Thermofluids - Bernoulli Applications

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.
Focus on the solution process rather than just obtaining the answer.

Relates pressure $P$ , speed $v$ , and height $h$ of two points in a steady streamline of fluid with density $ρ$ .
Equation:
$P1 + {1 \over 2} ρv1^2 + ρgh1 = P2 + {1 \over 2} ρv2^2 + ρgh2$
Alternative expressions:
- Specific energy (J/kg): $\frac{P1}{ρ} + \frac{V1^2}{2} + gz1 = \frac{P2}{ρ} + \frac{V2^2}{2} + gz2$
- Pressure (Pa): $P1 + ρ \frac{V1^2}{2} + ρgz1 = P2 + ρ \frac{V2^2}{2} + ρgz2$
- Head (m): $\frac{P1}{ρg} + \frac{V1^2}{2g} + z1 = \frac{P2}{ρg} + \frac{V2^2}{2g} + z2$
At any point on a streamline:
$\frac{P}{ρ} + \frac{V^2}{2} + gz = constant$

Most fluid problems use both mass conservation and Bernoulli's equations.
Mass conservation:
$\dot{m} = ρA\bar{v} = ρAV \ [kg/s]$
$ρ1A1V1 = ρ2A2V2$
$A1V1 = A2V2$
$V2 = \frac{A1}{A2} V1$
Bernoulli's equation (horizontal flow, $z$ constant):
$\frac{P}{ρ} + \frac{V^2}{2} = constant$
Example: Tank with a hole at the bottom:
- Velocity at the hole: $V_2 = \sqrt{2gH}$
- Mass flow rate: $\dot{m} = ρA_2 \sqrt{2gH}$