Chapter 2 Solute Transport In Biological SystemsSolution ------------------------------------------------------------------------------------------The solution to this problem requires knowledge of nonlinear curve fitting, a topic reviewedin appendix A. We first need to evaluate the experimental Dm/D or D0/D using the estimatedsolute bulk diffusivity.MW = 15000069000170001200180 D = 1.01310-4 (MW)-0.46 = 0421.00602.01147.03883.09294.0105Therefore (Dm/D)exp = 3086.01661.03487.04378.04842.0The solute radius is computed nexta = 3/143ANMW= 10-63903.03013.01889.00781.00415.0a) Find the effective pore radius r in the Renkin equationDm/D = (1 a/r)25395.009.21.21rarara= f(r)Since the expression depends on the ratio (a/r), we can multiply a by 107 and later divide the value of r obtained by 107. Therefore, let9a = 903.3013.3889.1781.0415.0We need an initial guess for r. Since the largest solute radius is 3.903(10-7), let r = 5.0.Follow the procedure in appendix A[Zj] = rfrfrfrfrf54321 = drrfdrrfdrrfdrrfdrrfdrrfdrrfdrrfdrrfdrrf54321)]()([)]()([)]()([)]()([)]()([ = 10-25856.00050.27505.68633.73399.5[Zj]T[Zj] = 10-2 5856.00050.27505.68633.73399.510-25856.00050.27505.68633.73399.5 = 0.0140(Dm/D)cal = 0038.00184.01208.04843.06955.0{D} = (Dm/D)exp (Dm/D)cal = 3086.01661.03487.04378.04842.0 0038.00184.01208.04843.06955.0 = 3047.01477.02279.00465.02113.010[Zj]T{D} = 10-2 5856.00050.27505.68633.73399.53047.01477.02279.00465.02113.0 =0.0052{A} = {[Zj]T[Zj] \[Zj]T{D}} = 0.0052/0.0140 = 0.3701The next r value is then r = 5 + 0.3701 = 5.3701. The steps are repeated until {A} <.005.Table 2.4-1 lists the Matlab program using the Gauss Newton method of nonlinear curvefitting to find r. The correlation and experimental values are plotted in Figure 2.4-3.______ Table 2.4-1 Matlab program to find r ______% Problem 2.8, Fournier%Mw=[180 1200 17000 69000 150000]';NA=6.023e23;a=1e7*(3*Mw/(4*pi*NA)).^(1/3);D=1.013e-4*(Mw).^(-.46);Dmexp=1e-6*[4.5 1.7 .4 .1 .13]';Dp=Dmexp./D;DmDexp=Dp;%f='((1-a/r).^2).*(1-2.1*a/r+2.09*(a/r).^3-.95*(a/r).^5)';% First guess for rr=5;dr=.01*r;% Note fr = DmDcalfor i=1:20 fr=eval(f);r=r+dr; frp=eval(f);r=r-dr; Zj=(frp-fr)/dr; ZjTZj=Zj'*Zj; ZjTD=Zj'*(DmDexp-fr); dA=ZjTD/ZjTZj; r=r+dA;if abs(dA/r)<.005, break, endendr=r/1e7;fprintf('r(cm) = %g\n',r)x=a/r;xmax=max(x);xmin=min(x);dx=(xmax-xmin)/25;xp=xmin:dx:xmax;fp=((1-xp).^2).*(1-2.1*xp+2.09*xp.^3-.95*xp.^5);plot(xp,fp,x,Dp,'o')xlabel('a/r');ylabel('Dm/D')legend('Correlation','Experiment')>> s2p8r(cm) = 6.29803e-007110 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.800.10.20.30.40.50.60.70.8a/rDm/DCorrelationExperimentFigure 2.4-2 Effective diffusivity in 5% agarose hydrogel.You can also use the MATLAB function fminbnd or fminsearch to curve fit the data. Youfirst must create a function that contains the expression to be minimized. In this case, theminimizing or objective function is[(Dm/D)exp (Dm/D)cal ]2Table 2.4-2 lists the function poren used in fminbnd. ______ Table 2.4-2 Matlab function for fminbnd ______function y=poren(r)a=[.415 .781 1.889 3.013 3.903];DmD=[.4842 .4378 .3487 .1661 .3086];x=a./r;DmDc=((1-x).^2).*(1-2.1*x+2.09*x.^3-.95*x.^5);y=sum((DmD-DmDc).^2);>> rm=fminbnd('poren',4,20)rm = 6.3134b) Find af and in the Brinkman equation12D0/D = K(a/r)r(a/r) = exp21faa2)(3111aaThe Gauss Newton method and the MATLAB function fminsearch (for more than oneparameter) sometimes will not converge. You need to use another method such as Marquardtmethod for better convergence. For the Brinkman equation, we can use MATHCAD i ..0 4Data indexgel diffusivity, cm2/sec solute molecular weightDmi.4.5 106.1.7 106.4 107.1 107.1.3 107MWi18012001700069000150000ai..3 MWi...4 6.023 102313107calculates solute radius=a0.4150.7811.8893.0133.903Di..1.103 104MWi.46calculates solute diffusivity, cm2/sec13Now work with the Brinkman-Ogston equations, assume 5%agarose gel gives a = .05 .05dm,,i af..Di11. ai.13. ai2exp. 1aiaf2SSE , af= 04iln Dmiln dm,,i af2 1af1initial guesses of the model parametersgivenSSE , af01 1afminerr , af= 0.571=af2.787Now make a plot to compare the data with the Brinkman-Ogston equation.The Brinkman-Ogston equations provide a good representation of the diffusivity datain the agarose gel.0 2 41 1081 1071 1061 105Dmidm,,i af,aiaiYou should notice that the values obtained from MATCAD must be corrected af = 2.78710-7 cm and = 0.571107
View Full Document