ALL CHAPTERS COVERED
SOLUTIONS MANUAL
, Solutions to Problems
Chapter 2
1-a
Organic phase benzoic acid concentration profile:
Starting with Eq. (2.4-3) and Fig. 2.1 pg. 24
(V1 + mV2 ) dx = RC A − (R + Sm )x
dt o
Subject to x = 0 at t = 0
Using the method of section 2.1
R + Sm R + Sm
R + Sm ∫ V1 + mV2 dt t
μ (t ) = such that μ (t ) = e =e V1 + mV2
V1 + mV2
Therefore the differential equation becomes
'
⎡ VR++mV
Sm
t⎤ RC Ao
R + Sm
V1 + mV2
t
⎢ xe 1 2
⎥ = e
⎢⎣ ⎥⎦ V1 + mV2
R + Sm
RC Ao −
x(t ) =
t
V1 + mV2
which solves to + k1e
R + Sm
where k1 is an arbitrary constant to be determined by the initial condition.
That is at t = 0, x = 0:
RC A0 RC Ao
0= + k1 ⇒ k1 = −
R + Sm R + Sm
RC Ao ⎡ t⎤
R + Sm
−
Therefore x(t ) = ⎢1 − e
V1 + mV2
⎥
R + Sm ⎢⎣ ⎥⎦
is the organic phase concentration profile for benzoic acid.
, 1-b
i) direct steady state solution to:
(V1 + mV2 ) dx = RC A − (R + Sm )x
dt o
RC Ao
= 0 . Therefore RC Ao = (R + Sm )x or
dx
means x=
dt R + Sm
RC Ao ⎡ t⎤
R + Sm
−
V1 + mV2
ii) by taking the limit as t → ∞ in x = ⎢1 − e ⎥ produce
R + Sm ⎢⎣ ⎥⎦
R + Sm
RC Ao − t
V1 + mV2
x= , sin ce lim e → 0.
R + Sm t →∞
1-c
yS RC Ao RC Ao
Let E = , then since x = and y = mx = m
RC Ao R + Sm R + Sm
RC Ao m 1 mS 1 1 R
Therefore E = S• = = = ,α=
R + Sm RC Ao R + Sm R
+1
α +1 mS
mS
1
i.e. E= for the steady-state process.
1+ α
2a.
, ⎧ y1 = mx1
Relationship for inter-stage concentration of acid: ⎨
⎩ y 2 = mx2
System Property T t + Δt
Stage 1 Stage 2 Stage 1 Stage 2
Flow rate of organic
R R R R
phase
Flow rate of aqueous
S S S S
phase
Volume of organic
V11 V12 V11 V12
phase
Volume of aqueous
V21 V22 V21 V22
phase
Input acid dx1
Concentration (Organic CA o x1 CA o x1 + Δt
phase) dt
Output acid dx1 dx 2
Concentration (Organic x1 x2 x1 + Δt x2 + Δt
phase) dt dt
Input acid dy 2
Concentration y2 0 y2 + Δt 0
(Aqueous phase) dt
Output acid dy1 dy 2
Concentration y1 y2 y1 + Δt y2 + Δt
(Aqueous phase) dt dt
Amount of acid in ⎛ dx ⎞ ⎛ dx ⎞
x1V11 x2V12 V11 ⎜ x1 + 1 Δt ⎟ V12 ⎜ x2 + 2 Δt ⎟
Organic phase ⎝ dt ⎠ ⎝ dt ⎠
Amount of acid in ⎛ dy ⎞ ⎛ dy ⎞
y1V21 y 2V22 V21 ⎜ y1 + 1 Δt ⎟ V22 ⎜ y 2 + 2 Δt ⎟
Aqueous phase ⎝ dt ⎠ ⎝ dt ⎠
R ⎛⎜ L ⎞ ⎛ kg ⎞
⎟, x 2 ⎜ ⎟
⎝ min ⎠ ⎝ L ⎠
CA O , R R , x1
Stage 1 Stage 2
S S
⎛ L ⎞
y1 ⎛kg⎞ S⎜ ⎟
y2⎜ ⎟ ⎝ min ⎠
⎝ L⎠
SOLUTIONS MANUAL
, Solutions to Problems
Chapter 2
1-a
Organic phase benzoic acid concentration profile:
Starting with Eq. (2.4-3) and Fig. 2.1 pg. 24
(V1 + mV2 ) dx = RC A − (R + Sm )x
dt o
Subject to x = 0 at t = 0
Using the method of section 2.1
R + Sm R + Sm
R + Sm ∫ V1 + mV2 dt t
μ (t ) = such that μ (t ) = e =e V1 + mV2
V1 + mV2
Therefore the differential equation becomes
'
⎡ VR++mV
Sm
t⎤ RC Ao
R + Sm
V1 + mV2
t
⎢ xe 1 2
⎥ = e
⎢⎣ ⎥⎦ V1 + mV2
R + Sm
RC Ao −
x(t ) =
t
V1 + mV2
which solves to + k1e
R + Sm
where k1 is an arbitrary constant to be determined by the initial condition.
That is at t = 0, x = 0:
RC A0 RC Ao
0= + k1 ⇒ k1 = −
R + Sm R + Sm
RC Ao ⎡ t⎤
R + Sm
−
Therefore x(t ) = ⎢1 − e
V1 + mV2
⎥
R + Sm ⎢⎣ ⎥⎦
is the organic phase concentration profile for benzoic acid.
, 1-b
i) direct steady state solution to:
(V1 + mV2 ) dx = RC A − (R + Sm )x
dt o
RC Ao
= 0 . Therefore RC Ao = (R + Sm )x or
dx
means x=
dt R + Sm
RC Ao ⎡ t⎤
R + Sm
−
V1 + mV2
ii) by taking the limit as t → ∞ in x = ⎢1 − e ⎥ produce
R + Sm ⎢⎣ ⎥⎦
R + Sm
RC Ao − t
V1 + mV2
x= , sin ce lim e → 0.
R + Sm t →∞
1-c
yS RC Ao RC Ao
Let E = , then since x = and y = mx = m
RC Ao R + Sm R + Sm
RC Ao m 1 mS 1 1 R
Therefore E = S• = = = ,α=
R + Sm RC Ao R + Sm R
+1
α +1 mS
mS
1
i.e. E= for the steady-state process.
1+ α
2a.
, ⎧ y1 = mx1
Relationship for inter-stage concentration of acid: ⎨
⎩ y 2 = mx2
System Property T t + Δt
Stage 1 Stage 2 Stage 1 Stage 2
Flow rate of organic
R R R R
phase
Flow rate of aqueous
S S S S
phase
Volume of organic
V11 V12 V11 V12
phase
Volume of aqueous
V21 V22 V21 V22
phase
Input acid dx1
Concentration (Organic CA o x1 CA o x1 + Δt
phase) dt
Output acid dx1 dx 2
Concentration (Organic x1 x2 x1 + Δt x2 + Δt
phase) dt dt
Input acid dy 2
Concentration y2 0 y2 + Δt 0
(Aqueous phase) dt
Output acid dy1 dy 2
Concentration y1 y2 y1 + Δt y2 + Δt
(Aqueous phase) dt dt
Amount of acid in ⎛ dx ⎞ ⎛ dx ⎞
x1V11 x2V12 V11 ⎜ x1 + 1 Δt ⎟ V12 ⎜ x2 + 2 Δt ⎟
Organic phase ⎝ dt ⎠ ⎝ dt ⎠
Amount of acid in ⎛ dy ⎞ ⎛ dy ⎞
y1V21 y 2V22 V21 ⎜ y1 + 1 Δt ⎟ V22 ⎜ y 2 + 2 Δt ⎟
Aqueous phase ⎝ dt ⎠ ⎝ dt ⎠
R ⎛⎜ L ⎞ ⎛ kg ⎞
⎟, x 2 ⎜ ⎟
⎝ min ⎠ ⎝ L ⎠
CA O , R R , x1
Stage 1 Stage 2
S S
⎛ L ⎞
y1 ⎛kg⎞ S⎜ ⎟
y2⎜ ⎟ ⎝ min ⎠
⎝ L⎠
