Blackman model

The model of Jassby and Platt is reported to best describe of the photosynthetic response to light (Chalker, 1980 Jassby and Platt, 1976). In this chapter the model of Jassby and Platt will be used to evaluate microalgal growth in photobioreactors, but also the simpler Blackman model. The Blackman model can be easily solved analytically and is well suited to highl%ht and discuss the specific aspects of phototrophic production processes. Besides these two models there are alternative models. Two models which are frequently used are the model of Webb and a Monod model ... 

Based on the Blackman model the compensation point Iph,c can be calculated. 

Figure 19 Specific rate of sugar production ql blue darkgray in the print version) lines) and the observable yield of sugar on photons absorbed n ph red dark gray in the print version) lines) as a function of photon flux density /ph- The solid lines follow the model of Jassby and Platt the dashed lines follow the Blackman model. Parameter values based on high-light acclimated Chlorella sorokiniana Ux=3.5 m mof T ph,m=0.10 molj fnolph qs,m = 1-25x 10 mols mol s /Mx=24 g moix .

When adopting the Blackman model a simple but useful expression for the volumetric productivity of photobioreactors can be derived. In the Blackman model a constant specific Hght absorption coefficient is used. In other words, the change of light spectrum by preferential absorption within a microaigai culture is not included. 

Figure 25 Schematic drawing of the light gradient inside microalgal cultures and the application of the Blackman model to calculate the average specific growth rate. The microalgal culture is subdivided into two zones (1) /ph(2)>/ph,s and (2) lph z)

The existence of a point of optimal productivity can be analytically deduced employing the Blackman model without spectral resolution. The relation between volumetric productivity and biomass concentration plotted in Fig. 26 is given by the following relation already introduced ... 

This relation can be integrated from time 0 to t provided the specific growth rate is constant and not dependent on the biomass concentration. The average specific growth rate in a microa al culture in a photobioreactor, however, is strongly dependent on the biomass concentration. This dependency is expressed in below relation which was already derived from the Blackman model. 

In this simple example the objective is to approximate the Blackman model for the specific growth rate (//) as a function of substrate concentration (S) ... 

Figure 2. Approximation results of the ME network to the Blackman model, (a) specific growth rate, (b) gating network outputs gi (-, solid line) and g2 (—, dash line).

Ceuleneer G. and Rabinowicz M. (1992) Mantle flow and melt migration beneath oceanic ridges models derived from observations in ophiolites. In Mantle Flow and Melt Generation at Mid-Ocean Ridges (eds. J. P. Morgan, D. K. Blackman, and J. M. Sinton). American Geophysical Union, Washington DC, vol. 71, pp. 123-154. 

For the sake of simplicity an alternative model will be introduced which wiU prove to be very usefiil when analyzing photobioreactor productivity. This one is based on Blackman kinetics (Blackman, 1905), and it follows the dashed lines in Fig. 12. According to this model the rate of photosynthesis increases Unearly with increasing photon flux density with the proportionality constant being a. When a saturating photon flux density of Iph,s is reached, photosynthesis reaches its maximal level and remains constant... 

In Fig. 13 the photosynthesis response according to both models (Blackman and Jassby Platt) is shown. As discussed before microalgae will... 

The variable depends on the photon flux density Iph as discussed in the previous section where the photosynthesis models of Jassby Platt and Blackman were introduced. Consequently, the specific growth rate fi can be written as a function of the photon flux density as illustrated in Fig. 16 ... 

In Fig. 18 the specific sugar production rate in the chloroplast q is plotted against the photon flux density Iph according to the model ofjassby and Platt (solid line) as well as the model of Blackman (dashed line). In the same figure also the specific photon absorption rate is plotted. The specific photon absorption increases linearly with the photon flux density according to the following relationship ... 

Below the calculation routines to calculate the average specific rate of sugar production in the chloroplast (photosynthesis) wiU be explained in more detail for the Blackman andjassby Platt models. In order to perform such calculations aU necessary biological parameters must be known for the microaigai species to be cultivated a Ts/ph,m qlm, Yk/,, m,. 

A complementary approach focuses on the kinetic aspect of biomass growth. The studies by Blackman (1905) and Monod (1949) opened the way for the description of growth using kinetic models, hi these models it is assumed that the concentration of a substrate and the affinity of the cell towards this substrate determine the rate of increase of biomass. 

It can be shown that the value of /x according to Monod kinetics approaches its asymptote too slowly to be a good representation of experimental data even in simple cases (cf. Fig. 5.18). The experimental model of Teissier (1936) and of Blackman kinetics (1905)... 

Figure 5.18. Comparison of three basic biokinetic models for substrate-limited growth in a conventional plot of /i versus s Monod equation (cf. Equ. 5.38), Blackman kinetics (cf. Equ. 5.44), and the three-constant (parameter) equation (cf. Equ. 5.47). (From Dabes et al., 1973.)...

If the constant K is very small we simply have the Monod equation, but if is very small the discontinuous function of Blackman kinetics is achieved. Model discriminations have shown that in a given confidence region of experimental error, the rival models cannot be distinguished (e.g., Boyle and Berthouex, 1974). Model functions with a minimum number of parameters are therefore preferred. 

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