12.11

A peristaltic pump delivers a unit flow (Q1) of a highly viscous fluid.The network is depicted in Fig. P12.11.  Every pipe section has the same length a diameter. The mass and mechanical energy balance can be simplified to obtain the flows in every pipe. Solve the following system of equations to obtain the flow in every stream.

Q3 + 2Q4 - 2Q2 = 0
Q5 + 2Q6 - 2Q4 = 0
3Q7 - 2Q6 = 0
Q1 = Q2 + Q3 == % -Q1 +  Q2 + Q3 = 0
Q3 = Q4 + Q5 == % -Q3 + Q4 + Q5 = 0
Q5 = Q6 + Q7 == % -Q5 + Q6 + Q7 = 0




Matlab:

% 7 6 5 4 3 2 1

A = [ ...

 0 0 0 2 1 -2 0;

 0 2 1 -2 0 0 0;

 3 -2 0 0 0 0 0;

 0 0 0 0 1 1 -1;

 0 0 1 1 -1 0 0;

 1 1 -1 0 0 0 0;

 0 0 0 0 0 0 1 ];

syms Q1 ;

B = [0; 0; 0; 0; 0; 0; Q1];

display(A)

display(B)

x = A\B;

display(x)

A =



     0     0     0     2     1    -2     0
     0     2     1    -2     0     0     0
     3    -2     0     0     0     0     0
     0     0     0     0     1     1    -1
     0     0     1     1    -1     0     0
     1     1    -1     0     0     0     0
     0     0     0     0     0     0     1



B =

  0
  0
  0
  0
  0
  0
 Q1


x =

Q7=  (8*Q1)/85
Q6=  (12*Q1)/85
Q5=  (4*Q1)/17
Q4=  (22*Q1)/85
Q3=  (42*Q1)/85
Q2=  (43*Q1)/85
         Q1