Wet sugar that contains one-fifth water by mass is conveyed through an evaporator in which 85.0% of the entering water is vaporized.
a) Taking a basis of 100 kg feed, calculate i) 𝑥𝑤, the mass fraction of water in the wet sugar leaving the evaporator, and ii) the ratio (kg H2O vaporized/kg wet sugar leaving the evaporator).
b) If 1000 tons/day of wet sugar is fed to the evaporator, how much additional water must be removed from the outlet sugar to dry it completely, and what annual revenue can be expected if dry sugar sells for $0.15/lbm?
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a)
i) To calculate the mass fraction of water in the wet sugar leaving the evaporator, we need to determine the mass of water in the wet sugar entering and leaving the evaporator.
Assuming 100 kg of wet sugar is fed to the evaporator:
- The initial mass of water in the wet sugar: (1/5) x 100 kg = 20 kg
- The mass of water vaporized: 85% of 20 kg = 17 kg
To find the mass of sugar leaving the evaporator, we subtract the vaporized water from the initial mass of wet sugar:
- Mass of sugar leaving the evaporator = 100 kg - 17 kg = 83 kg
The mass fraction of water, 𝑥𝑤, is calculated by dividing the mass of water by the total mass of wet sugar:
𝑥𝑤 = mass of water / total mass of wet sugar
=> 𝑥𝑤 = 17 kg / 83 kg ≈ 0.2051
ii) The ratio of water vaporized to wet sugar leaving the evaporator is calculated by dividing the mass of water vaporized by the mass of wet sugar leaving the evaporator:
Ratio = mass of water vaporized / mass of wet sugar leaving the evaporator
=> Ratio = 17 kg / 83 kg ≈ 0.2048
b)
To determine the additional water that must be removed from the outlet sugar to dry it completely, we first need to know the moisture content of dry sugar. Let's assume dry sugar has no moisture content.
If 1000 tons/day of wet sugar is fed to the evaporator, we convert it to kg/day:
1000 tons = 1000 tons * 2000 lbm/ton * 0.4536 kg/lbm = 907,200 kg/day (approx.)
The mass of sugar leaving the evaporator is equivalent to the mass of wet sugar fed to the evaporator:
Mass of sugar leaving the evaporator = 907,200 kg/day
To calculate the additional water that must be removed to dry the outlet sugar completely, we subtract the mass of dry sugar from the mass of wet sugar:
Additional water = Mass of wet sugar - Mass of dry sugar
=> Additional water = 907,200 kg/day - 907,200 kg/day = 0 kg/day
Since no additional water needs to be removed, the outlet sugar is already dry.
The annual revenue can be calculated by multiplying the mass of dry sugar by the selling price per pound, considering 365 days in a year:
Annual revenue = Mass of dry sugar * Selling price * 365 days
=> Annual revenue = 907,200 kg/day * $0.15/lbm * 365 days
=> Annual revenue = $49,756,400 (approx.)
Therefore, the annual revenue can be expected to be approximately $49,756,400.
To solve this problem, we'll break it down into steps:
a)
i) To find the mass fraction of water in the wet sugar leaving the evaporator (𝑥𝑤), we can start by calculating the mass of water vaporized in the evaporator.
Step 1: Calculate the mass of water vaporized.
- Since 85% of the entering water is vaporized, we can calculate the mass of water vaporized by multiplying the mass of water in the wet sugar by 0.85.
- Mass of water vaporized = 0.85 * (mass of water in wet sugar)
Step 2: Calculate the mass of wet sugar leaving the evaporator.
- Since wet sugar contains one-fifth water by mass, the mass of dry sugar can be calculated by subtracting the mass of water from the total mass of the wet sugar.
- Mass of wet sugar leaving the evaporator = 100 kg - (mass of water in wet sugar)
Step 3: Calculate the mass fraction of water in the wet sugar leaving the evaporator.
- The mass fraction of water (𝑥𝑤) is calculated by dividing the mass of water vaporized by the mass of wet sugar leaving the evaporator.
- 𝑥𝑤 = (mass of water vaporized) / (mass of wet sugar leaving the evaporator)
ii) To find the ratio of water vaporized to wet sugar leaving the evaporator (kg H2O vaporized/kg wet sugar), we need to divide the mass of water vaporized by the mass of wet sugar leaving the evaporator.
Step 1: Calculate the ratio of water vaporized to wet sugar leaving the evaporator.
- Ratio (kg H2O vaporized/kg wet sugar) = (mass of water vaporized) / (mass of wet sugar leaving the evaporator)
b)
To find how much additional water must be removed from the outlet sugar to dry it completely, and the annual revenue, we can follow these steps:
Step 1: Calculate the mass of water in the wet sugar fed to the evaporator.
- Given that 1000 tons/day of wet sugar is fed to the evaporator, we need to convert it to kilograms.
- Mass of wet sugar fed/day = 1000 tons/day * 1000 kg/ton
- Mass of water in the wet sugar fed to the evaporator = Mass of wet sugar fed/day * (mass fraction of water in the wet sugar)
Step 2: Calculate the mass of water remaining in the outlet sugar after evaporation.
- Since the wet sugar is dried completely, the mass of water remaining is 0.
Step 3: Calculate the additional water that needs to be removed from the outlet sugar to dry it completely.
- Additional water to be removed = Mass of water in the wet sugar fed to the evaporator - Mass of water remaining in the outlet sugar
Step 4: Calculate the annual revenue.
- Revenue = (Mass of dry sugar produced annually) * (price per unit of dry sugar)
Step 5: Convert the mass of wet sugar fed/day to annual mass.
- Multiply the mass of wet sugar fed/day by the number of days in a year.
Step 6: Calculate the mass of dry sugar produced annually.
- Mass of dry sugar produced annually = (Mass of wet sugar fed/year - Mass of additional water to be removed) * (1 - 𝑥𝑤)
Step 7: Calculate the annual revenue.
- Annual revenue = Mass of dry sugar produced annually * price per unit of dry sugar
Note: You can now substitute the given values to calculate the final results.