Step Function

1
Step Function

• Previous solution was limited to a steady-state
  approach. However, the changes in a defined space
  need to be examined on a dynamic basis.
• So, lets consider a design space in which a
  steady state is eventually achieved.
• What happens to concentrations in this transition
  with time, as steady state is approached?

2
Conc.
Time
Initiation of activity
3
Step Function

• Assume V, Flows and sources are defined and the
  space is well mixed
• (dC/dt)V FiCi FoCo rxV
• (dC/dt)V FiCi FC KCV
• We know the steady state
  (i.e., dC/dt 0) solution gives
• C Css FiCi/(Fo KV)
• Css (S)/(F KV), where S source flux

4
Css
Heres the steady-state concentration
Conc.
Css (S)/(F KV)
Time
Initiation of activity
5
What does the conc. do in transition?
Css
Conc.
Time
Initiation of activity
6
Step function

• Examine (C)t as Css is approached
• (dC/dt)V S FC KCV
• dC/dt S/V - FC/V KC
• S/V S(F/V K)/(V)(F/V K)
• S/V Css (F/V K) therefore
• dC/dt -(F/V K) (C - Css)

7
Step function

• Now, assume y C-Css dC/dt dy/dt
• y y0 exp -(KF/V), therefore
• Ct Css (C0- Css) exp (KF/V)(t)
• Css steady state conc.
• C0 initial background conc.

8
SmokeyBar Givens

• Room volume 500 m3
• People 50 smokers 50 non-smokers
• Cig. Emissions
• 1.4 x 50 x 2 140 mg/hr of HCHO _at_ t0
• HCHO oxidation
• rx -k HCHO, w/ K 0.4 hr-1

9
SmokeyBar Required

• Transient concentration of HCHO in ambient air,
  as steady state is approached, under dynamic
  conditions.

10
SmokyBar as step function

• Earlier, Css 0.117 mg/m3 for HCHO
• Lets assume C0 zero and the time-step of
  interest is 1 hour
• Ct Css (C0- Css) exp (KF/V)(t)
• Ct1hr 0.117 (1 exp 0.4 1000/500)1)
  0.106 mg/m3
• 0.045 _at_ 0.2hr 0.06_at_ 0.3 hr 0.082_at_0.5hr

11
Conc. profile is then an exponential increase
that is virtually at ss within 2 hours
0.117




Conc., mg/m3


2
1
Time,hr
Initiation of activity