Bernoulli's Equation — Question 10

Question 10

Mixing can conserve momentum while dissipating mechanical energy

Two parallel water streams merge into one fully mixed stream at the same elevation. Each incoming stream has mass flow $\dot m=0.50\,\mathrm{kg/s}$, with speeds $v_1=2.0$ and $v_2=6.0\,\mathrm{m/s}$ in the same direction. Assume uniform speeds at the selected inlet and far-outlet sections, equal ambient pressure there, and negligible net external force along the flow. There is no shaft work.

1. Use mass and momentum conservation to find the final speed. Compute incoming and outgoing kinetic-energy rates.

2. Find the lost mechanical power and explain why applying lossless Bernoulli separately from both inlets to the same final state is inconsistent.

Question 10 - Solution

Mixing can conserve momentum while dissipating mechanical energy

(a) Derivation and calculation

$$\dot m_f=2\dot m=1.00\,\mathrm{kg/s},\qquad2\dot m v_f=\dot m(v_1+v_2).$$$$\boxed{v_f=(2.0+6.0)/2=4.00\,\mathrm{m/s}}.$$$$\boxed{\dot E_{K,\mathrm{in}}=\tfrac12\dot m(v_1^2+v_2^2)=10.0\,\mathrm W}.$$$$\boxed{\dot E_{K,\mathrm{out}}=\tfrac12(2\dot m)v_f^2=8.00\,\mathrm W}.$$

(b) Result and interpretation

$$\boxed{P_{\mathrm{loss}}=10.0-8.00=2.00\,\mathrm W}.$$$$P_{\mathrm{loss}}=\frac{\dot m}{4}(v_1-v_2)^2\ge0.$$Lossless Bernoulli from both equal-pressure, equal-height inlets would demand $v_f=v_1$ and $v_f=v_2$ simultaneously. Irreversible mixing removes the mechanical energy while conserving the stated momentum.

Physical check

The loss vanishes when the two inlet speeds are equal, as expected for no velocity mismatch.