Modeling and Optimization of Lactic Acid Production using Cashew Apple Juice as Substrate
The fermentation was divided into two periods regarding the growth of the microorganism. The existence of two periods was based on the behavior of the sugar consumption and on the lactic acid production obtained experimentally. In the first period, the microorganism grows at a slow rate while adapting to the fermentation medium (Eq. 1). This period takes approximately 2 h and its end is marked by a 20±2% increase in biomass.
In the second period, the microorganism is fully adapted to the culture medium and presents a faster growth rate. The growth of the microorganism is inhibited by the product and also by the amount of sugars in the fermentation medium (Eq. 2), as observed in the experimental results that will be presented in the section Results and Discussion.
Where k1 is the growth rate constant during the adaptation stage (g/Lh); k2 is the growth rate constant (h−1); Kc is the limitation constant for sugars (g/L); kd is the rate constant for the death of microorganisms (h−1); Ke is the inhibition constant for sugars (g/L); Kp is the inhibition constant for lactic acid (g/L); P is the concentration of lactic acid (g/L); S is the concentration of sugars (g/L); X is the concentration of viable microorganisms (g/L).
The consumption of sugars was modeled assuming that sugars are consumed by the microorganism for its maintenance and for its reproduction. Lactic acid is a metabolite of the microbial energy
metabolism; thus sugars are converted into lactic acid. The consumption of sugars was also assumed to be inhibited by high lactic acid concentration.
Where Pcrit is the critical acid lactic concentration in the reaction medium for which the microbial activity diminishes steeply (g/L); α is the constant for sugars consumption during reproduction of microorganisms (g/g); β is the rate constant for sugars consumption constant during maintenance of the microorganisms (g/gh).
The production of lactic acid was modeled assuming that acid lactic is a product of the energy metabolism of the microorganism.
Where kP is the lactic acid production constant (g/g).
Equation 5 was added to the model to account for the amount of microorganisms that died or that became not viable during the fermentation.
Where, XD is the concentration of dead or not viable microorganism (g/L). The total amount of microorganisms (viable and not viable) was
calculated summing X and XD.
Please use the parameters in the given table
to estimate the parameters: Kc, Ke, Kp, Pcrit for the following three experimental data by using the Levenberg–Marquardt method and then also solve the differential equations by the adaptive step forth order Runge– Kutta method (do not use the 4th & 5th way to do this but use the method by doubling one single step into 2, e.g. marching one step by h/2 ==>and marching again by h/2 (sum up one h) instead of a single h step. (Reference: http://wenku.baidu.com/view/7f62170d4a7302768e993990.html )
All the estimated parameters for these three cases please should be tabulated in a table or displayed as we can see.
Please, write the codes and schemes by yourself, the built in functions for the Levenberg–Marquardt and Runge–Kutta methods are not allowed (but could use relative syntax to help construct the codes).
sugar concentration= 20 (g/L)
t (h)
biomass (g/L)
sugar (g/L)
lactic acid (g/L)
0
0.1796
17.7764
0.0000
1
0.2199
16.9752
0.0932
2
0.2489
16.8261
1.0435
3
0.2370
15.7826
2.7205
4
0.3727
14.7578
4.2484
5
0.4152
13.5093
6.3168
6
0.5985
11.0124
8.4224
7
0.7796
9.1677
10.4161
8
0.8981
6.5217
13.3789
9
0.9563
5.1615
13.8261
10
0.9339
2.9255
16.2298
11
1.0129
2.3106
17.1801
12
1.0949
0.6335
18.6149
sugar concentration= 40 (g/L)
t (h)
biomass (g/L)
sugar (g/L)
lactic acid (g/L)
0
0.2003
38.8539
0.0000
1
0.2229
37.6259
0.0630
2
0.2128
35.4849
1.1335
3
0.3350
35.4849
2.2985
4
0.5491
33.2179
4.1247
5
0.7418
28.0856
6.5491
6
0.9118
27.3300
9.0995
7
1.1045
26.3539
13.4446
8
1.3514
22.8904
17.2544
9
1.4975
19.1436
21.6940
10
1.6159
17.2544
25.6612
11
1.6398
13.2872
28.5894
12
1.6877
11.8703
31.5176
13
1.7985
10.0441
34.0680
19
1.7708
1.3224
39.4836
sugar concentration= 60 (g/L)
t (h)
biomass (g/L)
sugar (g/L)
lactic acid (g/L)
0
0.2087
58.0097
0
1
0.2403
56.5534
1.1165
2
0.2998
54.4660
2.7670
3
0.5097
50.8738
4.5631
4
0.8192
47.9612
6.4563
5
0.9345
45.8252
8.9320
6
0.9709
42.5243
11.9903
7
1.0801
38.5437
15.1456
8
1.1468
37.9126
18.3981
9
1.2015
34.3689
22.2330
10
1.2779
30.8738
25.1456
11
1.3471
25.6796
30.2913
12
1.4757
25.6796
34.9515
13
1.5777
22.1845
39.0777
14
1.7112
17.9612
43.3495
21
1.6578
8.1553
53.9320
24
1.8113
3.9320
55.9223
28
1.7294
1.7961
59.2718
After estimating the three sets of parameters for the data of the three experiments individually, please use these parameters and initial conditions to find the maximum production of lactic acid in these three cases.
Please plot 3 figures as Fig.1, 2, 3 (concentration vs time) by using our experimental data and predicted model function. And use the expression in Fig.4 to plot 3 figures for total, viable(living) and not viable(dead) cells when feeding sugar=20, 40, 60 g/L.