A glass bottle washing facility uses a well agi(Es) tated hot water bath at \(50^{\circ} \mathrm{C}\) with an open top that is placed on the ground. The bathtub is \(1 \mathrm{~m}\) high, \(2 \mathrm{~m}\) wide, and \(4 \mathrm{~m}\) long and is made of sheet metal so that the outer side surfaces are also at about \(50^{\circ} \mathrm{C}\). The bottles enter at a rate of 800 per minute at ambient temperature and leave at the water temperature. Each bottle has a mass of \(150 \mathrm{~g}\) and removes \(0.6 \mathrm{~g}\) of water as it leaves the bath wet. Makeup water is supplied at \(15^{\circ} \mathrm{C}\). If the average conditions in the plant are \(1 \mathrm{~atm}, 25^{\circ} \mathrm{C}\), and 50 percent relative humidity, and the average temperature of the surrounding surfaces is \(15^{\circ} \mathrm{C}\), determine (a) the amount of heat and water removed by the bottles themselves per second, \((b)\) the rate of heat loss from the top surface of the water bath by radiation, natural convection, and evaporation, \((c)\) the rate of heat loss from the side surfaces by natural convection and radiation, and \((d)\) the rate at which heat and water must be supplied to maintain steady operating conditions. Disregard heat loss through the bottom surface of the bath and take the emissivities of sheet metal and water to be \(0.61\) and \(0.95\), respectively.

Step by step solution

01

Calculate the amount of heat and water removed by the bottles

First, we need to determine the heat and water removed by the bottles as they leave the bath. The bottles are entering at \(25^{\circ} \mathrm{C}\) and leaving at \(50^{\circ} \mathrm{C}\). Given that there are 800 bottles per minute with a mass of \(150 \mathrm{~g}\) each and remove \(0.6 \mathrm{~g}\) of water as they leave the bath. We will convert the mass flow rate of bottles and water removed per second. To calculate the heat removed by the bottles: 1. Determine the heat capacity of the bottles: We can assume they are made of glass, which has a specific heat capacity of \(c_p \approx 840 \mathrm{J/(kg·K)}\) 2. Calculate the total mass flow rate of the bottles per second: m_dot = (number of bottles/minute * mass of bottle) / 60 3. Calculate the temperature difference: ΔT = T_exit - T_entry 4. Calculate the heat removed by the bottles: Q_dot = m_dot * c_p * ΔT To calculate the water removed by the bottles: 1. Calculate the total mass flow rate of water removed per second: m_dot_water = (number of bottles/minute * mass of water removed) / 60

02

Calculate the heat loss from the top surface of the water bath

Next, we need to calculate the heat loss from the top surface of the water bath by radiation, natural convection, and evaporation. First, determine the surface area of the top surface: A = 2 * 4 m² To calculate the heat loss due to radiation: 1. Determine the Stefan-Boltzmann constant: σ = 5.67×10^-8 W/(m²·K^4) 2. Calculate the emissivity of the water: ε_water = 0.95 3. Calculate the temperature of the water in Kelvin: T_water = 50 + 273.15 K 4. Calculate the temperature of the surrounding surfaces in Kelvin: T_surrounding = 15 + 273.15 K 5. Calculate the heat loss due to radiation: Q_dot_radiation = ε_water * σ * A * (T_water^4 - T_surrounding^4) To calculate the heat loss due to natural convection and evaporation, we can use the following empirical correlation: Q_dot_convection_evaporation = 230 * A * ΔT

03

Calculate the heat loss from the side surfaces of the water bath

Similarly, we need to calculate the heat loss from the side surfaces of the water bath by natural convection and radiation. First, determine the surface area of the side surfaces: A_side = (1 * 2 + 1 * 4) * 2 m² To Calculate the heat loss due to radiation: 1. Determine the emissivity of the sheet metal: ε_metal = 0.61 2. Calculate the heat loss due to radiation for side surfaces: Q_dot_radiation_sides = ε_metal * σ * A_side * (T_water^4 - T_surrounding^4) To calculate the heat loss due to natural convection for side surfaces, we can use a similar empirical correlation for the vertical plates: Q_dot_convection_sides = 8 * A_side * ΔT

04

Calculate the rate at which heat and water must be supplied

Finally, we need to determine the rate at which heat and water must be supplied to maintain steady operating conditions. To calculate the necessary heat input: Q_dot_input = Q_dot_removed_by_bottles + Q_dot_radiation + Q_dot_convection_evaporation + Q_dot_radiation_sides + Q_dot_convection_sides To calculate the necessary water input, we simply need to balance the mass flow rate of water being removed by the bottles: m_dot_water_input = m_dot_water_removed_by_bottles Now we have calculated the amount of heat and water removed by the bottles, the rates of heat loss from the top surface by radiation, natural convection, and evaporation, the rates of heat loss from the side surfaces by natural convection and radiation, and the rate at which heat and water must be supplied to maintain steady operating conditions.

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