Name: Time allowed: 30mn Problem (100pts) A cylindrical vessel is used as part o

Question

Name: Time allowed: 30mn Problem (100pts) A cylindrical vessel is used as part of a chemical reaction process. The volume of the fluid in the vessel is Vo and is to be maintained constant throughout the process. The time dependent feed Qmt)Qssin(ot) has a constant chemical mass fraction Xi, where Qo and the frequency o are constant. The density is constant and the same, for all fluids. (a) Determine the expression of the outflow Qot to ensure that the volume of liquid in the vessel does not change over time. (b) Determine the expression of the chemical mass fraction in the vessel, assuming that the fluid is well stirred such that the outflow is at the same mass fraction as the fluid in the vessel. The fluid in the vessel is initially pure with no chemical in it. All constants of integration must be determined

Explanation / Answer

ANSWER:
Volume of fluid in boat = Vo ( is steady)

Qin(t) = Qo*sin(wt), mass part xl

a)For the Vo to remain steady

Qout = Qin = Qo*sin(wt)

b) substance mass fraction in volume = x'

Mass additional at time t = xQin

Mass detached at time t = x'*Qout

Hence

dm = (x*Qin - x'*Qout)dt = Vo*dx'

dx'/(x' - x*Qin/Qout) = -Qout*dt/Vo

integrating from x' = 0 to x' = x'

ln(1 - x'*Qout/x*Qin) = -Qout*t/Vo

x' = (1 - exp(-Qout*t/Vo))*x*Qin/Qout

c) for x' to reach maximum

dx'/dt = 0

0 + Qout/Vo *exp(-Qout*t/Vo) = 0

Hence

t = inf

Hence, x' reaches utmost at time inf where

x' = x*Qin/Qout

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