The most widely used equation in thermal diffusion is discussed in what concerns its range of validity.
It is shown that when the separation curves are non-symmetrical with respect to the initial composition the error involved in the Hoffman and Emery equation when using a value of t=0.3 tr for the lower limit
of the time, may be significantly larger than those predicted by their original authors.
Experimental cases described in the literature are discussed on the light of previous analysis

Morgado, M. Fátima

Pinheiro, J. D. R. S.

Romero, Joaquim José Barbosa

Universidade do Minho: RepositoriUM

THE VALIDITY OF THE HOFFMAN AND EMERY EQUATION IN THERMAL DIFFUSION :·I. F • L. S. i•IORGADO J,de D.R.s. PINHEIRO J,J.B. ROMERO Centro de Quimica Pura e Aplicada da Universidade do Minho,Ilraga,Portugal ABSTRACT The most widely used equation in thermal diffusi9n is discussed in what concerns its range of validity. It is shown that when the separation curves are non--symmetrical with respect to the initial composition the error involved in the Hoffman and Emery equation when using a value of t=0.3 tr for the lower limit of the time, may be significantly larger than those predicted by their original authors. Experimental cases described in the literature are discussed on the light of previous analysis. INTRODUCTION Thermal diffusion techniques have been increasingly used not only to separate difficult mixtures but also as a means of testing kinetic theories of the liquid state(l). In many instances, though, discre-pancies have been observed between the experimental results and the established phenomenological thee-ries of the columns, leading sometimes to the intr£ duction of "correction factors" into the phenomeno-logical equations involved. Of these, one of the most widely used is the so-called Hoffman and Eme-ry equation which relates the degree of separation between the two ends of the column to the physical variables affecting the process, including the time. Hoffman and Emery(Z) claim that the usual simpli-fied form of their equation may be used with an error less than 1% for separation WÙÜŸVĚgreater than 0.3 tr' where tr is a relaxation time that measures the rate of approach of equilibrium, or, roughly, the time required for the degree of VŤŮŠŲŸĚtion to attain 70% of its steady-state value. This 34.1 lower limit of the time - 0.3 tr - has, however, been questioned by Pinheiro(3)who showed that the criteria to estimate this lower limit should also take into account the value of the parameter :\ - the "separation potential" of the system. Otherwise, the error introduced in the simplified form of the Hoffman and Emery equation for t=0.3. tr could 'be much larger than 1%. Pinheiro dis-cussed only symmetrical separations of equimolar mixtures. Yet, the vast majority of separations are non-symmetrical and non-equimolar, i.e., the value of is different from c - c o B and c I 0.5 (the subscripts mean, respectively: 0 - T top, B = bottom O= feed compositions)which would apparently reinforce Pinheiro's comments. The present paper is, then, an attempt to shed further light into the subject by considering firstlydifferent kinds of theoretical cases and discussing, finally, experimental situations re-ported in the literature. THE HOFFMAN AND EMERY EQUATION Derivation In a batch thermogravitational column, the con-centration profile along the column length as a function of the time is given by the following equation(4 ) 2 µ ac = - H (1 - 2c) ŸĚ+ K a c 3t az az7 with µ (1) (2) 8/4 c - Z/2 H • ll SpH & B (AT) 2 (2w) 3 6 , < T > u CJ) • CZ,8) • e Ce"- eco>.) Ÿ·Ěo>._ 12 8 Z/2 + • + >. (4) (S) 9 ! D u2 (6) (••• Nomenclature for phyalcal .. anlna of the va-rlablH) Equation (l) was linearized by Hajundar(S)who firs tly introdured the dimensionless variables z • H z -K-0 • H2t ;-K so that equation (l) becomes ac ac a2 c Ta" • - (l - 2c) -;;--z + --o a z2 and afterwards used the transformation (z e) • ! + ! all c , 2 p 3Z where tp • tp (Z, 0) (7) (8) (9) (10) (11) to obtain the following equation linearized in tp : 1L. ii_ a 0 ŸĚ (12) Hajundar integrated equation (10) uaing the Laplace Transform to get: 34.2 e - 1 2 n -b >. • ip. Cf- bo > + I 1 - C-1) • 0 n•l 2 2 2 c1 + nw > Cb 2 + n w ) r;:r- 0 --;:r n2 •2 nw (exp •-:y- e). aln-r >. ... (13) where >. 11 the adl .. nalonal lenath of the colUllll (for Z • t.) and - c 0 (14) Now, to determine the degree of separation A , i.e., the difference between the concentration at the top (cT) and bottom (c8) of the column, one evaluates a p /az from equation (13) substi-tutes this value into equation (10) and obtains c8 for z-e and cT for Z•>.. In doing so, the degree of separation is obtai-ned in terms of a sum of two terms, one of which involves a infinite series, as :-where c >. >. c0 >. (e 0 -l)(e - e ) + n b >. 2 (-1) e 0 - l • -7 ( + -b!>· -b ! 1 -(-l)n e 0 n2 n•l ĤĴĴĦĦĦĦŸŸĤĴĦŚĤŚĴĴĴĦĦĦĚ__ 2 2 ( l + n w ) T --;_r exp (-2 2 _1_ + !!,.._,!,__ 4 >.2 2 l + -2!:... 4 2 2 2 ( b0 + n w ŸĚ(15) (16) i1 the relaxation time. Hoffman and Emery claim that for times greater than 0.3 t all the terms of the infinite series r in equation (15) can be neglected with an error less than 1%. This rapid convergence enables the simplification of equation (15) to -tit e r) (17) where 600 is the steady-state degree of separation: COA A c A =(e -1) (e - e 0 ) 600 c ii e 0 (e 11 - 1) (18) and k3 is a coefficient given by 8 -2-1T 2 c, A c \ _TI_)(e o -l)(eil- e o) \2 (19) Equation (17) is the so-called Hoffman and Emery equation. Validity In obtaining equation (17), Hoffman and Emery in-troduced the limitation of t > 0.3 tr so that,and with an error less than 1% as claimed, a simplifEd form for the degree of separation could be ŸWWŠÙŪŤTĚHoffman and Emery have based their criterium on the analysis of the two most important factors concerning the convergence of the infinite series in equation (15) :- n and t/tr. The other param!:_ ters involved, b0 and A were neglected in the analysis. In what concerns b , it may be easily seen that, 0 in fact, its value is of no importance for the series convergence. Yet, for \ the same is not true since it affects remarkably the relative ma-gnitude of the terms of the series. To define new criteria for the lower limit of vali dity of the Hoffman and Emery equation it is ŪŤȘŤŸĚsary to analyse further the infinite series invol-ved in equation (15). In doing so it may be noti-34.3 ced that - Regardless of the value of c0, the even terms are positive and the odd terms are negative. Thus, the series may be thought as composed of two sub-series, one being positive (n even) and the other negative (n odd). - Both sub-series are rapidly convergent with n - The absolute values of the negative sub-series are much larger than those involved in the po-sitive sub-series and therefore the convergen-ce of the series may be studied with relation solely to the odd terms In neglecting all the terms beyond the first, an error is implicitly introduced, which may be defined as £ (%) 'f f ( ) n=2 n ------- x 100 (20) 'f nr-1 f (n) where f (n) goes for the argument of the infini-te series. Rejlresenting the ith_term of the series as fi(n) and noting that for n-even and n >3 the terms become negligible as compared to the first term. equation (20) becomes t: (%) f 3 (n) ------- x 100 f 1 (n) + f 3 (n) (21) But, since f 1 (n) » f 3 (n) one may si:'.1ply write f 3 (n) E (%) X 100 (22) f 1 (n) is therefore a measure of the error involved in neglecting all the terms of the series beyond the first. From equation (15) it is seen that its value will depend strongly on t/tr and \. Now, the influence of A may be analysed in two different ways, either a) Making E = 1% and evaluating t/tr vs. A or b) Making t/tr 0.3 and evaluating E vs. \ In this work an attempt is made to tackle both approaches since both are of interest:- Case b) to give an idea of the errors possibly committed in the past and case a) to provide some guide-lines for the future. SYMMETRICAL SEPARATIONS Symmetrical separations ocurr when the separation attained in the upper half of the WUŤŲÜŬŦŲŠẂÙWŠWÙŸĚnal column equals the separation in the lower halt, i.e.' (21) c - c T ·O Symmetrical separations have been reported for se-veral hydrocarbon mixtures( 6)for which the depen-dence of 6 on the composition is a parabola. For symmetrical separations the value of \ ŠVVŬȘÙŸĚted with a certain degree of separation may be obtained theoretically either through equation(l8l or more accurately using its definition(4) ln (24) ŴUŸŲŤĚ the subscript w indicates a steady-state condition, i . e. , 600 = c - c T B 00 The results obtained using equation (24) are shown in Fig 1 for several values of the initial composi_ tion c • 0 (Note that the curves obtained for c0 are exactly tlw same as obtained for 1-c ) ,, l·--... 1 •• FIGURE 1 - The relationship between 600 and,>. for various values of feed composition in symmetrical separations. 34.4 It is seen that regardless of the values of c0 or 600 involved the values of ;\ may vary from zero to infinite, although in pratice they re-main relatively small. In fact, equation (24) clearly shows that;\ only reaches high values if one of the extreme compo-sitions (cT or cB) approaches either zero or unity, u 11., FIGURE 2 - ŸẂVĦĚ t/tr for different values of '· for c0 0.2 and symmetrical se-paration Thus if, for instance, cB - 0, then cT - 2 c0 and equation (24) becomes 2 c /(1-2 c ) >. = ln ŸĤĤGÌGĤĤŸŸŸÌŸĚCB (25) clearly showing that when cB approaches zero, \ approaches infinite. One may also conclude that when c0 (24) simplifies to 2 ln 0.5 equation (26) It is now possible for different values of the initial composition c0, to evaluate a) E vs. \ for t/tr = 0.3 and b) t/tr vs.;\ for E 1% ÍĤĤĤŸ·ÌÍ ÍĚFIGURE 3 - E vs. t/tr for different values of A for c = 0.3 and symmetrical separation 0 Tbe results obtained are presented graphically in Figs.2 to 4 where it may be seen that in the vast majority of cases (for which A is not large, i. e. A< 5 say),the error E although larger than 1%, is comparatively small and most probably smaller than the usual experimental error. One may then conclude, that the Hoffman and Emery equation represents fairly accurately the symmetri cal separatiuns. " t/tr FlGuRE 4 - , vs. t/t fur Jiffc·r ... nt values of ), r for c0 = 0.4 and symmetrical separation NON - SYMMETRICAL SEPARATIONS Non-symmetrical separations, i.e., when 34.5 are, by far, the most common case in practice, though in many instances the "degree of assyme-try" (7) is smal 1. Theoretical discussion For non-symmetrical separations it is virtually impossible to use equation (24) to evaluate the value of ), associated with a given value of 600 mainly because there are several pairs of values (cT, cB) that may satisfy the contition 600 = cT-1 ... ____ , • -----ŸĚ. ., • •• $ •• . ,, FIGURE 5 - Ihe relationship between 600 and A for various values of feed composition in non-symmetrical separations To overcome this difficulty one may use equation (18) to determine the dependence of A on 600 for various values of c • The consequent results are 0 shown in Fig. S, where it may be noticed that (contrary to what happened with the symmetrical situation) the values of A may reach relatively high values, even though the end compositious may be far from zero or unity, FIGURE 6 - E vs, t/t for different values of \ r for c0= 0.1 and non-synunetrical separa-tion Not surprisingly the curves of c vs. t/t for r different values of \ and c0 (Figs. 6 to 10) show a quite distinct behavior. In fact, for t/tr 0.3 the corresponding values of c are larger than 1% even for moderate values of \ (or, which is the same, E = 1% only for t/t r CD .. CD .. CD " I) " CD " c II 1/;r FIGURE 7 - £ vs. t/t for different values of A r for c0 = 0.2 and non-synunetrical sepa-ration Experimental cases It is interesting to analyse the range of values of \ and c that have been involved in experimen-o tal separations reported in the literature in order to estimate (with the help of Figs. 5 to 34.6 10) the implicit values of E involved in such cases, FIGURE 8 - E vs, t/t for different values of A r for c0 = 0.3 and non-synunetrical sep!!_ ration A summary of this analysis is presented in Table 1 where one may clearly note that the va-lues of E for t/tr = 0.3 is, in some instances, of the order of 10% or more. This may explain some of the discrepancies that have been repor-ted when attempting to the experimental results to the Hoffman and Emery equation. " lltr 11 .. " " " II flGURE 9 - £ vs, t/t for different ẂŠŨẀŤŸĚ uf \ r for c0 = 0,4 and non-synunetrical separation GHĤĤĤŸĤĤĤĤĦĦĶĶĶĶĶĤĤĤĤĤĤ! \ Mon-Symm•lrocal MPQratOll-Co•O 5 I E {'I.) FIGURE 10 - E vs. t/t for different values of \ r for c0 = 0.5 and non-symmetrical sepa-ration CONCLUSIONS The use of the Hoffman and Emery equation to corre late the experimental separation vs. time curve in volves an error of the order of 1% for t/t = 0.3 r when the separation is symmetrical. For non-sy-mmetrical separations, the error involved for t/ /tr = 0.3 is generally quater than 1%, its value increasing with the value of \ and with the "degree of assymetry" of the curve relating the evolution of cT and cB with the time. It is therefore strongly recomended that for values of A greater than about 4-5 an inspection of the lower limit of time is carried out before using the Hoffman and Emery equation to correlate the experimental data, NOTATION b = 1/2 - c 0 0 B column dimension in the horizontal non--thermogravitational direction c molar fraction of the reference component c0 feed composition CT top composition cB botton composition D ordinary diffusion coefficient g accelaration of gravity 34.7 H thermogravitational transport coefficient k3 coefficient defined by eq. (19) K transport coefficient associated with the parasitic remixing K c transport coefficient associated with the convective remixing Kd transport coefficient associated with the ordinary back-diffusion L column length t time t r T ŸØĚrelaxation time absolute temperature temperature difference between the column walls <T> average absolute temperature Greek Letters a thermal diffusion factor E \l w temperature coefficient of density degree of separation = equilibrium degree of separation truncation error deiined by eq. ( 20 ) dimensionless length = 504 x MǾIĮŦŸØŇÍŴĞĚ4 number of moles per unit of column length viscosity one-half of the distance between the hot and cold walls molar concentration of the solution T A B L E I c: < 1% t/t = 0.3 Mixture Ref-ce r c t/t c: % 0 r n-dodecane/ carbon tetrachloride (8) 0.4846 7.65 0.98 9.02 n-heptane/cetane (9) 0.6659 2.31 0.36 1.52 n-octane/decane (9) 0.5459 1.37 0,32 1.17 iso-octane/n-octane (9) 0.4961 0.73 0.31 1.05 n-heptane/triptane (9) 0,4977 5.14 0.59 4.10 n-hexane/carbon tetrachloride (8) 0.6599 13.74 2.95 39.14 n-heptane/carbon tetrachloride (8) 0.6346 12.28 2.31 28.30 n-octane/carbon tetrachloride (8) 0.6133 8,25 1.13 11.34 cumene/cetene (10) 0.324 0.63 0.31 1.04 cumene/cetene (10) 0.324 0.89 0.31 1.08 cumene/cetene (10) 0.324 1.09 0.31 1.11 cumene/cetene (10) 0.324 1.17 0.32 1.13 cumene/cetene (10) 0.324 1.20 0.32 1.14 cumene/cetene (10) 0,317 0.77 0.31 1.06 cumene/cetene (10) 0.317 1.11 0.31 1.12 cumene/cetene (10) 0.317 1.85 0.34 1.33 cumene/cetene (10) 0.317 2.32 0,36 1.53 cumene/cetene (10) 0.317 2.41 0.37 1.58 cumene/cetene (10) 0.317 3.14 0.41 2.04 cumene/cetene (10) 0.317 3.30 0.42 2.17 benzene/n-heptane (8) 0.484 5.71 0.67 5.01 toluene/n-hexane (8) 0.5268 4,60 0.53 3.38 o-Xylene/n-hexane (8) 0.5816 5.33 0.62 4.45 benzene/ carbon tetrachloride (8) 0.6638 9.36 1.43 16.44 benzene/ carbon tetrachloride (9) 0,5211 6.09 0.72 5.67 benzyl alcohol/ethylenodiol (9) o. 7035 1.20 0.32 1.14 cumene/methyl-naphtalene (9) 0.5028 1.46 0,32 1.19 Folvene/chlorobenzene (9) 0.4974 l.ll 0.31 l.ll S-methylnaphtalene/a-methyl-naplitalene (9) 0.4936 0.52 0.30 1.03 cetane/benzene (9) o. 7670 1.48 0.33 1.22 cetane/toluene (9) 0.7332 0.61 0.31 1.04 34.8 T A B L E I (Cont.) Mixture Ref-ce cetanelm-xylene (9) n-heptanelbenzene (9) n-heptanelmethylcyclohexane (9) cyclohexanelcarbon tetrachlo ride (9) cyclohexaneln-hexane (8) REFERENCES (1) STORY, M. and TURNER, J. Trans. Far. Soc., ŸĦĚ1523, 349, (1969) (2) HOFFMAN, D. and EMERY J?, A.H., A.I.Ch.E. Journal .2_, 653 (1963). (3) PINHEIRO, J.de D.R.S., M.Sc. Thesis, Birmingham University (UK), 1974 (4) ROMERO, J.J.B., Rev. Fis.Quim. Eng. ULM, vol. II, .!_, Ser.A, 1 (1970) (5) MAJUMDAR, S.D., Phys. Rev. g, 844, (1955) (6) BOTT, T.R. and WHYSALL, M. J.Chem.EngnJapan !_, 167 (1974) (7) PINHEIRO, J,de D.R.S., Ph.D.Thesis, Birmingham University (UK), 1976 (8) NIKOLAEV, B., NIKOLAEV, N. and TUBIN, A. theor. J. Chem. Engn ŸĦĚ 144, (1977) (9) JONES, A.L. and MILBERGER, E., Ind. Eng. Chem. ŸĦĚ2689, (1953) (10) HOFFMAN, D., Ph.D. Thesis, Purdue University (USA), 1959 c 0 0. 7043 0.6228 0.5349 0.6471 0.4759 34.9 E < 17, tit = 0.3 r tit r £ % 0.73 0.31 1.06 6.42 0.78 6.64 5.16 0.59 4 .14 7 .96 1.08 10.95 7.95 1.04 9.78 CHEMP0R'78 INTERNATIONAL CHEMICAL ENGINEERING CONFERENCE University of Minho Braga, Portugal September 10-16 19 78 CHEMPOR'78 REUNIAO INTERNACIONAL DE ENGENHARIA QUiMICA Universidade do Minho Braga, Portugal Setembro 10-16 1978 

The validity of the Hoffman and Emery equation in thermal diffusion

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