If you have 10 lbs shrimp at 80 deg F how much ice would it take to reach 50 deg F in 10 minutes?
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To calculate the amount of ice needed to cool down the shrimp from 80°F to 50°F in 10 minutes, we need to consider the heat transfer equation and the specific heat capacity of both shrimp and ice.
The heat transfer equation can be expressed as:
Q = mcΔT
Where:
Q = amount of heat transferred
m = mass of the object
c = specific heat capacity of the object
ΔT = change in temperature
First, let's calculate the initial heat of the shrimp at 80°F:
Q1 = m1 * c1 * ΔT1
Where:
m1 = mass of the shrimp
c1 = specific heat capacity of the shrimp (assumed to be 1 calorie/gram°C)
ΔT1 = initial temperature difference (80°F - 32°F)
Since 1 pound is approximately equal to 454 grams, and 1 calorie is approximately equal to 4.18 joules, we can convert the units accordingly:
m1 = 10 lbs * 454 g/lb ≈ 4540 g
ΔT1 = (80°F - 32°F) * (5/9) ≈ 27.8°C = 27.8 K
c1 = 1 calorie/gram°C ≈ 4.18 joules/gram°C
Q1 = 4540 g * 4.18 joules/gram°C * 27.8 K
Now, let's calculate the heat needed to cool down the shrimp to 50°F:
Q2 = m1 * c1 * ΔT2
Where:
ΔT2 = final temperature difference (50°F - 32°F)
ΔT2 = (50°F - 32°F) * (5/9) ≈ 9.4 K
Q2 = 4540 g * 4.18 joules/gram°C * 9.4 K
To find the total heat transfer, we subtract Q2 from Q1:
Q_total = Q1 - Q2
Now, to find the amount of ice needed, we need to calculate the heat of fusion (latent heat) required to convert ice into water:
Q_fusion = m_ice * L_fusion
Where:
m_ice = mass of the ice
L_fusion = latent heat of fusion for water (334 Joules/gram)
Since we have 10 minutes to cool down the shrimp, we can assume that the cooling process occurs quickly, and the heat transfer rate is constant. Therefore, the cooling process can be treated as an isothermal process.
To calculate the cooling rate, we divide the total heat transfer by the time (in seconds):
Rate = Q_total / (10 minutes * 60 seconds/minute)
Finally, to determine the amount of ice needed, we divide the rate by the latent heat of fusion:
m_ice = Rate / L_fusion
Substituting the values into the equations will give you the exact amount of ice required.