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

Heat moves in three ways, and nearly every engineering heat problem mixes them: a furnace wall conducts heat, hot air convects it away, and a glowing element radiates it. Engineers size insulation, cooling fins, electronics heat sinks, kilns and building envelopes by putting numbers on these three **modes**. This lesson gives you one equation for each and shows when each one matters.

## Conduction: heat through a solid

Heat diffuses through a material from hot to cold. For steady, one-dimensional conduction through a flat wall, **Fourier's law** gives:

```text
q = k · A · (T_hot − T_cold) / L
```

- q is the heat transfer rate in watts (W).
- k is the **thermal conductivity** in W/(m·K): copper ≈ 400, steel ≈ 45–50, concrete ≈ 1.0–1.8, brick ≈ 0.7, glass ≈ 0.8–1.0, wood ≈ 0.12–0.16, mineral wool ≈ 0.035–0.04, still air ≈ 0.026.
- A is the area (m²) and L the thickness (m).

A temperature **difference** in kelvin equals the same difference in °C, so you can use either for ΔT.

**Worked example 1.** A brick wall (k = 0.72 W/(m·K)) is 0.2 m thick and 10 m² in area. The inside face is at 20 °C and the outside face at 0 °C.

```text
q = 0.72 × 10 × (20 − 0) / 0.2 = 720 W
```

## Convection: heat carried by a moving fluid

When a fluid flows past a surface, it carries heat away (or brings it in). **Newton's law of cooling** wraps all the messy fluid behaviour into one coefficient h:

```text
q = h · A · (T_surface − T_fluid)
```

- h is the **convection coefficient** in W/(m²·K). Rough ranges: still air (free convection) 2–25, forced air 25–250, forced water 100–20,000, boiling or condensing water 2,500–100,000.

h is not a material property: it depends on the fluid, its speed, the geometry and whether the flow is laminar or turbulent. Engineers get it from correlations based on dimensionless numbers (Nusselt, Reynolds, Prandtl) that you meet in a full heat transfer course.

**Worked example 2.** A 2 m² hot plate at 80 °C sits in a fan-driven airflow at 20 °C with h = 25 W/(m²·K).

```text
q = 25 × 2 × (80 − 20) = 3,000 W
```

## Radiation: heat as electromagnetic waves

Every surface above absolute zero emits thermal radiation; no medium is needed (that is how the Sun heats the Earth). A surface exchanging radiation with large surroundings loses net heat:

```text
q = ε · σ · A · (T_s⁴ − T_surr⁴)
```

- σ = 5.67 × 10⁻⁸ W/(m²·K⁴), the Stefan–Boltzmann constant.
- ε is the **emissivity**, between 0 and 1: polished aluminium ≈ 0.05, oxidised steel ≈ 0.8, paint, brick and skin ≈ 0.9.
- Temperatures **must** be absolute (kelvin), because of the fourth power.

**Worked example 3.** A 1 m² painted surface (ε = 0.9) at 350 K faces surroundings at 300 K.

```text
q = 0.9 × 5.67e-8 × 1 × (350⁴ − 300⁴) ≈ 352 W
```

Radiation grows very fast with temperature: at 1,000 K the same surface would lose over 50 kW per square metre. At room temperature it is often comparable to free convection, so do not ignore it.

## Which mode matters?

- Inside a solid: conduction only.
- Surface to a moving fluid: convection, plus radiation if the surface is hot or the convection is weak.
- Across a vacuum (spacecraft, vacuum flasks): radiation only.
- Most real problems are in **series**: inside air → wall → outside air. The next lesson handles that with thermal resistances.

## Common mistakes

- **Using °C in the radiation equation.** Use kelvin: 77 °C is 350.15 K.
- **Mixing up k and h.** k is in W/(m·K) and belongs to a material; h is in W/(m²·K) and belongs to a flow situation.
- **Using mm for thickness** with k in W/(m·K). Convert 200 mm to 0.2 m.
- **Forgetting that rates are in watts.** Energy over time is rate × time: 720 W for 24 h is 17.3 kWh.

## Recap

- Conduction: `q = k A ΔT / L` (Fourier).
- Convection: `q = h A ΔT` (Newton).
- Radiation: `q = ε σ A (T_s⁴ − T_surr⁴)`, in kelvin (Stefan–Boltzmann).
- Use real property data from handbooks or manufacturers for real work; these values are typical, not design values.

## Try it

On the Do tab, write Fourier's law in Python and compare conduction losses through different wall materials.


Compute heat flow by conduction through a wall, convection from a surface and radiation to the surroundings.

Conduction, convection, radiation

Heat Transfer - Conduction, Convection, and Radiation

Heat Transfer: Conduction, Convection, and Radiation

## Why it matters

Buildings account for a large share of energy use, and most of a building's heating and cooling load passes through its walls, roof and windows. Insulation is rated by **R-value**, and building energy codes set minimum R-values or maximum U-factors for each part of the envelope. Engineers, architects and energy auditors compare wall assemblies by adding up thermal resistances, exactly the way you add resistors in series in an electric circuit.

## Thermal resistance

Rewrite Fourier's law per unit area as "temperature difference divided by resistance":

```text
q/A = ΔT / R
```

- For a solid layer, `R = L / k` (thickness divided by conductivity).
- For a surface air film (convection), `R = 1 / h`.
- In SI, R is in **m²·K/W**. In the US building trade, R is in **ft²·°F·h/Btu**. To convert: `R_US = 5.678 × R_SI`. A product sold as "R-13" in the US is about R 2.3 in SI units.

Layers in **series** add:

```text
R_total = R_inside_film + R_1 + R_2 + ... + R_outside_film
U       = 1 / R_total            (W/(m²·K))
q       = U · A · ΔT             (W)
```

The **U-factor** (or U-value) is the inverse: lower U means better insulation. Windows are usually rated by U-factor, insulation by R-value.

## Worked example: two wall assemblies

Compare a 10 m² wall section with and without cavity insulation, for 20 °C inside and 0 °C outside. Typical surface film resistances are about 0.12 m²·K/W inside and 0.03 outside.

**Wall A, insulated:**

```text
inside air film                         0.120
12.5 mm gypsum board (k = 0.17)         0.0125 / 0.17 = 0.074
90 mm mineral wool (k = 0.04)           0.090 / 0.04  = 2.250
11 mm OSB sheathing (k = 0.13)          0.011 / 0.13  = 0.085
outside air film                        0.030
R_total ≈ 2.56 m²·K/W     U ≈ 0.391 W/(m²·K)
q = 0.391 × 10 × 20 ≈ 78 W
```

**Wall B, empty cavity** (an air space instead of insulation, R ≈ 0.17):

```text
R_total ≈ 0.12 + 0.074 + 0.17 + 0.085 + 0.03 ≈ 0.48 m²·K/W
U ≈ 2.09 W/(m²·K)     q ≈ 418 W
```

The insulation cuts the heat loss through this section by more than 80 %. Notice that the insulation layer provides almost all of wall A's resistance: thin layers of board and the air films matter much less.

## Real walls are not uniform

The example treats the wall as continuous layers. Real walls have studs, which conduct heat far better than mineral wool and form **thermal bridges**. A wood stud wall's whole-wall R can be 10–20 % lower than the centre-of-cavity value, and steel studs are far worse. Engineers use **parallel-path** or more detailed methods (from ASHRAE handbooks) to account for framing, and they also check for moisture: a wall that is warm on one side and cold on the other can condense water inside if the vapour control is wrong.

## Parallel paths in one step

When heat can flow through two different paths side by side (for example studs and insulation), add their **conductances** weighted by area fraction:

```text
U_wall = f_stud · U_stud_path + f_cavity · U_cavity_path
```

With 15 % studs, this is often enough for an estimate; detailed design uses software and code-approved values.

## Common mistakes

- **Adding U-values in series.** Add R-values in series; add U × area in parallel.
- **Mixing SI and US R-values.** Always state which units you mean.
- **Forgetting air films.** They are small for insulated walls but significant for single glazing.
- **Assuming more insulation always pays.** Doubling R halves the loss through that layer, so each extra centimetre saves less than the one before.

## Codes and professionals

Building energy codes (for example the IECC in many US states, or national codes elsewhere) set the required R-values and U-factors by climate zone. Local building officials enforce them, and envelope designs for real projects are prepared or reviewed by licensed architects or engineers. See ASHRAE for the technical handbooks: https://www.ashrae.org/ (Checked on 2026-10-23.)

## Recap

- `R = L/k` for a layer, `R = 1/h` for an air film; series resistances add.
- `U = 1/R_total`, `q = U A ΔT`.
- `R_US ≈ 5.678 × R_SI`.
- Thermal bridges and moisture make real walls more complex than the layer model.

## Try it

On the Do tab, add thermal resistances in Python and compare the two wall assemblies.


Add thermal resistances in series, convert R-values and U-factors, and compare the heat loss of two wall assemblies.

Insulation and R-values

Heat Transfer L6 p5 - R-Value and Thermal Resistance

Heat Transfer - Chapter 3 - Thermal Resistances in Parallel, Contact Resistance, R-Value

## Why it matters

Heat exchangers move heat from one fluid to another without letting them mix. They are everywhere: car radiators, home boilers, refrigerator condensers, power-plant condensers, the jackets on chemical reactors, pasteurisers in food and pharmaceutical plants, and the coolers on every large compressor. Knowing how they work, what the main types are, and how engineers rate them lets you read data sheets and follow design conversations, even before you design one yourself.

## The basic idea

A hot fluid and a cold fluid flow on opposite sides of a solid wall (a tube or a plate). Heat passes from the hot fluid, by convection, to the wall; through the wall by conduction; and from the wall to the cold fluid by convection. Those three resistances in series combine into an **overall heat transfer coefficient** U, in W/(m²·K), just like the wall in the previous lesson. **Fouling** (scale, biofilm, corrosion products) adds more resistance over time, so designers include a fouling allowance and plan for cleaning.

## Main types

- **Double-pipe:** one tube inside another. Simple, good for small duties and teaching labs.
- **Shell-and-tube:** a bundle of tubes inside a large shell. One fluid flows through the tubes, the other around them, guided back and forth by **baffles**. Rugged, handles high pressures and temperatures, easy to clean on the tube side. The workhorse of refineries, chemical plants and power stations.
- **Plate:** stacked, corrugated metal plates with gaskets (or brazed or welded) forming thin alternating channels. Very high U and compact, easy to expand by adding plates, common in HVAC, food and dairy. Limited by gasket pressure and temperature ratings.
- **Finned-tube / air-cooled:** fins increase the area on the air side, where h is low. Car radiators and air-conditioning coils.

## Flow arrangement

- **Parallel flow:** both fluids enter at the same end and flow the same way. The temperature difference is large at the inlet and shrinks along the exchanger; the cold outlet can never get hotter than the hot outlet.
- **Counterflow:** the fluids flow in opposite directions. The temperature difference stays more even, and the cold fluid can leave hotter than the hot fluid's outlet. For the same duty, counterflow needs less area, so designers prefer it whenever practical.
- **Cross-flow:** the fluids cross at right angles, typical of air-cooled coils. Shell-and-tube units with several passes behave between parallel and counterflow, handled with correction factors.

## Rating an exchanger: energy balance and LMTD

Two equations do most of the work.

**1. Energy balance.** With no losses, the heat lost by the hot fluid equals the heat gained by the cold fluid:

```text
Q = ṁ_h · c_p,h · (T_h,in − T_h,out) = ṁ_c · c_p,c · (T_c,out − T_c,in)
```

**2. Rate equation.** Because the temperature difference changes along the exchanger, it is averaged logarithmically:

```text
Q = U · A · ΔT_lm        ΔT_lm = (ΔT1 − ΔT2) / ln(ΔT1 / ΔT2)
```

ΔT1 and ΔT2 are the temperature differences between the two fluids at each end.

**Worked example.** In a counterflow exchanger, hot water cools from 90 °C to 60 °C while cold water warms from 20 °C to 40 °C. U = 500 W/(m²·K) and A = 10 m².

```text
Counterflow ends:  ΔT1 = 90 − 40 = 50 K     ΔT2 = 60 − 20 = 40 K
ΔT_lm = (50 − 40) / ln(50/40) = 10 / 0.2231 ≈ 44.8 K
Q = 500 × 10 × 44.8 ≈ 224 kW
```

When ΔT1 equals ΔT2 the formula gives 0/0; the LMTD then simply equals that common difference.

## The effectiveness–NTU method (in brief)

When outlet temperatures are unknown, engineers use the **effectiveness** ε (actual heat transfer divided by the maximum possible) and the **number of transfer units** NTU = UA / C_min, with charts or formulas for each flow arrangement. You will meet it in a full heat transfer course; the idea is that bigger UA relative to the smaller heat capacity rate means outlets closer to the ideal.

## Common mistakes

- **Using the arithmetic mean temperature difference** instead of the LMTD. It overestimates when the end differences are very unequal.
- **Pairing the wrong ends.** In counterflow, compare the hot inlet with the cold outlet, and the hot outlet with the cold inlet.
- **Ignoring fouling.** A clean-condition U can drop a lot in service.
- **Forgetting the pressure side.** Exchangers are pressure vessels; failures can release hot or hazardous fluids.

## Codes and safety

Industrial shell-and-tube exchangers are pressure equipment, designed and fabricated to codes such as the ASME Boiler and Pressure Vessel Code and TEMA standards, and inspected under local rules. Their design and any repairs belong to qualified, often licensed, engineers and certified fabricators. Treat the numbers here as learning examples, not design values.

## Recap

- A heat exchanger transfers heat across a wall: convection, conduction, convection, combined in U.
- Main types: double-pipe, shell-and-tube, plate, finned-tube.
- Counterflow is the most effective arrangement.
- Rate it with the energy balance and `Q = U A ΔT_lm`.

## Try it

This lesson has no activity. Check your understanding with the quiz on the Do tab.


Explain how shell-and-tube and plate heat exchangers work, why counterflow is preferred, and how the LMTD rates one.

Heat exchangers

Shell and Tube Heat Exchanger basics explained

Industrial Heat Exchangers Explained

How heat moves: conduction, convection and radiation, comparing wall assemblies with R-values and U-factors, and how heat exchangers work and are rated.