Wash-in, wash-out, runaway curve

[Ref: KB2:p259-260;TF8:p178-179]

Wash-in curve

Dependent variable rises towards a limiting value at a rate which decreases progressively in proportion to the gap between the variable and the limiting value.

i.e. Formula: y=(1 - e^-kx)

eg. concentration of pO2 when switched to 100% O2

Wash-out curve

Dependent variable declines towards zero at a rate directly proportional to its value

i.e. Formula: y= e^-kx

e.g. concentration of N2 when switched to 100% O2

Runaway curve

The dependent variable increases at a rate directly proportional to its value

i.e. Formula: y= e^kx

e.g. bacterial growth

Time constant and half life

Duration of negative exponential processes (e.g. wash-out curve) can be quantified by time constant or half life.

Time constant

... the time at which the process would have been complete if the initial rate of change had continued.

For a washout curve,

@ 1 time constant
--> 37% (63% drop)

@ 2 time constants
--> 13.5% (86.5% drop)

@ 3 time constants
--> 5% (95% drop)

@ 4 time constants
--> 1% (99% drop)

NB:

• 1 time constant
  = 1/e
  = 1/2.718 of maximum
  [WG21:p91]

Half life

...the time for the quantity to fall to half of its initial value

Conversion between half life and time constant

• 1 half life
  = 0.69 x time constant
• 1 time constant
  = 1.44 x half life