Thermofluids: Bernoulli Applications

Problem Solving

• Problem solving requires factual and procedural knowledge.

  • Factual knowledge: knowledge of "things."

  • Procedural knowledge: knowledge of "how to do things."

• Schema: a "specific type of problem."

• Improving problem solving:

  1. Increase domain knowledge.

  2. Learn schema for various problem types.

  3. Become more conscious of the problem-solving process.

• Problem-solving strategy:

  1. Understand the problem.

  2. Devise a plan.

  3. Carry out the plan.

  4. Look back and check the solution.

Bernoulli's Theorem Applications

• $P1 + \frac{1}{2} \rho v1^2 + \rho g h1 = P2 + \frac{1}{2} \rho v2^2 + \rho g h2$

• Specific energy (J/kg): $\frac{P1}{\rho} + \frac{V1^2}{2} + gz1 = \frac{P2}{\rho} + \frac{V2^2}{2} + gz2$

• Pressure (Pa): $P1 + \rho \frac{V1^2}{2} + \rho gz1 = P2 + \rho \frac{V2^2}{2} + \rho gz2$

• Head (m): $\frac{P1}{\rho g} + \frac{V1^2}{2g} + z1 = \frac{P2}{\rho g} + \frac{V2^2}{2g} + z2$

• At any point on a streamline: $\frac{P}{\rho} + \frac{V^2}{2} + gz = constant$

Bernoulli's Equation - Application

• Most fluid problems require using both mass conservation and Bernoulli equations.

• Mass conservation:

  • $\dot{m} = \rho A \overline{v} = \rho A V [kg/s]$

  • $\rho<em>1 A</em>1 V<em>1 = \rho</em>2 A<em>2 V</em>2$

  • $A<em>1 V</em>1 = A<em>2 V</em>2$

  • $V<em>2 = \frac{A</em>1}{A<em>2} V</em>1$

• Bernoulli equation:

  • $\frac{P}{\rho} + \frac{V^2}{2} + gz = constant$

  • $\frac{P}{\rho} + \frac{V^2}{2} = constant$

• Example: Tank with a hole at the bottom

  • Bernoulli Equation: $\frac{P<em>1}{\rho} + \frac{\overline{v}</em>1^2}{2} + gz<em>1 = \frac{P</em>2}{\rho} + \frac{\overline{v}<em>2^2}{2} + gz</em>2$

  • $P<em>1 = P</em>2 = P<em>{atm}$, $\overline{v}</em>1 = 0$

  • $\overline{v}<em>2 = \sqrt{2g(z</em>1 - z_2)} = \sqrt{2gH}$

  • $\dot{m} = \rho A<em>2 \overline{v}</em>2 = \rho A_2 \sqrt{2gH}$

General Points

• Water clock progress should tested before to ensure delivery to timeframe.