A mass balance (also called a material balance) is an accounting of material entering and leaving a system. Fundamental to the balance is the conservation of mass principle, i.e. that matter can not disappear or be created. Mass balances are used, for example, to design chemical reactors, analyse alternative processes to produce chemicals, in pollution dispersion models etc. 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.) The dynamic energy budget theory for metabolic organisation makes explicit use of time, mass and energy balances.

Introduction

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The mass that enters a system must ( conservation of mass principle) either leave the system or accumulate within the system, i.e.

$\backslash mathrm\{IN\}\; =\; \backslash mathrm\{OUT\}\; +\; \backslash mathrm\{ACC\}$

where IN denotes what enters the system, OUT denotes what leaves the system and ACC denotes accumulation within the system (which may be negative or positive). Mass balances are often developed for total mass crossing the boundaries of a system, but they can also focus on one element (e.g. carbon) or chemical compound (e.g. water). When mass balances are written for specific compounds, number of individuals in a population, etc. rather than for the total mass of the system, a production term (PROD) is introduced such that

$\backslash mathrm\{IN\}\; +\; \backslash mathrm\{PROD\}\; =\; \backslash mathrm\{OUT\}\; +\; \backslash mathrm\{ACC\}$

The production term may then describe chemical reaction rates, the difference between births and deaths, etc. PROD might be positive or negative, just as for ACC.

Mass balances are either Integral Mass Balances or Differential Mass Balances. An integral mass balance is a black box approach and focus on the overall behaviour of a system whereas a differential mass balances focuses on mechanisms within the system (which in turn affect the overall behavior).

To make an integral mass balance one must first identify system boundaries, i.e. how the system is connected to the rest of the world and how the rest of the world influence the system. In some systems the system boundaries are evident, e.g. for a tank reactor the walls of the tank are the system boundaries and the outer world influence the system through the inlet and outlet. For e.g. a forest soil, the choice of system boundary, e.g. if vegetation is considered external or internal to the system, depend on the focus of the study.

To make a differential mass balance one must also describe the interior of the system. In the simplest case the system is homogeneous (perfectly mixed).

Differential mass balances

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These two steps lead to a basic description of the system and its boundaries, in chemical engineering called a reactor model. The three most simple reactor models are:

1. Ideal (continuously stirred) Batch reactor
2. Ideal tank reactor, also named Continuously Stirred Tank Reactor (CSTR)
3. Ideal Plug Flow Reactor (PFR)

Ideal Batch reactor

A closed system. Many chemistry textbooks implicitly assumes 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

$\backslash mathrm\{IN\}\; +\; \backslash mathrm\{PROD\}\; =\; \backslash mathrm\{OUT\}\; +\; \backslash mathrm\{ACC\}$

$0\; +\; r\_\{\backslash mathrm\{A\}\}\; V\; =\; 0\; +\; \backslash frac\{dn\_\{\backslash mathrm\{A\}\}\}\{dt\}$

where r_A denote 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.

Example

In this example we will use the law of mass action to derive the expression for a chemical equilibrium constant.

Assume we have a closed reactor in which the following liquid phase reversible reaction occurs:

$a\backslash mathrm\{A\}\; +\; b\backslash mathrm\{B\}\; \backslash leftrightarrow\; c\backslash mathrm\{C\}\; +\; d\backslash mathrm\{D\}$

The mass balance for substance A becomes

$\backslash mathrm\{IN\}\; +\; \backslash mathrm\{PROD\}\; =\; \backslash mathrm\{OUT\}\; +\; \backslash mathrm\{ACC\}$

$0\; +\; r\_\{\backslash mathrm\{A\}\}\; V\; =\; 0\; +\; \backslash frac\{dn\_\{\backslash mathrm\{A\}\}\}\{dt\}$ As we have a liquid phase reaction we can (usually) assume a constant volume and since $n\_\{\backslash mathrm\{A\}\}=\; V\; *\; C\_\{\backslash mathrm\{A\}\}$ we get

$r\_\{\backslash mathrm\{A\}\}\; V\; =\; V\; \backslash frac\{dC\_\{\backslash mathrm\{A\}\}\}\{dt\}$

or

$r\_\{\backslash mathrm\{A\}\}\; =\; \backslash frac\{dC\_\{\backslash mathrm\{A\}\}\}\{dt\}$

In many text books this is given as the definition of reaction rate 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 action the forward reaction rate can be written as

$r\_1=k\_1*^a*^b$

and the backward reaction rate as

$r\_\{-1\}=k\_\{-1\}1*^c*^d$

The rate at which substance A is produced is thus

$r\_\{\backslash mathrm\{A\}\}\; =\; r\_\{-1\}\; -\; r\_1$

and since, at equilibrium, the concentration of A is constant we get

$r\_\{\backslash mathrm\{A\}\}\; =\; r\_\{-1\}\; -\; r\_1=\; \backslash frac\{dC\_\{\backslash mathrm\{A\}\}\}\{dt\}\; =0$

or, rearranged

$\backslash frac\{k\_1\}\{k\_\{-1\}\}=\backslash frac\{*^c*^d\}\{*^a*^b\}=K\_\{eq\}$

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

$\backslash mathrm\{IN\}\; +\; \backslash mathrm\{PROD\}\; =\; \backslash mathrm\{OUT\}\; +\; \backslash mathrm\{ACC\}$

$Q\_0*C\_\{\backslash mathrm\{A\},0\}\; +\; r\_A*V\; =\; Q*C\_\{\backslash mathrm\{A\}\}\; +\; \backslash frac\{dn\_\{\backslash mathrm\{A\}\}\}\{dt\}$

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 state where all state variables (temperature, concentrations etc.) remain constant ($\backslash mathrm\{ACC\}\; =\; 0$)

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, $\backslash mathrm\{PROD\}\; =0$ and since there is no outflow $Q=0$. The mass balance becomes

$\backslash mathrm\{IN\}\; +\; \backslash mathrm\{PROD\}\; =\; \backslash mathrm\{OUT\}\; +\; \backslash mathrm\{ACC\}$

$Q\_0*C\_\{\backslash mathrm\{A\},0\}\; +0\; =\; 0*C\_\{\backslash mathrm\{A\}\}\; +\; \backslash frac\{dn\_\{\backslash mathrm\{A\}\}\}\{dt\}$

or

$Q\_0*C\_\{\backslash mathrm\{A\},0\}=\; \backslash frac\{dC\_\{\backslash mathrm\{A\}\}V\}\{dt\}=V\; \backslash frac\{dC\_\{\backslash mathrm\{A\}\}\}\{dt\}\; +\; C\_\{\backslash mathrm\{A\}\}\backslash frac\{dV\}\{dt\}$

Using a mass balance for total volume, however, it is evident that $\backslash frac\{dV\}\{dt\}=Q\_0$and that $V=V\_\{t=0\}+Q\_0t$. Thus we get

$\backslash frac\{dC\_\{\backslash mathrm\{A\}\}\}\{dt\}=\backslash frac\{Q\_0\}\{(V\_\{t=0\}+Q\_0t)\}\backslash left(\; C\_\{\backslash mathrm\{A\},0\}-C\_\{\backslash mathrm\{A\}\}\; \backslash right)$

Note that there is no reaction and hence no reaction rate or rate law involved, and yet $\backslash frac\{dC\_\{\backslash mathrm\{A\}\}\}\{dt\}\backslash neq\; 0$. We can thus draw the conclusion that reaction rate can not be defined in a general manner using $\backslash frac\{dC\}\{dt\}$. One must first write down a mass balance before a link between $\backslash frac\{dC\}\{dt\}$ and the reaction rate can be found. Many textbooks, however, define reaction rate as

$v=\; \backslash frac\{dC\_\{\backslash mathrm\{A\}\}\}\{dt\}$

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 infinitesimal part of the tube and make a mass balance over that using the ideal tank reactor model. That mass balance is then integrated over the entire reactor volume. In numeric solutions, e.g. when using computers, the ideal tube is often translated to a series of tank reactors.

More complex problems

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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 transfer rates may be important in the mathematical description of a system, especially in heterogeneous systems.

As the chemical reaction rate depends on temperature it is often necessary to make both an energy balance (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 conduction.

See also

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• chemical reactor
• chemical engineering
• chemical equilibrium
• mass action
• conservation of mass
• Continuity equation
• Bioreactor

Mass | Physical chemistry | Chemical engineering

Balans (scheikunde)