Tapered Pipe

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Determine the ratio of the diameters input?

Stuck on this question:

A duct needs to be designed so that it has a flow rate of an incompressible fluid of 0.04m/s which is increased through a gradually tapering pipe to 2m/s. Using the continuity of mass and hence the continuity of volume equations determine the ratio of the diameters of input and output diameters of the duct.

(any help appreciated, also working out to help me understand would be greatly appreciated and rewarded)
Thanks pointy

Working formula is

Q = VA = constant

where

Q = flow rate
V = velocity of fluid at any given section
A = cross sectional area of any given section

Since Q = constant, then

0.04(A1) = 2(A2)

where

A1 and A2 represent the cross sectional areas where the fluid velocities are 0.04 m/sec and 2 m /sec, respectively.

Since

A = π/4(D^2), then the flow continuity equation becomes

0.04(π/4)(D1)^2 = 2(π/4)(D2)^2

where

D1 and D2 represent the diameters of the cross sections where the fluid velocities are 0.04 m/sec and 2 m /sec, respectively.

Simplifying the above,

(D1/D2)^2 = 2/0.04 = 50

(D1/D2) = 5*sqrt 2

Hope this helps.

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