The Equation of Continuity — Question 2

Question 2

Conservation can determine an unknown branch direction

At a steady water junction, branch A brings in $8.0\,\mathrm{L/s}$ and branch B carries out $3.0\,\mathrm{L/s}$. In branch C the mean speed magnitude is $2.0\,\mathrm{m/s}$ through area $40\,\mathrm{cm^2}$, but its direction is unknown. A fourth branch D is also present. No storage occurs.

1. Find the flow in D for each possible direction in C.

2. If an indicator shows D is an outlet, which direction in C is physically consistent? Find D's speed if its area is $50\,\mathrm{cm^2}$.

Question 2 - Solution

Conservation can determine an unknown branch direction

(a) Derivation and calculation

$$|Q_C|=A_C|v_C|=(40\times10^{-4})(2.0)=.00800\,\mathrm{m^3/s}=8.0\,\mathrm{L/s}.$$Take inward flow positive:$$Q_A+Q_B+Q_C+Q_D=0,\quad Q_A=8,quad Q_B=-3.$$$$C\ \text{out}:\ Q_D=-[8-3-8]=\boxed{+3\,\mathrm{L/s}};$$$$C\ \text{in}:\ Q_D=-[8-3+8]=\boxed{-13\,\mathrm{L/s}}.$$

(b) Result and interpretation

An outlet D selects inward C. Then$$\boxed{|v_D|=|Q_D|/A_D=.0130/.00500=2.60\,\mathrm{m/s}}.$$$$Q_{\mathrm{in}}=8+8=16,\quad Q_{\mathrm{out}}=3+13=16\,\mathrm{L/s}.$$Assign directions before combining flow rates; magnitudes alone do not conserve signed flux.

Physical check

Both possible branch assignments satisfy conservation, but only one matches the outlet indicator.