 In this module we will talk about the kinetics of fed batch culture as I have already told you in our previous modules that fed batch have two strategy one with variable volume and other is fixed volume. But as concerned the kinetics of variable volume so the kinetics in variable volume discussed in detail by different scientist as reported by Dunn and Moore 1975 and pert discuss in detail in 1974 and publish his research work in 1974 1975 and 1979. So we will discuss in detail with respect to the pert 1975 and 1979. So when we talk about the fed batch with variable volume just consider a batch culture in which growth is limited by the concentration of one substrate which we called as limiting substrate. So the biomass at any point with respect to time can be discussed by this equation Xt. So the biomass concentration at time t when we want to know that either addition of the substrate is needed or not then X0 is the amount of the biomass which we add in term of inoculum. So that biomass at time not mean quantity of the inoculum and why is the yield in result of fermentation either the yield in term of biomass either yield in term of product. So SR minus S so SR here represent with the initial substrate concentration so R can be denoted by the reservoir. So the amount of the substrate present in the reservoir mean in a fermenter vessel at the time of the start. So minus S so this small s represent the residual substrate so at time t. So this whole equation see the biomass concentration is equal to the biomass concentration as an inoculum plus increase at the yield in the biomass with respect to the initial substrate concentration and the amount of the substrate which remain in the vessel at the time of the fed batch strategies. So this equation suppose we assume either our strategy that we only add the substrate when there is no more because when we say there is no more availability of the substrate in the fermentation vessel then we add that. So that is the limiting substrate when we say the limiting substrate it mean that substrate is now become fully utilized by that organism so we can say that the S is equal to 0. So in that condition when we say that S become residual substrate become 0 no more availability of that substrate. So at that say we can say that maximum growth is achieved so no more increase in the specific growth rate of the fermentation vessel. So at that time we can say that maximum biomass has already been achieved. So we can write this equation that X maximum approximately equal to Y yield and SR. So if we can say that maximum biomass can be produced until and unless a substrate is present in the medium. So if we say that X is equal to X maximum a medium is fed started so the medium feed is started such that the dilution is less than mu max. So here we would be very critical. So when we start the feeding in the fermentation vessel we say that if our fed batch fermentation is going on and we start feeding our substrate when there is no more residual substrate S not as we can say that S is equal to 0. So when we add any substrate into the medium so we have to make that what quantity of the substrate in what rate we are adding mean the flow rate of the addition. So when we add something into the fermentation two parameters are very critical one is the flow rate and other is as much as the flow rate will be then that also act as the dilution of the medium. So when you add something into the medium if there are already the amount of the biomass that has already produced that become diluted and then we can say that so at the substrate consumed so we can say that the F flow rate SR is the substrate which we are adding that will be equal to the specific growth rate that will be the biomass concentration over the yield. So you can say here that F is the flow rate and X is the total biomass concentration in the culture and Xv is basically you can say that X is the total biomass concentration with respect to the total volume of the fermentation because volume is very critical. So here we are talking about different parameter with respect to the variable volume. So as the volume will be varied then the concentration of the biomass can be varied with respect to the time t. So when we say that and we are adding some substrate and our strategy is that we only add different substrate in such a way in having the flow rate that as we add different substrate that will be utilized by that organism. So we can say that ds over dt is equal to 0. So we can say that when there is no change in the substrate concentration and we are adding the substrate in such a way that will be equal to the utilization of the utilization of the we can say that by the microorganism or the organism present in the fermentation vessel. So the addition and the utilization become equal so that we can say that so when by the utilization of the substrate and then we can say that gradually the increase of the volume what happened. So during that time there will be a continuous increase in the biomass. So when we totally calculate the change in biomass concentration so the increase in the biomass with respect to the volume that remains same. So we can say that dx over dt is equal to 0 that is only possible when our flow rate which effect on the dilution of the fermentation vessel. And the dilution inside the fermentation vessel become equal with the specific growth rate which we called as mu. So if we say that mu is equal to d means specific growth rate of the organism become equal to the dilution. So then we can say that there is no more change in the biomass concentration of the fermentation vessel. So in fat batch when such conditions then that is known as quasi steady state. So in fat batch culture this is very important state.