Material Balances#

Introduction#

Suppose we have a process producing a critical component in a next-generation battery, a biofuel, or an mRNA vaccine. To model this process, we break it into two pieces, a system (where the chemistry or physical change is occurring) and its surroundings (system and surroundings are called the universe). Previously, we introdcued the idea of a stuff balance; equations that can be used to track the amount of stuff in a system. But what is stuff?

To exlore this question, let’s expand upon our previous discussion of stuff balances, and consider a particular type of stuff, namely the mass and moles of chemical compounds. For example, the mass (or number of moles) of active mRNA being produced by a system. Mass (or moles) are types of material balances, they allow us to track the amount of material in a system as time evolves.

In this lecture:

Species and Total Mass Balance Equations#

Suppose we interested in tracking the mass of each chemical component in a mixture of \(\mathcal{M}\) components for a chemical reaction process e.g., the starting material, the final product and perhaps some by-products flowing into and from a chemical reactor unit.

Open species mass balances#

If we are interested in the amount of each chemical species, we can let stuff equal the mass of each of the chemical components; this choice gives the open species mass balance (Definition 4):

Definition 4 (Open Species Mass Balance)

Let \(m_{i}\) denote the mass of chemical component \(i\) in the system (units: mass e.g., grams g). Further, denote the number of chemical components in the system we want to track as \(\mathcal{M}\), and the set of streams flowing into (or from) the system as \(\mathcal{S}\), where each stream \(s\in\mathcal{S}\) has a direction parameter \(\nu_{s}\in\left[-1,1\right]\).

If stream \(s\) enters the system \(\nu_{s} = +1\), however is stream \(s\) exits the system then \(\nu_{s} = -1\).

Then, the mass of chemical component \(i\) in the system as a function of time (units: g) is described by an open species mass balance equation:

(8)#\[\sum_{s\in\mathcal{S}}\nu_{s}\dot{m}_{s,i} + \dot{m}_{G,i} = \frac{dm_{i}}{dt} \qquad\forall{i}\in\mathcal{M}\]

The quantity \(\dot{m}_{s,i}\) denotes the mass flow rate of component \(i\) in stream \(s\) (units: g \(i\)/time), \(\dot{m}_{G,i}\) denote the rate of generation of component \(i\) in the system (units: g \(i\)/time), and \(dm_{i}/dt\) denotes the rate of accumlation of mass of component \(i\) in the system (units: mass of \(i\) per time).

At steady state, the accumulation term vanishes; thus, the the steady state species mass balances are given by:

(9)#\[\sum_{s\in\mathcal{S}}\nu_{s}\dot{m}_{s,i} + \dot{m}_{G,i} = 0 \qquad\forall{i}\in\mathcal{M}\]

where we have one balance per chemical species.

Open total mass balances#

Of course, we don’t need to track the individual mass of each chemical compound in a mixture of \(\mathcal{M}\) compounds; instead, we can track the total mass of all \(\mathcal{M}\) components collectively. This choice gives the open total mass balance (Definition 5):

Definition 5 (Open Total Mass Balance)

Let \(m_{i}\) denote the mass of chemical component \(i\) in the system (units: mass e.g., grams g). Further, denote the number of chemical components in the system we want to track as \(\mathcal{M}\), and the set of streams flowing into (or from) the system as \(\mathcal{S}\), where each stream \(s\in\mathcal{S}\) has a direction parameter \(\nu_{s}\in\left[-1,1\right]\).

If stream \(s\) enters the system \(\nu_{s} = +1\), however, if stream \(s\) exits the system then \(\nu_{s} = -1\).

Then, the total mass of all chemical components in the system as a function of time (units: g) is described by an open total mass balance equation:

(10)#\[\sum_{s\in\mathcal{S}}\nu_{s}\dot{m}_{s,T} = \frac{dm_{T}}{dt}\]

The quantity \(\dot{m}_{s,T}\) denotes the total mass flow rate of stream \(s\) (units: g/time), and \(dm_{T}/dt\) denotes the rate of accumlation of total mass in the system (units: g/time).

At steady-state, the accumulation terms vanish in Eqn (10); thus, the steady state total mass balance equation is given by:

(11)#\[\sum_{s\in\mathcal{S}}\nu_{s}\dot{m}_{s,T} = 0\]

Relationship between the species and total mass balance#

To go from a species perspective i.e., where we track the mass of each chemical component in a mixture of \(\mathcal{M}\) components, to the total mass balance we use the Law of Conservation of Mass. The Law of Conservation of Mass says that mass can neither be created nor destroyed, just rearranged or transformed.

Observation 3 (Total Mass Balance using the Law of Conservation of Mass)

Suppose we have a system with the chemical species set \(\mathcal{M}\) that has dimension \(|\mathcal{M}|\). Further suppose we write a species balance around each of the elements of the species set \(\mathcal{M}\), where we index the species from \(j=1,2,\dots,|\mathcal{M}|\):

\[\begin{split}\begin{eqnarray} \sum_{s\in\mathcal{S}}\nu_{s}\dot{m}_{s,1} + \dot{m}_{G,1} &=& \frac{dm_{1}}{dt}\\ & \vdots & \\ \sum_{s\in\mathcal{S}}\nu_{s}\dot{m}_{s,|\mathcal{M}|} + \dot{m}_{G,|\mathcal{M}|} &=& \frac{dm_{|\mathcal{M}|}}{dt}\\ \end{eqnarray}\end{split}\]

Then, we can sum the individual \(j=1,2,\dots,|\mathcal{M}|\) species mass balances to arrive at the total mass balance equation, i.e., we take advantage of the summation relationship(s):

(12)#\[m_{T} = \sum_{i\in\mathcal{M}}m_{i}\]

and

(13)#\[\dot{m}_{s,T} = \sum_{i\in\mathcal{M}}\dot{m}_{s,i}\]

for some stream \(s\). Let’s use the summation relationships in Eqn. (12) and Eqn. (13) to explore each term of the total mass balance:

  • Accumulation terms: The accumulation term for the total mass balance is simply the time-derivative of the sum of the accumulation terms for each component:

\[\frac{dm_{T}}{dt} = \frac{d}{dt}\left(\sum_{i\in\mathcal{M}}m_{i}\right)\]

  • Transport terms: The total mass flow rate in stream \(s\) can be calculated using Eqn. (13):

\[\sum_{s\in\mathcal{S}}\nu_{s}\dot{m}_{s,T} = \sum_{s\in\mathcal{S}}\nu_{s}\left(\sum_{i\in\mathcal{M}}\dot{m}_{s,i}\right)\]

  • Generation terms: There are no generation terms in the total mass balance. However, the Law of Conservation of Mass says that mass is not created or destroyed, just transformed from one form to another; if we consume component \(j\), then the amount of mass consumed must show up in the other components. Thus, the sum of the individual species mass generation terms must be equal to zero:

\[\sum_{i\in\mathcal{M}}\dot{m}_{G,i} = 0\]

Mass fraction#

Often it is not convenient to use the mass of component \(i\) directly when doing calculations. Instead, we use an analogous quantity called the mass fraction:

Definition 6 (Mass fraction)

Suppose we have a stream (or system) composed of the species in the set \(\mathcal{M}\). Then the mass flow rate of stream \(s\) can be written as:

\[\dot{m}_{s,T} = \sum_{i\in\mathcal{M}}\dot{m}_{s,i}\]

Dividing both sides by \(\dot{m}_{s,T}\) gives:

\[ \sum_{i\in\mathcal{M}}\left(\frac{\dot{m}_{s,i}}{\dot{m}_{s,T}}\right) = 1\]

or

\[ \sum_{i\in\mathcal{M}}w_{s,i} = 1\]

where \(w_{s,i} \equiv \dot{m}_{s,i}/\dot{m}_{s,T}\) is the mass fraction of component \(i\) in stream \(s\). An analogous expression can be written for the mass fraction of component \(i\) inside the system.

Using Definition 6, we can re-write the species mass balance in terms of the mass fraction of component \(i\):

(14)#\[\sum_{s\in\mathcal{S}}\nu_{s}\dot{m}_{s}w_{s,i} + \dot{m}_{G,i} = \frac{d}{dt}\left(w_{i}m_{T}\right)\qquad\forall{i}\in\mathcal{M}\]

where \(w_{s,i}\) denotes the mass fraction of component \(i\) in stream \(s\), \(w_{i}\) denotes the mass fraction of component \(i\) in the system, \(\dot{m}_{s}\) denotes the mass flow rate of stream \(s\) and \(\dot{m}_{G,i}\) denotes the rate of mass generation for species \(i\).

Species and Total Mole Balance Equations#

Mass-based units are generally convenient for systems that do not involve chemical reactions. However, it is often easier to use mole-based units when chemical reactions occur. The principles of mole-based balance equations are similar to their mass-based equivalents, i.e., we can write species mole balances or total mole balances, and the terms play an equivalent role as their mass-based equivalents.

Remark 1 (Total moles are not conserved)

There is one key difference between a mass and mole description of the material in a system: total moles are not conserved in the presence of generation terms in a system.

Open species mole balances#

If we are interested in the number of moles of each chemical species, we can write the open species mole balance (Definition 7):

Definition 7 (Open Species Mole Balance)

Let \(m_{i}\) denote the number of moles of chemical component \(i\) in the system (units: \(\star\)moles, e.g., mmol or \(\mu\)mol). Further, denote the number of chemical components in the system we want to track as \(\mathcal{M}\), and the set of streams flowing into (or from) the system as \(\mathcal{S}\), where each stream \(s\in\mathcal{S}\) has a direction parameter \(\nu_{s}\in\left[-1,1\right]\).

If stream \(s\) enters the system \(\nu_{s} = +1\), however is stream \(s\) exits the system then \(\nu_{s} = -1\).

Then, the number of moles of chemical component \(i\) in the system as a function of time is described by an open species mass balance equation:

(15)#\[\sum_{s\in\mathcal{S}}\nu_{s}\dot{n}_{s,i} + \dot{n}_{G,i} = \frac{dn_{i}}{dt} \qquad\forall{i}\in\mathcal{M}\]

The quantity \(\dot{n}_{s,i}\) denotes the mole flow rate of component \(i\) in stream \(s\) (units: \(\star\)mol \(i\)/time), \(\dot{n}_{gen,i}\) denote the rate of generation of component \(i\) in the system (units: \(\star\)mol \(i\)/time), and \(dn_{i}/dt\) denotes the rate of accumlation of the number of moles of component \(i\) in the system (units: \(\star\)mol \(i\)/time).

At steady state, all the accumulation terms vanish; thus, the steady state system of species mole balances is given by:

(16)#\[\sum_{s\in\mathcal{S}}\nu_{s}\dot{n}_{s,i} + \dot{n}_{G,i} = 0 \qquad\forall{i}\in\mathcal{M}\]

Open total mole balances#

Of course, we may not need to track the individual number of moles of each chemical compound in the species set \(\mathcal{M}\). Instead, we can track the total number of moles of all the components the species set \(\mathcal{M}\). This choice gives the open total mole balance (Definition 8):

Definition 8 (Open Total Mole Balance)

Let \(n_{i}\) denote the number of moles of chemical component \(i\) in the system (units: \(\star\)mol e.g., \(\mu\)mol or mmol, etc). Further, denote the number of chemical components in the system we want to track as \(\mathcal{M}\), and the set of streams flowing into (or from) the system as \(\mathcal{S}\), where each stream \(s\in\mathcal{S}\) has a direction parameter \(\nu_{s}\in\left[-1,1\right]\).

If stream \(s\) enters the system \(\nu_{s} = +1\), however, if stream \(s\) exits the system then \(\nu_{s} = -1\).

Then, the total number of moles of all chemical components in the system as a function of time is described by an open total mole balance equation:

\[\sum_{s\in\mathcal{S}}\nu_{s}\dot{n}_{s,T} +\dot{n}_{G,T} = \frac{dn_{T}}{dt}\]

The quantity \(\dot{n}_{s,T}\) denotes the total mole flow rate of stream \(s\) (units: \(\star\)mol/time), \(\dot{n}_{gen,T}\) denotes the total generation rate (units: \(\star\)mol/time), and \(dn_{T}/dt\) denotes the rate of accumlation of total moles in the system (units: \(\star\)mol/time).

At steady state, the accumulation terms vanish; thus, the steady state total mole balance is given by:

(17)#\[\sum_{s\in\mathcal{S}}\nu_{s}\dot{n}_{s,T} + \dot{n}_{G,T} = 0\]

Relationship between the species and total mole balance#

To go from a species perspective, i.e., where we track the moles of each chemical component in a species set \(\mathcal{M}\), to the total mole balance, we sum the species mole balances.

However, unlike the mass frame of reference, we generally know that total moles are not conserved in the presence of the generation terms; thus, the sum of the generation terms is not equal to zero.

Observation 4 (Total Mole Balances)

Suppose we have a system with the chemical species set \(\mathcal{M}\) that has dimension \(|\mathcal{M}|\). Further suppose we write a species mole balance around each of the elements of the species set \(\mathcal{M}\), where we index the species from \(j=1,2,\dots,|\mathcal{M}|\):

\[\begin{split}\begin{eqnarray} \sum_{s\in\mathcal{S}}\nu_{s}\dot{n}_{s,1} + \dot{n}_{G,1} &=& \frac{dn_{1}}{dt}\\ & \vdots & \\ \sum_{s\in\mathcal{S}}\nu_{s}\dot{n}_{s,|\mathcal{M}|} + \dot{n}_{G,|\mathcal{M}|} &=& \frac{dn_{|\mathcal{M}|}}{dt}\\ \end{eqnarray}\end{split}\]

Then, we can sum the individual \(j=1,2,\dots,|\mathcal{M}|\) species mole balances to arrive at the total mole balance equation, i.e., we take advantage of the summation relationship(s):

(18)#\[n_{T} = \sum_{i\in\mathcal{M}}n_{i}\]

and

(19)#\[\dot{n}_{s,T} = \sum_{i\in\mathcal{M}}\dot{n}_{s,i}\]

for some stream \(s\). Let’s use the summation relationships in Eqn. (18) and Eqn. (19) to explore each term of the total mole balance:

  • Accumulation terms: The accumulation term for the total mole balance is simply the sum of time-derivatives of the the accumulation terms of each component:

\[\frac{dn_{T}}{dt} = \frac{d}{dt}\left(\sum_{i\in\mathcal{M}}n_{i}\right)\]

  • Transport terms: The total mole flow rate in stream \(s\) can be calculated using Eqn. (19):

\[\sum_{s\in\mathcal{S}}\nu_{s}\dot{n}_{s,T} = \sum_{s\in\mathcal{S}}\nu_{s}\left(\sum_{i\in\mathcal{M}}\dot{n}_{s,i}\right)\]

  • Generation terms: Unlike the total mass balance, there are generation terms in the total mole balance (becuase moles are not conserved):

\[\dot{n}_{G,T} = \sum_{i\in\mathcal{M}}\dot{n}_{G,i}\]

Mole fraction#

Often it is not convenient to use the moles of species \(i\) directly when doing calculations. Instead, we use an analogous quantity called the mole fraction:

Definition 9 (Mole fraction)

Suppose we have a stream (or system) composed of species in the set \(\mathcal{M}\). Then the total mole flow rate of stream \(s\) can be written as:

\[\dot{n}_{s,T} = \sum_{i\in\mathcal{M}}\dot{n}_{s,i}\]

Dividing both sides by \(\dot{n}_{s,T}\) gives:

\[ \sum_{i\in\mathcal{M}}\left(\frac{\dot{n}_{s,i}}{\dot{n}_{s,T}}\right) = 1\]

or

\[ \sum_{i\in\mathcal{M}}x_{s,i} = 1\]

where \(x_{s,i} \equiv \dot{n}_{s,i}/\dot{n}_{s,T}\) is the mole fraction of component \(i\) in stream \(s\). An analogous expression can be written for the mole fraction of component \(i\) inside the system.

When dealing with liquid streams, the symbol \(x_{\star}\) is used for mole fraction; however, for vapor streams, we use the symbol \(y_{\star}\).

Using Definition 9, we can re-write the species mole balance in terms of the mole fraction of component \(i\):

(20)#\[\sum_{s\in\mathcal{S}}\nu_{s}\dot{n}_{s}x_{s,i} + \dot{n}_{G,i} = \frac{d}{dt}\left(x_{i}n_{T}\right)\qquad\forall{i}\in\mathcal{M}\]

where \(x_{s,i}\) denotes the mole fraction of component \(i\) in stream \(s\), \(x_{i}\) denotes the mole fraction of component \(i\) in the system, \(\dot{n}_{s}\) denotes the mole flow rate of stream \(s\) and \(\dot{n}_{G,i}\) denotes the rate of mole generation for species \(i\) (units: mole per time).

Open Extent of Reaction#

Finally, let’s discuss the generation terms. Generation terms describe the impact of chemical reactions; theoretically, we can describe reactions using a mass or mole basis. However, typically we operate on a mole basis when dealing with chemical reactions.

Let’s explain how far the reaction has proceeded using the extent of reaction. Later, we will develop kinetic models, which describe how the rate of a chemical reaction changes with the concentration of the reactants (and sometimes products) of a reaction. However, for now, we’ll use the extent. The open extent of reaction describes how far a chemical reaction has proceeded toward completion in an open system:

Definition 10 (Open Extent of Reaction)

Suppose we have chemical species set \(\mathcal{M}\); further, suppose the chemical species in set \(\mathcal{M}\) participate in the chemical reaction set \(\mathcal{R}\). Then, the species generation rate \(\dot{n}_{G,i}\) can be written in terms of the open extent of reaction:

(21)#\[\dot{n}_{G,i} = \sum_{r\in\mathcal{R}}\sigma_{ir}\dot{\epsilon}_{r}\]

where \(\dot{\epsilon}_{r}\) denotes the open extent of reaction \(r\) (units: mol per time). The quantity \(\sigma_{ir}\) denotes the stoichiometric coefficient for species \(i\) in reaction \(r\):

  • If \(\sigma_{ir}>0\) then species \(i\) is produced by reaction \(r\), i.e., species \(i\) is a product of reaction \(r\)

  • If \(\sigma_{ir}=0\) then species \(i\) is not connected to reaction \(r\)

  • If \(\sigma_{ir}<0\) then species \(i\) is consumed by reaction \(r\), i.e., species \(i\) is a reactant of reaction \(r\).

Putting these ideas together, we can rewrite the dynamic species mole balance as:

(22)#\[\sum_{s\in\mathcal{S}}\nu_{s}\dot{n}_{s,i} + \sum_{r\in\mathcal{R}}\sigma_{ir}\dot{\epsilon}_{r} = \frac{dn_{i}}{dt}\qquad\forall{i\in\mathcal{M}}\]

and the dynamic total mole balance as:

(23)#\[\sum_{s\in\mathcal{S}}\nu_{s}\dot{n}_{s,T} + \sum_{i\in\mathcal{M}}\sum_{r\in\mathcal{R}}\sigma_{ir}\dot{\epsilon}_{r} = \frac{dn_{T}}{dt}\]

where the double summation term over the species and the reaction sets denotes the total rate of generation, \(\dot{n}_{G,T}\) (units: mole per time).

Summary#

In this lecture, we introduced open mass and mole balances. These equations can be used to describe the amount of material in a system; thus, they are often collectively called material balances.

  • We first introduced species mass balances that can be used to account for the mass of species in the chemical species set \(\mathcal{M}\).

  • Next, we used the Law of Conservation of Mass to compute a total mass balance equation, which is the sum of the species mass balances. Because of the conservation of mass principle, the species generation terms must sum to zero when in the mass frame of reference.

  • We then introduced total and species mole balances. We used these equations to track the number of moles in a system, either at the individual level of the \(j=1,2,\dots,|\mathcal{M}|\) chemical components in the species set \(\mathcal{M}\), or the total number of moles.

  • Finally, we introduced the open extent of reaction. The open extent of reaction describes how far a chemical reaction has gone in an open system.