Biomass production rate equations

Then determine the net biomass production rate rg using the equation... 

Rate Isotherms. Depending on substrate concentration, three thermal sensitivity patterns may be seen in both batch and continuous reactors when the net biomass production rate (as specified by equations (lib) and (15) and Figures 3, 4, and 6) is plotted... 

The rate Isotherms for metabolite excretion (as specified by equations (11b) and (17) and Figures 3 and 4) show the same three thermal sensitivity patterns as the net biomass production rate Isotherms. Thus, the rate of metabolite excretion may also have an optimum temperature that shifts to higher values as substrate concentration rises. 

In the present case there are seven flows, and Equ. 2.11 specifies four equations between the flows represented in the matrix of Equ. 1.8. Hence, only three flows are independent variables (cf. Sect. 1.2). Which kind of flows to be chosen for measurement depends on the possibilities for experimental determination. The knowledge, for example, of the respiratory quotient and the ratio of oxygen consumption to substrate consumption allows direct estimation of the biomass production rate and the product formation rate. This conclusion from the application of balancing is of the greatest importance in situations where process variables, for example, X, are very difficult to measure, which is the case in penicillin fermentation (Mou and Cooney, 1983). 

C02 exists under anaerobic conditions in wastewater. They also found that typically 50% of the C02 was produced by the sulfate-reducing bacteria, the other half by the fermenting biomass. However, the net production rate of Ss was typically about 70% of the total produced Ss by anaerobic hydrolysis [Equation (7.10)]. Hence, this equation may, even in a reduced form, be valuable for the estimation of the production of readily biodegradable substrate under anaerobic conditions. 

Concentration of A Arrhenius constants Arrhenius constant Constant in equation 5.82 Surface area per unit volume Parameter in equation 5.218 Cross-sectional area Concentration of B Stoichiometric constants Parameter in equation 5.218 Concentration of gas in liquid phase Saturation concentration of gas in liquid Concentration of G-mass Concentration of D-mass Dilution rate DamkOhler number Critical dilution rate for wash-out Effective diffusion coefficient Dilution rate for maximum biomass production Dilution rate for CSTF 1 Dilution rate for CSTF 2 Activation energy Enzyme concentration Concentration of active enzyme Active enzyme concentration at time t Initial active enzyme concentration Concentration of inactive enzyme Total enzyme concentration Concentration of enzyme-substrate complex with substance A... 

The oscillatory behavior of product-inhibited cultures cannot simply be described by a common inhibition term in the equation for the biomass growth. A better description must include an indirect or delayed effect of the product ethanol on the biomass growth rate as indicated in experiments. The decay rate pmaa was introduced to account for the accumulation of the inhibitory product pyruvic acid. Other more mechanistic, structured models can be formed that relate to the internal key-compound e. In these, the inhibitory action of ethanol is accounted for in the inhibition of the key-compound e formation. Mathematically, however, these two model descriptions are equivalent, except that the key-compound e is washed out as a part of the biomass in continuous cultures and the rate constant //ma55 does not vary. Our proposed indirect inhibition model provides a good qualitative description of the experimental results shown in Figure 7.25. 

Because of the complex functional interactions in lake ecosystems, the limiting factor concept needs to be applied with caution. We should distinguish between rate-determining factors (an individual nutrient, temperature, light, etc.) that determine the rate of biomass production and a limiting factor where a nutrient determines in a stoichiometric sense (equation 3) the maximum possible biomass standing crop. 

For most cell lines it is found to increase with the specific growth rate. Thus the specific rates of lactate production can be expressed as a function of p by Equation 4.3.4, which contains two parameters the non-growth-associated specific lactate production rate, mLac (mmol lactate 10 cells h ), and the lactate to biomass stoichiometric yield, (mmol lactate 10 cells). A similar expression is often applicable to the specific rate of ammonia production (Equation 4.3.6) and antibody secretion (Equation 4.3.7). 

The rates of biomass production and metabolite excretion can be shown to saturate In substrate concentration If Bg Is assumed to adjust much more slowly to changes In S than B does. So, on the time scale of changes In Bg, B 0, and from equation (8) ... 

The first term In equation (14), representing the gross rate of biomass production, Is Identical with the function Monod (25) originally adopted "to express conveniently the relation between exponential growth rate and concentration of an essential nutrient." Such a rectangular hyperbolic function has been derived many times from various reaction mechanisms (26-30). but none has addressed the present case of continuous culture systems where y j and K have been observed to vary with temperature and dilution rate. 

Maximum-Velocity Coefficients Vr, Vm. According to equations (lib) and (15c), the maximum-velocity coefficient for biomass production depends on the microscopic rate coefficients as follows ... 

At this point, it would appear that one can account for most of the observed effects of temperature and dilution rate on the macro-coefficients by simply assuming temperature dependencies such as those shown in Figure 3 for the micro-coefficients in eqtiatlons (20) - (27). These equations with the parameter values specified in Figure 3 will now be used to analyze the thermal sensitivity of the net biomass production and metabolite excretion rates. 

The shapes of the plots for substrate consumption and biomass production, as well as both the entries in the underlying spreadsheet and equations (D) and (H) of this illustration, indicate that not only is the rate of growth of the microorganism exponential but so too is the rate at which the mass of substrate present declines. 

Finally, the determination of the mean growth rate allows the mass balance equation, here for hiomass, to be solved (Eq. 11). The variable PFD in sunhght conditions means that the irradiance field inside the culture bulk and the resulting local and mean volumetric growth rates vary continuously, and hence steady state carmot he assumed in Eq. (11). This implies solving the transient form of the mass balance equation. Once the time course ofbiomass concentration has been determined, the corresponding biomass productivity can be calculated, as well as surface productivity P (g m day ) which is a useful variable to extrapolate to land-area production (Eq. 2). 

X represents the biomass, P the product, and 5 the glucose (or substrate) concentration (kg/m ). For a clear insight into the first equation in particular it is useful to remember that p is the specific biomass growth rate in h (see Section... 

L-Phenylalanine. L-phenylalanine is produced via fermentation using a mutant Brevibacterium lactofermentum 2256 (ATCC No. 13869) known as No. 123 [2]. The rate equations for biomass (bacteria, X), substrate (mainly glucose, S), and product (L-phenylalanine, P) are described by Monod kinetics. 

IJA is the rate of decomposition of detritus in environment A kA is the kinematical coefficient of vertical diffusion is the velocity of nutrient assimilation by the photosynthetic process per unit of phytoplankton production ef is the proportional part of the eth radionuclide that is chemically analogous to B6 A on substrate A H is the rate of input flow of the eth radionuclide 7) is the rate of exchange with the environment p is that part of biomass losses due to exchange that transforms into nutrients (Legendre and Legendre, 1998) and f3v is upwelling velocity. Equation (6.1) is the basic element of block NM. 

Although the yields and total growth rate are useful parameters, it is the state variables like the concentrations of biomass, substrate and product, and the culture parameters like the specific rates of growth, substrate uptake, etc., that provide a complete description of the bioreactor. One attempt in estimating these variables from R and the yields consisted of integrating the governing differential equation with known initial conditions and the measured values for R and the yields (9). For a batch reactor, for example, b was estimated by integrating... 

Figure 15.6. Photosynthesis and respiration, (a) A well-balanced ecosystem may be characterized by a stationary state between photosynthetic production, P (rate of production of organic material) and heterotrophic respiration, R (rate of destruction of organic matter). Photosynthetic functions and respiratory functions may become vertically segregated in a lake or in the sea. In the surface waters the nutrients become exhausted by photosynthesis, (b) The subsequent destruction (respiration) of organism-produced particles after settling leads to enrichment of the deeper water layers with these nutrient elements and a depletion of dissolved oxygen. The relative compositional constancy of the aquatic biomass and the uptake (P) and release (R) of nutritional elements in relatively constant proportions (see equation 3) are responsible for a co-variance of carbon, nitrate, and phosphate in lakes (during stagnation period) and in the ocean an increase in the concentration of these elements is accompanied by a decrease in dissolved oxygen, (c, d) The constant proportions AC/AN/AP/AO2 typically observed in these waters are caused by the stoichiometry of the P-R processes.

Rating index for liquefaction is given in equation (2) which is defined as product of liquid yield and higher heating value of pyrolysis liquids divided by mole of hydrogen required per mole of carbon in biomass liquids for upgrading to premium fuel. 

Using the aforesaid procedure, the RIs for the other biomass (whose product properties are not experimentally studied) have been calculated. The RI values obtained are normalised to obtain NRI as per equation 5 and are presented in Table 2. The values of the maximum rate of devolatilisation is obtained from the actual TGA curves. (These may also be calculated with a high level of accuracy using the chemical analysis of biomass as described in steps (ii) and (iii) in Section 3). The last column indicates the remarks regarding their utilisation or end-product based on the ranking systems suggested. 

A comprehensive model of a membrane bioreactor has been developed by Moueddeb et al [5.103] for a simple irreversible reaction A B. The goal of the model was to describe their experimental reactor system, which was described earlier in Chapter 4. The model equations were established by taking into account the effect of the biomass on the permeate flow rate in the annular volume. The mass balance equations for the substrate (A) and the product (B) in cylindrical coordinates, utilized by Moueddeb et al [5.103] are given as ... 

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