A
mass balance (also called a material balance) is an application of
conservation of massThe law of conservation of mass/matter, also known as principle of mass/matter conservation is that the mass of a closed system will remain constant over time, regardless of the processes acting inside the system. A similar statement is that mass cannot be created/destroyed, although it may be...
to the analysis of physical systems. By accounting for material entering and leaving a system,
mass flowMass flow, also known as Mass transfer and bulk flow, is the movement of substances at equal rates or as a single body. For example, blood circulation, transport of water and assimilates in xylem vessels and phloem tubes of plants. This relies upon the cohesion of water molecules to each other and...
s can be identified which might have been unknown, or difficult to measure without this technique. The exact
conservation lawIn physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....
used in the analysis of the system depends on the context of the problem but all revolve around mass conservation, i.e. that
matterThe term matter traditionally refers to the substance that all objects are made of. One common way to identify this "substance" is through its physical properties; a common definition of matter is anything that has mass and occupies a volume...
cannot disappear or be created spontaneously.
Mass balances are used widely in
engineeringEngineering is the discipline, art and profession of acquiring and applying technical, scientific and mathematical knowledge to design and implement materials, structures, machines, devices, systems, and processes that safely realize a desired objective or inventions.The American Engineers' Council...
and environmental analyses. For example mass balance theory is used to design
chemical reactorIn chemical engineering, chemical reactors are vessels designed to contain chemical reactions. The design of a chemical reactor deals with multiple aspects of chemical engineering. Chemical engineers design reactors to maximize net present value for the given reaction...
s, analyse alternative processes to produce chemicals as well as in
pollutionPollution is the introduction of contaminants into an environment that causes instability, disorder, harm or discomfort to the ecosystem i.e. physical systems or living organisms . Pollution can take the form of chemical substances, or energy, such as noise, heat, or light...
dispersion models and other models of physical systems. Closely related and complementary analysis techniques include the
population balancePopulation balance equations have been introduced in several branches of modern science, mainly in branches with particulate entities. This includes topics like crystallization, aerosol engineering, biology , polymerization, etc. PBEs define how populations of separate entities develop in specific...
,
energy balanceEnergy balance has the following meanings in several fields:* In physics, energy balance is a systematic presentation of energy flows and transformations in a system. Theoretical basis for an energy balance is the first law of thermodynamics according to which energy cannot be created or destroyed,...
and the somewhat more complex
entropyEntropy is a concept of information maintaining great importance in physics, chemistry, and information theory...
balance. These techniques are required for thorough design and analysis of systems such as the refrigeration cycle.
In environmental monitoring the term
budget calculations is used to describe mass balance equations where they are used to evaluate the monitoring data (comparing input and output, etc.) In biology the
dynamic energy budgetThe Dynamic Energy Budget theory aims to identify simple quantitative rules for the organization of metabolism of individual organisms that can be understood from basic first principles...
theory for metabolic organisation makes explicit use of time, mass and energy balances.
Introduction
The general form quoted for a mass balance is
The mass that enters a system must, by conservation of mass, either leave the system or accumulate within the system .
Mathematically the mass balance for a system without a chemical reaction is as follows:
In the absence of a
chemical reactionA chemical reaction is a process that leads to the transformation of one set of chemical substances to another. They are studied by chemists under a field of science called chemistry. Chemical reactions can be either spontaneous, requiring no input of energy, or non-spontaneous, often coming about...
the amount of any chemical species flowing in and out will be the same; This gives rise to an equation for each species in the system. However if this is not the case then the mass balance equation must be amended to allow for the generation or depletion of each chemical species. Note that the one term (depletion or generation) is used in the equation, which will be negative for depletion and positive for generation. This modified equation can be used not only for reactive systems, but for population balances such as occur in particle mechanics problems. The amended equation is given below; Note that it simplifies to the earlier equation in the case that the generation term is zero.
- In the absence of a nuclear reaction
In nuclear physics and nuclear chemistry, a nuclear reaction is the process in which two nuclei or nuclear particles collide to produce products different from the initial particles...
the number of atoms flowing in and out are the same, even in the presence of a chemical reaction
- To perform a balance the boundaries of the system must be well defined
- Mass balances can be taken over physical systems at multiple scales.
- Mass balances can be simplified with the assumption of steady state
A system in a steady state has numerous properties that are unchanging in time. The concept of steady state has relevance in many fields, in particular thermodynamics. Steady state is a more general situation than dynamic equilibrium. If a system is in steady state, then the recently observed...
, where the accumulation term is zero
Illustrative example
At this point a simple example shall be given for illustrative purposes. Consider the situation whereby a
slurryA slurry is, in general, a thick suspension of solids in a liquid.-Examples of slurries:Examples of slurries include:* A mixture of water and cement to form concrete* A mixture of water, gelling agent, and oxidizers used as an explosive...
is flowing into a settling tank to remove the solids in the tank, solids are collected at the bottom by means of a
conveyor beltA belt conveyor consists of two or more pulleys, with a continuous loop of material - the conveyor belt - that rotates about them. One or both of the pulleys are powered, moving the belt and the material on the belt forward. The powered pulley is called the drive pulley while the unpowered pulley...
partially submerged in the tank, water exits via an overflow outlet.
In this example we shall consider there to be two species, solids and water. The species are concentrated in each of the output streams, that is to say that the water to solid ratio at the water overflow outlet is higher than at the slurry inlet and the solids concentration at the exit of the conveyor belt is higher than that at the slurry inlet.
Assumptions
- Steady state
- Non-reactive system
Analysis
The slurry inlet composition has been measured by sampling the inlet and has a composition (by mass) of 50% solid and 50% water, with a mass flow of 100 kg per minute, the tank is assumed to be operating at steady state, and as such accumulation is zero, so input and output must be equal for both the solids and water. If we know that the removal efficiency for the slurry tank is 60%, then the water outlet will contain 20Kg/min of solids (40% times 100Kg/min times 50% solids). If we measure the flow rate of combined solids and water the water outlet to be 60Kg per minute then the amount of water exiting via the conveyor belt is 10Kg/min. This allows us to completely determine how the mass has been distributed in the system with only limited information and using the mass balance relations across the system boundaries
Mass Feedback (Recycle)
Mass balances can be performed across systems which have cyclic flows. In these systems output streams are fed back into the input of a unit for often for further reprocessing.
Such systems are common in grinding circuits, where materials are crushed then sieved to only allow a particular size of particle out of the circuit and the larger particles are returned to the grinder. However recycle flows are by no means restricted to solid mechanics operations, they are used in liquid and gas flows as well. One such example is in
cooling towerCooling towers are heat removal devices used to transfer process waste heat to the atmosphere. Cooling towers may either use the evaporation of water to remove process heat and cool the working fluid to near the wet-bulb air temperature or rely solely on air to cool the working fluid to near the ...
s, where water is pumped through the cooling tower many times, with only a small quantity of water drawn off at each pass (to prevent solids build up) until it has either evaporated or exited with the drawn off water.
The use of the recycle aids in increasing overall conversion of input products, which is useful for low per-pass conversion processes, for example the
Haber processThe Haber process, also called the Haber–Bosch process, is the nitrogen fixation reaction of nitrogen gas and hydrogen gas, over an enriched iron catalyst, to produce ammonia. The Haber process is important because ammonia is difficult to produce on an industrial scale, and the fertilizer...
.
Differential Mass balances
A mass balance can also be taken
differentiallyCalculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental...
. The concept is the same as for a large mass balance, however it is performed in the context of a limiting system (for example, one can consider the limiting case in time or, more commonly, volume). The use of a differential mass balance is to generate
differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s that can be used to provide an understanding and effective modelling tool for the target system.
The differential mass balance is usually solved in two steps, firstly a set of governing differential equations must be obtained, and then these equations must be solved, either analytically or, for less tractable problems, numerically.
A good example of the applications of differential mass balance are shown in the following systems:
- Ideal (stirred) Batch reactor
- Ideal tank reactor, also named Continuous Stirred Tank Reactor (CSTR)
- Ideal Plug Flow Reactor
The plug flow reactor model is used to describe chemical reactions in continuous, flowing systems. The PFR model is used to predict the behaviour of chemical reactors, so that key reactor variables, such as the dimensions of the reactor, can be estimated...
(PFR)
Ideal Batch reactor
A closed system. Many chemistry textbooks implicitly assume that the studied system can be described as a batch reactor when they write about reaction kinetics and chemical equilibrium.
The mass balance for a substance A becomes
where r_A denotes the rate at which substance A is produced, V is the volume (which may be constant or not), n_A the number of moles (n) of substance A.
In a fed-batch reactor some reactants/ingredients are added continuously or in pulses (compare making porridge by either first blending all ingredients and the let it boil, which can be described as a batch reactor, or by first mixing only water and salt and making that boil before the other ingredients are added, which can be described as a fed-batch reactor). Mass balances for fed-batch reactors become a bit more complicated.
Reactive Example
In this example we will use the law of
mass actionIn chemistry, the law of mass action is a mathematical model that explains and predicts behaviors of solutions in dynamic equilibrium. It can be described with two aspects: 1) the equilibrium aspect, concerning the composition of a reaction mixture at equilibrium and 2) the kinetic aspect...
to
derive the expression for a
chemical equilibriumIn a chemical process, chemical equilibrium is the state in which the chemical activities or concentrations of the reactants and products have no net change over time. Usually, this would be the state that results when the forward chemical process proceeds at the same rate as their reverse reaction...
constant.
Assume we have a closed reactor in which the following liquid phase reversible reaction occurs:
The mass balance for substance A becomes
As we have a liquid phase reaction we can (usually) assume a constant volume and since we get
or
In many text books this is given as the definition of
reaction rateThe reaction rate or rate of reaction for a reactant or product in a particular reaction is intuitively defined as how fast a reaction takes place...
without specifying the implicit assumption that we are talking about reaction rate in a closed system with only one reaction. This is an unfortunate mistake that has confused many students over the years.
According to the law of
mass actionIn chemistry, the law of mass action is a mathematical model that explains and predicts behaviors of solutions in dynamic equilibrium. It can be described with two aspects: 1) the equilibrium aspect, concerning the composition of a reaction mixture at equilibrium and 2) the kinetic aspect...
the forward reaction rate can be written as
and the backward reaction rate as
The rate at which substance A is produced is thus
and since, at equilibrium, the concentration of A is constant we get
or, rearranged
Ideal tank reactor/Continuously stirred tank reactor
An open system. A lake can be regarded as a tank reactor and lakes with long turnover times (e.g. with a low flux to volume ratio) can for many purposes be regarded as continuously stirred (e.g. homogeneous in all respects). The mass balance becomes
where Q_0 and Q denote the volumetric flow in and out of the system respectively and C_A_0 and C_A the concentration of A in the inflow and outflow respective. In an open system we can never reach a chemical equilibrium. We can, however, reach a
steady stateA system in dynamic equilibrium is a particular example of a system in a steady state. In a steady state the rate of inputs is equal to the rate of outputs so that the composition of the system is unchanging in time...
where all
state variables (temperature, concentrations etc.) remain constant
Example
Consider a bathtub in which we have some bathing salt dissolved. We now fill in more water, keeping the bottom plug in. What happens?
Since there is no reaction, and since there is no outflow . The mass balance becomes
or
Using a mass balance for total volume, however, it is evident that and that . Thus we get
Note that there is no reaction and hence no
reaction rateThe reaction rate or rate of reaction for a reactant or product in a particular reaction is intuitively defined as how fast a reaction takes place...
or rate law involved, and yet . We can thus draw the conclusion that reaction rate can not be defined in a general manner using . One
must first write down a mass balance before a link between and the reaction rate can be found. Many textbooks, however, define reaction rate as
without mentioning that this definition implicitly assumes that the system is closed, has a constant volume and that there is only one reaction.
Ideal Plug Flow Reactor (PFR)
An open system with no mixing along the reactor but perfect mixing across the reactor. Often used for systems like rivers and water pipes if the flow is turbulent. When a mass balance is made for a tube, one first considers an
infinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
part of the tube and make a mass balance over that using the ideal tank reactor model. That mass balance is then
integratedIntegration is an important concept in mathematics which, together with differentiation, forms one of the main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integralis defined informally...
over the entire reactor volume to obtain:
In numeric solutions, e.g. when using computers, the ideal tube is often translated to a series of tank reactors, as it can be shown that a PFR is equivalent to an infinite number of stirred tanks in series, but the latter is often easier to analyze, especially at steady state.
More complex problems
In reality, reactors are often non-ideal, in which combinations of the reactor models above are used to describe the system. Not only chemical reaction rates, but also
mass transferMass transfer is the transfer of mass from high concentration to low concentration. The phrase is commonly used in engineering for physical processes that involve molecular and convective transport of atoms and molecules within physical systems...
rates may be important in the mathematical description of a system, especially in
heterogeneousHeterogeneous is an adjective used to describe an object or system consisting of multiple items having a large number of structural variations. It is the opposite of homogeneous, which means that an object or system consists of multiple identical items...
systems.
As the chemical
reaction rateThe reaction rate or rate of reaction for a reactant or product in a particular reaction is intuitively defined as how fast a reaction takes place...
depends on temperature it is often necessary to make both an
energy balanceEnergy balance has the following meanings in several fields:* In physics, energy balance is a systematic presentation of energy flows and transformations in a system. Theoretical basis for an energy balance is the first law of thermodynamics according to which energy cannot be created or destroyed,...
(often a heat balance rather than a full fledged energy balance) as well as mass balances to fully describe the system. A different reactor models might be needed for the energy balance: A system that is closed with respect to mass might be open with respect to energy e.g. since heat may enter the system through
conductionIn heat transfer, conduction is the transfer of thermal energy between neighboring molecules in a substance due to a temperature gradient. It always takes place from a region of higher temperature to a region of lower temperature, and acts to equalize temperature differences. Conduction takes...
.
See also
- Bioreactor
A bioreactor may refer to any device or system that supports a biologically active environment. In one case, a bioreactor is a vessel in which is carried out a chemical process which involves organisms or biochemically active substances derived from such organisms. This process can either be...
- Chemical reactor
In chemical engineering, chemical reactors are vessels designed to contain chemical reactions. The design of a chemical reactor deals with multiple aspects of chemical engineering. Chemical engineers design reactors to maximize net present value for the given reaction...
- Chemical engineering
Chemical engineering is the branch of engineering that deals with the application of physical science , and life sciences with mathematics, to the process of converting raw materials or chemicals into more useful or valuable forms...
- Chemical equilibrium
In a chemical process, chemical equilibrium is the state in which the chemical activities or concentrations of the reactants and products have no net change over time. Usually, this would be the state that results when the forward chemical process proceeds at the same rate as their reverse reaction...
- Conservation of mass
The law of conservation of mass/matter, also known as principle of mass/matter conservation is that the mass of a closed system will remain constant over time, regardless of the processes acting inside the system. A similar statement is that mass cannot be created/destroyed, although it may be...
- Continuity equation
A continuity equation in physics is a differential equation that describes the transport of some kind of conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved, a vast variety of physics may be described with continuity equations.Continuity...
- Continuous stirred-tank reactor
- Energy balance
Energy balance has the following meanings in several fields:* In physics, energy balance is a systematic presentation of energy flows and transformations in a system. Theoretical basis for an energy balance is the first law of thermodynamics according to which energy cannot be created or destroyed,...
- Mass action
In chemistry, the law of mass action is a mathematical model that explains and predicts behaviors of solutions in dynamic equilibrium. It can be described with two aspects: 1) the equilibrium aspect, concerning the composition of a reaction mixture at equilibrium and 2) the kinetic aspect...
External links