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Matlab code:

%this program iterates k to minimize the error in the check equationclose all; clear all; clc;%constantsg=9.81; %this defines the acceleration of gravityDens=1000; %the density of waterGamma=g*Dens; %the specific gravity of water

%the design point parametersN=1500; %the speed of the pump impellerQ=0.03; %the volumetric flow rateH=60; %the pump head

%the calculation procedure%(1) first we calculate the shaft diameterN_s=(N*Q^0.5)/(H^0.75); %the specific speed of the pumpEtta_0=0.625; %from stepanoff graphOmega=2*pi*N/60;Torq=(Gamma*Q*H)/(Etta_0*Omega);S=2*10^6; %assumed allowable shear

D_sh=(Torq/(pi*S/16))^(1/3); %the desired shaft diameterD_shcor=1.2*D_sh %a corrected value for the shaft diameter

%(2) we calculate the impeller dimensionsK_cm2 = 0.083; %from stepanoff diagramK_cm1 = 0.115; %from stepanoff diagramCM_1 = K_cm1*((2*g*H)^0.5); %the flow velocity at inletCM_2 = K_cm2*((2*g*H)^0.5); %the flow velocity at outletEtta_vol = 0.96; %assumption for the vol efficiencyQL = (Q/Etta_vol)-Q; %the leakage flowD_hf = 1.3*D_shcor; %assumption for the fronthub diameterD_hb = 1.35*D_shcor; %assumption for the backhub diameter

Ceye = 0.91*CM_1; %assumption for the velocity at the eyeDeye = ((4*Q)/(pi*Ceye)+D_hf^2)^0.5 %the eye diameterD1 = 1.1*Deye %assumption for the impeller inlet diameterepsilon1 = 0.85; epsilon2 = 0.925; %assumption for the contraction coefficientsat inlet and exitb1 = (Q+QL)/(pi*D1*epsilon1*CM_1); %the width of the impeller at inletDr = D_hf+Deye-D_sh; %the wear-ring diameterEtta_m = 0.96; %assumption for the mechanical efficiencyEtta_h = Etta_0/(Etta_vol*Etta_m) %hydraulic efficiencyt = 0.003 %assumption for the thickness

%now we iterate over k and phi to find the optimum design parameters%that give a consistant design for the pumpphi_e = 0.99:0.001:1.02; %the range for the speed ratiok = 0.65:0.001:0.68; %the range for the pressure coefficienterror = 100*ones(length(k),length(phi_e)); %the error holder matrix

forj = 1:length(phi_e)phi = phi_e(j);U2 = phi*(2*g*H)^0.5; %the impeller velocity at exitD2 = (60*U2)/(pi*N); %the impeller outlet diameter

%(3) calculating the flow anglesU1 = (pi*D1*N)/60; %the impeller velocity at inlet

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beta1 = atand(CM_1/U1); %the angle at inlet

fori = 1:length(k)Hth_inf = H/k(i); %the theoretical headCu2 = (g*Hth_inf)/U2; %the tangential velocity at exitbeta2 = atand(CM_2/(U2-Cu2)); %the angle at exitslipf = k(i)/Etta_h; %assumption for the slip factorCu2_cor = slipf*Cu2; %the corrected tangential velocitybeta2_cor = atand(CM_2/(U2-Cu2_cor)); %the corrected value for the angle

%(4) calculating the number of blades and actual contraction coefficientsbetam = (beta1+beta2)/2; %used incalculating ZZ_calc = 6.5*((D1+D2)/(D1-D2))*sin(betam); %the calculated no of bladesZ = double(uint8(Z_calc)+1);%(5) refining the parameters and dimensions to find the errorr1 = D1/D2;epsi = 1.1*(1+sin((pi/180)*beta2_cor))*r1;Cp = epsi*(2/Z)*(1/(1-r1^2));r2 = CM_2/(2*tan((pi/180)*beta2_cor));RHS = r2+(r2^2+g*(H/Etta_h)*(1+Cp))^0.5;LHS = U2;

error(i,j) = (RHS-LHS)/LHS;end

end[C1,I1] = min(error);[C,I] = min(C1);error(I1(I),I);

%(6) calculating the final parameters for the designphi = phi_e(I) %the speed ratiok_e = k(I1(I)) %the pressure coefficientU2 = phi*(2*g*H)^0.5; %the impeller velocity at exitD2 = (60*U2)/(pi*N) %the impeller outlet diameter

b2 = (Q+QL)/(pi*D2*epsilon2*CM_2); %the impeller width at exitU1 = (pi*D1*N)/60; %the impeller velocity at inletC1 = CM_1; %assuming radial flow at inlet "CU1 = 0"beta1 = atand(CM_1/U1); %the angle at inletbeta1_cor = beta1+2 %accounting for pre-rotationHth_inf = H/k_e; %the theoretical headCu2 = (g*Hth_inf)/U2; %the tangential velocity at exitbeta2 = atand(CM_2/(U2-Cu2)); %the angle at exitslipf = k_e/Etta_h %assumption for the slip factorCu2_cor = slipf*Cu2; %the corrected tangential velocitybeta2_cor = atand(CM_2/(U2-Cu2_cor)) %the corrected value for the anglebetam = (beta1+beta2)/2; %used incalculating ZZ_calc = 6.5*((D1+D2)/(D1-D2))*sin(betam); %the calculated no of bladesZ = double(uint8(Z_calc)+1)epsilon1_cor = (pi*D1-Z*(t/sin((pi/180)*beta1_cor)))/(pi*D1) %actual valueepsilon2_cor = (pi*D2-Z*(t/sin((pi/180)*beta2_cor)))/(pi*D2) %actual valueb1_act = (Q+QL)/(pi*D1*CM_1*epsilon1_cor) %the actual impeller width at inletb2_act = (Q+QL)/(pi*D2*CM_2*epsilon2_cor) %the actual impeller width at exit

%(5) refining the parameters and dimensionsr1 = D1/D2;diameters_ratio = 1/r1epsi = 1.1*(1+sin((pi/180)*beta2_cor))*r1;Cp = epsi*(2/Z)*(1/(1-r1^2));

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r2 = CM_2/(2*tan((pi/180)*beta2_cor));RHS = r2+(r2^2+g*(H/Etta_h)*(1+Cp))^0.5;LHS = U2;error_percentage = abs((RHS-LHS)/LHS)

NS=6; %number of samplesR1=D1/2R7=D2/2

dR=(R7-R1)/NS; %sampling distanceR=@(x) (x*dR)+R1;

dcm=(CM_2-CM_1)/NS; %slope CMCM=@(x) (x*dcm)+CM_1;arr=R(0:6); %sample form the root diameter to the outerdiameterarr2=CM(0:6); %CM form the root diameter to the outer diameteru=@(x) 2*pi*R(x)*N/60; %blade velocity @ samplearr1=u(0:6);w1=sqrt((C1^2)+(U1^2)); %calculate W1w2=sqrt((CM_2^2)+((U2-Cu2_cor)^2)); %calculate W2

dw=(w2-w1)/NS; %slope ww=@(x) (x*dw)+w1;arr3=w(0:6);beta=@(x) asind(CM(x)/w(x)); %beta calc.arr4=[beta(0) beta(1) beta(2) beta(3) beta(4) beta(5) beta2];rho1=@(x) R(x).^2;rho2=@(x) cosd(beta(x))*(R(x));R12u=rho1(1)-rho1(0);R12den=2*(rho2(1)-rho2(0));R12=R12u/R12den;R23u=rho1(2)-rho1(1);R23den=2*(rho2(2)-rho2(1));

R23=R23u/R23den;R34u=rho1(3)-rho1(2);R34den=2*(rho2(3)-rho2(2));R34=R34u/R34den;R45u=rho1(4)-rho1(3);R45den=2*(rho2(4)-rho2(3));R45=R45u/R45den;R56u=rho1(5)-rho1(4);R56den=2*(rho2(5)-rho2(4));R56=R56u/R56den;R67u=rho1(6)-rho1(5);R67den=2*((R(6)*cosd(beta2_cor))-rho2(5));R67=R67u/R67den;disp(' R1 R2 R3 R4 R5 R6 R7')disp(arr)disp(' CM1 CM2 CM3 CM4 CM5 CM6 CM7 ')disp(arr2)disp(' w1 w2 w3 w4 w5 w6 w7 ')disp(arr3)disp(' bet1 beta2 beta3 beta4 beta5 beta6 beta7 ')disp(arr4)disp(' R1a Rab Rbc Rcd Rde Ref Rf2 ')arr5=[R12 R23 R34 R45 R56 R67];disp(arr5)

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Output:

Z = 16 slipf = 0.9585

D_shcor = 0.0925 Etta_h = 0.6782

Deye = 0.1584

D1 = 0.1743

t = 0.0030

phi = 1.0140

k_e = 0.6500

D2 = 0.4430

beta1_cor = 18.0814

beta2_cor = 16.1352

epsilon1_cor =0.7175

epsilon2_cor = 0.8759

b1_act = 0.0202

b2_act =0.0090

diameters_ratio = 2.5419

error_percentage = 0.0335

R1 Ra Rb Rc Rd Re R2

0.0871 0.1095 0.1319 0.1543 0.1767 0.1991 0.2215

CM1 CMa CMb CMc CMd CMe CM2

3.9457 3.7627 3.5797 3.3967 3.2137 3.0307 2.8478

w1 wa wb wc wd we w2

14.2443 13.5781 12.9119 12.2458 11.5796 10.9134 10.2473

beta1 betaa betab betac betad betae beta2

16.0814 16.0881 16.0956 16.1039 16.1131 16.1235 18.0038

R1a Rab Rbc Rcd Rde Rfe2

0.1024 0.1257 0.1490 0.1723 0.1957 0.2190