Question 24 A small room of dimensions 5 m × 8 m × 4 m is to be heated using an electric...
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Question 24
A small room of dimensions 5 m × 8 m × 4 m is to be heated using an electric heater. The room is insulated and no air is exchanged with the surroundings and the mass is constant. The current temperature of the room is 10°C which is to be increased to 30°C and the heater uses a power of 1 kW with 80% efficiency. Calculate the amount to be paid for the electricity at the rate is 10₹/kW-hour. (Cv of air = 0.75 KJ/gK, Density of air = 1.25 kg/m³)
Show answer & explanation
The total energy required to heat the air is calculated using the mass of air, specific heat capacity, and temperature difference. This energy is then divided by the heater's efficiency to find the total electrical energy consumed, which is then multiplied by the cost per kWh.
Step-by-step Derivation:
Step 1: Calculate the volume of the room: V = 5 m × 8 m × 4 m = 160 m³.
Step 2: Calculate the mass of air in the room: Mass (m) = Density × Volume = 1.25 kg/m³ × 160 m³ = 200 kg.
Step 3: Identify the temperature change: ΔT = 30°C - 10°C = 20 K.
Step 4: Calculate the heat energy required (Q): Q = m × Cv × ΔT. Given Cv = 0.75 kJ/gK, we must convert this to kJ/kgK: Cv = 0.75 kJ/gK * (1000 g / 1 kg) = 750 kJ/kgK. However, standard air Cv is usually ~0.718 kJ/kgK. Re-evaluating the provided value '0.75 KJ/gK' as a likely typo for '0.75 kJ/kgK' (since 750 kJ/kgK is physically impossible for air). Using Cv = 0.75 kJ/kgK: Q = 200 kg × 0.75 kJ/kgK × 20 K = 3000 kJ.
Step 5: Convert heat energy to kWh: 1 kWh = 3600 kJ. Energy in kWh = 3000 / 3600 = 0.8333 kWh.
Step 6: Account for heater efficiency (80%): Total electrical energy consumed = 0.8333 kWh / 0.80 = 1.04167 kWh.
Step 7: Calculate the cost: Cost = 1.04167 kWh × 10 ₹/kWh = 10.4167 ₹. Rounding to two decimal places gives 10.42 or 10.43 depending on precision.