 In this module, we will talk about some basic principles of the growth kinetics inside the continuous culture. We have already discussed the different aspects of continuous culture with respect to the dilution and flow rate etcetera, but here we will discuss in detail that how it happen. So, continuous culture or a continuous cultivation at steady state is possible only when all factors which we called as contributing factors to the accumulation of the biomass are exactly balanced by all factors which we called as contributing factors. So, this equation in which is cell added to the system, cell removed from the system, cell produced through the growth and cell consumed in result of death. So, when we say that cell added to the system, equation we can say that f x naught over v. So, here f is the flow rate and x naught is the amount of the biomass which we add as in oculum. So, v is the volume of the medium inside the fermentor. We can say that when we start the fermentation, that volume of the fermentor at the time of in oculum addition. So, then there is a f x, f is the flow rate and x is the biomass concentration when the system is on and when we check that. So, v is again the volume and then is a mu x, mu x is the which we called as dx over dt. The change in biomass concentration with respect to time and the A x, A x referred to that when there is a fermentation process that there are different stages. Some cells have achieved their last stage which we called as death stage. And cells become dead due to some toxic secretions. So, if we have this equation which we called as equation 3. So, these are the different 4 situations. What is the biomass which we are adding as in term of in oculum and then the actual biomass concentration and then the specific growth rate and then the death rate. So, that is basically the input inside the fermentation vessel while we say that there is a continuous culture. So, these different aspects we have in our mind. So, we have here that when we say that dx over dt is equal to 0, then we called as when this dx over dt is equal to 0, we referred in our previous module this kind of state is a steady state. So, in a previous slide you have seen that equation which we called as the various situation inside the continuous culture. So, according to that if we assume that the death as compared to the growth is very less. Here we see that A which is the death rate is very very less as compared to the growth rate. So, then we can say and then we can assume that Ax the total number of the death of the cells is very less we can ignore it. So, if we say that x naught is equal to 0 mean the amount of the biomass added as an inoculum that mostly considered 0 negligible amount as compared to the total biomass concentration inside the fermentation vessel. So, then we can say that so that condition that can also be ignored. So, hence when we say that by this equation then we can have a fourth equation we can say that mu specific growth rate is equal to flow over volume which has become equal to D. So, as we have already discussed that in a steady state that is only when mu is equal to D but when we say that mu is equal to D we are ignoring the biomass concentration as an inoculum and the death rate and the death of the cells happening during the fermentation conditions. So, the specific growth rate of the population within the continuous fermenter is determined by the dilution rate as we have discussed in detail in a previous module. So, the chemo state operated with the sterile field so when we say that we are feeding something and there is no more addition of the substrate out when we are adding something from the outside and we are feeding some substrate it is assumed that should be the sterile mean we are not having more addition of the biomass. So then we say that x is x naught is equal to 0 then we say that the feed or the substrate and the dilution which we have adding that is a very in a sterile conditions. So, in such case we can say that mu is equal to 0 that is very controlling factor of that but if we are the feed is not sometime having sterilized then that can disturb the whole continuous culture. So, when we say that limiting nutrient balance for the chemo state conditions so as we have seen that when we are talking about the basic principles of that so when we say that the limiting nutrient balance for a chemo state so we can see here in a previous slide we see that that is the situation when we are only focusing the biomass concentration. But if we also focused on the substrate concentration during the continuous culture so the input minus output and then minus consumed so what we have added into that and what we have harvest that and then some amount of the substrate during the fermentation period will be consumed by the organism so that is then there is the overall accumulation of that substrate. So this can be expressed by different letters that ds naught minus ds minus mu max over yield coefficient is equal to ds over dt the overall change in substrate concentration divided by the time that is the overall change so we can as we have already seen that d is basically equal to flow rate over volume and then s naught and s are the substrate concentration and then x is the dry-sul mass and y x over s is the yield coefficient with respect to the substrate so if this equation 5 we can say that if the product formation other than the cell so our product is not the cell then we can say in a chemostat study state the equation 5 can be transformed into d is equal to s naught minus s because equation 5 when our product is associated with the biomass so this equation equation 6 is that equation when we say that that the product is not associated with the biomass so d is equal to s naught minus s is equal to mu max over y over x s so if we substitute the this equation 6 when we have the substitution of the equation 4 into this equation so then we have the equation like this x is equal to y x over s into x naught minus x because we have just put the value of equation 4 into this equation when we say that there is a growth yield that is basically dependent over the limiting nutrient concentration then by rearranging the previous equation we have this equation which is basically simple equation of the rearrangement and reshaping of the monod equation so we say that if we rearrange that previous equation by just having equation 7 to 8 then we have this equation that is basically the reshaping of the previous equation so when we having more reshaping of 7 and 8 equation after all we have this last equation which we called as x is equal to yield coefficient and s naught and that so by this if we know the critical delusion rate and the chaos value then we can determine the biomass concentration in continuous culture.