 8404: silicone surfactant Surfactant Definition
Surfactant is a complex naturally occurring substance made of six lipids (fats) and four proteins that is produced in the lungs. It can also be manufactured synthetically. (Goldsmidth AG).
* Blowing Agents:
-- water = a chemical blowing agent; the reaction with MDI leads to the formation of [CO.sub.2]:
-- R-NCO + [H.sub.2]O [right arrow] -R-NHCOOH [right arrow] -R-[NH.sub.2] + [CO.sub.2][up arrow]
-- n-pentane = a physical blowing agent, nowadays employed as a substitute for CFCs and HCFCs (with O.D.P. = 0) in the production of low density rigid foams
-- methyl formate = a physical blowing agent not yet used, although it is less inflammable in·flam·ma·ble
1. Easily ignited and capable of burning rapidly; flammable. See Usage Note at flammable.
2. Quickly or easily aroused to strong emotion; excitable. than n-pentane. It may combine with water to produce formic acid formic acid or methanoic acid (mĕth'ənō`ĭk), HCO2H, a colorless, corrosive liquid with a sharp odor; it boils at 100.7°C; and solidifies at 8.4°C;. , which reacts with isocyanate itself (8). Because of the scarce humidity of the system, we will omit these reactions.
The physical and chemical properties of these blowing agents are reported in Table 1 (8-10).
Formulations and Foam Preparation
Several different "sandwich" polyurethane specimens were prepared in a pilot plant. This laboratory system comprises two tanks, one for the polyols mixture (i.e. polyols, surfactant, catalyst and blowing agent) and one for the isocyanate. The temperature in the tanks is kept uniform and constant by means of a circulating fluid and by a stirrer which also prevents the formation of crusts. The appropriately dosed reactants are introduced into a high thermal capacity thermal capacity: see heat capacity. mold through a high speed mixing-head, which assures utmost homogeneity Homogeneity
The degree to which items are similar. . After preliminary thermostabilization at 28[degrees]C, the mold is filled with the mixture, closed and set under a press.
The foam formulations used in these experiments, where only the blowing agents are changed, are reported in Table 2. The amount of blowing agent is calculated in order to obtain density values in open molds equal to 0.1 g/[cm.sup.3], in accordance with industrial formulations.
After having prepared all the specimens, structural rigid foams are conditioned at 70[degrees]C for 24 hours Adv. 1. for 24 hours - without stopping; "she worked around the clock"
around the clock, round the clock before cutting them in order to test their physical-mechanical properties.
Of, relating to, or being a reversible thermodynamic process that occurs without gain or loss of heat and without a change in entropy. temperature rise is the method used to obtain kinetic data, i.e. the reaction rate, activation energy activation energy, in chemistry, minimum energy needed to cause a chemical reaction. A chemical reaction between two substances occurs only when an atom, ion, or molecule of one collides with an atom, ion, or molecule of the other. [E.sub.a], the frequency factor A, the heat of reaction [delta][H.sub.r], and the order of reaction n.
The temperature variation with time is measured by using copper-constantan thermocouple and it is recorded using a data acquisition system. The thermocouple is held in the middle of the mixture foaming into an open box (base: 25 X 25 cm, height 30 cm). The energy balance per unit polymer weight may be expressed as:
[c.sub.p] * dT/dt = (-[delta][H.sub.r]) dX/dt - U * (T - [T.sub.o]) * S + [h.sub.v] * dL/dt (1)
This equation takes into account the fact that the temperature increase is the result of exothermic exothermic /exo·ther·mic/ (-ther´mik) marked or accompanied by evolution of heat; liberating heat or energy.
ex·o·ther·mic or ex·o·ther·mal
1. heat generated by the chemical urethane urethane (yoor´ithān´),
n ethyl carbamate used as an anesthetic agent for laboratory animals, formerly used as a hypnotic in humans. bond formation reaction (also urea bonds if water is used), of heat dissipation Noun 1. heat dissipation - dissipation of heat
chilling, cooling, temperature reduction - the process of becoming cooler; a falling temperature through the mold and vaporization and condensation of solvents, if physical blowing agents are involved in the structural foam production process.
Evaporation evaporation, change of a liquid into vapor at any temperature below its boiling point. For example, water, when placed in a shallow open container exposed to air, gradually disappears, evaporating at a rate that depends on the amount of surface exposed, the humidity is assumed to start when the boiling point boiling point, temperature at which a substance changes its state from liquid to gas. A stricter definition of boiling point is the temperature at which the liquid and vapor (gas) phases of a substance can exist in equilibrium. of the blowing agent in the mixture is reached (cream temperature [T.sub.B]), and terminating at the gel point ([X.sub.gel]). Under this hypothesis, the following conditions may be described:
dL/dt = 0 when: T [less than] [T.sub.B] and X [greater than] [X.sub.gel]
In addition, when T [greater than] [T.sub.B]: dL/dT * dT/dt where dL/dT is the gradient of the experimental curve, obtained by relating the blowing agent mass fraction in the polyurethane mixture with the boiling temperature ([T.sub.B]) of the same mixture. This can be calculated by preparing different formulations with increasing initial amounts of physical blowing agents ([w.sub.B]) and measuring the initial boiling temperature (1). The experimental curves are shown in Fig. 1 and may be correlated with the following equation:
n-pentane: [w.sub.B] = 0.0080 + 14.668 exp exp
2. exponential (-T([degrees]C)/6.516) methyl formate:
[w.sub.B] = 0.0031 exp (35.751/(T([degrees]C) - 20.236))
The energy balance (Eq 1) for adiabatic conditions is reduced to:
[c.sub.p] . d[T.sub.ad]/dt = (-[delta][H.sub.r]) dX/dt + [h.sub.v] . dL/dt (2)
By integrating Eq 2, the resulting equation using the limits T = [T.sub.0] at t = 0 and T = [T.sub.ad] at a general time t, we obtain the adiabatic temperature profile with time from the experimental one.
[T.sub.ad] = [T.sub.exp] + [[[integral].sup.t].sub.0] U' . ([T.sub.exp] - [T.sub.0] . dt
where: U' = U . S/[c.sub.p] (3)
The heat of reaction (-[delta][H.sub.r]) is given by the following expression, where the evaporation of the blowing agent is also taken into account:
(-[delta][H.sub.r]) = [c.sub.p] . [delta][T.sub.ad] + [h.sub.v] . ([L.sub.0] - [L.sub.f] (4)
Ultimately, by using the general form for the reaction rate (Eq 5), from Eq 2 we obtain the final expression (Eq 6), which carry out all kinetic parameters ([E.sub.a], A, n) related to the three different systems prepared with water, n-pentane and methyl formate. The results are reported in Table 3.
dX/dt = k . [(1 - X).sup.n] (5)
lnk = ln ([c.sub.p] . d[T.sub.ad]/dt - [h.sub.v] . dL/dt/(-[delta][H.sub.r]) . [(1 - X).sup.n]) = lnA - [E.sub.a]/[RT.sub.ad] (6)
For these calculations, an average value of [c.sub.p] = 0.427 cal/g [degrees]C is assumed, in agreement with values given by the Modem Plastics Encyclopedia (11) for cast polyurethane materials.
The value of 'n' is chosen to be equal to 2, because when the left side of Eq 6 is plotted against 1/T, a fairly straight line is obtained (within a satisfactory temperature range) for all the blowing agents.
The value of conversion at gel point ([X.sub.gel]) is obtained experimentally. Figure 2 shows the variation of conversion X with time, which may be easily calculated by using the well-known general form of the kinetic constant (as a function of [E.sub.a] and A) or by integration of Eq 5, respectively, if the reaction order n is chosen equal to 2.
Experiments in Closed Molds
By using the same kinetic parameters of the previous open-mold tests, as an equal amount of blowing agent is employed and the reactants are unchanged, we may obtain the profile of kinetic constant k and conversion x in closed-mold experiments. At first, the temperature variation in the center of the mold is recorded by means of a copper-constant an thermocouple inserted through the side hole into the high capacity aluminum flat mold measuring 210 mm X 130 mm X 10 mm, and used for all the experimental studies reported. The mixture containing polyols, catalyst, surfactant and blowing agents is mixed with isocyanate (using the same pilot plant) and poured into the cavity of the mold, which is locked into position and set under a press.
The results are reported in Fig. 3, which shows the variation of conversion X with time for the three different blowing agents. In closed molds, the maximum value of X is about 20% lower than the X values obtained in open molds because of the choice of a lower initial temperature (28[degrees]C vs. 32[degrees]C) and a smaller amount of reactants poured in the mold.
The mathematical model
Of, relating to, or indicating equal or constant temperatures.
having the same temperature. and adiabatic boundary conditions. As known, a large amount of heat is generated in a short time during structural foam polyurethane reaction. Therefore, a temperature gradient temperature gradient
The rate of change of temperature with displacement in a given direction from a given reference point.
temperature gradient will exist inside the mold, with the temperature near the mold walls differing from the temperature in the foamed core, where the influence of heat conduction Heat conduction or thermal conduction is the spontaneous transfer of thermal energy through matter, from a region of higher temperature to a region of lower temperature, and hence acts to even out temperature differences. is not so strong. As a matter of fact, the difference of temperature between the inner and the exterior layer of foamed material is an essential element for the formation of the high skin density. As it seems to be convenient to define a differential material control volume of thickness dz, the heat conduction equation could be written as:
[rho] . [c.sub.p] . (dT/dt) =
[lambda] . [d.sup.2]T/d[z.sup.2] + (1 - [w.sub.B0]) . [rho] . [delta][H.sub.r] . dX/dt + [rho] . [h.sub.v] . dL/dt (7)
From this equation it is quite simple to realize that in any volume element, the temperature changes are the result not only of exothermic polymerization and blowing agent evaporation but also of heat conduction with neighboring neigh·bor
1. One who lives near or next to another.
2. A person, place, or thing adjacent to or located near another.
3. A fellow human.
4. Used as a form of familiar address.
v. layers. In this case, we consider only the temperature gradient perpendicular to the main mold walls as significant (foaming direction z). For simplicity, the specific heat [c.sub.p], the thermal conductivity [lambda], the reaction heat and the density of the system are assumed to be reasonably constant. Actually, thermal conductivity [lambda], according to according to
1. As stated or indicated by; on the authority of: according to historians.
2. In keeping with: according to instructions.
3. equation reported by Marciano (3), changes with foam density. We will assume an average value (equal to [10.sup.-4] cal/cm s [degrees]C) for [lambda] related to the average foam density equal to 0.4 g/[cm.sup.3]. It is also assumed that the mold wall temperature is constant, because of the great thermal capacity of the mold.
The energy balance is solved using an implicit finite difference method In mathematics, more precisely in numerical analysis, finite differences play an important role, they are one of the simplest ways of approximating a differential operator, and are extensively used in solving differential equations. , according to which Eq 7 is expanded to Eqs 8 and 9:
dT/dt = ([T.sub.m,n+1] - [T.sub.m,n])/[delta]t (8)
[d.sup.2]T/d[z.sup.2] = ([T.sub.m+1,n+1] - 2 * [T.sub.m,n+1] + [T.sub.m-1,n+1]/[delta][z.sup.2] (9)
The whole time period (t - [t.sub.0]) is divided in N little steps equal to [delta]t = (t - [t.sub.0])/N and the distance between the center and the wall of the mold (the mold is assumed to be totally symmetrical) is divided in M steps equal to [delta]z = 0.5/M. [delta]z is considered constant because we have ignored the convective effects in the heat balance equation. In this way, the index n is related to the time coordinate and the index m to the spatial coordinate. A general solution of the differential equation differential equation
Mathematical statement that contains one or more derivatives. It states a relationship involving the rates of change of continuously changing quantities modeled by functions. will be the generic temperature T(z,t) = T(m[delta]z, n[delta]t). We employ the implicit form (the 2nd order derivative is written as a function of the (n + 1)th and not the nth time coordinate) in order to choose the integration steps [delta]z and [delta]t independently. The temporal and spatial derivatives written in the above form, if substituted into the energy balance equation, lead to a system of linear differential equation (Math.) an equation which is of the first degree, when the expression which is equated to zero is regarded as a function of the dependent variable and its differential coefficients.
See also: Linear :
[T.sub.m,n] = -A*[T.sub.m-1,n+1] + [1 + 2 . A] . [T.sub.m,n+1] - A . [T.sub.m+1,n+1] - [delta]t.[B.sub.n+1] (10)
A = [lambda] .[delta]t/[rho] . [c.sub.p] . [delta][z.sup.2]
and [B.sub.n+1] = (-[delta][H.sub.r])/[c.sub.p] . k(t) . [[1 - X(t)].sup.2] + [h.sub.v]/[c.sub.p] . dL/dt
The boundary conditions are defined as follows:
[T.sub.m,o] = [T.sub.o] for n = 0 and m = 1,...,M + 1
[T.sub.m+1,n] = [T.sub.o] for m = M + 1 and n = 0,...,N
The system of linear equations, written as a matrix, will be solved with the Gauss method, in particular with the Thomas algorithm, by using an iterative it·er·a·tive
1. Characterized by or involving repetition, recurrence, reiteration, or repetitiousness.
2. Grammar Frequentative.
Noun 1. technique: the mth temperature value is calculated by backward substitutions from [T.sub.M+1,n+1] to [T.sub.m,n+1] with m = M,...,1 and the unknowns obtained from previous iteration One repetition of a sequence of instructions or events. For example, in a program loop, one iteration is once through the instructions in the loop. See iterative development.
(programming) iteration - Repetition of a sequence of instructions. are substituted into the equations in the subsequent step.
At the gel time we have also calculated the foam density distribution inside the mold according to the equation reported below (3):
[rho] = 1/([w.sub.Bo] - [w.sub.B]) . RT/P . [M.sub.B] + (1 - [w.sub.Bo]/[[rho].sub.pol] + [w.sub.B]/[[rho].sub.B] (11)
The blowing agents in the vapor phase are supposed to act as an ideal gas. Then, the value of pressure, which appears in Eq 11, is assumed to be equal to the blowing agent vapor pressure vapor pressure, pressure exerted by a vapor that is in equilibrium with its liquid. A liquid standing in a sealed beaker is actually a dynamic system: some molecules of the liquid are evaporating to form vapor and some molecules of vapor are condensing to form liquid. corresponding to each temperature values at gel time. In this paper, [[rho].sub.pol]. is assumed to be equal to 1.16 g/[cm.sup.3] and this is the same value of skin density assumed for the outer liner.
RESULTS AND DISCUSSION
Figures 4. 5 and 6 show the simulation results of temperature distribution with the three different blowing agents. At 50 s the temperature profile is fairly flat while, after this time, a temperature gradient appears and a maximum value is reached at the foamed inner core (z = 0). In order to check the validity of the model applied here, we need to compare analytical results with experimental data; Figs. 7, 8 and 9 show the temperature increasing at the core, where the experimental values were recorded. The correspondence between real data and values predicted by the model is satisfactory (especially if we keep into account the fact that mean values were assumed for most variables), particularly in relation to the methyl formate blown specimen test. A three-dimensional graph is also reported as an example (Fig. 10), showing the temperature profile as a function of temporal and, at the same time, spatial coordinates.
With the solution of Eq 11, the density distribution at gel time is represented in Fig. 11. Density profiles are only related to tests prepared with physical blowing agents, because Eq 11 is not valid for water-blown systems. Considering the temperature and density distribution inside the mold, the specimens obtained with n-pentane and methyl formate as blowing agents reveal similar behavior with good skin thickness and high skin density: at the moment, from these tests, methyl formate, like n-pentane, appears to be a satisfactory alternative blowing agent.
Then, in order to verify these density values predicted by the model, the skin and core density values are measured by cutting out a rectangular piece of the skin (1 mm thick) and one of the inner layer (5 mm thick). The results reported in Table 4 show that there is good agreement between experimental and theoretic data for physical foaming agents. As foreseeable, the use of [H.sub.2]O as a foaming agent provided foams with a lower skin density, thus greatly affecting the physical-mechanical properties of the specimens.
Characterization of Structural PU Foams
In the final part of this work, the mechanical behavior of structural foams was examined. As known, the mechanical properties of these materials depend quite strongly on their density. So a number of different "sandwich" polyurethane specimens were prepared with the pilot plant by changing the pouring time in order to obtain foams with different densities.
* Superficial Hardness
In accordance with ASTM ASTM
American Society for Testing and Materials D 2240 standards, a Shore D instrument was used to measure surface hardness; this instrument comprises a conic tip that penetrates into the surface of structural foams more or less deeply depending on the hardness and compactness of the skin layer. The superficial hardness reported in Fig. 12 is an average value calculated after having tested about ten different areas on the same specimen. Figure 12 shows the clear inferiority of water-blown systems, attested as known by a lower skin thickness. Instead n-pentane and methyl formate reveal a satisfactory outer layer thickness, testified by the fairly high surface hardness values.
* Flexural Strength and Elastic Modulus
Specimens are cut into a rectangular shape and with a particular size in accordance with ASTM D 790 standards. The results of the test are shown in Figs. 13 and 14. In the first Figure, flexural strength values versus average density are reported, while the second Figure shows elastic modulus. From these diagrams, n-pentane and methyl formate turn out to be excellent blowing agents because the rate of increase of the mechanical properties with respect to average density is much greater than foams obtained by using [H.sub.2]O.
* Impact Test
This test is carried out in accordance with ASTM D 256 Standards: measurements of impact strength are made on a Charpy pendulum. The results obtained by changing specimen density are reported in Fig. 15. This chart shows that water is a poor blowing agent when compared to n-pentane and methyl formate; the latter agents managed to reach greater performance ratings See benchmark. , in accordance with previous tests.
In this paper, a study of the effects of employing new blowing agents, as alternatives to harmful CFCs and HCFCs, is presented. With the determination of kinetic data, based on the Marciano model, the most important kinetic parameters for the formation of polyurethane integral skin rigid foams are calculated. A simplified numerical computation, which involves spatial and temporal variations, is also carried out. The distribution of temperature and the density profile through the mold thickness obtained by this theoretical model is in good agreement with experimental values, which reveal the presence of a satisfactory skin thickness when physical blowing agent foamed systems are used (i.e. n-pentane and methyl formate). These data are confirmed by the physical-mechanical characterization of specimens. Every test reveals the highest mechanical properties for the two above mentioned systems, when compared to water-blown foams, that are closely connected with an outer skin layer formation. This indicates the fac t that methyl formate, here tested as a blowing agent for integral skin polyurethane foams, shows as good qualities as the better-known n-pentane, but it is characterized by a lower flammability flam·ma·ble
Easily ignited and capable of burning rapidly; inflammable.
[From Latin flamm level.
A = frequency factor [l/mole.s]
[c.sub.p] = average specific heat [cal/g.K]
[E.sub.a] = activation energy [cal/g]
[h.sub.v] = latent vaporization heat of physical blowing agents per unit polymer mass [cal/g]
[delta][H.sub.r] = heat of reaction per unit polymer mass [cal/g]
[L.sub.B] = pondered ratio of liquid blowing agent
[L.sub.B,f]= pondered ratio of liquid blowing agent at [T.ad,max]
[L.sub.B,O] = initial pondered ratio of liquid blowing agent
[M.sub.B] = molecular weight of blowing agent [g/mole]
n = order of reaction
P = pressure in the mold [atm]
R = gas constant [cal/mole.K]
S = heat transfer specific area [[cm.sup.2]/g]
t = temporal co-ordinate [s]
T = experimental temperature [K]
[T.sub.ad] = adiabatic temperature [K]
[T.sub.B] = boiling temperature of blowing agent in the mixture [K]
[T.sub.B,O] = boiling temperature of pure blowing agent at [P.sub.atm] [K]
[T.sub.o] = ambient temperature Outside temperature at any given altitude, preferably expressed in degrees centigrade. [K]
[delta][T.sub.ad] = [T.sub.ad,max] - [T.sub.o]
U = global heat transfer coefficient The heat transfer coefficient is used in calculating the convection heat transfer between a moving fluid and a solid in thermodynamics. The heat transfer coefficient is often calculated from the Nusselt number (a dimensionless number). [cal/[cm.sup.2].s.K]
U' = U.S/[C.sub.P] [[s.sup.-1]]
[W.sub.B,O] = initial pondered fraction of blowing agent
[x.sub.B] = molar molar /mo·lar/ (mo´lar)
1. pertaining to a mole of a substance.
2. a measure of the concentration of a solute, expressed as the number of moles of solute per liter of solution. Symbol M, , or mol/L. fraction of blowing agent in liquid phase
X = conversion
z = spatial co-ordinate [cm]
[lambda] = average thermal conductivity of foamed polymer [cal/cm.s.[degrees]C]
[rho] = average density of foamed polymer [g/[cm.sup.3]]
[[rho].sub.pol]. = density of the liquid solution without blowing agent [g/[cm.sup.3]]
[[rho].sub.B] = density of physical blowing agent [g/[cm.sup.3]]
(1.) A. J. Rojas, J. H. Marciano, and R. J. J. Williams For other persons of the same name, see John James Williams.
John James Williams (born April 1, 1949 in Nantyffyllon), known universally as J.J. Williams, is a former Welsh rugby union player who gained thirty caps for Wales as a winger. , Polym. Eng. Sci, 22,840(1982).
(2.) J. H. Marciano, A. J. Rojas, and R. J. J. Williams, Polymer, 23, 1479 (1982].
(3.) J. H. Marciano, M. M. Reboredo, A. J. Rojas, and R. J. J. Williams, Polym. Eng. Sci, 26, 717(1986).
(4.) L. J. Lee and C. W. Macosko, Int. J. Heat Mass Transfer, 23, 1479 (1980).
(5.) V. K. Gupta and D. V. Khakhar, Polym. Eng. Sci., 39, 164 (1999).
(6.) H. Yokono, S. Tsuzuku, Y. Hira, and M. Gotoh, Polym. Eng. Sci., 25, 959 (1985).
(7.) E. Broyer and C. W. Macosko. AIChE J., 22, 268 (1976).
(8.) M. Modesti, N. Baldoin, and F. Simioni, Eur. Polyrn. J., 34, 1233 (1998).
(9.) Handbook of Chemistry and Physics, 53rd Ed., tables C-407, C-304. D-167. D-157 (1972-73).
(10.) M. Modesti, F. Simioni, and S. A. Rienzi, International Conference: CFCs, The Day After, p. 811, Padova, Italy (1994).
(11.) J. Agranoff, Modern Plastics Encyclopedia, 52, 488, McGraw-Hill, New York New York, state, United States
New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of (1975-76).