A juice processing operation uses steam to heat the fruit juice up to a temperature of 96̊C prior to packaging it in drink boxes. They need to heat 3,250 kg of apple juice (10.5% solids) from an initial storage temperature of 39̊F to its heat treatment temperature. How much saturated steam at 617.8 kPa would be needed for this task (to the nearest kg)? In this case, the condensate is to be discharged from the process without any further cooling to recover any of its heat and is returned to the “steam plant” where it is reheated and turned back into steam.
How much steam is required per batch of product (see description in Part a) if the condensate is discharged from the process after being cooled to 75̊C to use some of its heat in the process? Express your answer to the nearest kg
How much steam is saved by recovering some of the heat in the condensate (i.e., the difference between the process as outlined in part a as compared to the process in part b). Express your answer to the nearest of kg of steam.
To calculate the amount of steam required for the task described in the question, we need to calculate the energy required to heat the apple juice.
1. Convert the initial storage temperature from Fahrenheit to Celsius:
39̊F = 3.9̊C
2. Calculate the temperature difference:
ΔT = Final temperature - Initial temperature
= 96̊C - 3.9̊C
= 92.1̊C
3. Convert the mass of apple juice from kg to g:
3,250 kg = 3,250,000 g
4. Calculate the energy required to heat the apple juice:
Energy = mass × specific heat capacity × temperature difference
Specific heat capacity of apple juice: 4186 J/(kg·°C)
Energy = 3,250,000 g × 4186 J/(g·°C) × 92.1̊C
Next, we need to calculate the energy provided by the steam:
5. Convert the pressure of the saturated steam from kPa to Pa:
617.8 kPa = 617,800 Pa
6. Determine the heat transfer capacity of the steam using steam tables or steam properties calculator. Let's assume it is 2025 kJ/kg.
7. Calculate the mass of steam required:
Energy provided by steam = Energy required to heat the apple juice
Mass of steam × heat transfer capacity of the steam = Energy required to heat the apple juice
Mass of steam = Energy required to heat the apple juice / heat transfer capacity of the steam
To determine the results for Part a, we can follow the above calculations.
For Part b, we need to consider that the condensate is discharged from the process after being cooled to 75̊C. We'll need to calculate the energy saved by recovering some of the heat from the condensate. Let's assume the specific heat capacity of the condensate as 4.18 J/(g·°C).
8. Calculate the energy saved:
Energy saved = Mass of condensate × specific heat capacity of condensate × temperature difference
Mass of condensate = Energy saved / (specific heat capacity of condensate × temperature difference)
Finally, to determine the difference in steam saved between Part a and Part b, we need to calculate the mass of steam saved in Part b compared to Part a.
Let's plug in the values and calculate the results.
To calculate the amount of steam required, we need to calculate the heat required to heat the apple juice from 39̊F to 96̊C.
Step 1: Convert the initial storage temperature from Fahrenheit to Celsius.
39̊F = (39 - 32) x 5/9 = 3.9̊C
Step 2: Calculate the temperature difference.
Temperature difference = (96 - 3.9) = 92.1̊C
Step 3: Convert the mass of apple juice from kg to liters.
1 kg of apple juice = 1 liter
3,250 kg of apple juice = 3,250 liters
Step 4: Calculate the heat required to heat the apple juice.
Heat (Joules) = mass (kg) x specific heat capacity (J/kg·K) x temperature difference (°C)
Specific heat capacity of apple juice = 4,190 J/kg·K (assumed value for water)
Heat (Joules) = 3,250 kg x 4,190 J/kg·K x 92.1̊C = 1,197,162,750 J
Step 5: Convert the heat from Joules to kilocalories.
1 kilocalorie = 4186.8 J
Heat (kcal) = 1,197,162,750 J / 4186.8 J/kcal = 285,946.6 kcal
Step 6: Convert the heat from kilocalories to kg of steam.
1 kg of steam = 540 kcal
Steam (kg) = 285,946.6 kcal / 540 kcal/kg = 529.5 kg
Therefore, to heat 3,250 kg of apple juice from 39̊F to 96̊C, approximately 530 kg of saturated steam at 617.8 kPa would be needed.
For part b, where the condensate is discharged from the process after being cooled to 75̊C, we need to recalculate the amount of steam required.
Step 7: Calculate the temperature difference.
Temperature difference = (96 - 75) = 21̊C
Step 8: Calculate the heat required to heat the apple juice.
Heat (Joules) = mass (kg) x specific heat capacity (J/kg·K) x temperature difference (°C)
Heat (Joules) = 3,250 kg x 4,190 J/kg·K x 21̊C = 286,736,050 J
Step 9: Convert the heat from Joules to kilocalories.
Heat (kcal) = 286,736,050 J / 4186.8 J/kcal = 68,526.4 kcal
Step 10: Convert the heat from kilocalories to kg of steam.
Steam (kg) = 68,526.4 kcal / 540 kcal/kg = 127 kg
Therefore, if the condensate is discharged from the process after being cooled to 75̊C, approximately 127 kg of steam would be required per batch of product.
To calculate the steam saved by recovering some of the heat in the condensate, we need to find the difference between the amount of steam required in part a and part b.
Steam saved (kg) = Steam required in part a - Steam required in part b
Steam saved (kg) = 530 kg - 127 kg = 403 kg
Therefore, by recovering some of the heat in the condensate, approximately 403 kg of steam would be saved.