3.(25pts) Particle emissions in a house. The power goes out in yobind paricle en

Question

3.(25pts) Particle emissions in a house. The power goes out in yobind paricle eni night, so you decide to light a few candles. These candles havhe mg/h. Your apartment has a ventilation flow of 1 m/min to deposition, which acts similar to a reaction) of 0.2hnd loss terms (volume 1 a combined particle emission rate ofE have a first order (a) Write the differential equation, and define the souwour housim (b) What is the steady state particle concentration in your (c) Extra eredit: Write the to this the concentration after 1 hour of candle-burning (in rm n after 1 hotte- varying solution to this problem, and calculaste the particle grn').

Explanation / Answer

3. volume of aartment = V = 100 m^3

particel emiision rat eof candles m'in = 1 mg/h

for the apartment flow rate

first order loss rate = k = 0.2 per h

a. let concentration of particels in the room at time t be c

then

from loss term

dc/dt = -kc

decrease in concentration in time dt

dc = -kc*dt

at time t, mass of particles inside the room = cV

then

mass added in time dt, from input term = dm

dm = m'in*dt = 1000 dt ( dm in ug and t in hr)

increase in concentration = dm/V

hence

dc/dt = 1000/V - 0.2c

b. at steady state, dc/dt = 0

1000/V = 0.2c

c = 50 ug/m^3

c. dc/dt = 1000/v - 0.2c = 0.2(1000/0.2 - c) = 0.2(5000 - c)

dc/(5000 - c) = 0.2dt

integrating from c = 0 to c = t, t = 0 to t = t

ln(1 - c/5000) = -0.2t

c = 5000(1 - e^(-0.2t))

at t = 1 hr

c = 5000(1 - e^(-0.2))

c = 906.3462 ug/m^3

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