Heat Transfer: Quick Revision

A brief note on Heat Transfer. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Conduction: $H = \dfrac{kA\,\Delta T}{L}$; resistance $\dfrac{L}{kA}$, adding in series.
• Convection: a moving fluid carries the heat; warm fluid rises.
• Radiation: $P = e\sigma AT^4$; net $e\sigma A(T^4 - T_0^4)$; $\lambda_mT = b$. Good absorbers are good emitters.
• Newton's law of cooling: rate $\propto$ excess; $T = T_0 + (T_s - T_0)e^{-kt}$.

The facts at a glance

A worked example

A hot water pipe $2.2\ \text{cm}$ across and $10\ \text{m}$ long is wrapped in $2.0\ \text{cm}$ of lagging ($k = 0.035\ \text{W m}^{-1}\text{K}^{-1}$). The water is $50\ \text{K}$ warmer than the air. Treating the lagging as a flat layer, how much heat is lost?

1. Area. Halfway through the lagging the circumference is $\pi \times 0.042\ \text{m}$, so $A = \pi \times 0.042\ \text{m} \times 10\ \text{m} = 1.32\ \text{m}^2$.
2. Heat flow. $H = 0.035\ \text{W m}^{-1}\text{K}^{-1} \times 1.32\ \text{m}^2 \times \dfrac{50\ \text{K}}{0.020\ \text{m}} = 115\ \text{W}$.
3. Over a day. $115\ \text{W} \times 86\,400\ \text{s} = 9.9 \times 10^{6}\ \text{J}$, about $2.8\ \text{kWh}$.

Practice with new numbers

A program to try

The program works out the lagged pipe.

Mistakes to avoid

• Celsius in Stefan's law. Use kelvin.
• Forgetting the surroundings' radiation. The net loss is $e\sigma A(T^4 - T_0^4)$.
• Thickness in centimetres. Change to metres.
• Adding conductivities in series. Add the resistances.
• Cooling by equal steps. The excess falls by equal fractions.

Check yourself

1. Heat flows through a $2.0\ \text{m}^2$ slab of brick $0.10\ \text{m}$ thick, with faces $12\ \text{K}$ apart ($k = 0.60\ \text{W m}^{-1}\text{K}^{-1}$). How many watts?

2. A body's peak wavelength is $2.0\ \mu\text{m}$. What is its temperature?

3. By what factor does a body's radiated power grow if its kelvin temperature rises by half?

4. Why is the inside of a vacuum flask silvered?

5. A body $30\ \text{K}$ above the room cools at $1.5\ \text{K/min}$. How fast at $10\ \text{K}$ above?