Genre: Information/Explanation

Paper 26 : The application of Clifford algebra to calculations of multicomponent chemical composition . Dr John P. Fletcher , Chemical Engineering and Applied Chemistry , School of Engineering and Applied Science , Aston University , Aston Triangle , Birmingham B4 7 ET , U. K. Email : J. P. F letcher @ aston . a c . u k Abstract In a mixture of chemical compounds , the significant variable is often the proportion of molecules , normally expressed as a mole fraction in a particular sample . The fractions are constrained to add to one , so any change such as the addition of some more of any chemical component causes all of the mole fractions change in a nonlinear way . Lasenby et al [ 1 ] have applied Clifford Algebra to the problem of projection and shown that problems can be made linear and results obtained . In this paper a similar approach is used to show that some nonlinear problems , such as conversion between molar and mass basis , become linear when projected into a space of increased dimension . When this is done using the V product defined by Miralles et al this [ 2 ] can be done using a Clifford algebra with all positive signature . Examples are given of the results of the formulation . 1 Introduction Calculation of multicomponent chemical composition is necessary in many calculations in chemical engineering and chemistry . It is complicated by the fact the some of the calculations are on the basis of the mass of each component present , and others on the basis of the relative number of molecules . Different molecules differ in relative mass , expressed as the relative molecular mass , or molecular weight . The proportions of a mixture are often expressed as a set of fractions which are constrained to add to one . Both the mass fractions and the mole fractions are nonlinear functions of the total number of molecules of each component present . Conversion between them is a nonlinear operation . These calculations are often necessary in phase equilibrium calculations , where it is desired to calculate the compositions of a vapour phase and a liquid phase , or two liquid phases , which are in equilibrium . In general the two phases will be of different chemical composition . The calculation is often made difficult by problems of modelling the nonlinear behaviour of the properties of the chemical components . Such calculations are a vital feature of chemical engineering design , There are advantages to be gained if the basis part of the calculation can be made linear . This paper shows that the calculations needed for phase equilibrium calculations have some similarity in form to the projection calculations discussed by Lasenby et al [ 1 ] . They have shown that some nonlinear problems are linear when projected to an extra dimension using a Clifford algebra . In this paper consideration is given to applying these methods to the formulation of the algebra of multicomponent chemical composition using the same techniques . One issue in these calculaltions is the appropriate choice of the Clifford algebra to use . In this , use has also been made of the methods of Mirrales et al [ 2 ] who define a V product which is effectively a means of using one basis on Clifford objects defined in a different basis . This opens up the possibility of carrying out the projection calculations of Lasenby et al [ 1 ] while using a Clifford basis with all positive signature . These ideas have been applied to the calculation of the transformation from the mass basis to the mole basis , with particular emphasis on the formulation of phase equilibrium calculations . 2 Projection Lasenby et al [ 1 ] describe the use of a projective space for representation of invariants in computer vision . They introduce a vector X in a Clifford algebra of signature ( 1 , 3 ) with basis vectors with the signatures Taking the product of X with the basis vector 4 gives an expression in terms of a three dimensional vector x where x is defined in terms of the bivectors of the original algebra . This author considered this representation as possible way to represent the multicompent chemical composition , and has adapted it for the modelling of multicomponent chemical composition . 3 The V product Mirralles et al [ 2 ] discuss the algebra of signature change among Clifford algebras of the same total dimension . The example which is discussed in their paper . For the case of the Clifford ( 4 ) algebra they take two objects of any grade A and B with Clifford product AB . They define a Vee or V product such that A B is equivalent to the multiplication of objects defined in Clifford ( 1 , 3 ) . For two vector objects u and v in Clifford ( 4 , 0 ) the definition is where e 0 is chosen from among the basis vectors of the algebra . For theis definition the symmetric part is given by in comparision with the Clifford product They show that the unsymmetric part is the same as for the Clifford product . The definition extends to the grades of the multigrade objects as follows . This author considered this as an alternative for the representation of the multicompent chemical composition , and has applied it for the modelling of multicomponent chemical composition . 4 Multicomponent chemical composition 4.1 Projection Model The analysis of the multicomponent chemical composition follows the algebra of Lasneby et al [ 1 ] , using the Clifford ( 3 , 1 ) algebra rather than the Clifford ( 1 , 3 ) . Suppose that a closed chemical system contains a mass M i of each of 3 chemical components . Define an object M in a Clifford ( 3 , 1 ) vector space with orthogonal basis vectors i as follows The system is defined such that This will be referred to as the full system . A reduced coordinate system can be constructed by taking the geometric product of an object in the system with 0 . This will be called the reduced system . If the bivectors in this relation are taken as defining a three dimensional Clifford ( 3 , 0 ) algebra Then an object m in this new space is given by where are the mass fractions for the system . An object D in the full system is defined as follows . Then the dot product of D with M gives which represents the mass balance and the relation that the x i sum to unity . The advantage of this procedure is that M is linear in the components whereas m is nonlinear , but can be easily constructed from it . Thus 4.2 Conversion between mass and molar basis . To scale to units proportional to the numbers of molecules , using the relative molecular mass r i of each component , then a new vector N in the full system can be constructed as follows , using the inverses of the relative molecular masses then if M 0 = M 1 + M 2 + M 3 This is a linear operation , which in matrix terms can be written , with M 0 = M 1 + M 2 + M 3 , as and defined as The molar fraction vector n can then be found from N using the same method as before . where with are the molar fractions in the system . This can be expressed in terms of N i , the components of N , as This conversion is quite general , and the reverse operation can be written as If the two matrices are combined , the result is a matrix which transforms an object such as M into itself . This is defined to be I. and clearly 4.3 Vapour-Liquid Equilibrium The usual chemical engineering formulation of vapour liquid equilibrium is for each of the components , where x i are the liquid molar fractions and y i are the vapour molar fractions . K i is called the equilibrium constant and is in general a function of temperature , pressure and all the molar fractions in both phases . Defining objects for the two phases as previously , where X i and Y i are on a mloar basis , and then converting the equilibrium expression to this basis results in the expression Summing over all the components which implies that the K i are not independent . It follows that and This means that the Y i are in the same proportions as the K i X i and the following calculation yields some linear multiple of Y. so that , using the previous notation , 4.4 Flash calculation One application of these equations is to a flash calculation where a fixed amount of material is available , which will be distributed between the two phases . If that material is designated by Z such that then The matrix form of this is This is invertible as where This provides a formulation of the flash calculation which can be used for an iterative solution given models for the dependence of K i on the variables . 4.5 Chemical Reaction If a chemical reaction occurs transforming a vector A in the full molar system to a vector B then where Q is the effect of the reaction . If on a molar basis Q 1 moles of component 1 become Q 2 moles of component 2 then so that 4.6 Volume and thermodynamic functions Functions whose value is proportional to the amount of material , termed extensive functions , can be expressed in terms of the contribution of each component . For example the volume V of a liquid can be expressed in terms of the molar numbers N i of a vector N and the partial molar volumes V i as This can be represented in the full system as and the value of the property can be extracted as This can be extended to intensive properties such as density where and expressing with the value of the property given by 5 Vee product model The development of this model is very similar to the previous model , using a different Clifford basis . Suppose that a closed chemical system contains a mass M i of each of the components . Define an object M in a Clifford ( 4 , 0 ) vector space with orthogonal basis vectors i as follows The system is defined such that with The bivectors in this relation are taken as defining a three dimensional Clifford ( 0 , 3 ) algebra Then an object m in this new space is given by where are the mass fractions for the system as before . If an object D in the full system is defined as follows . then the symmetric part of the V product of D with M choosing 0 as the special basis vector gives which is the mass balance equation as in the projection model . Thus the alternative of the Clifford ( 4 , 0 ) basis can be used and still have the same relationship among the components . One difference is that the reduced basis now has an all negative signature . That is not a problem because it is the sum of the mass fractions which is significant in calculation and not their vector norm . The only calculation which depends upon the signature of the algebra is the extraction of an extensive property which now becomes 6 Generalisation The examples have been constructed using three chemical components . The method can be generalised to more components by increasing the number of basis vectors . No calculation makes use of any property of the Clifford Algebra which changes as the dimension increases . 7 Conclusions The projection method can be used to formulate the equations of multicomponent chemical composition , including phase equilibrium and chemical reaction , and both extensive and intensive properties which can be calculated from contributions from each component . This can be done either using the approach of Lasenby et al [ 1 ] or that of Miralles et al [ 2 ] . 8 References [ 1 ] J. Lasenby , E. Bayro-Corrochano , A. N. Lasenby and G. Sommer . " Geometric Algebra : a Framework for Computing Invariants in Computer Vision " Proceedings of the International Conference on Pattern Recognition ( ICPR ' 96 ) , Vienna . [ 2 ] D. Mirrales , J. M. P arra , and J. Vaz , " Signature Change and Clifford Algebras " , International Journal of Theoretical Physics , vol. 40 ( 1 ) pp. 227-238 , ( 2001 ) John Fletcher May 2001 , updated March 2003 . Aston University , Aston Triangle , Birmingham B4 7ET , United Kingdom Telephone : +44 ( 0 ) 121 204 3000 Fax : +44 ( 0 ) 121 333 6350 Copyright © 10.25.06 Aston University