**2.4 Mathematical Definition of Yield**

Mathematically, cell yield can be defined as

where DX represents change in cell concentration
and DS represents change in substrate concentration.
The subscript X/S indicates the basis of yield - cell
on the basis of substrate. This notation comes in handy when we need to
calculate yield based on more than one substrate. Examining the above and
comparing with **growth reaction**, one notes that the yield defined
here corresponds to a mass-based stoichiometric coefficient.

Taking the limit of Eq(2-2) as DS approaches zero,

The absolute sign is used to eliminate the negative value of the derivative. Note that dS is negative, because substrate is consumed. Yield is always reported as a positive value.

The above definition of yield can be applied to product, P on the basis of substrate consumed. Thus,

Similarly product yield based on cell will be expressed as,

In general, yield of the species, i, based on species, j, can be calculated from

From the above it is clear that we can combine two different yields which have a common species as

Example 2-2

Batley (1979) reported aerobic gowth of yeast on ethanol as:

Calculate Y

_{X/E}, Y_{X/O2}, Y_{X/NH3}on mass basis.

SolutionMW of cell = 12 +1.704 +(14) (0.149) + (16) (0.408) = 22.32

MW of ethanol = (2) (12) + 5 + 16 + 1 = 46

Yield of yeast based on ethanol of about 0.5 is consistent with the observation that roughly one half of the substrate is converted to cell mass aerobically. If yield on a carbohydrate source is significantly less than 0.5, it is likely that medium formulation is inadequate to support good growth.

Example 2-3

Yeast grown on glucose is described by

Calculate the following for a design requiring 50 g/L of yeast in a batch reactor of 100,000 liters.

Nutirent media concentration for glucose and ammonium sulfate.

Calculate Y

_{X/S}and Y_{X/O2}Calculate total oxygen required

Determine oxygen uptake rate (g O

_{2}L^{-1}h^{-1})when cell concentration increases at a rate of 0.7 g L^{-1}h^{-1},

SolutionTotal cell mass to be produced is = (10

^{5}L) (50 g L^{-1}) = 5000 kg