Before considering what model to use you should consider what (mathematically) you are trying to optimize. You said you want "the fastest flow rate with the minimum cloudiness in the effluent water." that's two objectives to optimize. While you have two separate objectives it's impossible to say whether a very slow filter providing perfect water is better than a very fast filter that produces cloudy water (or any configuration between these two extremes).

The way in which you combine these two objectives may need a separate question entirely, a good approach could depends a lot on the specifics of what a filter system can achieve.

Once you have a a single objective to measure the filter's performance you'll have to consider how to model the performance (call it $P$) as it varies with parameters of the filter.

If you just use a linear model you may run into problems. For example, if $P$ depends on two variables: How coarse the filter is ($X$), and how many layers of filtration there are ($Y$).

From a few experiments you may get a formula like $P=-3X+4Y+7$

As expected, a less coarse filter gives better results and more layers of filtration gives better results.

If you try to optimize this for a maximum $P$ you will find that the maximum occurs at $X=-\infty$ and $Y=\infty$. A linear model like this will always give extreme values for the optimum.

Often the extreme values of parameters perform poorly in practice. To capture the fact that very small and very large values of $X$ and $Y$ give bad performance but moderate values give good performance you can use a second order model. This will give you a performance formula like this:

$P=c_0+c_1X+c_2Y+c_3X^2+c_4Y^2+C_5XY$

If you data displays a trend of moderate values of $X$ and $Y$ outperforming extreme values then this kind of performance model can help you find an optimum.

You may want to ask another question here once you know your measure of performance and the various parameters which affect it. People can give more directed advice; here I'm just aiming to show the problem with linear modeling.

Before considering what model to use you should consider what (mathematically) you are trying to optimize. You said you want "the fastest flow rate with the minimum cloudiness in the effluent water." that's two objectives to optimize. While you have two separate objectives it's impossible to say whether a very slow filter providing perfect water is better than a very fast filter that produces cloudy water (or any configuration between these two extremes).

The way in which you combine these two objectives may need a separate question entirely, a good approach could depends a lot on the specifics of what a filter system can achieve.

Once you have a single objective to measure the filter's performance you'll have to consider how to model the performance (call it $P$) as it varies with parameters of the filter.

If you just use a linear model you may run into problems. For example, if $P$ depends on two variables: How coarse the filter is ($X$), and how many layers of filtration there are ($Y$).

From a few experiments you may get a formula like $P=-3X+4Y+7$

As expected, a less coarse filter gives better results and more layers of filtration gives better results.

If you try to optimize this for a maximum $P$ you will find that the maximum occurs at $X=-\infty$ and $Y=\infty$. A linear model like this will always give extreme values for the optimum.

Often the extreme values of parameters perform poorly in practice. To capture the fact that very small and very large values of $X$ and $Y$ give bad performance but moderate values give good performance you can use a second order model. This will give you a performance formula like this:

$P=c_0+c_1X+c_2Y+c_3X^2+c_4Y^2+C_5XY$

If your data displays a trend of moderate values of $X$ and $Y$ outperforming extreme values then this kind of performance model can help you find an optimum.

You may want to ask another question here once you know your measure of performance and the various parameters which affect it. People can give more directed advice; here I'm just aiming to show the problem with linear modeling.