- General equation of mass balance
- Simplification
- Example of use: fish in the river
- Types
- Differential balance
- Comprehensive balance
- Example exercise
- References
The material balance is the count of the components that belong to a system or process under study. This balance can be applied to almost any type of system, since it is assumed that the sum of the masses of such elements must remain constant at different measurement times.
The component can be understood as marbles, bacteria, animals, logs, ingredients for a cake; and in the case of chemistry, molecules or ions, or more specifically, compounds or substances. So the total mass of the molecules entering a system, with or without a chemical reaction, must remain constant; as long as there are no leakage losses.
Rock pile: a literal example of balanced matter. Source: Pxhere.
In practice there are innumerable problems that can affect the balance of matter, in addition to taking into account various phenomena of matter and the effect of many variables (temperature, pressure, flow, agitation, size of the reactor, etc.).
On paper, however, the mass balance calculations must match; that is, the mass of the chemical compounds must not disappear at any time. Taking this balance is analogous to balancing a pile of rocks. If one of the masses gets out of place, everything falls apart; in this case, it would mean that the calculations are wrong.
General equation of mass balance
In any system or process, it must first be defined what its boundaries are. From them, it will be known which compounds enter or leave. This is especially convenient if there are multiple process units to consider. When all units or subsystems are considered, then we speak of a general mass balance.
This balance has an equation, which can be applied to any system that obeys the law of conservation of mass. The equation is as follows:
E + G - S - C = A
Where E is the amount of matter that enters the system; G is what is generated if a chemical reaction occurs in the process (as in a reactor); S is what comes out of the system; C is what is consumed, again, if there is a reaction; and finally, A is what is accumulated.
Simplification
If there is no chemical reaction in the system or process under study, G and C are worth zero. Thus, the equation looks like:
E - S = A
If the system is also considered in a steady state, without appreciable changes in the variables or flows of the components, it is said that nothing accumulates inside it. Therefore, A is worth zero, and the equation ends up simplifying further:
E = S
In other words, the amount of matter that enters is equal to that which leaves. Nothing can be lost or disappear.
On the other hand, if there is a chemical reaction, but the system is in a steady state, G and C will have values and A will remain zero:
E + G - S - C = 0
E + G = S + C
Meaning that in a reactor the mass of the reagents that enter and of the products that they generate in it, is equal to the mass of the products and reagents that leave, and of the reagents consumed.
Example of use: fish in the river
Suppose you are studying the number of fish in a river, whose banks come to represent the boundary of the system. It is known that on average 568 fish enter per year, 424 are born (generate), 353 die (consume), and 236 migrate or leave.
Applying the general equation we have then:
568 + 424 - 353 - 236 = 403
This means that per year 403 fish accumulate in the river; that is to say, per year the river gets richer with fish. If A had a negative value, it would mean that the number of fish is decreasing, perhaps due to negative environmental impacts.
Types
From the general equation it can be thought that there are four equations for different types of chemical processes. However, the mass balance is divided into two types according to another criterion: time.
Differential balance
In the differential material balance we have the quantity of the components within a system at a given time or moment. Said mass quantities are expressed in units of time, and therefore, represent speeds; for example, Kg / h, indicating how many kilometers enter, leave, accumulate, generate or consume in an hour.
For there to be mass flows (or volumetric, with the density at hand), the system must generally be open.
Comprehensive balance
When the system is closed, as happens with the reactions carried out in intermittent reactors (batch type), the masses of its components are usually more interesting before and after the process; that is, between the initial and final times t.
Therefore, quantities are expressed as mere masses and not velocities. This type of balance is made mentally when using a blender: the mass of the ingredients that enter must be equal to that left after turning off the engine.
Example exercise
It is desired to dilute a flow of a 25% methanol solution in water, with another of a 10% concentration, more dilute, in such a way that 100 Kg / h of a 17% methanol solution is generated. How much of both 25% and 10% methanol solutions must enter the system per hour to achieve this? Assume the system is in a steady state
The following diagram exemplifies the statement:
Flow diagram for the mass balance of the dilution of the methanol solution. Source: Gabriel Bolívar.
There is no chemical reaction, so the amount of methanol entering must be equal to the amount leaving:
E Methanol = S Methanol
0.25 n 1 · + 0.10 n 2 · = 0.17 n 3 ·
Only the value of n 3 · is known. The rest are unknowns. To solve this equation of two unknowns, another balance is needed: that of water. Making the same balance for water, we have:
0.75 n 1 · + 0.90 n 2 · = 0.83 n 3 ·
The value of n 1 · is solved for water (it can also be n 2 ·):
n 1 · = (83 Kg / h - 0.90n 2 ·) / (0.75)
Substituting then n 1 · in the mass balance equation for methanol, and solving for n 2 · we have:
0.25 + 0.10 n 2 · = 0.17 (100 Kg / h)
n 2 · = 53.33 Kg / h
And for n 1 · simply subtract:
n 1 · = (100- 53.33) Kg / h
= 46.67 Kg / h
Therefore, per hour, 46.67 kg of 25% methanol solution and 53.33 kg of 10% solution must enter the system.
References
- Felder and Rousseau. (2000). Elementary principles of chemical processes. (Second edition.). Addison Wesley.
- Fernández Germán. (October 20, 2012). Definition of mass balance. Recovered from: industriaquimica.net
- Balances of matter: industrial processes I.. Recovered from: 3.fi.mdp.edu.ar
- UNT Regional College La Plata. (sf). Material balance.. Recovered from: frlp.utn.edu.ar
- Gómez Claudia S. Quintero. (sf). Material balances.. Recovered from: webdelprofesor.ula.ve