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.)

TS1 = 85 C

xS1 = 0.5000

TS2 = 85 C
xS2 = 0.0 ( saturated liquid water at 85 C )

Steam Table Data at 85 C :
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T = 85 C
Psat = 57.83 kPa
hf = 355.90 kJ / kg
hfg = 2296.0 kJ / kg
hg = 2651.9 kJ / kg

Get hS1 for the steam from the quality equation :
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hS1 = hf + ( xS1 ) ( hfg )

hS1 = 355.90 + ( 0.5000 ) ( 2296.0 ) = 1503.9 kJ / kg

hS2 = hf at 85 C = 355.90 kJ / kg

Delta hS = hS2 - hS1

Delta hS = 355.9 - 1503.9 = - 1148 kJ / kg

qS = QS / mS = Delta hS = - 1148 kJ / kg

Now get QAC :
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QAC = ( mAC ) ( CpAC ) ( TAC2 - TAC1 )

QAC = ( 150 kg ) ( 3.651 kJ / kg - C ) ( 65.0 C - 4.0 C )

QAC = 33410 kJ

You have QS = - ( QAC ) :
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QS = - 33410 kJ

mS = QS / qS

mS = ( - 33410 ) / ( - 1148 kJ/kg )

mS = 29.1 kg <-------------------------

To solve this problem, we need to calculate the heat transfer required to heat the cider and then use the specific heat capacity of the cider and the latent heat of steam to determine the mass of steam needed.

Step 1: Calculate the heat transfer required to heat the cider.
The formula to calculate the heat transfer is: Q = m * Cp * ΔT
where:
Q = heat transfer (in kJ)
m = mass (in kg)
Cp = specific heat capacity (in kJ/kg°C)
ΔT = change in temperature (in °C)

In this case, the initial temperature of the cider (T1) is 4°C, and the final temperature (T2) is 65°C.

ΔT = T2 - T1
ΔT = 65°C - 4°C
ΔT = 61°C

Q = m * Cp * ΔT

Step 2: Calculate the heat transfer needed to convert steam to water.
The formula is: Q = m * L
where:
Q = heat transfer (in kJ)
m = mass (in kg)
L = latent heat (in kJ/kg)

In this case, the quality of steam (x) is given as 50%, which means that 50% of the mass is steam and the remaining 50% is water.

Step 3: Write the energy balance equation.
To find the mass of steam needed, we equate the heat transfer required to heat the cider (Q1) to the heat transfer needed to convert steam to water (Q2).

Q1 + Q2 = 0

Step 4: Solve the equations.
Substituting the formulas and values into the energy balance equation, we have:

m * Cp * ΔT + m * L = 0

Now, let's plug in the given values:
Cp = 3.651 kJ/kg°C
ΔT = 61°C
L = 280.18 kJ/kg

m * 3.651 * 61 + m * 280.18 = 0

Simplifying the equation, we have:
221.911m + 280.18m = 0
501.091m = 0
m ≈ 0 kg

Based on the equation above, we can understand that the mass of steam required to heat the cider is approximately 0 kg. This result suggests that there might be an error in the calculation or missing information. Please double-check the given values and ensure no information has been omitted.