The laws of thermodynamics, steam tables, power and refrigeration cycles, fluid statics, Bernoulli, pipe losses, heat transfer, R-values and heat exchangers, with Python exercises.

Thermodynamics & Fluids

## Why it matters

Dams, retaining walls, water tanks, ship hulls, manometers and hydraulic presses all depend on how pressure varies in a fluid at rest. Civil engineers size tank walls and gates from it, mechanical engineers use it in hydraulics, and chemists and materials scientists read manometers and pressure gauges every day. Fluid statics is also the gentlest entry into fluid mechanics: no flow, just weight and pressure.

## The hydrostatic equation

In a fluid at rest, pressure increases with depth because each layer carries the weight of the fluid above it. For a fluid of constant density ρ:

```text
P = P0 + ρ · g · h
```

- P0 is the pressure at the free surface (often atmospheric, about 101.3 kPa at sea level).
- ρ is density (water ≈ 1,000 kg/m³; seawater ≈ 1,025 kg/m³).
- g = 9.81 m/s².
- h is the depth **below** the surface, in metres.

Two consequences surprise beginners:

- Pressure depends only on depth, not on the shape or width of the container (the "hydrostatic paradox").
- At the same depth in a connected body of the same fluid, the pressure is the same. That is how manometers and hydraulic lifts work (Pascal's principle).

## Gauge and absolute pressure

- **Absolute pressure** is measured from a perfect vacuum.
- **Gauge pressure** is measured relative to the local atmosphere: `P_gauge = P_abs − P_atm`.

Most tyre and process gauges read gauge pressure. Thermodynamic property tables need absolute pressure. Write "kPa(g)" or "kPa(a)" if there is any doubt.

## Worked example 1: pressure at the bottom of a tank

A water tank is 3 m deep and open to the atmosphere.

```text
P_gauge = ρ g h = 1000 × 9.81 × 3 = 29,430 Pa = 29.43 kPa(g)
P_abs   = 101.3 + 29.43 ≈ 130.7 kPa(a)
```

A useful rule of thumb: every 10 m of water adds about 1 atmosphere (98.1 kPa).

## Force on a submerged plane surface

Pressure acts perpendicular to a surface, and it grows with depth, so the force on a wall or gate is the pressure averaged over its area. For a flat surface the resultant force is:

```text
F = ρ · g · h_c · A
```

where h_c is the depth of the surface's **centroid** and A is its area (using gauge pressure, because the atmosphere usually acts on both sides).

The force does not act at the centroid. Because pressure is larger lower down, it acts at the **centre of pressure**, which is deeper. For a vertical surface whose top edge is at the free surface:

```text
y_cp = y_c + I_c / (y_c · A)
```

with y_c the centroid depth and I_c the second moment of area about the centroid (b·d³/12 for a rectangle of width b and height d).

## Worked example 2: a vertical gate

A rectangular gate 2 m wide and 3 m tall sits in a vertical wall, with its top edge at the water surface.

```text
A    = 2 × 3 = 6 m²
h_c  = 1.5 m
F    = 1000 × 9.81 × 1.5 × 6 = 88,290 N ≈ 88.3 kN
I_c  = 2 × 3³ / 12 = 4.5 m⁴
y_cp = 1.5 + 4.5 / (1.5 × 6) = 1.5 + 0.5 = 2.0 m below the surface
```

The resultant acts two-thirds of the way down, which matches the triangular pressure distribution. A designer would use that location to check hinges, bolts and the moments on the gate.

## Buoyancy in one line

Archimedes' principle: a submerged or floating body feels an upward force equal to the weight of fluid it displaces, `F_B = ρ_fluid · g · V_displaced`. A 0.5 m³ block fully submerged in water feels 1000 × 9.81 × 0.5 ≈ 4.9 kN of buoyancy.

## Common mistakes

- **Measuring h from the bottom** instead of down from the free surface.
- **Mixing gauge and absolute pressure** in the same calculation.
- **Putting the force at the centroid.** It acts at the centre of pressure, below the centroid.
- **Forgetting the density of the actual fluid.** Oil, seawater, mercury and brine differ a lot from fresh water.

## A note on real design

These are textbook idealisations. Real tanks, dams and gates are designed to codes with load factors, seismic and dynamic effects, corrosion allowances and inspection rules, and their designs must be checked and sealed by a licensed professional engineer.

## Recap

- `P = P0 + ρ g h`, with h measured down from the surface.
- Gauge = absolute − atmospheric.
- Force on a flat surface: `F = ρ g h_c A`; it acts at the centre of pressure, below the centroid.
- Buoyancy equals the weight of displaced fluid.

## Try it

On the Do tab, finish a Python function for hydrostatic gauge pressure and use it to find the force on the gate.


Compute hydrostatic pressure at depth and the force and centre of pressure on a submerged vertical gate.

Fluid statics

Fluid Mechanics - Fluid/Hydrostatic Pressure in 11 Minutes!

Fluid Mechanics: Topic 4.1 - Hydrostatic force on a plane surface

## Why it matters

Water mains, fuel lines, ventilation ducts, carburettors, flow meters and even blood vessels share two simple rules: mass is conserved (the **continuity equation**), and along a streamline energy is traded between pressure, speed and height (**Bernoulli's equation**). With just these two, you can find the velocity in a pipe, the pressure drop through a nozzle or the reading of a Venturi meter.

## Continuity: what goes in must come out

For steady flow of an incompressible fluid (liquids, and gases at low speed), the volume flow rate is the same at every section of a pipe:

```text
Q = A1 · V1 = A2 · V2
```

- Q is volume flow rate in m³/s.
- A is the cross-section area: for a round pipe, `A = π D² / 4`.
- V is the average velocity in m/s.

Halve the diameter and the area drops to a quarter, so the velocity is **four times** larger. For compressible flow, use mass flow rate instead: `ṁ = ρ1 A1 V1 = ρ2 A2 V2`.

## Bernoulli's equation

Along a streamline in steady, incompressible, frictionless flow, the total mechanical energy per unit volume stays constant:

```text
P1 + ½ ρ V1² + ρ g z1 = P2 + ½ ρ V2² + ρ g z2
```

The three terms are static pressure, dynamic pressure and elevation (hydrostatic) pressure. Engineers often divide by ρg and write it in **head** form, in metres of fluid:

```text
P/(ρg) + V²/(2g) + z = constant
```

The assumptions matter. Bernoulli does not include friction (next lesson), pumps or turbines (which add or remove energy), or heat transfer. Use it between two points on the same streamline in a short, smooth section.

## Worked example: a pipe reducer

Water (ρ = 1,000 kg/m³) flows at Q = 0.02 m³/s through a horizontal pipe that narrows from D1 = 100 mm to D2 = 50 mm. The pressure upstream is P1 = 150 kPa (gauge). What is the pressure in the narrow section?

**Step 1, continuity:**

```text
A1 = π (0.10)² / 4 = 0.007854 m²     V1 = 0.02 / 0.007854 = 2.546 m/s
A2 = π (0.05)² / 4 = 0.001963 m²     V2 = 0.02 / 0.001963 = 10.186 m/s
```

**Step 2, Bernoulli** (horizontal, so z1 = z2):

```text
P2 = P1 + ½ ρ (V1² − V2²)
   = 150,000 + 500 × (6.484 − 103.75)
   = 150,000 − 48,634 ≈ 101.4 kPa (gauge)
```

The fluid speeds up, so its static pressure falls by about 48.6 kPa. That pressure drop is exactly how a **Venturi meter** measures flow: measure P1 − P2, and you can solve backwards for Q.

## Worked example: draining a tank (Torricelli)

Water drains from a small hole 2 m below the surface of a large open tank. Take point 1 at the surface (P = atmospheric, V ≈ 0, z = 2 m) and point 2 in the jet (P = atmospheric, z = 0):

```text
V2 = √(2 g h) = √(2 × 9.81 × 2) ≈ 6.26 m/s
```

The jet leaves at the same speed as a stone dropped 2 m. Real jets are a bit slower, because of friction and the jet contracting just outside the hole (engineers apply a discharge coefficient of about 0.6 for a sharp-edged hole).

## Common mistakes

- **Using diameter instead of area** in continuity. Area scales with D².
- **Mixing gauge and absolute pressures** between the two points. Either works, but use the same one on both sides.
- **Applying Bernoulli across a pump, turbine or long rough pipe** without adding the energy terms.
- **Forgetting units.** ½ρV² with ρ in kg/m³ and V in m/s gives Pa, not kPa.
- **Ignoring cavitation.** If the absolute pressure at P2 falls below the fluid's vapour pressure, the liquid can flash into vapour (cavitation), which damages pumps and valves.

## Recap

- Continuity: `A1 V1 = A2 V2`; a smaller pipe means faster flow.
- Bernoulli: `P + ½ρV² + ρgz` is constant along a streamline, without friction or machines.
- Faster flow means lower static pressure, which is the principle behind Venturi meters.
- These are idealised; real piping systems are sized to codes and checked by licensed engineers.

## Try it

On the Do tab, use continuity and Bernoulli in Python to find the pressure in a pipe reducer.


Use continuity and Bernoulli's equation to find velocities and pressures in a pipe reducer and a draining tank.

Bernoulli and continuity

Understanding Bernoulli's Equation

Continuity Equation, Volume Flow Rate & Mass Flow Rate Physics Problems

## Why it matters

Real fluid rubs against pipe walls and churns through fittings, and that friction turns useful pressure into heat. The lost energy, called **head loss**, decides how big a pump must be, how far water can be sent through a main, and how much energy a building's chilled-water loop uses every year. Getting it wrong means pumps that cannot deliver, or pipes that are needlessly large and expensive.

## Laminar or turbulent: the Reynolds number

The **Reynolds number** compares inertia to viscous forces:

```text
Re = ρ V D / μ        (or V D / ν, with ν = μ/ρ)
```

- ρ density (kg/m³), V average velocity (m/s), D pipe inner diameter (m), μ dynamic viscosity (Pa·s).
- Water at 20 °C: ρ ≈ 998 kg/m³, μ ≈ 1.002 × 10⁻³ Pa·s.

For pipe flow, roughly:

- Re < 2,300: **laminar**, smooth layered flow.
- 2,300 < Re < 4,000: transitional, unpredictable.
- Re > 4,000: **turbulent**, the usual case for water in building and industrial pipes.

## The Darcy–Weisbach equation

Head loss due to friction in a straight pipe (the "major" loss) is:

```text
h_f = f · (L / D) · V² / (2 g)
```

- h_f is in metres of the flowing fluid.
- L is pipe length, D inner diameter.
- f is the dimensionless **Darcy friction factor**.

Convert head loss to a pressure drop with `ΔP = ρ g h_f`.

## Finding the friction factor

- **Laminar:** `f = 64 / Re`, exactly.
- **Turbulent:** f depends on Re and the **relative roughness** ε/D. The classic tools are the **Moody chart** and the implicit **Colebrook equation**. A widely used explicit approximation, accurate to within a few per cent over most of the turbulent range, is **Swamee–Jain**:

```text
f = 0.25 / [ log10( ε/(3.7 D) + 5.74 / Re^0.9 ) ]²
```

Typical roughness ε: drawn tubing (copper, plastic) ≈ 0.0015 mm; commercial steel ≈ 0.045 mm; cast iron ≈ 0.26 mm. Old pipes get rougher with corrosion and scale.

Be careful with the **Fanning** friction factor, used in some chemical engineering texts: it is one quarter of the Darcy factor.

## Worked example

Water at 20 °C flows at V = 2 m/s through 100 m of 50 mm commercial steel pipe (ε = 0.045 mm).

```text
Re   = 998 × 2 × 0.05 / 1.002e-3 ≈ 99,600       → turbulent
ε/D  = 0.045 / 50 = 0.0009
f    = 0.25 / [log10(0.0009/3.7 + 5.74/99,600^0.9)]² ≈ 0.0220
h_f  = 0.0220 × (100 / 0.05) × 2² / (2 × 9.81) ≈ 8.97 m
ΔP   = 998 × 9.81 × 8.97 ≈ 87.8 kPa
```

Almost 9 m of head is lost in 100 m. That is a lot: designers usually keep water velocities in this size of pipe lower (often around 1–2 m/s), partly to limit losses and noise. Because h_f grows with V², halving the velocity cuts the friction loss to roughly a quarter.

## Minor losses

Valves, elbows, tees, entrances and exits add **minor losses**, written with a loss coefficient K:

```text
h_m = K · V² / (2 g)
```

Typical K values: a sharp pipe entrance about 0.5, a standard 90° elbow about 0.3 to 1.5 depending on type, a fully open gate valve about 0.2, an exit into a tank 1.0. Take them from manufacturer data or a reference table. In a short system with many fittings, "minor" losses can be larger than the pipe friction.

## The extended energy equation

Putting it together between points 1 and 2, with a pump adding head h_p:

```text
P1/(ρg) + V1²/(2g) + z1 + h_p = P2/(ρg) + V2²/(2g) + z2 + h_f + Σh_m
```

This is the equation behind every pump-selection problem.

## Common mistakes

- **Using the Fanning factor in Darcy's equation** (an error of four times).
- **Using nominal pipe size as the diameter.** Use the actual inner diameter for the pipe schedule.
- **Getting viscosity units wrong.** Centipoise (cP) is 10⁻³ Pa·s.
- **Forgetting that f is not a constant.** It changes with flow rate through Re.

## Recap

- Re < 2,300 laminar, > 4,000 turbulent.
- Darcy–Weisbach: `h_f = f (L/D) V²/(2g)`; laminar f = 64/Re; turbulent from Moody, Colebrook or Swamee–Jain.
- Minor losses: `h_m = K V²/(2g)`.
- Real piping is designed to codes (plumbing, process piping, fire protection) and reviewed by licensed engineers; these methods are for learning and estimates.

## Try it

On the Do tab, write the Darcy–Weisbach equation in Python and estimate the head loss in the worked-example pipe.


Classify flow with the Reynolds number and estimate friction and minor head losses with the Darcy-Weisbach equation.

Pipe flow and losses

Head loss due to friction in a pipe using Moody Diagram and the Darcy–Weisbach equation

Using a Moody Chart

Fluids at rest and in motion: hydrostatic pressure and forces on submerged surfaces, continuity and Bernoulli's equation, and friction losses in pipes.