INDUSTRIAL AND ENGINEERING CHEMISTRY
498
GRAPHICAL WETHODS
The problem is to know Q1,Q,, Q3, etc., and SI,SB,Sa,etc., in Equations 16 and 17 (types 1 and 2 , respectively) and how t o obtain PI,P,, Pt, etc., by graphical means. The following methods have been devised for the two types of equations: FOREQUATIONS OF TYPE1. On any graph paper, assign the whole length or suitable length of abscissa t o A B as unity, as indicated in Figure 1. Ignoring the above assignment, plot points &, Sz, S3,etc., arranged in ascending order, on the abscissa by choosing a suitable scale. It is obTious that if 81, 81, 8 3 , etc., are arranged in ascending order, Q1, Q2, Q 3 , etc., will be in random order. Plot point a so that its horizontal ordinate equals SI and its vertical distance from A B equals Ql. Connect points Aa and prolong t o intercept line BB’ a t D. Draw line CD parallel to AB. Plot point b so t h a t its horizontal distance equals Ss and its vertical distance from line CD equals Q2. Connect Cb which cuts line BB‘ a t F . The process is repeated until the last point (d in Figure 1) is plotted and the final intercept on line BB‘, I , is obtained. Connect AI which intercepts horizontal lines CD, EF, and GH a t C’, E’, and G‘, respectively. Points A’, D’, and F’ are their corresponding vertical projections on lines A B , CD, and EF. Then AA’, C‘D’, E‘F‘, and G‘H mill equal PI, Pz, P3, and P4 required, if they are read off from the graph, according to the assignment t h a t A B = 1. The proof is simple. Since A B is assigned unity (or any constant), it is obvious from simple geometry that BD, DF, F H , and HI equal Ql/Sl,Q2/S2,Q3/S3,and Q4/S4,respectively, and BI equals z(Q/S). Again, by simple geometry, it is apparent t h a t AA‘, C‘D‘, E’F‘, and G‘H equal (Ql/Sl)/J (&dX2)/’ ( & 3 / 8 3 ) /’ (&/S) and (Q4 ”4) z(Q/S)or PI,P,, Pa, and Pd, respectively.
’
(&/s),
horizontal ordinate equal to SZ. Draw line EF passing b’ and repeat the process until the last point 1 is obtained. Join A I , then AA’, C’D’, E’F’, and G’H, read from the graph considering A B = 1, will give the fractions P I , Pu,P3, and P 4 sought. The proof is simple and similar to that for type 1.
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Vol. 35, No. 4
A
D
A’
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Figure 2
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Although only a quaternary example has been chosen for Figures 1 and 2 , it is obvious that they will serve any polycomponent system equally well. The methods may seem ’complicated. They are simple to operate after the fundamentals involved are mastered. The methods should prove useful for such cases as a quaternary system in which compositions of two components are fixed while the other two vary.
G
NOMEUCLATURE
V
I IF’
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C
P
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A‘
SI
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53
54
molecular weight p density W = weight fraction V = volume fraction X = mole fraction U = part byweight C‘ = numberofmoles c”’ = part by volume Q = a quantitj nhich represents TY, V , X , C , ,c‘; or C” S = a quantity which represents Mj p , ( X / p ) or unity P = a quantity nhich represents W , V, ot X Subscripts 1, 2, 3, 4, etc., = components 1, 2, 3, 4, etc., respectively, of polycomponent system = =
COXTRIRUCION 476 from t h e Department of Chemistry, Cniverslty of Pittsburgh.
Figure 1
FOREQUATIONS OF TYPE2 . As in type 1, consider A B unity. Ignore the assignment temporarily and plot 81,S,, 8 3 , and S 4 in ascending order on abscissa A B (Figure 2). Plot point a on line BB’ so that Ba = Q1. Join Aa and locate point a’ on it so that its horizontal ordinate equals SI. Draw line CD passing through a’ parallel to AB. Similarly, locate h on BB‘ so that Db = &2 and locate point b‘ on line Ch Mith a
Pure Hydrocarbons from Petroleum (Correction) In this article in the February issue there is an error of referencing. On page 250, first column, second paragraph, eighth line, “Table I1 (14)” should be “Table I1 (15)”. This may seem a minor error, but is called to the reader’s attention fiince it is followed by considerable disc&ion and cit,Rtion of data from JOHW GRISWOLD that, reference.
