A heat exchanger is used to warm apple cider using steam as the heat source. The cider is heated from an initial temperature of 4°C to a final temperature of 65°C. The steam enters the heat exchanger as 50% quality steam and exits as water condensate at 85°C. Calculate the mass of steam required to heat 150 kg of cider. (For the cider, assume the Cp=3.651 kJ/kg°C, and latent heat=280.18 kJ/kg.)

29.1

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To calculate the mass of steam required to heat the cider, we need to consider two separate processes:

Process 1: Heating the cider from 4°C to the boiling point (100°C).

Process 2: Condensing the steam and heating the cider from the boiling point (100°C) to the final temperature of 65°C.

Let's calculate the mass of steam required for each process.

Process 1: Heating the cider from 4°C to the boiling point (100°C):

The energy required to heat the cider can be calculated using the specific heat formula:
Q = mcΔT

where Q is the energy absorbed, m is the mass of the cider, c is the specific heat, and ΔT is the temperature change.

Using the given information:
m = 150 kg (mass of cider)
c = 3.651 kJ/kg°C (specific heat of cider)
ΔT = 100°C - 4°C = 96°C (temperature change)

Q1 = mcΔT
= 150 kg × 3.651 kJ/kg°C × 96°C
= 528,264 kJ

Process 2: Condensing the steam and heating the cider from the boiling point (100°C) to the final temperature of 65°C:

To condense the steam, we need to release the latent heat of vaporization.

The energy required to condense the steam can be calculated as:
Q2 = mL

where Q2 is the energy absorbed, m is the mass of steam, and L is the latent heat of vaporization.

Using the given information:
Q2 = mL
= 150 kg × 280.18 kJ/kg
= 42,027 kJ

Now, let's calculate the mass of steam required for Process 2.

The energy released by the steam as it condenses and heats the cider is equal to Q2, so we can equate it to the energy absorbed in Process 1, Q1.

Q2 = Q1

mL = mcΔT

m = (mcΔT) / L

m = (528,264 kJ) / (280.18 kJ/kg)

m ≈ 1885.7 kg

Therefore, the mass of steam required to heat 150 kg of cider is approximately 1885.7 kg.