TestAS · Engineering
Heat, temperature, efficiency and energy balances of machines and heating systems.
testas.de lists this topic for the Engineering module. In the test it comes as testlets: a short text, table or formula, then 4 to 8 questions you answer from it.
Topic list: testas.de
Full exam-length testlets from the Kollego course, generated the same way as our mocks. Answer from the material, then open each answer to check.
Heating water and expansion of materials
Heat is energy transferred because of a temperature difference. The heat Q needed to raise the temperature of a substance is proportional to its mass m, its specific heat capacity c and the temperature change ΔT: Q = m·c·ΔT. Water has a high specific heat capacity of about 4,180 J/(kg·K), which is why it is used in heating systems and cooling circuits: it can carry a lot of energy with a moderate change in temperature. A temperature difference of 1 K is the same as a difference of 1 °C.
Q = m·c·ΔT ΔL = α·L0·ΔT t = Q ÷ P
Most materials expand when heated. For a rod or beam, the change in length ΔL is proportional to its original length L0, the temperature change ΔT and the linear expansion coefficient α of the material. Steel has α ≈ 12 × 10⁻⁶ per kelvin, aluminium about 23 × 10⁻⁶ per kelvin. The effect is small for a short piece but important for long structures: bridges have expansion joints, railway rails are laid with gaps or pre-stressed, and pipelines contain loops that can flex. If expansion is prevented, large forces arise that can buckle rails or crack concrete.
Materials that are joined together must expand similarly. Steel and concrete have almost the same coefficient, which is one reason why reinforced concrete works. A bimetallic strip, made of two metals with different coefficients bonded together, bends when heated, because one side expands more than the other; it was long used in thermostats and switches.
Heat is transferred by conduction, convection and radiation. Metals conduct heat well, while air and insulating materials such as mineral wool conduct it poorly, which is why buildings are insulated with materials that trap air.
In practice, a kettle does not transfer all its electrical energy to the water: some heats the kettle itself and some is lost to the surrounding air. Real heating times are therefore somewhat longer than the ideal calculation suggests, and engineers include an efficiency factor when they size heating elements or estimate energy consumption. The same kind of calculation is used to size the heat exchangers in buildings and the cooling systems of engines and computers.
| Quantity | Value |
|---|---|
| Specific heat capacity of water | 4,180 J/(kg·K) |
| Linear expansion coefficient of steel | 12 × 10⁻⁶ per K |
| Linear expansion coefficient of aluminium | 23 × 10⁻⁶ per K |
1. How much heat is needed to warm 2 kg of water from 20 °C to 80 °C?
C: 501.60 kJ
2 × 4,180 × 60 J.
2. A 2 kW kettle with no losses heats this water. About how long does it take?
A: About 4.2 minutes
501,600 J ÷ 2,000 W ≈ 251 s.
3. By how much does a 100 m steel bridge lengthen when its temperature rises by 40 K?
A: 48.0 mm
12 × 10⁻⁶ × 100 m × 40 = 0.048 m.
4. An aluminium beam of the same length is heated by the same amount. Compared with steel, it expands…
A: almost twice as much
23 vs 12 × 10⁻⁶.
5. Why does a bimetallic strip bend when heated?
B: Its two metals expand by different amounts
One side becomes longer than the other.
6. Why can steel and concrete be combined in reinforced concrete?
C: They expand by almost the same amount
Similar coefficients prevent cracking.
Heat engines and heat pumps
Thermodynamics studies heat, work and energy. Its first law is the conservation of energy: the heat supplied to a system equals the increase in its internal energy plus the work it does. Its second law states that heat flows spontaneously only from a hotter to a colder body, never the other way round, and that it is impossible to convert heat completely into work in a cyclic process. In more general terms, the entropy of an isolated system, a measure of its disorder, never decreases.
A heat engine, such as a steam turbine or a car engine, takes heat from a hot source, converts part of it into work and releases the rest as waste heat to a colder environment. The second law sets an upper limit on its efficiency, the Carnot efficiency, which depends only on the absolute temperatures of the hot and cold reservoirs. A large temperature difference allows a high efficiency, which is why power stations operate at high steam temperatures. Real engines reach much less than the Carnot limit because of friction and other irreversible processes; modern combined-cycle gas power plants reach about 60 percent, car engines typically 25 to 40 percent.
η_Carnot = 1 − T_cold ÷ T_hot (temperatures in kelvin) COP_heat pump = Q_heat ÷ W
A heat pump works in reverse. Using electrical work, it moves heat from a cold source, such as the outside air or the ground, into a warmer building. This does not violate the second law, because work is supplied. The coefficient of performance (COP) is the heat delivered divided by the electrical energy used. A COP of 4 means that 1 kWh of electricity delivers 4 kWh of heat: 1 kWh from the electricity and 3 kWh taken from the environment. The smaller the temperature difference between source and building, the higher the COP; heat pumps are therefore most efficient with underfloor heating, which needs only low flow temperatures, and in mild weather. A refrigerator is also a heat pump, moving heat from its interior to the kitchen.
Temperatures in thermodynamic formulas must always be absolute temperatures in kelvin: T in K = θ in °C + 273.
| Case | Description |
|---|---|
| 1 | A steam turbine works between 600 K and 300 K. |
| 2 | A heat pump with a COP of 4 heats a house that needs 12,000 kWh of heat per year. |
| 3 | An engine takes in 1,000 J of heat per cycle and does 300 J of work. |
1. What is the Carnot efficiency in case 1?
D: 50%
1 − 300 ÷ 600.
2. How much electricity does the heat pump in case 2 use per year?
B: 3,000 kWh
12,000 ÷ 4.
3. How much heat does the heat pump take from the environment per year?
D: 9,000 kWh
12,000 − 3,000.
4. How much waste heat does the engine in case 3 release per cycle?
C: 700 J
1,000 − 300, by the first law.
5. Why is a heat pump more efficient in mild weather?
A: The temperature difference it must bridge is smaller
COP rises as the difference falls.
6. Which statement follows from the second law of thermodynamics?
B: Heat cannot be converted completely into work in a cycle
Some waste heat is unavoidable.
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