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 degrees celcius to a final temperature of 65 degree celcius. The steam enters the heat exchanger as 50% quality steam and exits as water condensate at 85 degrees celcius. Calculate the mass of steam required to heat 150 Kilograms of cider. (For the cider, assume the Cp = 3.651 KJ/Kg degrees celcius and latent heat = 280.18 KJ/Kg)
To calculate the mass of steam required to heat the cider, we can use the energy balance equation:
Qcider = Qsteam
Where Qcider is the energy required to heat the cider and Qsteam is the energy transferred by the steam.
The energy required to heat the cider can be calculated using:
Qcider = mcider * Cp * ΔTcider
Where mcider is the mass of the cider, Cp is the specific heat capacity of the cider, and ΔTcider is the change in temperature of the cider.
In this case, mcider = 150 kg, Cp = 3.651 kJ/kg°C, and ΔTcider = (65 - 4) = 61°C.
Now let's calculate the energy transferred by the steam, Qsteam.
Qsteam = msteam * (hfinal - hinitial)
Where msteam is the mass of the steam, hfinal is the enthalpy of the condensate water at 85°C, and hinitial is the enthalpy of the 50% quality steam.
First, let's calculate the enthalpy of the condensate water:
hfinal = hliquid + (hf * x)
Where hliquid is the enthalpy of the liquid water at 85°C, hf is the latent heat of vaporization, and x is the quality of the water (since it has completely condensed, x = 0).
To find the values of hliquid and hf, we can use steam tables or steam properties software, as these values depend on the pressure of the system. Let's assume a typical pressure of 1 atmosphere.
Using steam tables, we find:
hliquid at 85°C = 334.61 kJ/kg
hf = 280.18 kJ/kg
Therefore, hfinal = 334.61 + (280.18 * 0) = 334.61 kJ/kg
Next, let's calculate the enthalpy of the 50% quality steam:
hinitial = hliquid + (hf * x)
Where x = 0.5, as the steam is a 50% quality steam.
Using steam tables, we find:
hinitial = 334.61 + (280.18 * 0.5) = 474.29 kJ/kg
Now we can calculate the energy transferred by the steam:
Qsteam = msteam * (hfinal - hinitial)
We don't know the mass of the steam (msteam), so let's keep it as a variable for now.
Now we can equate the energy transferred by the steam to the energy required to heat the cider:
mcider * Cp * ΔTcider = msteam * (hfinal - hinitial)
Substituting the known values:
150 * 3.651 * 61 = msteam * (334.61 - 474.29)
Therefore:
msteam = (150 * 3.651 * 61) / (334.61 - 474.29)
Calculating this expression, we find:
msteam = 72.69 kg
The mass of steam required to heat 150 kilograms of cider is approximately 72.69 kilograms.