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)#
