Wet sugar, containing one-fifth water by mass is conveyed through an evaporator in which 85.0% of the entering water is vaporized. Taking a Basis of 100 kg of feed to the evaporator
A barrel of oil contains about 6 million Btu and about 1000 Btu is needed to evaporate one lbm of water. Find the current cost of a barrel of oil (document your source) and determine the cost of fuel per 1 lbm to dry the sugar. How does your cost change if the first stage drying (to 85% of the water is evaporated) is only 80% thermally efficient and the remaining drying is 50% efficient?
After calculation, I figure out that I need to find the cost to dry 200 tons of water to get sugar. However, when "the first stage drying (to 85% of the water is evaporated) is only 80% thermally efficient and the remaining drying is 50% efficient", I feel like it just costs the same amount of money because we still need to dry 200 tons of water. However, i am not sure if it is right?
Any help is greatly appreciated
We can break the problem down into multiple parts:
1. First, we determine the amount of water that needs to be evaporated.
2. Then, we find the heat required to evaporate the water.
3. Finally, we calculate the cost of fuel to provide the required heat.
Let's do this step by step:
1. Amount of water to evaporate:
Mass of wet sugar = 100 kg
Mass fraction of water in wet sugar = 1/5
Mass of water in wet sugar = (1/5) * 100 kg = 20 kg
The evaporator removes 85% of the entering water:
Mass of water evaporated = 0.85 * 20 kg = 17 kg
2. Heat required to evaporate water:
1 kg of water = 2.205 lbm (pounds mass)
17 kg of water = 37.485 lbm
1 lbm of water requires 1000 Btu to evaporate:
Heat required to evaporate 17 kg (37.485 lbm) of water = 37,485 Btu
3. Calculate the cost of fuel
As of October 2021, the cost of a barrel of oil is around $80 USD (source: https://www.macrotrends.net/1369/crude-oil-price-history-chart). Therefore:
1 barrel of oil = 6,000,000 Btu
1 Btu of oil = $80 / 6,000,000 = $0.0000133
Cost of fuel to evaporate 37,485 Btu = 37,485 Btu * $0.0000133 = $0.498
Now, let's consider the change in cost based on efficiency:
First-stage drying efficiency = 80% (0.8)
Remaining drying efficiency = 50% (0.5)
The heat required in both stages will be the same due to needing to evaporate the same amount of water, but the energy input into the system will differ based on efficiency.
First stage required energy input:
37,485 Btu / 0.8 = 46,856.25 Btu
Remaining stage required energy input:
37,485 Btu / 0.5 = 74,970 Btu
Total energy input with efficiency included = 46,856.25 + 74,970 = 121,826.25 Btu
Cost of fuel with efficiency considered:
121,826.25 Btu * $0.0000133 = $1.62
So, the cost to dry the sugar increases to $1.62 with the efficiency changes considered.
To determine the cost of fuel per 1 lbm to dry the sugar, we need to calculate the amount of fuel required to evaporate the water.
First, let's calculate the amount of water in 100 kg of feed. Since the wet sugar contains one-fifth water by mass, the amount of water in the 100 kg feed is 100 kg * (1/5) = 20 kg.
Next, we need to convert the mass of water to pounds. Since 1 kg is equal to approximately 2.20462 pounds, the amount of water in pounds is 20 kg * 2.20462 lbs/kg = 44.0924 lbs.
We are given that 1000 Btu is needed to evaporate 1 lbm of water. Therefore, the total amount of energy required to evaporate the 44.0924 lbs of water is 44.0924 lbs * 1000 Btu/lbm = 44,092.4 Btu.
To find the cost of fuel per 1 lbm, we need to know the cost of a barrel of oil. The price of oil is variable and can change over time. To find the current cost of a barrel of oil, you can refer to financial news websites, government energy agencies, or oil price tracking websites. For the purpose of this explanation, let's assume the current cost of a barrel of oil is $60.
Given that a barrel of oil contains about 6 million Btu, we can calculate the cost of fuel per 1 lbm using the following formula:
Cost of fuel per 1 lbm = (Cost of a barrel of oil / Energy content of a barrel of oil) * Energy required to evaporate 1 lbm of water.
Substituting the values, we have:
Cost of fuel per 1 lbm = ($60 / 6,000,000 Btu) * 44,092.4 Btu = $0.73487
Therefore, the cost of fuel per 1 lbm to dry the sugar is approximately $0.73487.
Now, let's consider the scenario where the first stage drying is only 80% thermally efficient and the remaining drying is 50% efficient. In this case, the energy required to evaporate the water will increase due to the reduced efficiency.
To calculate the new energy required, we can divide it into two stages:
1. First stage drying (to 85% of the water is evaporated) at 80% efficiency:
Energy required for first stage drying = (Energy required to evaporate 44.0924 lbs of water) / (0.85 * 0.80)
2. Remaining drying (15% of the water remaining) at 50% efficiency:
Energy required for remaining drying = (Energy required to evaporate 15% of the initial water) / 0.50
The total energy required will be the sum of these two stages.
To find the new cost of fuel per 1 lbm, we use the same formula as before, but using the new energy value calculated.
Keep in mind that the exact cost will depend on the efficiency factors and the specific energy costs, so these values are for illustrative purposes.
Remember to check the current cost of a barrel of oil from reliable sources to get accurate results.
To find the cost of fuel per 1 lbm to dry the sugar, we need to calculate the amount of fuel required.
First, let's calculate the amount of water in 100 kg of wet sugar. Since the sugar contains one-fifth water by mass, it means the remaining four-fifths is sugar. Therefore, the water content in 100 kg of wet sugar is (1/5) * 100 kg = 20 kg.
Next, we need to determine the amount of water that needs to be evaporated. Given that 85% of the entering water is vaporized, the amount of water to be evaporated is (85/100) * 20 kg = 17 kg.
Using the conversion factor of 1 lbm of water requiring 1000 Btu, we can calculate the total energy required to evaporate 17 kg of water: 17 kg * 2.20462 lbm/kg * 1000 Btu/lbm = 37,521 Btu.
To determine the cost of fuel per 1 lbm to dry the sugar, we need to divide the total energy required (37,521 Btu) by the energy content of one barrel of oil (6 million Btu). Thus, the cost of fuel per 1 lbm to dry the sugar is (37,521 Btu / 6,000,000 Btu) * the cost of one barrel of oil.
The current cost of a barrel of oil will depend on various factors and can vary over time. To find the most up-to-date cost, it is best to consult reliable sources such as financial news websites or energy market websites.
Regarding the efficiency question, if the first stage drying is only 80% thermally efficient and the remaining drying is 50% efficient, the total energy required would increase. The calculation would be as follows:
First stage drying efficiency: 80%
Remaining drying efficiency: 50%
Energy required with modified efficiency:
Total energy required = (37,521 Btu / 0.8) + (37,521 Btu / 0.5)
To find the updated cost of fuel per 1 lbm to dry the sugar, divide the total energy required by 6 million Btu and multiply by the cost of one barrel of oil.