Mass Balance
Benno Rahardyan
FTSL-ITB
Mass Balance
The law of conservation of
mass states that mass can
neither be produced nor
destroyed
Mass at time t + dt = mass at time t
+ mass that entered from t to t + dt
- mass that exited from t to t + dt
+ net mass of chemical produced from
other compounds by reactions
between t and t + dt
Unit : mass
• It is usually more convenient to work
with values of mass flux
the rate at which mass enters or leave
the systems.
(mass at time t + dt) / dt =
(mass entering from t to t+dt) / dt
- (mass exiting from t to t+dt) / dt
+ (net chemical production from t to t+dt)/dt
Unit : mass/time
Mass accumulation rate =
Mass flux in – mass flux out + net rate of
chemical production
or
dm/dt = min – mout + m reaction
The control volume
• A mass balance is only meaningful in
terms of specific region of space,
which has boundaries across which in
terms min and mout are determined.
• This region is called the control
volume.
• control volume has boundaries over
which min and mout can be calcultated.
CMFR
• Completely mixed flow reactor 
control volume
• Example : CSTR (continously stirred
tank reactor)
dm/dt = d (VC) / dt
m : mass
V : volume
C : concentration
• Steady state : condition that no longer
change with time.
• The concentration and hence the mass
within the control volume remains
constant.
dm/dt = 0
When the mass in the control volume
vary with time  mass balance will be
non steady state
Discussion
• For each of the following mass balance
problems, determine whether a steady
state or non steady state mass balance
would be appropriate
• A mass balance on chloride (Cl-)
dissolved in a lake. Two rivers bring
chloride into the lake, and one river
removes chloride. No significant
chemical reactions occur, as chloride
is soluble and non reactive. What is the
annual average concentration of
chloride in the lake?
• A degradation reaction within a wellmixed tank is used to destroy a
pollutant. Inlet concentration and flow
are held constant, and the system has
been operating for several days. What
is the pollutant concentration in the
effluent, given the inlet flow and
concentration and the first-order
decay rate constant?
• The source of pollutant in previous
problem is removed, resulting in an
isntantaneous decline of the inlet
concentration to zero. How long would
it take until the outlet concentration
reaches 10% of its initial value?
• Max flux in
min = Q in x C in
m=QxC
mass/time = volume/time x mass/volume
• Max flux out
mout = Q out x C out
m out = Q out x C
mass/time = volume/time x mass/volume
• Net rate of chemical reation
– Net rate of production of a compound from
chemical or biological reactions.
– Mass/time
– Positive or negative
– Usulally expressed in terms of concentration
– Mrxn = V x (dC/dt)
• Conservative compound
– dC/dt = m reaction = o
• Zero order decay : the rate of loss of
the compound is constant
– dC/dt equals – k
– Mrxn equals - Vk
• First order decay
– Rate of losses of the compound is directly
proportional to its concentration
– dC/dt equals – kC, for such compound Mrxn
equals - VkC
approaches
• Draw a schematic diagram of situation
• Write a mass balance equation
dM/dt = Min – Mout + Mrxn
• Determine whether the problem is steady
state or not (dm/dt = 0) or non steady state
(dm/dt = V dC/dt)
• Determine whether the compound being
balanced is concervative (mrxn = 0) or non
conservatives (mrxn must be determined
based on the reaction kinetics.
Batch Reactor
• The reactor that has no inlet or outlet
is termed as batch reactor.
Min = 0 and M out = 0
dM/dt = M rxn
VdC/dt = V (dC/dt) (reaction only)
dC/dt = dC/dt (reaction only)
Example
• First order decay r = - kC
 dC/dt = - kC
or Ct / Co = e-kt
Plug Flow Reactor
• Model the chemical transformation of
compound as they are transported in
systems resembling pipes.
• Because the velocity (v) of the fluid in
the PFR is constant, time and
downstream distance x are
interchangeable and
t = x/v
dm/dt = min – mout + mrxn
V(dC/dt) = 0 – 0 + V(dC/dt) reaction only
Min and mout are set equal to zero
because there is no mass exchange
across the plug boundaries.
In the case of first order decay
V(dC/dt) reaction only = - VkC
and
VdC/dt = - VkC
Which results in Ct/Co = exp (-kt)
In PFR or length L
t = L/ v = L x A/Q
 Ct/Co = exp (-kV/Q)
Retention time and other
expression form V/Q
Retention time, detention time, and
residence time
t = V/Q
Example of Steady State CMFR
with Conservative Chemical Mixing
• A pipe from a municipal wastewater
treatment plant discharges 1.0 m3/s of
poorly treated effluent containing 5.0
mg/L of phosphorus compunds
*reported as mgP/L) into a river with a
upstream flow rate of 25 m3/s and a
background phosphorus concentration
of 0.010 mgP/l. What is the resulting
concentration of phosphorus (in mg/L)
in the river just downstream of the
plant outflow?
dm/dt = min – mout + mrxn
= rQin - rQout + 0
Qin =Qout
Qd=Qu+Qe = 26 m3/s
0= (CuQu+CeQe) – CdQd + 0
Cd =(CuQu+CeQe) / Qd
=(0.010 mg/L)(25 m3/s)+(5.0 mg/L)(1.0m3/s)
------------------------------------------------------------26 m3/s
Example Steady State CMFR with
First Order Decay
• The CMFR is used to treat an industrial
waste product, using a reaction that
destroys the pollutant according to
first order kinetics, with k = 0.216/day.
The reactor volume is 500 m3, the
volumetric flow rate of the single inlet
and exit is 50 m3/day and inlet
pollutant concentration is 100 mg/L.
What is the outlet concentration after
treatment?
dm/dt = min – mout + mrxn
0 = QCin-QC-VkC
C = Cin x Q / (Q + kV)
C = Cin x 1 / (1 + kV/Q)
100 mg/L x 50 m3/day
C = ---------------------------------50 m3/day + (0.216/day)(500 m3)
Example Non Steady State CMRF
with First Order Decay
• The manufacturing process that
generates the waste in previous
example has to be shut down, and
starting at t=0, the concentration Cin
entering the CMFR is set to 0. What is
the outlet concentration as a function
of time after the concentration is set to
0? How long does it take the tank
concentration to reach 10% of its
initial, steady-state value?
dm/dt = min – mout + mrxn
VdC/dt = 0- QC- kCV
dC/dt = - (Q/V + k)C
To determine C as a function of time
[Int Co Ct] (dC/dt) = ln (C)- ln (Co)
[Int 0t] - (Q/V + k)C = - (Q/V + k)t
ln (C/Co) = - (Q/V + k)t
Ct/Co = exp [- (Q/V + k)t]
• Ct = 32 mg/L
x exp[- 50 m3/day/500 m3 + 0.216/day)t]
= 32 mg/L x exp (-0.316/day x t)
Ct/C0= 0.10
0.10 = exp (-0.316/day x t)
ln(0.10) = -2.303 = (-0.316/day x t)
t = 7.3 days
Non Steady State CMFR
Conservative Substance
• The CMFR reactor is filled with clean
water prior to being started. After start
up, a waste stream containing 100
mg/L of concervative pollutant is
added to the reactor at a flow rate of
50 m3/day. The volume of the reactor
is 500 m3. What is the concentration
exiting the reactor as a function of
time after it is started?
dm/dt = min – mout + mrxn
VdC/dt = QCin-QC + 0
dC/dt = -(Q/V)(C-Cin)
y = (C-Cin)
dy/dt = dC/dt – d(Cin /dt).
Cin constant  d(Cin /dt) = 0
dy / dt = dC/dt
dy/dt = - Q/V x y
[int y(0)  y (t) ] dy/y = ln (y(t)/ y(0) )
[int (0)  (t) ] – Q/V dt = - (Q/V) t
(y(t)/ y(0) ) = exp (- (Q/V) x t)
C - Cin = - Cin exp (- (Q/V) x t)
C = Cin (1+ exp (- (Q/V) x t))
t  : exp (- (Q/V) x t)  0  C  Cin
Example Required Volume in PFR
• Determine the volume required for a
PFR to obtain the same degree of
pollutant reduction as in first example.
Assume that the flow rate and first
order decay rate constant are
unchanged (Q= 50 m3/day, k =
0,216/day)
Cout/Cin = 32/100 = 0.32
Cout/Cin = exp [- (kV/Q]
0.32 = exp [-(0.216/day V / 50 m3/day)
V = ln (0.32) x 50 m3/day/-0.216/day
= 264 m3
Example of Retention Time in a
CMFR and PFR
• CMFR
t = V/Q = 500 m3/50 m3/day = 10 days
• PFR
t = V/Q = 264 m3/50 m3/day = 5.3 days
Homework :
Comparison of CMFR and PFR Performance
Determine Cout given Cin, V, Q and k
• Determine Cout/Cin
V = 100 L, Q + 5.0 L/s, k = 0.05/day
Determine V, given Cin ,Cout , Q and k
• Determine V
Cout/Cin = 0.5 Q=5.0 L and k=0.05/s
Selamat Menunaikan
Ibadah Shaum
(bagi yang menjalankannya)
Semoga sukses dengan
Ramadhannya