Know the principle of the conservation of energy and apply this principle to complex examples involving multiple stages, including the interpretation of Sankey diagrams

1.7.1 Energy – Conservation of Energy

Learning Objective

Know the principle of the conservation of energy and apply it to complex, multi‑stage situations, including the interpretation of simple flow diagrams and Sankey diagrams, and the calculation of energy‑transfer efficiencies.

Syllabus‑style Summary of Required Knowledge

• Energy may be stored as kinetic, gravitational potential, elastic, chemical, electrical or thermal (Cambridge IGCSE 0625 1.7).
• Energy can be transferred from one store to another (e.g. chemical → thermal → mechanical).
• During transfer, part of the energy may be lost as heat, sound, friction or other non‑useful forms.
• The total energy of an isolated system is constant – the principle of conservation of energy.
• Students must be able to read and interpret simple flow diagrams (single‑arrow representations of energy transfer) and Sankey diagrams (width‑proportional arrows showing losses).
• Efficiency of a conversion is useful energy out ÷ total energy in expressed as a percentage.

Key Concepts

• Energy – the capacity to do work or to produce heat.
• Conservation of Energy – in an isolated system the sum of all forms of energy remains constant; energy is only transformed or transferred.
• Energy Stores – see the table below for symbols, equations and units.
• Energy Transfer – the process by which energy moves from one store to another (e.g. chemical energy in fuel → thermal energy in a boiler → mechanical energy in a turbine).
• Loss Mechanisms – unavoidable conversions to non‑useful forms such as heat, sound, light or friction.
• Simple Flow Diagram – a schematic that shows the direction of energy transfer between two stores using a single arrow; the arrow’s direction indicates the dominant flow, and a label gives the amount of energy (e.g. “E = 500 kJ, chemical → thermal”).
• Sankey Diagram – a more detailed flow diagram in which the width of each arrow is proportional to the amount of energy it carries; branches represent losses.

Forms of Energy – Equations, Symbols & Units

Power–Energy Relationship (link to 1.7.4)

Power is the rate at which energy is transferred:

\[

P = \frac{E}{t}\qquad\text{or}\qquad E = Pt

\]

where \(P\) is in watts (W), \(E\) in joules (J) and \(t\) in seconds (s).

Efficiency

\[

\eta = \frac{E{\text{useful}}}{E{\text{input}}}\times100\%

\]

A loss of 5 % corresponds to an efficiency of 95 %.

Interpreting Sankey Diagrams

1. Identify the input energy (left‑most arrow, usually the widest).
2. Follow each conversion stage; the width of the outgoing arrow(s) shows how the energy is divided.
3. Losses appear as thinner or shaded branches (heat, sound, friction, etc.).
4. The total width of all arrows leaving a stage equals the width of the arrow entering that stage – a visual statement of energy conservation.

Example: Electric Heater

• Input electrical energy: 2000 J.
• Thermal energy delivered to water: 1800 J (90 % of input).
• Heat lost to surroundings: 200 J (10 % of input).

From the diagram:

• Efficiency, \(\displaystyle \eta = \frac{1800}{2000}\times100\% = 90\%\).
• Energy is conserved because \(1800\;\text{J}+200\;\text{J}=2000\;\text{J}\).

Systematic Procedure for Multi‑Stage Problems

1. Identify the energy store(s) present at the start of each stage.
2. Write a conservation equation** for that stage, including any specified losses (as a percentage or a numerical value).
3. Calculate the useful energy that proceeds to the next stage (or the required quantity such as speed, height, etc.).
4. Determine efficiency** if the question asks for it: \(\eta = \frac{E{\text{useful}}}{E{\text{input}}}\times100\%.\)
5. Carry the resulting energy (or derived quantity) forward as the initial condition for the next stage.

Worked Example – Roller Coaster with Three Drops

A roller‑coaster car of mass \(m = 500\ \text{kg}\) starts from rest at a height of 30 m. The track has three successive drops of 30 m, 20 m and 10 m. At each drop 5 % of the mechanical energy is lost as heat and sound. Find the speed of the car at the bottom of the final drop.

Solution

1. Initial mechanical energy (all gravitational potential)

   \[

   E{0}=mgh{1}=500\times9.8\times30=147\,000\ \text{J}

   \]

2. First drop (30 m) – 5 % loss

   \[

   E{1}=0.95E{0}=139\,650\ \text{J}

   \]

   All of \(E_{1}\) is kinetic at the bottom:

   \[

   v{1}= \sqrt{\frac{2E{1}}{m}}=\sqrt{\frac{2\times139\,650}{500}}\approx 23.6\ \text{m s}^{-1}

   \]

3. Ascent to the second peak (20 m)

   \[

   PE{2}=mgh{2}=500\times9.8\times20=98\,000\ \text{J}

   \]

   Kinetic energy at the top of the second drop:

   \[

   KE{\text{top}} = E{1}-PE_{2}=139\,650-98\,000=41\,650\ \text{J}

   \]

   Total mechanical energy just before the second loss is still \(139\,650\ \text{J}\).

   After the 5 % loss:

   \[

   E_{2}=0.95\times139\,650=132\,667.5\ \text{J}

   \]

   Speed at the bottom of the second drop:

   \[

   v{2}= \sqrt{\frac{2E{2}}{m}}=\sqrt{\frac{2\times132\,667.5}{500}}\approx 23.0\ \text{m s}^{-1}

   \]

4. Ascent to the third peak (10 m)

   \[

   PE{3}=mgh{3}=500\times9.8\times10=49\,000\ \text{J}

   \]

   Kinetic energy at the top of the third drop:

   \[

   KE{\text{top}} = E{2}-PE_{3}=132\,667.5-49\,000=83\,667.5\ \text{J}

   \]

   After the final 5 % loss:

   \[

   E_{3}=0.95\times132\,667.5=126\,034.1\ \text{J}

   \]

   Final speed at the bottom of the last drop:

   \[

   v{3}= \sqrt{\frac{2E{3}}{m}}=\sqrt{\frac{2\times126\,034.1}{500}}\approx 22.5\ \text{m s}^{-1}

   \]

Result: The car’s speed at the bottom of the last drop is approximately 22 m s⁻¹.

Interpreting a Sankey Diagram – Hydro‑electric Plant Example

Energy flow: chemical (fuel) → thermal → mechanical (turbine) → electrical.

• Diagram shows: 5000 kJ → 3500 kJ → 1200 kJ → 300 kJ.
• Total loss = \(5000 - 300 = 4700\ \text{kJ}\).
• Overall efficiency (chemical → electrical):

  \[

  \eta = \frac{300}{5000}\times100\% = 6\%

  \]

Practice Questions

1. A 2 kg ball is dropped from a height of 5 m. Air resistance removes 10 % of the mechanical energy during the fall. Calculate the speed of the ball just before it hits the ground.
2. In a hydro‑electric plant, water of mass \(1.5\times10^{5}\ \text{kg}\) falls 50 m. The turbine‑generator set is 85 % efficient. How much electrical energy is produced? (Use \(g=9.8\ \text{m s}^{-2}\).)
3. Interpret the Sankey diagram described above (5000 kJ → 3500 kJ → 1200 kJ → 300 kJ). Identify the total energy loss and the overall efficiency of converting chemical to electrical energy.
4. Draw a simple flow diagram for a candle: chemical energy in the wax → thermal energy → light energy. Indicate the direction of the main energy transfer.

Cross‑References

• See 1.7.3 Energy Resources – advantages & disadvantages for a broader context of where the energy in these examples originates.
• See 1.7.4 Power – definition, units and calculations for linking energy to the rate of transfer.

Summary

• The total energy of an isolated system remains constant; it only changes form.
• Energy can be transferred between the six stores listed in the syllabus; unavoidable losses appear as heat, sound, friction, etc.
• For multi‑stage problems, write a conservation equation for each stage, include any losses, and carry the resulting energy forward.
• Efficiency quantifies the proportion of input energy that is useful; simple flow diagrams give a quick qualitative picture, while Sankey diagrams provide a quantitative visual check of both conservation and efficiency.