The Equation of Continuity — Question 1

Question 1

A diameter constraint is a squared-area constraint

Water flows steadily through a full circular pipe at volume rate $Q=1.20\times10^{-3}\,\mathrm{m^3/s}$. The inlet diameter is $40\,\mathrm{mm}$. The outlet is required to have mean speed no greater than $6.0\,\mathrm{m/s}$. Treat water as incompressible.

1. Find the inlet mean speed and the minimum permitted outlet diameter.

2. If the outlet diameter were half the inlet diameter, what would its speed be? Would it meet the requirement?

Question 1 - Solution

A diameter constraint is a squared-area constraint

(a) Derivation and calculation

$$A=\pi D^2/4,\quad\boxed{\bar v_1=\frac{4Q}{\pi D_1^2}=\frac{4(.00120)}{\pi(.0400)^2}=.954930\,\mathrm{m/s}}.$$$$\bar v_2\le6.0\Rightarrow A_2\ge Q/6.0=2.00\times10^{-4}\,\mathrm{m^2}.$$$$\boxed{D_{2,\min}=\sqrt{\frac{4Q}{6.0\pi}}=.0159577\,\mathrm m=15.96\,\mathrm{mm}}.$$

(b) Result and interpretation

$$D_2=D_1/2\Rightarrow A_2=A_1/4\Rightarrow\boxed{\bar v_2=4\bar v_1=3.81972\,\mathrm{m/s}}.$$$$3.81972<6.0\Rightarrow\text{the requirement is met}.$$Speed scales inversely with squared diameter because continuity uses area.

Physical check

The proposed 20 mm outlet is larger than the minimum 15.96 mm.