# 8.4: Biomass Production

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Biological compounds are produced by living cells, and the design and analysis of biological reactors requires both macroscopic balance analysis and kinetic studies of the complex reactions that occur within the cells. Given essential nutrients and a suitable temperature and pH, living cells will grow and divide to increase the cell mass. Cell mass production can be achieved in a chemostat where nutrients and oxygen are supplied as illustrated in Figure \(\PageIndex{1}\). Normally the system is charged with cells, and a start-up period occurs during which the cells become accustomed to the nutrients supplied in the inlet stream. Oxygen and nutrients pass through the cell walls, and biological reactions within the cells lead to *cell growth* and the creation of new cells. In Figure \(\PageIndex{2}\) we have illustrated the process of cell division in which a single cell (called a *mother cell*) divides into two *daughter cells*. In Figure \(\PageIndex{3}\) we have identified species \(A\) and \(B\) as *substrates*, which is just another word for nutrients and oxygen. Species \(C\) represents *all the species* that leave the cell, while species \(D\) represents *all the species* that remain in the cell and create cell growth. The details of the enzyme-catalyzed reactions that occur within the cells are discussed in Sec. 9.2.

To analyze cell growth in a chemostat, we need to know the rate at which species \(D\) is produced^{5}. In reality, species \(D\) represents many chemical species which we identify explicitly as \(F\), \(G\), \(H\), etc.

The appropriate mass balances for these species are given by

\[\frac{d}{dt} \int_{\mathscr{V}(t)}\rho_{F} dV + \int_{\mathscr{A}(t)}\rho_{F} (\mathbf{v}_{F} -\mathbf{w})\cdot \mathbf{n} dA =\int_{\mathscr{V}(t)}r_{F} dV \label{36a}\]

\[\frac{d}{dt} \int_{\mathscr{V}(t)}\rho_{G} dV + \int_{\mathscr{A}(t)}\rho_{G} (\mathbf{v}_{G} -\mathbf{w})\cdot \mathbf{n} dA =\int_{\mathscr{V}(t)}r_{G} dV \label{36b}\]

\[\frac{d}{dt} \int_{\mathscr{V}(t)}\rho_{H} dV + \int_{\mathscr{A}(t)}\rho_{H} (\mathbf{v}_{H} -\mathbf{w})\cdot \mathbf{n} dA =\int_{\mathscr{V}(t)}r_{H} dV \label{36c}\]

\[\text{etc.} \label{36d}\]

Here \(\mathscr{V}(t)\) represents the control volume illustrated in Figure \(\PageIndex{3}\) and \(\mathscr{A}(t)\) represents the surface area at which the speed of displacement is \(\mathbf{w}\cdot \mathbf{n}\).

In order to develop the macroscopic balance for the *total density of cellular material*, we simply add Eqs. \ref{36a} - \ref{36d} to obtain

\[\begin{align} {\frac{d}{dt} \int_{\mathscr{V}(t)}\rho_{D} dV + \int_{\mathscr{A}(t)}\rho_{F} (\mathbf{v}_{F} -\mathbf{w})\cdot \mathbf{n} dA + \int_{\mathscr{A}(t)}\rho_{G} (\mathbf{v}_{G} -\mathbf{w})\cdot \mathbf{n} dA } \nonumber\\ {+ \int_{\mathscr{A}(t)}\rho_{H} (\mathbf{v}_{H} -\mathbf{w})\cdot \mathbf{n} dA + \text{ etc.}=\int_{\mathscr{V}(t)}r_{D} dV } \label{37} \end{align} \]

Here we need to be very clear that \(\rho_{D}\) represents the *total density* of the cellular material and that this density is defined by

\[\rho_{D} =\rho_{F} + \rho_{G} + \rho_{H} + \text{ etc.} \label{38}\]

In addition, we need to be very clear that \(r_{D}\) represents the *total mass rate of production* of cellular material, and that this mass rate of production is defined by

\[r_{D} =r_{F} + r_{G} + r_{H} + \text{ etc.} \label{39}\]

There are other molecular species in the system illustrated in Figure \(\PageIndex{2}\); however, we are interested in the rate of growth of cellular material, thus \(r_{D}\) is the quantity we wish to predict.

Returning to Equation \ref{37}, we note that terms such as \((\mathbf{v}_{G} -\mathbf{w})\cdot \mathbf{n}\) are *negligible* everywhere except at the entrance where cellular material may enter the chemostat, and at the *exit* where the product leaves the system. It is reasonable to assume that all the species associated with the cellular material move with the same velocity at the entrance and exit, and this allows us to express Equation \ref{37} as

\[\frac{d}{dt} \int_{\mathscr{V}(t)}\rho_{D} dV + \int_{\mathscr{A}_{ e} }\rho_{D} \mathbf{v}_{D} \cdot \mathbf{n} dA =\int_{\mathscr{V}(t)}r_{D} dV \label{40}\]

where \(A_{ e}\) represents the area of the entrance and exit. Here we need to be very clear that this result is based on the plausible assumption that all the velocities of the species remaining in the cell are the same

\[\mathbf{v}_{F} = \mathbf{v}_{G} = \mathbf{v}_{H} = \text{ etc.} \label{41}\]

and we have identified this *common velocity* by \(\mathbf{v}_{D}\). For the typical chemostat, it is reasonable to ignore variations in the control volume and to assume that the velocities at the entrance and exit are constrained by

\[\mathbf{v}_{D} \cdot \mathbf{n}= \mathbf{v}_{\ce{H2O}} \cdot \mathbf{n} , \quad \text{ at } A_{ e} \label{42}\]

so that Equation \ref{40} takes the form

\[\underbrace{\mathscr{V} \frac{d\langle \rho_{D} \rangle }{dt} }_{\begin{array}{c} \text{rate of accumulation} \\ \text{of cellular material} \\ \text{ in the chemostat} \end{array}}+\underbrace{ \langle \rho_{D} \rangle_{2} Q_{2} }_{\begin{array}{c} \text{rate at which} \\ \text{cellular material leaves } \\ \text{ the chemostat} \end{array}}-\underbrace{ \langle \rho_{D} \rangle_{1} Q_{1} }_{\begin{array}{c} \text{rate at which} \\ \text{ cellular material enters} \\ \text{ the chemostat} \end{array}} = \underbrace{ \langle r_{D} \rangle \mathscr{V} }_{\begin{array}{c} \text{rate of production} \\ \text{of cellular material} \\ \text{ in the chemostat} \end{array}} \label{43}\]

It is the rate of reaction term on the right hand side of this result that is important to us since it represents the *mass rate of production of cellular material* in the chemostat. Rather than work directly with this quantity, there is a tradition of using the *rate of production of cells* to describe the behavior of the chemostat. We define the *average mass of a cell* in the chemostat by

\[m_{cell} =\left\{\begin{array}{c} \text{average mass} \\ \text{ of a cell} \end{array}\right\}=\frac{\left\{\begin{array}{c} \text{mass of cellular material} \\ \text{ per unit volume} \end{array}\right\}}{\left\{\begin{array}{c} \text{number of cells} \\ \text{per unit volume} \end{array}\right\}} \label{44}\]

and we represent the number of cells per unit volume by

\[\langle n\rangle =\left\{\begin{array}{c} \text{number of cells} \\ \text{per unit volume} \end{array}\right\} \label{45}\]

This allows us to express the *mass of cellular material per unit volume* according to

\[\langle \rho_{D} \rangle =\langle n\rangle m_{cell} \label{46}\]

Given these definitions, we can divide Equation \ref{43} by the constant, \(m_{cell}\), to obtain a macroscopic balance for the number density of cells.

\[\mathscr{V}\frac{d\langle n\rangle }{dt} + \langle n\rangle_{2} Q_{2} - \langle n\rangle_{1} Q_{1} =\left({\langle r_{D} \rangle / m_{cell} } \right)\mathscr{V} \label{47}\]

Here we have assumed that average mass of a cell in the chemostat is *independent of time*, and this may not be correct for transient processes. In addition, Equation \ref{46} is based on the assumption that all of species \(D\) is contained within the cells. This is consistent with the cellular processes illustrated in Figure \(\PageIndex{2}\); however, that illustration *does not take into account* the process of cell death^{6}. Because of cell death, Equation \ref{46} represents an over-estimate of the number of cells per unit volume.

Traditionally, one assumes that the volumetric flow rates entering and leaving the chemostat are equal so that Equation \ref{47} simplifies to

\[\frac{d\langle n\rangle }{dt} + \langle n\rangle_{2} \left({Q / \mathscr{V}} \right) - \langle n\rangle_{1} \left({Q / \mathscr{V}} \right)={\langle r_{D} \rangle / m_{cell} } \label{48}\]

This represents a governing differential equation for cells per unit volume, \(\langle n\rangle\); however, it is the cell concentration at the exit, \(\langle n\rangle_{2}\), that we wish to predict, and this prediction is usually based on the assumption of a *perfectly mixed* system as described in Sec. 8.1. This assumption leads to \(\langle n\rangle_{2} =\langle n\rangle\) and it allows us to express Equation \ref{48} in the form

\[\underbrace{ \frac{d\langle n\rangle }{dt} }_{accumulation} + \underbrace{ \langle n\rangle \left({Q / \mathscr{V}} \right) }_{outflow} - \underbrace{ \langle n\rangle_{1} \left({Q / \mathscr{V}} \right) }_{inflow} = \underbrace{\left({\langle r_{D} \rangle / m_{cell} } \right)}_{production} \label{49}\]

In previous sections of this chapter the quantity, \(\mathscr{V}/Q\), was identified as the *mean residence time* and denoted by \(\tau \). However, in the biochemical engineering literature, the tradition is to identify \(Q/\mathscr{V}\) as the *dilution rate* and denote it by \(D\). Following this tradition we express Equation \ref{49} in the form

\[\frac{d\langle n\rangle }{dt} + \left(\langle n\rangle - \langle n\rangle_{1} \right)D={\langle r_{D} \rangle / m_{cell} } \label{50}\]

where the term on the right hand side should be interpreted as

\[{\langle r_{D} \rangle / m_{cell} } =\left\{\begin{array}{c} \text{number of cells} \\ \text{ produced per unit} \\ \text{volume per unit time} \end{array}\right\} \label{51}\]

This rate of production, caused by biological reactions, is traditionally represented as

\[{\langle r_{D} \rangle / m_{cell} } ={\mu } \langle n\rangle \label{52}\]

so that our governing differential equation for the cell concentration takes the form

\[\frac{d\langle n\rangle }{dt} + \left(\langle n\rangle - \langle n\rangle_{1} \right)D={\mu } \langle n\rangle \label{53}\]

The quantity, \({\mu }\), is known as the *specific growth rate* and if \({\mu }\) is known one can use this result to predict \(\langle n\rangle\) as a function of time.

For many practical applications, there are no cells entering the chemostat, thus \(\langle n\rangle_{1}\) is zero and we are dealing with what is called a *sterile feed*. For a sterile feed, the cell concentration is determined by the following governing equation and initial condition

\[\frac{d\langle n\rangle }{dt} + \langle n\rangle D={\mu } \langle n\rangle \label{54a}\]

IC. \[\langle n\rangle =n_{ o} , \quad t=0 \label{54b}\]

Here \(n_{ o}\) represents the initial concentration of cells in the chemostat and this is usually referred to as the *inoculum*. If we treat the specific growth rate, \({\mu }\), is a constant, the solution of the initial value problem for \(\langle n\rangle\) is straightforward and is left as an exercise for the student.

The steady-state form of Equation \ref{54a} is given by

\[\left(D - {\mu }\right)\langle n\rangle =0 \label{55}\]

and this indicates that the steady state can only exist when \(\langle n\rangle =0\) or when \(D={\mu }\). The first of these is of no interest, while the second suggests that a steady-state chemostat might be rather rare since adjusting the *dilution rate*, \(D=Q/\mathscr{V}\), to be exactly equal to the *specific growth rate* might be very difficult if the specific growth rate were *constant*. However, a little thought indicates that the specific growth rate, \({\mu }\), must depend on the concentration of the nutrients entering the chemostat, must depend on the concentration of the nutrients entering the chemostat, thus \(\mu\) can be controlled by adjusting the input conditions.

When the substrate \(B\) is present in excess, the rate of cell growth can be expressed in terms of the concentration of species \(A\) in the extracellular fluid, \(\langle c_A \rangle\), a reference concentration \(K_A\), and other parameters according to

\[\mu = \mathscr{F} ( \langle c_A \rangle , K_A, \text{ other parameters}) \label{56}\]

If a specific growth rate has the following characteristics

\[\mu= \begin{cases} 0 & \left\langle c_{A}\right\rangle \rightarrow 0 \\

\mu_{\max } & \left\langle c_{A}\right\rangle >> K_{A} \end{cases} \label{57}\]

it could be modeled by what is known as Monod’s equation^{7}

Monod’s equation: \[{\mu }= \frac{\mu_{ max} \langle c_{A} \rangle }{K_{A} + \langle c_{A} \rangle } \label{58}\]

The parameter \(K_A\) is sometimes referred to as the “half saturation” since it represents the concentration at which the growth rate is half the maximum growth rate, \(\mu_{max}\). It should be clear that there are many other functional representations that would satisfy Equation \ref{57}; however, the form chosen by Monod^{8} has been used with reasonable success to correlate macroscopic experimental data.