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8.10: Untitled Page 199

  • Page ID
    18332
  • Chapter 8

    so that Eq. 8‐40 takes the form

    d 

    D

    V

      Q

    D 2

    2

    dt

    

    

    rate at which

    rate of accumulation

    cellular material leaves

    of cellular material

    the chemostat

    in the chemostat

    (8‐43)

      Q

    r V

    D 1 1

    D

    

    

    rate at which

    rate of production

    cellular material enters

    of cellular material

    the chemostat

    in the chemostat

    It is the 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

    mass of cellular material

    average mass

    per unit volume

    m

     

     

    (8‐44)

    cell

    of a cell

    number of cells

    per unit volume

    and we represent the number of cells per unit volume by

    number of cells

    n  

    (8‐45)

    per unit volume

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

        nm

    (8‐46)

    D

    cell

    Given these definitions, we can divide Eq. 8‐43 by the constant, m

    , to obtain a

    cell

    macroscopic balance for the number density of cells that takes the form dn

    V

      nQ   nQ

    r m

    V

    (8‐47)

    2

    2

    1 1

    D cell

    dt

    Here we have assumed that the average mass of a cell in the chemostat is independent of time, and this may not be correct for transient processes. In addition, Eq. 8‐46 is based on the assumption that all of species D is contained within the cells. This is consistent with the cellular processes illustrated in

    Transient Material Balances

    373

    Figure 8‐9; however, that illustration does not take into account the process of cell death (Bailey and Ollis, 1986). Because of cell death, Eq. 8‐46 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 Eq. 8‐47 simplifies to dn

      nQ V   nQ V

      r m

    (8‐48)

    2 

    1 

    D

    cel

    dt

    l

    This represents a governing differential equation for cells per unit volume,  n ; however, it is the cell concentration at the exit,  n , that we wish to predict, and 2

    this prediction is usually based on the assumption of a perfectly mixed system as described in Sec. 8.1. This assumption leads to  n   n and it allows us to 2

    express Eq. 8‐48 in the form

    dn

      n  Q V    nQ V

    r m

    (8‐49)

    1 

    D cell

    dt

    

    

    

    

    

    

    outflow

    inflow

    production

    accumulation

    In previous sections of this chapter the quantity, V / Q , was identified as the mean residence time and denoted by  . However, in the biochemical engineering literature, the tradition is to identify Q / V as the dilution rate and denote it by D.

    Following this tradition we express Eq. 8‐49 in the form

    dn

      n   nD   r m

    (8‐50)

    1 

    D

    ce

    dt

    ll

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

    number of cells

    r m

      produced per unit

    (8‐51)

    D

    cell