3. In an air heating system, the outside air at t_(1)=-15^circ C is heated in a duct heater at p=const to 60^circ C How much heat must be expended to heat 1000m^3 of air? Consider the heat ca- pacity of the air as constant. Take the air pressure as 760 mm Hg.

To solve this problem, we need to use the formula for the heat required to change the temperature of a gas:<br /><br />Q = m * Cp * ΔT<br /><br />Where:<br />- Q is the heat required (in Joules)<br />- m is the mass of the air (in kg)<br />- Cp is the specific heat capacity of air (in J/kg·K)<br />- ΔT is the change in temperature (in K)<br /><br />Given information:<br />- Initial temperature, t1 = -15°C = 258.15 K<br />- Final temperature, t2 = 60°C = 333.15 K<br />- Volume of air, V = 1000 m³<br />- Air pressure, p = constant (not given the value, but we can assume it to be 1 atm or 101.325 kPa)<br /><br />Step 1: Calculate the mass of the air.<br />Density of air, ρ = p / (R * T)<br />Where:<br />- R is the specific gas constant for air, R = 0.287 kJ/kg·K<br />- T is the absolute temperature in K<br /><br />ρ = 101.325 kPa / (0.287 kJ/kg·K * 333.15 K) = 1.186 kg/m³<br /><br />Mass of air, m = ρ * V = 1.186 kg/m³ * 1000 m³ = 1186 kg<br /><br />Step 2: Calculate the change in temperature.<br />ΔT = t2 - t1 = 333.15 K - 258.15 K = 75 K<br /><br />Step 3: Calculate the specific heat capacity of air.<br />Cp = 1.005 kJ/kg·K<br /><br />Step 4: Calculate the heat required.<br />Q = m * Cp * ΔT<br />Q = 1186 kg * 1.005 kJ/kg·K * 75 K<br />Q = 89,427.5 kJ<br /><br />Therefore, the heat required to heat 1000 m³ of air from -15°C to 60°C is 89,427.5 kJ.