Dimensional Analysis, New Approach

DIMENSIONAL ANALYSIS , NEW APPROACH Naji m O . Salim AL-Gazali Civil Engrg. Departmellt, Baby/oil UniversNy Ra fn H. AL-Sn\l a ili Dept. ofEll vi/'ollme~l(al Engrg.. Col/ege of Eng/g., Baghdad Univ. Abstra c t A new method for dimcnsiO Il;Ji analys is thnt Co1n be easily progra mmed was developed. T he method call be used with significant s<lving ofHme, ,1fld avoiding hand calculations. The method \V,IS checked by using an illustrative example, I II trod uctio n Dimensional homogeneity provides clues to the form of the relat ion connecting the quant ities involved in a physical problem. Knowledge of the fo rm of the relation is valullblc in suggesting the pattern that a series of experi ments could most usefully take. The search fo r the correct form of the relation is well known as dimensional analysis. The methods o f the dimensional analysis are of very wide applications. Those methods are especially valuable in the study of those phenomena that are too complex fo r complete theoretical treatment. Such complexity results fro m a large number of independent variables affecting the phenomena, or fro m a mathematically complicated form of the expressions relating those variables. The mathemati cal relat ionship connecting the magnitudes of quantities affecting a particular phenomena can be pu t in the general form: Q, = <I>(Q,.Q,.Q,.... ) in which : Ql "" the magni tude of the quantity of special interest. Q2,Q.~, ... = the magnitudes of other quanti ties. Provid that the fUllction is a continues one, it may be expressed as fo ll ows : QI = K I Q ~I Q~I Q~I ... + K 2 Q;~ Q!! Q~~ ... + ... (series expansion theory) where K),K 2,K),.. arc numerics (dimenlionless). By the principle of dimentional homogeneity, all terms in the series must have the same di mension. Morever, ·the dimensio n of(Q,) must be the same as that of each term of series . Thus [Q,]= [Q; Q: Q: J As it is known Rayleigh's (AL-Bazaz and AL-Khazraji, 199 7), Bukingham's (Streeter, 1975), Langhaar's (Langhaar, 1951), Barr's (Barr, 197 1) and Hunsaker and Rightmire's (Streeter, 1975) rr.ethods are used fo r dimensional analysis. All these methods need band calculations. As a result, error may occur al any slep of calculations as well as these methods are lime consuming, especially whe n the number of variab les is large. The aim of this paper is to fo rmu late a method Ihat can be progf<'ull med to save time and avoid hand calculat ions. In order to int roduce the methods superiority, a general example is taken and it is solved by alll11ethods mentioned abovc wi th brief explana.!ion. r··· , 0 ,xl 10 0l;oi l""oidJ pgkJ 1 JL ii<.h ok Relyieh method (1892) Suppose that the initial relasionship between variables is <I>(A,B,C,D,E,F,G) ~ O where : A(M L"IT"'), B(M L'" P ), C(M L"I T"I) , D (M L T'" a" I), E(a), F(M T'" a"l) & G(L) and it is required to obtain (A) equation. We seek an equation of the form A ~ t1>(B ,C,D,E,F,G) the function is expressible as aseries of terms each of the form : K Bb CC Dd Ee Ff Gg A = Kf Bbl eel Ddt E el FI L G!;l + K2 Bb2 ee2 Ddl Ee2 F/2 Gt2 + ... [ series expansion theory] as each must have the same dimension, we need co nsider one term only [A] ~ [B' CO D' E' FI G'] [ ML"I r'J equating the exponents af M : I ~ b + c + d + f ""." ... ."" ... "".. ... .""..."" ."".. ".. .. "."... "... "." .. ."".. .. ... .... .. ....... ". (1) equating the exponents of L : - 1 ~ -2b -c + d + g .. "... "... ""." .. ,,,,,,,, .. , , , .,, , .... .."." .. "" .. .. ."... """ ... .,,, ... ,,.,, .. , .. ,,,,,,,, (2) equating the exponents of T : -2 ~ -2b - c -Jd - J f .. .. .. .. ........ "" ........ ".. "................. .. .... .... "...... .. ....... (J) oce : equating the exponents o ~ -d + e - f ........ ...... "...... "............ "" .. ".. "" .... "............................ "...... (4) With (6) unknowns but only four equations, a complete soluti on is not possible. However. we can determine four of the exponents in terms of remaining two. The choice of exponents to remain unknowns is arbitary, but let us select d and f . .... "... ."." ." .. "... ......... ."...... .. .. .... ... ."............ , (5) ............. "" ................... "........ "" ..................... (6) """" .... (7) g~ 1-( 2d therefore + Jf 2 ) ................................................... .................. (8) r·- , 0 ...~! + r,_ $1 o l e _~ +.'1... 1 ,,.J 10 '""" ,.....".,. D <I, E",+r, F r, G pgkI , J4 ad, iii,. . _l~'_":~l. 1 A=BG Bukigham 1t T heo rem This theory as formulated by Bukingham in 1915, states that if a physical phenomenon involves (n) quantities and if these quantities involve (m) fundamental dimensions, then the quantities can be arranged into (n·m) dimentionless parameters. Let Ql. Q2•... Q~ be the quantities involved, then ~(Q"Q" ... Q") = 0 If the number of the fundamental di mensions in the above (n) quantities is m, then the relation $(11:1. 1t1 •..•• 1tn."') CI 0 Where each 1t-term is a non-dimensional product involving the above quantities . Each of the it-terms should be exist. Returning to the previous example. Suppose that the repeating variables are A,B,C and D. no of variables (n) = 7 no, of basic dirnentions(m) = 4 no, of dimenio nless groups (1t ) = 7-4 :: 3 1[1 = A· I BioI eel Ddl E A a2 Bb2 eel Dd2 F Tt~ "" Ad at..1 C"~ Dd~ G [ itl]'" [A, I Shl eel Ddl E] ;[2 = [M(lLo TUG"]" [ML'I r1r l [ML-21'2JIoI [ML'I TIt [ML r~ G'l t [9] equating the exponents of M : O=al +bl +cl +d .......... .. equating the exponents of L : O= -al-2b l- cl +d l .............. .. equating the exponents orT : .... ....... .(1 ) ...... ............. (2) 0=-2a l -2b l -cl-3dl ............................................................ (3) equating the exponents ofS : 0=I·dl .............................. ......................................................... (4) solvillg the four equations gives a I = -4, b I "" 2, c l ::::: I, d I "" I then Ttl = A '~ ll2 e 0 E r· .... The same procedure can be followed to obtain 1t2 TCJ =A =A I Dol F B G 'ltJ) "" 0 TCl and ".Jd , J4 iW4 db.. TCl :- a -I ¢I(1t l, 1tl, :. 41( oI,a.I ' ......"... 0 , ,.I 10 A·F BD A. S '; C·l D 'l • G )-0 AB " - Langhaar's meth od (ma t rix oper ation procedure) I . Put the repeating variables on the left in the dimentional matrix as shown in Fig. I . 2. Constru ct the aux.illary matrix using the following steps. a. Write the lower Irigaular matrix and put the assumed variables (H, I,J,K) and dimensions as in Fig. 2. b. Find exponents of dimensions to correspond wi th the base dimenions of the vari ab les in the fi rst four columns from upper to lower as shown in Fig. 3. c. Find the dimensions of the remai ning columns using the dimensions to the left as shown in Fig. 4. 3. Put the repeati ng variables to the left in terms of the,. new variables (H, I,J ,K) and wri te the unit matrix as shown in Fig. 5. 4. Obtain the dimensio n of the remaining variables fro m down to up as shown in Fig . 6. 5. Obtai n the dimensionless groups by making all the unrepeated variables as a Ilumerator and the reprating variabl es raised to their exponents as a denominater as follows : e (E F~) ~O A "13- 2 C- I O- I ' A -IBO ' A B- l A I -I M L T 0 -2 ( II) MLTO ( I) LTO ;~go J{ 0 {1l)ML O ' rl 0· (I) V ro· (J )TO· (I{) 0·' ( I-I )M L·' Y ' O {I> L ' (J)T (I{lO" 0 I 0 II " I -2 -2 0 C 0 0 0 I I I I 0 I I 0 0 0 0 I I 0 0 0 0 0 0 0 A - II D .. III I 0 C - IIJ D .. 111 01 J O' K 0 0 A- H I 0 0 0 B " HI 0 C- IIJ O" lI r 1 J .'J( 0 0 I 0 II -3 0 -3 -I I -I 0 0 0 I 0 G I II D I I I 0 0 0 0 F I -I -I I 0 I I E 0 I Fig. I Dimensional m:lI ri:l: Fig. 2 I I ·2 ·1 I I -2 -I I 0 Fi l;.3 0 0 0 1 -I -I -I II -I 1 0 O. 0 0 Fig04 (a uxillury m a l ,olx) Fi:':o5 0 I I 0 0 0 0 -2 0 I II 0 0 I -I -I 4 -I I I -I 0 0 I II Fi~o (, Dan" s Method (Proportionnlities Method) ¢leB, A. C, D. E. F) = 0 (Initial co ndition) I) Reduce the dynamic terms to kinetic terms by dividing them by a variable containing M in its dimensions. Let this variable be A in the previous example, then write the dimensions of each new variable as follows: ~ (B/A, CIA, D/A, FIA, E, G) L'] T L 1T 1S'l Lyle-I e L 2) Multiply each of the third (D/A) and fourth eFtA) terms by the nfth term (E) to eliminate from the dimensions, then write the dimensions of each new variable as a follows : $(BIA, CIA, DElA, FElA, G) 3) In order to eliminate T, multiply terms three (DE/A) and four (FEtA) by term two (CIA). then write the dimensions or each new variable as follows: $ (BIA ,DECIA', FECIA' , G) L'I L2 L L 4) In order to have"dimension less g roups, mu ltiply term one (B/A) by (G), divide the square root of term two (DEC/ A2) over (G) and divide term three (FEC/A 2) over (G), Result is the folowing : ~( Jf5EC BG ' A FEC) = 0 A G ' Al G for the sake of comparison with other methods, the di mensionless groups are rearr<lllged , This arrangement can be done sice all the terms are dimensionless. The rearranged groups are as rollows : ~(A~' A'G EC 'A'Gr;EC /'\ A-I BD C:::::=::I Q AG ,JDC -A'--G=-I-,-F.j"D""'C'-) = 0 A~ & S'2 C l D'I thererore ~ , ' Aie,:=-=0:- $( A Note : C- Q == Dimensions are equal to Hunsa ker and rightmirc method, (1947) A ra pid method (streeter, 1975) for obtaining 1t parameters, developed by Hunsaker and right mire uses the repeating variables as primary quantities and solve for M,L,T and in terms of them , Returning to the previouse example: e G= L E=S CIA=T F =MT 'S" L = G S = E T - C I A M = FEC'IA' ()=ML'~T '~= FEel A) * G ': · c-~ A '~ FEC =--;::-i=-G: A A'G FEC D 'Ttl = FG ~Q FG AB G' EC Q G A B,1 A ' BD T he new method To explain the new method leI us take the previous example : ProcctillJ"C 1· W rite the "Di mentional M atrix" such that the repeating variables arc pur to the as shown in Fig. 7. 2· A. divide all first row elements by the firs t number all the len (in this case it is cq unl 10 I ) . B. Make the elements remained on the fi rst column equal to zero by usi ng gausselimination or any other method . The result is as shown in Fig. S. 3, A. Divide all second row clements by the second number on the leO (i n this case it is equal 10 (-I)) . B. Make the elements remai ned for the second column equal to zero. The result is as shown in Fig. 9 , 4. Repeal the same procedure for row 3, the result is as shown in Fig. 10, and fo r row 4 the result is as shown in Fig. II , fig. 11 is known as a solut ion matrix . 5. As the number of variables m = 7 and the number of repealing variables n = 4, the number of dimensionless groups is q = III • n = ?; 4 "" 3. These dimensionless groups eM be obtained by making all the un repeated variables as a neumerator and repea ling variables raised to thei r exponents as a dcnomilli\lor as follows ; G)-0 AB" Note - If there is zero element on the matrix diagonal during transforming it to unit matrix change the positions of the repeating variables in the dimensional matrix. A M 1 L -I -2 T e 0 1 0 0 0 1 A U C D B 1 . -2 -2 0 1 -I 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 C 1 -I -1 0 1 0 1 0 1 0 1 0 0 0 1 0 0 0 1 0 D 1 1 -3 -I 1 2 -I -I 3 -2 -I -I 4 -2 -I -I 0 0 0 1 E F 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 1 4 -2 -I -I -3 -I 1 1 -I -I 2 -I -1 -1 3 -1 -I -I -I 1 0 1 .. G 0 1 0 0 0 1 0 0 1 -I 0 0 1 -1 0 0 1 -1 0 0 Fig. 7 Dimensional Mntdx Fig. 8 Fig. 9 -Fig. to Fig. 11 Soluti on Matri x Concl usion T he following conclusion can be drowned : The developed method is easily programmed with exact resu lts. Moreover, it requires much less compu tation efTorts. than other available methods. Acknow lcdcgements Thanks are due to Dr. Salah Tawfeck, Assis. Professor of civil Engineering, Babylon University, and Dr, Kassim H. Jaiul, Associate Prof. ofeivil Engineering. Babyloll University, fo r their encouragement , reading and comments concerned with thi s paper. References 1- AL-Bazaz, S.T. and AL-Khazraji, K., ( 1997)," Analysis of Experimental Data", Babylon University Press, Babylon. 2- Barr, D.l.H.V., ( 1971), "The Proportionali ti es Method of Oimenlional Analysis", Jornual of tile Frank li n fnstitute, Vol. 292. 3- 13ridgnmn, P.W., (1963), "Di mcnsional Ana lysis". Yalc University Press, Ynlc. 4· Lnaghaar. H.L.• (1951), "Dimensional Analysis and Theory of Models", Wiley, New York. . 5- Massey, B.S. (1971), 'Units, Dimensional Analysis and Physical Similarity', Van Nastrand, Reinhold. ~ Streeter, V.L.• and Wdcy, E.B,. (1975). "Fluid Mcchanics". McGraw-Hill,Tokyo. Japan. APPENDIX I PROGRAM: The following is a simple program since it is assumed .that there is no uro elements on the matrix diagonal during transfonning it to unit matrix, CLS INPUT 'NO. OF ROWS'; N INPUT "NO. OF COLUMNS"; M DIM A(N,M), F(N), AI(N,M) PRINT "INPUT MATRIX ELEMENTS" FOR 1=1 TO N: FOR 1=1 TO M INPUT A(I,J) : NEXT 1,1 FOR 1=1 TON: FOR 1=1 TOM AI(~I) =A(I,I) I A(I,I) :NEXT I FOR]=I TOM A(I,J) =AI(I,J) : NEXT I FORK=l TON IF KmTTllEN ISO F(K) =-A(K,I) FORZ= I TOM A(K,Z) = A(K,Z) + F(K) • A(I,Z) : NEXT Z 150NEXTK NEXT I FOR 1=1 TO N: FOR 1=1 TO M PRINT A(I.I); NEXT I PRINT : NEXT I ~I r--' ,y,,-WI .,J. ....".. <l.-. t.<. -""" ........' .....,. ,_ >';'...,1. .;= ""'" ,;. .; " .' _ . t..., .. _.:,; ,,,__ ·>o;.:J,,,U""-.:.Ii.J.> 0:1....- .." ~J >! -:T' ~• .. vv . View publication stats '_>,;..L> ..s'f,J- ,,,<, r----