Theory Residence time distribution

an rtd curve reasonably well-mixed reactor

the theory of residence time distributions begins 3 assumptions:

the incompressibility assumption not required, compressible flows more difficult work , less common in chemical processes. further level of complexity required multi-phase reactors, separate rtd describe flow of each phase, example bubbling air through liquid.

the distribution of residence times represented exit age distribution,



e
(
t
)


{\displaystyle e(t)}

. function



e
(
t
)


{\displaystyle e(t)}

has units of time , defined such that

∫

0


∞


e
(
t
)

d
t
=
1


{\displaystyle \int _{0}^{\infty }e(t)\,dt=1}

.

the fraction of fluid spends given duration,



t


{\displaystyle t}

inside reactor given value of



e
(
t
)
d
t


{\displaystyle e(t)dt}

.

the fraction of fluid leaves reactor age less




t

1




{\displaystyle t_{1}}

is

∫

0



t

1




e
(
t
)

d
t


{\displaystyle \int _{0}^{t_{1}}e(t)\,dt}

.

the fraction of fluid leaves reactor age greater




t

1




{\displaystyle t_{1}}

is

∫


t

1




∞


e
(
t
)

d
t
=
1
−

∫

0



t

1




e
(
t
)

d
t


{\displaystyle \int _{t_{1}}^{\infty }e(t)\,dt=1-\int _{0}^{t_{1}}e(t)\,dt}

.

the average residence time given first moment of age distribution:

t
¯



=

∫

0


∞


t
⋅
e
(
t
)

d
t


{\displaystyle {\bar {t}}=\int _{0}^{\infty }t\cdot e(t)\,dt}

.

if there no dead, or stagnant, zones within reactor






t
¯





{\displaystyle {\bar {t}}}

equal



τ


{\displaystyle \tau }

, residence time calculated total reactor volume , volumetric flow rate of fluid:

τ
=


v
v




{\displaystyle \tau ={\frac {v}{v}}}

.

the higher order central moments can provide significant information behavior of function



e
(
t
)


{\displaystyle e(t)}

. example, second central moment indicates variance (




σ

2




{\displaystyle \sigma ^{2}}

), degree of dispersion around mean.

σ

2


=

∫

0


∞


(
t
−



t
¯




)

2


⋅
e
(
t
)

d
t


{\displaystyle \sigma ^{2}=\int _{0}^{\infty }(t-{\bar {t}})^{2}\cdot e(t)\,dt}

the third central moment indicates skewness of rtd , fourth central moment indicates kurtosis (the peakedness ).

one can define internal age distribution



i
(
t
)


{\displaystyle i(t)}

describes reactor contents. function has similar definition



e
(
t
)


{\displaystyle e(t)}

: fraction of fluid within reactor age of



t


{\displaystyle t}





i
(
t
)
d
t


{\displaystyle i(t)dt}

. shown danckwerts, relation between



e
(
t
)


{\displaystyle e(t)}

,



i
(
t
)


{\displaystyle i(t)}

can found mass balance:

i
(
t
)
=


1
τ



(
1
−

∫

0


t


e
(
t
)
 
d
t
)


e
(
t
)
=
−
τ



d
i
(
t
)


d
t





{\displaystyle i(t)={\frac {1}{\tau }}\left(1-\int _{0}^{t}e(t)\ dt\right)\qquad e(t)=-\tau {\frac {di(t)}{dt}}}