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clear;
clc;
// Illustration 2.4
// Page: 99
printf('Illustration 2.4 - Page: 99\n\n');
// solution
// Mass Transfer into a Dilute Stream Flowing Under Forced Convection in a Circular Conduit
n = 6; // [number of variables]
// Variables Symbols Dimensions
// Tube diameter D L
// Fluid density row M/L^3
// Fluid viscosity u M/(Lt)
// Fluid velocity v L/t
// Mass diffusivity D_AB L^2/t
// Mass-transfer coefficient kc L/t
// To determine the number of dimensionless parameters to be formed, we must know the rank, r, of the dimensional matrix.
// The dimensional matrix is
DM = [0,0,1,1,0,0;1,1,-3,-1,2,1;-1,-1,0,0,-1,-1];
[E,Q,Z ,stair ,rk]=ereduc(DM,1.d-15);
printf("Rank of matrix is %f\n\n",rk);
//The numbers in the table represent the exponent of M, L, and t in the dimensional expression of each of the six variables involved. For example, the dimensional expression of p is M/Lt; hence the exponents are 1, -1, and -1
// From equation 2.16
i = n-rk; // [number of dimensional groups]
// Let the dimensional groups are pi1, pi2 and pi3
// Therefore pi1 = (D_AB)^a*(row)^b*(D)^c*kc
// pi2 = (D_AB)^d*(row)^e*(D)^f*v
// pi3 = (D_AB)^g*(row)^h*(D)^i*u
// Solving for pi1
// M^0*L^0*t^0 = 1 = (L^2/t)^a*(M/L^3)^b*(L)^c*(L/t)
// Solution of simultaneous equation
function[f]=F(e)
f(1) = 2*e(1)-3*e(2)+e(3)+1;
f(2) = -e(1)-1;
f(3) = -e(2);
funcprot(0);
endfunction
// Initial guess:
e = [0.1 0.8 0.5];
y = fsolve(e,F);
a = y(1);
b = y(2);
c = y(3);
printf("The coefficients of pi1 are %f %f %f\n\n",a,b,c);
// Similarly the coefficients of pi2 and pi3 are calculated
// Therefore we get pi1 = kc*D/D_AB = Sh i.e. Sherwood Number
// pi2 = v*D/D_AB = P_ed i.e. Peclet Number
// pi3 = u/(row*D_AB) = Sc i.e. Schmidt Number
// Dividing pi2 by pi3 gives
// pi2/pi3 = D*v*row/u = Re i.e. Renoylds number
printf('The result of the dimensional analysis of forced-convection mass transfer in a circular conduit indicates that a correlating relation could be of the form\n Sh = function(Re,Sc)\n which is analogous to the heat transfer correlation \n Nu = function(Re,Pr)');
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