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To get the proper lime dosage it is necessary to know the initial total carbonate (CT) in the system. e.g CT, from the initial Alkalinity and pH of Marcala and Cuatro Comunidades. We can find the CT based on equation 1, and because we assume this is a closed system, the CT will not change during the process, so that the relationship between Alkalinity and pH can be also measured with equation 1:
equation 1:

{latex}
\large
$$
ANC = C_T (\alpha _1 + 2\alpha _2 ) + OH^ - - H^ +
$$
{latex}

In the next step, considering the flow rate of the plant will also change, the team would like to use the ratio between the lime feeder flow rate and the plant flow rate to get a more practical function

{latex}
\large
$$Ratio = {Q_{feed}}/{Q_{plant}}$$
{latex}

When this flow ratio changes, lime concentration will also change, and we can acquire the relationship between this flow ratio and the concentration of hydroxide ion in raw water from this mass balance equation,
equation 2:

{latex}
\large
$$
OH_{Balance}^ - = {\textstyle{{[OH^ - ]_{added} Q_{feed} + [OH^ - ](Q_{Plant} - Q_{feed} )} \over {Q_{feed} + Q_{Plant} }}}
$$
{latex}

The relationship between OH- concentration and alkalinity can be calculated with equation 3:
equation 3

{latex}
\large
$$
ANC_{final} = ANC_{initial(Carbonate)} + OH_{feed(Lime)}^ - - H_{feed(Alum)}^ +
$$
{latex}

The premise of using the above equation is that we have to know the equivalent of proton created by adding aluminum sulfate, which can be measured from the difference of initial alkalinity and final alkalinity from our data.     
Based on this data and these equations  we can make a model showing the change of pH and alkalinity as the function of flow ratio between lime feeder and plant. The model which follow these relationships in Marcala and Cuatro Comunidades are shown in figure 1: 

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