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Chapter 3. Matter balances without chemical reaction.
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Notes and solved problems of topic 3 of the subject of fundamentals of chemical processes.
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Chapter 3Mass balances without chemical
reaction: application to problems
with recirculation, purge and bypass
Enrique Roca Bordello
Department of Chemical Engineering
Outline
Chapter 3. Mass balances without chemical reaction
1. General balance equation
2. Material/Mass balance: introduction, objectives and nomenclature
3. General equation of macroscopic mass balance without chemical reaction
4. Strategy for solving balance problems
5. Mass balances without chemical reaction in steady-state systems
• Application to recirculation, purge and bypass
6. Mass balances without chemical reaction in unsteady-state systems
, 1. General balance equation
Input to Output from Generation in Consumption in Accumulation
the system
from t1 to t2 ‐ the system
from t1 to t2
+ the system
from t1 to t2 ‐ the system
from t1 to t2
= in the system
from t1 to t2
=
Accumulation Amount in Amount in
in the system
from t1 to t2
= system at
t = t2 ‐ system at
t = t1
INPUT – OUTPUT GENERATION = ACCUMULATION
I–O G=A
•Input/Output: this corresponds to the flow of property that crosses the limits of the
system in a given time.
•Generation: this corresponds to the amount of property that appears (+) or
disappears (-) within the system in a given time.
•Accumulation: this corresponds to the amount of property within the system after
a time t, so it can be positive or negative if the amount increases or decreases over
time, respectively.
1. General balance equation
General considerations
•Each balance term is usually expressed per unit time (kg/s, J/s, etc.).
•The accumulation term only appears in unsteady-state systems, since in steady-
state systems, the properties are independent of time.
•The generation term in mass and energy balances only appears if there are
chemical reactions within the system.
, 1. General balance equation
Mathematical expression of the macroscopic balance
d
1·QL1 2 ·QL 2 ·S Gm ·V ( ·V )
dt m
• π1 is the average property concentration in the input stream.
• QL1 is the input volumetric flow rate (Q1 = v1·s1).
• π2 is the average property concentration in the output stream.
• QL2 is the output volumetric flow rate (Q2 = v2·s2).
• ϕ is the property flux density entering/leaving the system per unit area and time.
• S is the surface area through which property enters/exits.
• Algebraic sum in which inputs are positive and outputs are negative.
• Gm = amount of property generated/consumed per unit volume and time.
• V = system volume.
• πm is the average property concentration in the system.
Macroscopic balance vs. Microscopic balance
2. Mass balances: introduction and objectives
•Mass balances are essential for the preparation of a project, the verification of
facilities in operation or the determination of the performance of processes.
•The final objective of carrying out a macroscopic mass balance is to know the flow
rates and compositions of the different inlet and outlet streams of a system, as well
as the total quantities and average composition inside the system at a given
moment.
•The macroscopic mass balances are a generalization of the law of conservation
of mass (Lavoisier's principle): "mass is neither created nor destroyed, it is
only transformed."
• Its mathematical complexity is low:
• For balances in steady state systems, algebraic equations are obtained,
usually linear.
• For balances in non-steady state systems, first-order differential equations
are obtained, with time being the independent variable.
, 2. Mass balances: introduction and objectives
m Mass flow rate (kg/s) M Mass (kg)
mj Mass flow rate of component j (kg/s) Mj Mass of component j (kg)
n Molar flow rate (kmol/s) N System mass in molar units (kmol)
nj Molar flow rate of component j Nj Component j mass in molar units
(kmol/s) (kmol)
QL Volumetric flow rate (m3/s) V Volume (m3)
xj Mole fraction of component j wj Mass fraction of component j
t Time (s) T Temperature (ºC)
v Rate (m/s) vj Rate of component j (m/s)
Density (kg/m3) j Density of component j (kg/m3)
c Molar concentration (kmol/m3) cj Molar concentration of component
j (kmol/m3)
rj Molar reaction rate of component j (kmol/m3·s)
Rj Mass reaction rate of component j (kg/m3·s)
3. General equation of mass balance without chemical reaction
INPUT – OUTPUT = ACCUMULATION
I–O=A
d
1·QL1 2 ·QL 2 ·S ( m ·V )
dt
Assuming conduction negligible versus convection macroscopic mass balance
without chemical reaction
1·Q L1 2 ·Q L 2 d ( m ·V )
dt
• The macroscopic mass balance can be applied to:
• The whole system overall mass balance.
• A component component mass balance.
• An atom.
• A molecule.
• An ion.
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