Using Q equals mass times specific heat times temperature rise, then finding heat-up time.
This example shows how much energy it takes to heat a cylinder of water and how long an immersion heater takes to do it. The numbers are chosen for the worked example.
The formula. Q = m x c x change in T, where m is mass, c is specific heat capacity and change in T is the temperature rise. For water, 1 litre is about 1 kg and c is about 4.2 kJ per kg per °C. Metals need far less: copper's specific heat is about 0.39 kJ per kg per °C and steel's about 0.45, so warming a kilogram of water takes roughly eleven times the energy of warming a kilogram of copper by the same amount.
The situation. A 120 litre cylinder (so about 120 kg of water) is heated from 10 °C to 60 °C. The temperature rise is 60 - 10 = 50 °C. A kelvin is the same size as a degree Celsius, so a temperature difference has the same number in both units: this 50 °C rise is also 50 K. Adding 273 converts a temperature to kelvin (10 °C is about 283 K), never a difference.
Step 1 — energy required. Q = 120 x 4.2 x 50 = 25,200 kilojoules (kJ), which is 25.2 megajoules.
Step 2 — convert to kWh (optional). 1 kWh is 3600 kJ, so 25,200 / 3600 = 7.0 kWh.
Step 3 — heat-up time. A 3 kW immersion heater delivers 3 kJ every second. Time = energy / power = 25,200 kJ / 3 kJ per second = 8400 seconds, which is 140 minutes, or 2 hours 20 minutes.
Takeaway. Energy scales with both the amount of water and the temperature rise, and time is just energy divided by power. This is why a larger cylinder or a colder start takes noticeably longer to recover. Real systems also lose some heat, so allow a margin, and follow current Part L guidance for efficiency.
Changing state takes far more. Melting and boiling take in energy without any rise in temperature; this energy is called latent heat. Melting 1 kg of ice at 0 °C takes about 334 kJ, the same energy that would warm 1 kg of water from 0 °C to about 80 °C (334 ÷ 4.2 is about 80). Turning 1 kg of water at 100 °C into steam takes about 2256 kJ, more than five times the 420 kJ that warms it from 0 °C to 100 °C (4.2 × 100 = 420). The energy is given back when the steam condenses.