Question

Water is to be heated from \(10^{\circ} \mathrm{C}\) to \(80^{\circ} \mathrm{C}\) as it flows through a \(2-\mathrm{cm}\)-internal-diameter, \(13-\mathrm{m}\)-long tube. The tube is equipped with an electric resistance heater, which provides uniform heating throughout the surface of the tube. The outer surface of the heater is well insulated, so in steady operation all the heat generated in the heater is transferred to the water in the tube. If the system is to provide hot water at a rate of \(5 \mathrm{~L} / \mathrm{min}\), determine the power rating of the resistance heater. Also, estimate the inner surface temperature of the pipe at the exit.

Short answer

Answer: The mass flow rate of the water is 5/60 kg/s. 2. What is the heat transfer rate in kW? Answer: The heat transfer rate is 24.5 kW. 3. What is the power rating of the resistance heater? Answer: The power rating of the resistance heater is 24.5 kW. 4. What is the inner surface temperature of the pipe at the exit in °C? Answer: The inner surface temperature of the pipe at the exit is approximately 80.53 °C.

Step-by-step solution

Find the mass flow rate of the water

To find the mass flow rate of the water, we need to convert the flow rate given in Liters per minute to a mass flow rate in kg/s. We'll use the density of the water, which is typically 1000 kg/m³. Flow Rate = 5 L/min = 5/1000 m³/min = (5/1000)*(1/60) m³/s Now convert the flow rate to mass flow rate using the density of water: Mass Flow Rate = (5/1000)*(1/60) m³/s * 1000 kg/m³ = 5/60 kg/s

Calculate the heat transfer rate

To calculate the heat transfer rate, we can use the formula: Heat Transfer Rate = Mass Flow Rate * Specific Heat Capacity * Temperature Change where Specific Heat Capacity of water (C_p) = 4.18 kJ/kg·K Temperature Change = 80 - 10 = 70 K Heat Transfer Rate = (5/60 kg/s) * (4.18 kJ/kg·K) * 70 K = 24.5 kW

Determine the power rating of the resistance heater

Since all the heat generated in the heater is transferred to the water in the tube, the power rating of the resistance heater is equal to the heat transfer rate. Power Rating of Heater = Heat Transfer Rate = 24.5 kW

Estimate the inner surface temperature of the pipe at the exit

We can estimate the inner surface temperature of the pipe at the exit using the heat transfer rate. Let's denote the heat transfer coefficient as h and the temperature difference between the inner surface of the pipe and the exit temperature of the water as ΔT. We can use the formula: Heat Transfer Rate = h * (π * Diameter * Length) * ΔT Assuming a heat transfer coefficient (h) of 5000 W/m²·K: 24.5 kW = (5000 W/m²·K) * (π * 0.02 m * 13 m) * ΔT Solve for ΔT and add to the exit temperature of the water (80 °C): Inner Surface Temperature at Exit = ΔT + 80 = 80.53 °C The inner surface temperature of the pipe at the exit is approximately 80.53 °C. In conclusion, the power rating of the resistance heater is 24.5 kW, and the inner surface temperature of the pipe at the exit is approximately 80.53 °C.