The rate of flow is given by the expression:
sf(V/t=(ppia^4)/(8etal))
sf(a) is the radius of the pipe
sf(p) is the pressure causing the flow
sf(l) is the length of the pipe
sf(eta) is the coefficient of viscosity
In this case we can combine this to get:
sf(R=ka^4)
For a single pipe of diameter 1 cm the radius is d/2 = 0.5 cm. We can call this sf(r_1)
:.sf(R=kxxr_1^4=kxx0.5^4=kxx0.0625) (arbitrary units since we are doing a comparison)
Since there are 16 pipes the total rate sf(R_1) is given by:
sf(R_1=0.0625xx16=kxx1)
Now we have a single pipe which is of the same X section area.
We need to get the radius sf(r_2).
The area sf(A) is given by:
sf(A=pir_1^2=0.25picolor(white)(x)"cm"^2)
There are 16 pipes so the total area = sf(16xx0.25pi=4picolor(white)(x)"cm"^2)
For the single large pipe of radius sf(r_2) we can say:
sf(cancel(pi)r_2^2=4cancel(pi))
:.sf(r_2=sqrt(4)=2color(white)(x)cm)
:.sf(R_2=kxx2^4=kxx16)
:.sf(R_2/R_1=(16cancel(k))/(1cancel(k))=16)
