Hi I H «HHIWIHH i ‘t “It — ._—_.—— —_—— — ,_____ — ___—_._— __—_# — — ____——— — _——_— ___—— ___—_——— —___— NJ .‘. I ‘ga \ \ '5..- .- - ’ I‘D“ . .. .. ,u.‘ "E. ,_ ‘- >1. :. . o ;, . I. . o. o _J . \fi 0 u. - a a ‘ . Is ‘ :. 1 ~ A «£565, A 5-. n 2.. ‘.V 5| ‘— 1‘. '~ ‘ -| .\ ‘. v. I , .v 0 QOH‘ ~ ‘ O 5 “ - . 5 0 . . . -r q . _ - u 4.. ~ ~ I .~ I I; . - a - ‘ .‘ .~ .2 C \ - x .- I... 4 v . a u. . b .- ant-- gums Mll‘lllllHllllllllllHllilIHllJHIlllllllllllUIHIIIHIHI Q 3 1293 10421 9302 9 Q i 3 . ‘4! 1 - 4 'w v .’ é a? ., . J." y é :JTI‘R‘ ' "' ‘5, 'J‘t“ \‘lefi ABSTRACT A COMPUTER SIMULATION OF RIGID POLYURETHANE FOAM FORMATION BY P. V. S. R. Krishnam Raju A computer simulation of polyurethane foam forma- tion, based on available literature data, is presented in this work. The model describes the prepolymer, or 2-step, process for producing foam. A brief description of foam formation is presented and followed by the chemistry of selected model compounds. The kinetics of the model com- pounds are given in a mathematical form which is useful for modeling the actual reaction and heat transfer steps in the foaming process. A relation between extent of reaction, polymer chain length, and time of reaction is derived. A relation between the number average molecular weight and chain length is also derived. The kinetics of the cross- linking and foaming reactions are presented in a mathemati- cal form. A heat balance is made on the system to determine the time necessary to evaporate the blowing agent and the adiabatic temperature rise in the system. A computer program was written based on all of the above mathematical equations. After the results are obtained a comparison between P. V. S. R. Krishnam Raju experimentally observed data and the computer simulation shows that the models are correct in their essential fea- tures. This is believed to be the first successful computer simulation of a 2-step polyurethane foam formation process which includes: (1) the prepolymer formation step, (2) the subsequent polymerization, crosslinking, and gel stages of the reaction; and (3) the heat transfer and blowing agent evaporation steps. A COMPUTER SIMULATION OF RIGID POLYURETHANE FOAM FORMATION BY P. V. S. R. Krishnam Raju A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Chemical Engineering 1975 To my family. ii ACKNOWLEDGMENT The writer would like to express his sincere appre- ciation to Dr. Robert F. Blanks for his guidance and assis- tance in the completion of this work. Appreciation is also extended to Fedders Corporation for giving certain data. The writer also wishes to thank other faculty members and fellow students in the Department of Chemical Engineering for their help. iii TABLE OF CONTENTS Page INTRODUCTION . . . . . . . . . . . . . . . . . . . . . 1 THEORY OF FOAM FORMATION O O 0 O 0 O O O O O O O O O O 4 CHEMISTRY OF FOAM FORMATION . . . . . . . . . . . . . . 7 KINETICS O O 0 O O O O O O O O O O O O O O O O O O O O 9 Reaction of Isocyanate With Alcohol . . . . . . . . . 9 Reaction of Isocyanate With Amine . . . . . . . . . . 13 Reaction of Isocyanate With Ureas . . . . . . . . . . l4 Reactions of Diisocyanate With Diol, Diamine and Diurea o o ‘0 o o o o o o o o o o o o o o o o o 15 DEVELOPMENT OF FINAL MATHEMATICAL EQUATIONS . . . . . . 18 Mathematical Treatment of Prepolymer . . . . . . . . 18 Molecular Weight of the Prepolymer . . . . . . . . 19 Mathematical Treatment of the Crosslinking Step . . . 22 COMPUTER MODEL . . . . . . . . . . . . . . . . . . . . 24 DISCUSSION OF THE RESULTS . . . . . . . . . . . . . . . 29 SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . 33 APPENDIX A . . . . . . . . . . . . . . . . . . . . . . 34 APPENDIX B . . . . . . . . . . . . . . . . . . . . . . 36 NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . 44 REFERENCES . . . . . . . . . . . . . . . . . . . . . . 48 INTRODUCTION Unlike many other great discoveries in chemistry, polyurethanes are not the outgrowth of an accidental find- ing, but are the result of painstaking and systematic efforts to develop new polymers. Polyurethanes are among the most recent additions to the many commercially impor- tant classes of polymers. The term polyurethane is more one of convenience than of accuracy, since these polymers are not derived by polymerizing a monomeric urethane molecule, nor are they usually polymers containing pri- marily urethane groups. The polyurethanes include those polymers which contain a significant number of urethane groups regardless of what the chemical structure of the rest of the molecule may be. The polyurethane foams in rigid types were devel- oped during 1941-1945 by Bayer (3) and flexible types were reported first in 1952 by Hochtlen (14). The urethane foam industry developed first vfijj1 diisocyanate-polyester combinations and more recently with diisocyanate-polyether combinations which are now used in largest volume. The polyester systems employed a 'One Shot' technique in which polyester, diisocyanate, foaming agent, catalysts and foam stabilizers are all mixed in one step, and permitted to l foam. The first commercial use of polyethers employed a 'Pre Polymer' process, in which the polyether and diisocyanate were first reacted to form a prepolymer which was subsequently mixed with a catalyst, blowing agent, and stabilizers to produce foam. By the end of 1958, a one shot process for polyether foams was developed. This newer technology, and especially the very low foam den- sity available from it, marked a further economic improve- ment in this industry. At present the industrial technique of polyure— thane foam formation is at an advanced stage. Unfortu- nately the details of the process are not very well described in the literature. This might be due to the heavy industrial competition in this field. Furthermore, partly due to the complexity of the process, a detailed mathematical analysis of the kinetics of the polymeriza- tion and foam formation is not available. This lack makes process analysis, trouble shooting, process scale-up, and plant design more of an art than a science. The main goal of this work was to treat the kinet- ics and foam formation as a step-by-step process, so that the prepolymer, foaming and crosslinking reactions can be represented by mathematical equations. It was expected that the application of computer techniques to the mathe- matics of such a problem would prove fruitful, and that a computer simulation of the foaming process, in its entirety, would result. Because of the diverse types and large numbers of polyurethane foam formulations and processes in use, a general model was not developed. Instead a simulation was based upon a specific two step prepolymer process for producing polyurethane foam from tolylene diisocyanate and a diol using triethylene diamine as a catalyst and a Freon blowing agent. Variations in the reactant type, catalyst, or blowing agent may be incorporated in the program by appropriate modifications. The system chosen to model is of commercial importance. Some literature data for this system exists and makes possible comparison of the data with simulation predictions. THEORY OF FOAM FORMATION An understanding of the formation of rigid poly- urethane foams involves consideration of the organic chemistry of the polymerization reactions leading to gas formation and molecular growth. In this section a brief description of foam formation along with the prepolymer step is presented. This is done in order that the reader can better understand and appreciate the organic chemistry and kinetics of the polymerization reactions which are presented in the next sections. Two types of processes are generally used for producing rigid urethane foams. They are 'One Shot' and 'prepolymer' processes. The computer program of this work is based on the prepolymer process. In the prepolymer process the reaction of iso- cyanate with hydroxyl terminated low molecular weight polymer is completed first. The amount of isocyanate present is in excess, so that when the reaction is com- pleted, the prepolymer formed will contain only isocyanate end groups. O 0 II I! 2R(NCO)2 + H0 WWW OH = OCN - R - NHCO WNW OCHN - R - NCO cfiisoqguate fwdnxqi iaxammauetxmmhfiund 'Uumdmmxd. pnqxflymer (A) pohwmm R - Alkyl or Aryl Group This reaction is exothermic and is catalysed by amines and tin compounds. The isocyanate terminated prepolymer is then foamed in further reaction using a catalyst, and cross- linking and blowing agents. In rigid urethane foams 'flurocarbon' is generally used as a blowing agent. The crosslinking reactions are exothermic, so the flurocarbon evaporates due to the heat liberated. In rigid urethane foams, the catalyst is chosen so that there is a balance between the exothermic crosslinking reactions and endo-a thermic flurocarbon evaporation. The gas evolution and the polymer growth must be matched so that the gas is entrapped efficiently to produce a closed cell foam with good insulat- ing prOperties and the resulting polymer has the right strength at the end of the gas evolution to maintain its volume without collapse or gross shrinkage. In the model 'triethylene diamine' is used as a crosslinking agent. It has been experimentally shown that this substance acts both as a catalyst and a crosslinking agent (10,11). The isocyanate terminated polymer will react with triethylene diamine to form substituted urea. Flurocarbon and triethylene diamine are added to the pre- polymer to start the foaming and crosslinking reactions. 0 0 II II ZOCNWWWNCO + H2N - R - NH2 = OCNWWWNHCHN - R - NHCHNWWWNCO prepolymer diamine (B) The reaction is exothermic. It is also a chain extension reaction. Since the reactivity of the isocyanate group is high, even when it is part of a polymer chain, and the chain also contains active hydrogen atoms, crosslinking occurs with the virtual elimination of the isocyanate groups (6). The substituted urea will react with excess isocyanate to form a crosslink as indicated below (16). O ZOCNWWWNHCHN - R - NHCHNWWWNCO + OCNWWWNCO OCNmmmNHCHN - R - NCHNmmmNco T” = Biuret (C) Crosslink g H T-H o f = O OCNWNWNHCHN - R - NCHNWWWNCO O The crosslinking does not occur unless excess isocyanate is present. This is the main reason for using excess isocyanate in the prepolymer step. CHEMISTRY OF FOAM FORMATION Urethane foam formation is the result of a series of rather complex chemical reactions leading to the forma- tion of many chemical bonds other than the urethane groups. Instead of describing the exact reactions that take place during the foam formation, the reactions of model com- pounds are described in this section. This approach pro— vides enough information to develOp a kinetic model. The reaction of an isocyanate with a hydroxyl compound produces urethane. RNCO + R'OH = RNHCOR' (D) Isocyanate also reacts with amine to form substituted urea. RNCO + R"NH2 = RNHCHNR" (E) Isocyanate also reacts with substituted urea and urethane to give a Biuret and Allophanate linkage, respectively. 0 RNCO + RNHCHNR" = RNCHNR" CONHR (F) Biuret 0 N I! RNCO + RNHCOR' = RNCOR' CONHR (G) Allophanate The last two reactions lead to branching and crosslinking. All the reactions described are exothermic. The reaction 'G' leading to the Allophanate formation is neglected in the model due to the fact that it is many times slower than the reaction 'F' which is considered to be slow by itself (16). The relative rates of these isocyanate reactions in uncatalysed dilute systems is approximately one for reaction with urethane, to one hundred for a reaction with substituted urea, to about four hundred for a reaction with alcohol (16). The rates of these reactions are also influ- enced by the electronic structure of the reactants and steric hindrance. One should bear in mind the possibility that rela- tive rate data obtained in dilute solution may not be an accurate guide to rates in non-solvent systems, where the reaction medium changes markedly, as in foam formation. With these limits, however, the reactions of model com- pounds provide a starting point from which to build an understanding of urethane foam chemistry. KINETICS It was mentioned earlier that, in foam formation, one should ensure a prOper balance between the exothermic crosslinking reactions and the endothermic flurocarbon evaporation. To obtain a computer simulation it is neces- sary to develOp a thorough understanding of the kinetics of foam formation. The kinetics of the reaction of iso- cyanate with alcohol, amine, and urea are described sep- arately. Next these kinetics are extended to describe the reactions of diisocyanate with diol, diamine, and diurea, the actual compounds in the foaming system. A. Reaction of Isocyanate With Alcohol The mechanism and kinetics of the reaction of isocyanate with alcohol have been studied more thoroughly than those of any other isocyanate reactions. The first quantitative study of isocyanate, hydroxyl reaction was that of Davis and Farnum (8). A more detailed study of this reaction was reported by Baker and coworkers (2). The uncatalysed reaction between a monoisocyanate and alcohol is, K0 R - NCO + R'OH '* RNHCOOR' where K0 is the uncatalysed reaction rate constant. 9 10 The mechanism for the above reaction as proposed by Baker and Gaunt (2) is K1 9 R - N = C = O + R' - O - H a R - N = C - O (___ K2 . ° Complex R' — O - H S 9 R - N = C - O . ! K3 I U . + R - 0 - H + RNHCOOR + R OH R'-O-H S Here the alcohol acts as a catalyst in producing the complex. The rate of disappearance of isocyanate is given by d(NCO) dt = K1(NCO)(OH) - K2 (Complex) where ( ) refers to the concentration of the particular com- ponent present in the parenthesis. The rate of formation and disappearance of Complex is given by d(COEEleX) = K1(NCO)(OH) _ K2(Complex) - K3(OH)(Complex) At steady state, d(Complex) = O; :.(Comp1ex) _ K1(NCO)(OH) dt (K2+K3(OH)) 11 For the entire reaction, _ d(NCO) _ d(urethane) _ ———dt - dt — Ko(NCO) (0H) A slight manipulation gives the uncatalysed reaction rate constant as KO = KlK3(OH)/(K2 + K3(OH)) The kinetics for the base catalysed reaction of isocyanate and alcohol are next considered. K RNCO + R'OH + B -* RNHCOOR' + B where B is the catalyst. The reaction mechanism proposed by Baker et a1. (2) is, K1' 9 ?'R-N=C-O R - N = C = O + B . K ' 3 + R'OH -* RNHCOOR' + B By the same type of mathematical treatment as done for uncatalysed reaction, the reaction rate constant for the catalysed reaction is obtained R Kl'K3'(B)/(K2' + K3'(OH)) 12 The experimentally observed rate constant for the catalysed reaction of isocyanate with alcohol is the sum of the rate constants for the uncatalysed and catalysed reactions. K Ko + K exp + K1 K3 (B)/(K2' + K3'(OH)) K1K3(OH)/(K2 + K3(OH)) The catalytic function of alcohol raises doubt whether or not any reaction occurs by truly an uncatalysed mechanism. If the reaction is not catalysed by an external catalyst, it was experimentally shown that alcohol does act as a catalyst (9). However, in a catalysed system, the uncata- lysed reaction rate constant is of a very small order of magnitude compared to the catalytic reaction rate constant. The overall reaction rate constant can therefore be simpli- fied as K = Ku + KC(B) exp (If one assumes that the overall reaction rate constant is independent of hydroxyl concentration.) where Ku is the uncatalysed reaction rate constant. Kc is the catalytic coefficient for the particular catalyst. The above assumption that the overall reaction rate con- stant is independent of hydroxyl concentration was verified by Baker and Holdsworth (2). 13 B. Reaction of Isocyanate With Amine Systematic studies of the reaction of isocyanate with amine were carried out by Craven and Baker (2). Upon the basis of their observations they suggested the follow- ing mechanism. 1. Spontaneous Reaction: K1 ArNCO + ArNH2 —;9 Complex 1 2 K3 Complex 1 + ArNH -* (ArNH)2CO + ArNH 2 2 sub urea 2. Product Catalysed Reaction: K3' Complex 1 + (ArNH)2 CO -* 2(ArNH)2CO Ar-*alkyl or aryl radical,en:steady state; d(Complex) dt =0 Then a slight mathematical manipulation gives, Ka = K (K (NH ) + K '(sub urea))/ . 1 3 2 3 (K2 + K3(NH2) + K3 (sub urea)) where Ka is the overall rate constant for the isocyanate amine reaction. For the initial stages of the reaction when the concentration of substituted urea is very small 14 Ka = K1K3(NH2)/(K2 + K3(NH2)) If K2 is also small and can be neglected compared to K3(NH2) then Ka can be assumed to be constant for the initial stages of the reaction. For the final stages of the reaction when the concentration of the substituted urea is large compared to that of amine then, _ I Since K2 is much smaller than K3' (sub urea) then Ka can be assumed to be constant for the final stages of the reaction. On the whole the rate constant for the amine, isocyanate reaction may be assumed to be constant for the entire reaction. C. Reaction of Isocyanate With Ureas Because of the relatively high temperatures required and the many side reactions possible at these temperatures, the reaction between isocyanate and substi- tuted urea is a difficult one to study kinetically with satisfactory results. Though the initial reaction leads to biuret, the active hydrogens in this compound are as active as those in the original substance, so many side reactions take place. Dissociation of biuret and other 15 products may occur at higher temperatures. Kinetic data about this reaction are not available in the literature so the rate constant for this reaction is assumed to be constant for the entire reaction. D. Reactions of Diisocyanate With Diol, Diamine and Diurea The kinetics of these reaction have to be con- sidered, since these are the main ingredients of the foam- ing system. The reactions of diisocyanates with diol are more complicated kinetically than are-those of monoisocyanates previously described. The initial reactivity of diiso- cyanate is similar to that of monoisocyanate substituted by an activating group, in this case a second isocyanate group. As soon as one isocyanate group has reacted with alcohol, the remaining isocyanate group has a reactivity similar to that of a monoisocyanate substituted by a urethane group. A urethane group in meta or para position has only a nUle activating effect, much less than an isocyanate group in meta or para position. So the reactivity of a diisocyanate having both isocyanate groups on one aromatic ring should decrease significantly as the reaction passes approximately fifty percent completion. An example illustrating these effects is 2, 4-tolylene diisocyanate. 16 NCO NCO The most reactive group should be the 4-position isocyanate, 'which is activated by 2-position isocyanate group. The 2-position group has similar activation initially by the 4-position group but compensating deactivation by the l-position methyl group. After the 4-position isocyanate group has reacted with alcohol, the 2-position isocyanate group is even less reactive than initially because of strong deactivation by the l-methyl group, far overshadow- ing the very slight activating tendency of the 4—position urethane group. This example was verified experimentally (15). The data from the experiments for the reaction of diisocyanates with n-butanol are: Isocyanate Reactant K x 102 K x 103 2,6-Tolylene Diisocyanate 1.5 2.46 2,4-Tolylene Diisocyanate 3.3 2.77 To illustrate the above-discussed material a com- puter program was written for the formation of uncatalysed urethane polymer from diisocyanate and diol. One program was run taking into consideration that the rate of reaction decreases significantly as the reaction crosses fifty percent l7 completion. The results obtained are compared with another set of results on the same system but on the assumption that both isocyanate groups have equal reactivity through- out the entire reaction. The results from both the above cases are then compared with the values obtained experi- mentally by Bailey et al. (1), on the same system. All the results are tabulated in Appendix A. From the table in Appendix A, one may conclude that the results obtained by taking into consideration the unequal reactivity of isocyanate groups are in close agree— ment with the experimental values. The reactions of diisocyanate with diamine are considered next. Since in the foam formation, diamine is added to isocyanate terminated prepolymer, where the isocyanate groups are not on the same aromatic ring, but are far' apart, both time isocyanate groups have equal reactivity toward diamine. Therefore the rate constant for this reaction is the same for both the isocyanate groups. Similar reasoning applies to the reaction of diisocyanate with diurea, where the reaction rate constant is the same for both the isocyanate groups. DEVELOPMENT OF FINAL MATHEMATICAL EQUATIONS The kinetic expressions derived so far must be put in a mathematical form which allows solution with the computer. This section deals with the mathematical treat- ment of various expressions. Relations between extent of reaction and chain length and number average molecular weight and chain length are also derived for bifunctional monomers . A. Mathematical Treatment of Prepplymer The prepolymer reaction can be represented simply as: KO Diol + Diisocyanate = Urethane (OH) (NCO) In the deve10pment of the kinetics for the hydroxyl, isocyanate reaction it was concluded that the overall reac- tion rate constant is dependent only on the catalyst con- centration. Therefore fixing the catalyst concentration fixes the overall reaction rate constant. The decrease in the reaction rate constant as the reaction crosses fifty percent completion will be taken into account in the com- puter program. Thus, the prepolymer step can be represented 18 19 in mathematical form as: d (NCO) _ _ _ d(OH) dt — KC(OH)(NCO) - dt (H) Knowing Ko and initial concentrations of diol and diiso- cyanate the above equation can be solved on the computer using 'Eulers method.’ The Eulers algorithm is of the form Yi = Y(xo) + h t(xo,Y(xo)) Yi = new value of Y Y(xo) old value of Y h = step size t(xo,Y(xo))= derivative of Y with respect to x at the old value of Y The algorithm is repeated for subsequent steps until Yi exceeds some desired upper limit. The smaller the step increment, the better are the results. The solution would give the change in reactant concentrations with respect to time. Then, from these, extent of reaction can be calcu- lated with respect to time. The extent of reaction is defined as the fraction of OH groups that have reacted at a particular time. B. Molecular Weight of the Prepolymer The molecular weight of the prepolymer is of prime concern from the practical viewpoint, for unless the 20 polymer is of sufficiently high molecular weight, it will not have desirable strength characteristics. It is there- fore important to consider the change in polymer molecular weight with reaction time. The residue from each diol and diisocyanate (sep- arately) in the polymer chain is termed as a structural unit. The repeating unit in the chain consists of one structural unit. The number average degree of polymeriza- tion Xn is defined as the average number of structural units per polymer chain. Thus Xn is simply given as the total number of monomer molecules initially present divided by the total number of molecules present at time 't'. Xn = (M)O/(M) The number average molecular weight Mn, defined as the total weight of a polymer sample divided by the total number of molecules in it, is given by Mn = M0 Xn (I) where M0 is the mean molecular weight of one structural unit. In order to determine Mn, Xn should be known. A relation between Xn and extent of reaction is derived below, for a reaction containing only bifunctional monomers. For the polymerization of the bifunctional monomers HONWWOH and OCNWWWNCO where OCNWWWNCO is present in excess, the number of OH.and NCO groups is given by NA and NB, 21 respectively. NA and NB are equal to twice the number of HOWAAOH and OCNWWWNCO molecules, respectively. The stoichiometric imbalance 'r' of the two functional groups is given by r = NA/NB. The ratio 'r' is always defined so as to have a value equal to or less than unity, but never greater than unity. The total number of monomer molecules is given by (NA + NB) 2 = NA(1 + l/r)/2 The extent of reaction 'p' is introduced here and defined as the fraction of OH groups which have reacted at a particular time. The fraction of NCO groups that have reacted is given by 'rp'. The fraction of unreacted OH and NCO groups is (l—p) and (l—rp), respectively. The total number of polymer chain ends is given by the sum of the total number of unreacted OH and NCO groups. Since each polymer chain has two chain ends, the total number of polymer molecules is one-half the total number of chain ends or (NA(1 - p) + NB(1 - rp))/2 The number average degree of polymerization is the total number of HOWWWOH and OCNWWWNCO molecules initially present divided by the total number of polymer molecules. Xn = NA(1 + l/r)/2/(NA(1 - p) + NB (1 - rp))/2 (l + r)/(l + r - 2rp) (J) 22 The extent of reaction 'p' may be calculated as a function of time and temperature from the previously discussed polymerization kinetics. At some critical chain length, or equivalently at some critical extent of reaction, chain branches in the system reach a concentration such that crosslinking and gel formation begins. This critical extent of reaction, 'Pc', marks the onset of the cross— linking stage of the process. Prepolymers are formed by reacting to extents of reaction of less than 'Pc'. C. Mathematical Treatment of the Crosslinking Step The crosslinking reactions taking place in the foam formation can be simplified as K1 Isocyanate + Amine = Sub Urea K2 Isocyanate + Sub Urea = Biuret In the develOpment of the kinetics of the above reactions it was concluded that the reaction rate constants for the various reactions can be assumed to be constant for the entire reaction. Also from the kinetics these equations can be mathematically represented as, d(NCO) _—dt__ = -Kl(NCO)(amine) - K2(sub urea)(NCO) (K) d(amine) _ _ . _——dt—_— — Kl(NCO)(am1ne) (L) 23 d(sub urea) dt = K1(NCO)(amine) - K2(sub urea)(NCO) (M) d(bigiet) = K2(sub urea)(NCO) (N) Knowing K initial concentrations of diisocyanate and 1' K2' diamine, the equations can be solved on the computer using 'Eulers method.‘ The solution gives the rate of variation of the concentration of the reacting species with respect to time. The heat evolved during crosslinking is calcu- lated by knowing the moles of sub urea and biuret formed, with respect to time. The amount of heat required to evap- orate the flurocarbon can be calculated knowing the weight of the flurocarbon and its enthalpy of vaporization. The temperature rise in the system is obtained from a heat balance. COMPUTER MODEL The computer program was written for a 'prepolymer' process for producing rigid polyurethane foam, with the specific system comprised of (1) diol, (2) diisocyanate, (3) triethylene diamine, (4) flurocarbon. The reasons for selecting the particular system were mentioned earlier. The three variables in the computer program are (1) weight of diol, (2) weight of diamine, (3) required foam density. For a given foam density and given weight of diol and diamine, the computer program calculates the time required for prepolymer formation and the foam time.* From the results obtained one can judge whether the particular for- mulation is suitable for foaming or not. The computer program is broadly divided into two parts, one describing the prepolymer step and the second describing the crosslinking and foaming reactions. The calculations for the prepolymer reaction are done in the following manner on the computer. 1. For a given weight of diol the equivalent amount of diisocyanate required is calculated. The calcu- lated weight is then corrected for the specified NCO/OH ratio. *Defined on page 29. 24 25 2. Concentrations of diol and diisocyanate are calculated and the ratio of initial concentrations is found. 3. The mathematical equation (H) representing the prepolymer formation is solved using Eulers method, with very small time increments. 4. From the heat balance, the maximum temperature increase in the mixture is obtained, assuming an adiabatic reaction system. 5. The kinetic chain length and number average molecular weight are determined using relations (J) and (I). 6. The change of reaction rate constant with temperature is taken into account using an Arrhenius equation, K = A exp (AE/RT). 7. The decrease in the overall reaction rate constant is taken into account once the extent of reac- tion crosses fifty percent, (as discussed for tolylene diisacyante in the kinetic section). The crosslinking reactions are solved on the com- puter in the following order: 1. Weight of amine is calculated from the per- centage amine charged. The weight of surfactant is also calculated. 2. The weight of Freon required is calculated for a given density of foam, using the relation from the lit- erature (4): 26 F = 28.21/D0°9 flurocarbon % where F D II required foam density, lbs/cu. ft. 3. The concentration of amine and isocyanate in the foaming mixture is calculated. 4. The mathematical equations representing the crosslinking reactions (K,L,M,N) are solved on the computer by using Eulers method. 5. The amount of heat liberated is calculated from the amount of biuret and substituted urea formed. 6. The heat required for Freon to evaporate is obtained. 7. The temperature rise in the foam mixture is calculated under the assumption that there are no heat losses from the system. 8. The change of reaction rate constants with respect to temperature is taken into account using an Arrhenius equation. The computer model was developed assuming that the following quantities are known; these are also the input parameters for the computer program: 1. Hydroxyl number of diol 2. Weight of diol 3. Density of diol 4. Molecular weight of diol 27 5. Percentage of isocyanate in 2,4-tolylene diisocyanate 6. Density of diisocyanate 7. Molecular weight of diisocyanate 8. Density of diamine 9. Density of substituted urea 10. Heat evolved during the formation of one mole of sub urea 11. Heat evolved during the formation of one mole of biuret 12. Heat of vaporization of Freon at its boiling point 13. Rate constant for diol-diisocyanate reaction 14. Rate constant for diisocyanate-diamine reaction 15. Rate constant for diisocyanate-diurea reaction 16. Required foam density 17. Mean molecular weight of a structural unit 18. Desired ratio of NCO/OH 19. Density of Freon 20. Required percentage of diamine 21. Specific heat of foaming mixture For the simulation model all of these quantities were obtained from the literature. Typical values are in computer output in Appendix B. The program is written in such a way that the heat liberated during the crosslinking reactions is initially utilised toward heating the foaming mixture, until the boil- ing point of Freon is reached. At the boiling point of Freon, the mixture remains at that temperature until all the 28 Freon has evaporated. This is the outcome of the assump- tion that there are no heat losses. In actual foaming Operation, generally cooling is provided. In such a sit- uation the heat removed could be simply added to the exist- ing program. The computer program along with its output for one particular run is attached in Appendix B. DISCUSSION OF THE RESULTS The computer program was tested for variations in required foam density and percentage of trirethylene diamine added for a given weight of diol. All the results are not presented here, but three of the results for different runs are given in Table 1. In order to interpret the results given in Table 1, one must know the definitions of cream time and foam time. They are defined as follows: Cream time: It is the time required for the solu- tion to get supersaturated with the gaseous blowing agent. Foam time: It is the time required for the foam to reach its maximum volume. From the literature it is known (16) that for a rigid polyurethane foam, the cream time is approximately ten seconds and the foam time is approximately 60-120 seconds. The simulation provides reasonable estimates for the quanti- ties as shown in the table. Most of the foam formulations used in industry contain a triol or a quadrol. These com- pounds give a highly crosslinked structure to the foam, compared with the diols. Since in the model only diols were used, there were no data available in the literature for this particular system, to compare the details of the results. 29 30 m.mma ms .. o.H m.H m m.mw OH com o.N N.H N n.5m mum calms o.m m.H H mucoowm «:oflpmameou mccoomm mucoomm mcflsd .um .so\.mna mam mm nomwm 0» mafia Emmno mafia Emom ucmonm Suamcmo Boom cam Hmahaommum map How mafia H magma 31 The data obtained from one of the local firms (12) which manufacture rigid urethane foam by the prepolymer process, using flurocarbon as a blowing agent and diamine as a catalyst, are as follows: Foam Density Foam Time Cream Time lbs/cu.ft. Seconds Seconds 1.5 to 1.6 80-85 2-3 These data agree very well with the results predicted by the computer program for that particular foam density. However, the amount of catalyst present in the industrial formula- tion was not known (12). The results of the second computer run indicate that the particular foam formulation represented by the input data does foam, but it takes a very long time. To reduce the time necessary for foaming, the percentage of diamine would have to be increased. The results of the third computer run indicate that the particular foam formulation does not attain the required foam density. The blank in the foam time means that the blowing agent did not evaporate completely. This is due to the fact that sufficient heat was not evolved to evaporate all the blowing agent. This indicates that not many cross- links were formed, which is the outcome of using less diamine. The prepolymer step results are in agreement with the values available in the literature. 32 During the final stages of the crosslinking, when the chains start becoming stiffer time rate of collision of the molecules may be much slower than when the chains were free to move around. This was not taken into consid- eration in the model. This might result in a slightly longer time requirement for the crosslinking step than pre- dicted by the program. SUMMARY AND CONCLUS IONS A plausible computer simulation of a 2—step, rigid polyurethane foam process was develOped successfully. The results of the simulation appear to affirm the reliability of the essential features in the model. Polyurethane foam formation is the result of a series of rather complex chem- ical reactions, with simultaneous rapid changes in the tem- perature and viscosity of the medium. Predictions of the model are dependent on the accuracy of kinetic rate constants available, as the reaction medium changes. The rate con- stants that were used in the simulation were obtained from dilute solution literature data. These may not be a reliable guide, where the reaction medium changes markedly; however, they appear to give reasonable first estimates for the pro- cess simulations run. With the availability of more accurate rate data, and the amounts of heat evolved during the various reactions, one would be in a better position to judge the model. How- ever even now with the limited knowledge of the rate data, one could easily say that the model is predicting the results in the right direction, and in the right order of magnitude. The next step in developing the model would involve experi- mental studies aimed at providing some of the missing quanti- ties mentioned above. 33 APPENDIX A 34 mm.m hm.wm oo.mm ow mm.v mm.om oo.mm om mH.m vm.vm oo.vm ow oa.ma hm.mm oo.mm om va.mm vm.vv oo.ww om mh.mm mh.mm oo.mm OH w w w musom mumcm>00mflflo Umuommucb mumcm>00mflflo pmuommHCD mumcmmoomflflo pmuommHCD mafia mmsouw mumcm>00mH mo apa>fluomwm Hmswm mmsouo mumcm>00mH mo >uw>fiuomwm HmsquD fl XHDmem¢ AHV.HM um mmaflom mo mosam> HmucmEHummxm 35 APPENDIX B 36 UNT VERSION OCT 73 A 17 16 12/09/76 war- HM (NV H» mm 0"" PM“ NVV 1?va NM N5‘ 37 FROORAN URTHANE (INPUT,OUTPUI) ...1N1$ PROORAH TESTS THE MODEL FOR THE FORNAIION 0F PLOYUQETHANE FOAMS REAL' HHOH.HHNCO. KO. NRA.NO. K1, x2, HHFREFN ...................REAO IN VARIABL ES........................... PEAD 1.0HNO. °E°NCO,HOH, H0, <1, K2 ,DDH, HHOH, DNCO, HHNGO. K0, HHA HEODAF 10$.HSU3U,H1, OSNSTTquATIO, 0F,?A, 5° HEAT 1 FORMAT (7F10. A) 30.......OOOOIOOOOOOPRINI INPUI VARIABLESOOOOOOOOOOOOOOOOOOO... D530 . 000 FRINr 2.0HNO. NON, OOH. HHOH, 353NCO ,nNcO,wNNcO,OA,HuA 2 FOQHAT (1H1,%Ox,:INpur VARtaaLFs , 1III’IOXQ.HYCQOXYL NU‘A'JF R 01- "TOL = ‘927XQF1506' leylflxg‘NEIC4T OF OIOL = ‘,39¥,F15.6 FORAHSF, ~ 3//.1Ox.FOFNSITv CF DIOL = ‘,37!,F15.6.'GKAHS/LITER*. u//,1ox,-HOLFOOLAR WEIGHT OF 310L = F.2ax.F1s.6. 5//,1Ox,FFFRCFNIAcE OF Tscchan = ‘g27X,F15.6,‘PERCENT¥, e//.11x.FOFNsxrv CF ISOCYNATC = *.32X,F15.6,‘GRANS/tITER‘p 71/,19X.'HOLECULAP HEIGHT OF rsocharz = ‘.22X.F15.o, O/I,1JX.FOF:131rv OF A11*E = ‘.36X,F15.5.‘GRAHS/LITER‘, 91/.1Ox,FNOLFcuLAF HEIGHT OF «NINE = F.25x.F15.6) PRINT 30.os.Nsu9n,H3,HF,K o 3°1E??3§I 11/.1OX,FOFNSIIY OF SJSSTITUTEO UREA: F,25X.F15.6.FGRAHS/ §éisigxizéfifl EVOLVED BY THE FORMATION OF one HOLE OF SUBUREA = ., 21£6§EBF‘HEAI’EVOLVED FOR ONE HOLE OF OIORFT FORMED = F.3x, F15.6, 61/.1OXIFHFAT OF VAPORISATION OF FFFON = F .19x F15. 6.*KCAL ggé612§,‘RATE CONSTANT FOR OH Nco REACTION' = ,15x .F15. 6.‘L/MOLE. FRer AO.K1.K2.OFNSITY H0 RATIO, OF, PA, SPHEAT #0158313{E(géé19x,*9ATE OONsraNr FOR Nco Naz REACTION = '14X.F13.6, géécgoéééRgré éONSTANr FOR NCO SUBUREA REACTION = .1ox. F15.6,FL1 h//.ifix:'RFOUIRFO FOAM DENSITY = 23x, F15. 6,’LBS/CU. FT. ; 51/31ox.FwFAN NOLFOOLAR WEIGHT OF qu SIROOIURAL UNITS = ,rx.F15.6 6.//.10X.‘RATIO OF NCO/0H = F.3ax,F1a.e. 7//.10X.‘DENSIIY OF FREON = *,36X.F15.6, FLOS/cu. FT. 81/,10X,'PERCFNT AHINE = v.3«x.F15.6.*°FRcsNT c 9/1.1OX.FSFFOF1c HEAT OF FOAHING MIXTURE = F.1}x, F15. 5.FcAL/H. c.F) 5......EQUIVALENT HEIGHTS OF DIOL AND IS OCYANATE......... ENOH=56.1’1090./0HNO ; EHNCO= h2.‘100./PERNCO . ‘ G . ‘ . Goo...THE HEIGHT 0F ISOCYANATE REQJIRED FOR A GIVEN HEIGHT 0F DIOLoooooo THNCO=(HOHIEHOH)'EHNCO EoooooooooooTOTAL HEIGHT 0F ISOCYAVATE 2EOUIQEO........oooooooo HNCO=THNCO‘RATTO THEIGHT= HOH+HNCO c EOOOOOOOGOOOOOOOVOLU”ES 0F OIOL AN) ISOCYA\IATEOOOOOOOOOOOOOOO VOH=HOHIDOH vNco= NNOO/ONOO c . VOL: VOH+VNCO C........INI14L CONCENTRATIONS OF OtOL ANO IsochNArE........ COOH=NOH11NNOHFVOL1F2. O . CONGO: HNCO/(WNNCO‘VOLT‘Zofl _ . 1 0......OOOOOOORAIIO 0F INTIAL CONCENrRATIOVSOOOOOOOOO RICONC=COOHICONCO DO" 38 RUNT VERSION OCT 73 A 17 16 1210917“ .207 22k NNNNNNNNM @mmmmtrbr commaQNHEru NhFMVNHO N‘MH’OO‘ UWWVWR”* 277 «mu p 0h 12 h 316 322 332 rlwfluuéuu 6900mm“ Nfluombm 'nun non nun‘nnn C O fvt‘ zu1Huu Kflhfi L L CONCEHT ATIOV 0‘ N 0F INITIAL CONCENTRA c . go. .0... O I. I. OSTORE INTIAL CONCENIRATIOT'IS. . ...... ..... GOH =COOH CNGO= CONCO ' . G g poo-ocooooo.....oCALCULATIONS F0! THE FORMATION OF PREPOLYHERo.o....o PPINT 301 ' . . 301 FO¥SAT 1.56 ‘PREPOLYNER FORMATION REAGTIONS‘) 9R 200 FORHAT IO 'TIHE'.13X,‘UOHDT‘313XO'DNCODT"14XO'CNCU'915x. 1 00” g X C’Q17xz' T) T F A PRINT 201 FOR A F EFONOSF.7x FHOLES/LITER.SEC.4.ux.*HOLESILITER.SEc.' 1 . ITEFF. BXo‘HOLES/LITER‘) O ZINC VAOIABLESOOooooooooooooo__ 4 O X 9’ E 5 Es IA (1H X 200 1 l 7 12X E T8 201 T (I X, S ‘MCL IL INT LI ‘flflfl DdHHuztdd. ooxmamcuu nuuz :mumqqmom I‘d-LUDHIUWO u- H o