Continuous Cultivation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 618

TABLE E12-1.2 Parametric Estimation ResultsdCont'd

Error 2 or
( SLS C ) 2

S C calculated
from Eqn
(E12-1.5) , g/L

X c calculated
from Eqn
(E12-1.8) , g/L

Error
weighting
factor, u

2
D ( XLX c ) 2

S ,
mg/L

X ,
mg/L

u

D, h L 1

0.55 211.9521 341.5247

0.575 246.5724 322.1526

0.6 298 299 290.2169 297.37 63.23377 1

0.625 346.9428 264.7917

0.65 423.668 220.3428

0.675 533.2326 156.4519

0.7 702 59 702.4604 57.29264 3.12706 1

0.710082 3.4E-07

P ðS S C Þ 2 u 2 þðX X c Þ 2

800

q

¼ 16:43404

The strategy here is to make the error between Eqns (E12-1.5) and (E12-1.8) and the exper-
imental data minimum. Before we look at the error between the model prediction and the data,
we should examine the potential errors in the data set. Examining the data set, we can see that
the errors of S and X are not uniform. While the errors for all the data on biomass concentration
X may be regarded uniform (e.g.

1), however, the errors of S may be 1 for S > 100 and 0.1

for S 100. Therefore, when we calculate the error we need to scale the errors to the same level

so that the final parameters are the best estimates from the data. A scaling factor

<

is thus
needed to include the errors of the substrate concentrations. Table E12-1.2 shows the results.
When minimizing

u

q

P ðS S C Þ 2 u 2 þðX X c Þ 2

by altering m max , K S ,YF X/S , and k d , we obtain

0.8223 h 1 , 0.0304 h 1 , d m max ¼ K S ¼ 88.30 mg/L, YF X/S ¼ 0.6129 g-cells/g-S, k d ¼

q

P ðS S C Þ 2 u 2 þðX X c Þ 2

.
Now we are still missing the maintenance coefficient m S . To compute the maintenance
coefficient, we need to go back to definition Eqn (11.27) or

¼ 16:434

YF S = S m net þ m s r S ¼ X (E12-1.9)

When carrying out mass balance, we have used

YF S = S m net þ m s

r S ¼ YF S = S m G X ¼ X (E12-1.10)

Therefore,

h

YF S = S m G k d þ m s i

r S ¼ YF S = S m G X ¼ X (E12-1.11)

That is

m ¼ YF S = S k d ¼ k d =YF X = S (E12-1.12)

s