Activated Sludge : Wasting from F/M ratio

As you may know, I am trying to collect EVERY WATER AND WASTEWATER TREATMENT PLANT OPERATIONS MATH PROBLEM THERE IS. It is my goal to collect them, and publish them with worked solutions and answers. Then, when we take certification or employment exams, we will NEVER be surprised by never having seen a particular type of problem! (I am receiving some, but I would love to have more!!!) This month's Operator's Notebook originated from a question from a Wastewater Treatment Plant Operator in North Carolina, who asked along with his submission: "I was told that one could calculate an activated sludge wasting rate from the (following) formula. How can that possibly be?"

(This is an excellent question, and a perfect example of the type of questions I need to have all of you send in!)

The formula he passed along is as follows:

I will answer his question by providing an example, using our wastewater treatment plant's data here in Northern California. (No I have not yet retired from the "day job" of being the water and wastewater treatment superintendent!)

Question: For an activated sludge WWTP, determine the Activated Sludge wasting rate in gal/min units, given the following information:

Plant process flow rate: 8.2 MGD
Pri Effluent BOD: 85 mg/L
Mixed Liquor Per Cent volatile (MLVSS/MLSS): 77%
MLSS Concentration: 3,500 mg/L
WAS concentration: 7,200 mg/L
2.5 MG aeration basin (AB) capacity
Desired F/M ratio = 0.11

We will first find the number of pounds per day of BOD entering the activated sludge aeration basins:

Next we calculate the number of pounds of Mixed Liquor Volatile Suspended Solids in the aeration basins:

We then get to find the number of pounds of MLVSS to waste. We will insert his formula here at this time:

I will expand "his formula" and then show you how and why it works, and then return to our sample problem. Here we go:

BUT HOW DID WE GET TO THAT FORMULA which is supposed to give us the amount of MLVSS to waste? Thats really "the question." Lets logically derive the formula. If we want to have a specified number of pounds of MLVSS, then any amount above or below that value will have to be modified to make our desired value occur in the aeration basins. Therefore, if we setup a formula to find the number of pounds of MLVSS that we need to waste, we would start with the amount we desired to have. In our case we will call this "lb MLVSS desired." From this value we will subtract the amount we DO NOT want to have added to our desired amount. (Obviously, this will happen, as the BOD comes into the secondary activated sludge process and is converted into microbes!) So we will call these new, and additional pounds of MLVSS "Waste lbs, MLVSS." We will now start to setup a formula with these two values:

We now have on the left hand side the pounds that we will be wasting, and on the right side of the equal sign we have the desired number of pounds of MLVSS we want to have in the aeration basins. On the right hand side of the equal sign it stands to reason, if we have exactly what we want for a target amount in the aeration basins, we will NOT waste solids from the system. The next case says that we are "creating MLVSS", and we must waste some. If we then reason that we must therefore subtract the "additional pounds" we are creating from our base line target value, we will have the waste pounds of MLVSS on the right side of the equal sign and equal to those we wish to waste on the left side! How do we do that? Here is where that ratio comes into play! All we need to do is to take the number of pounds of incoming BOD/day, and divide it by the ratio that we wish to hold, and presto, we have a value to subtract! As we do this, please focus on the denominator; the "Desired F/M." The key is in the units.... (my students hear this probably all to often, which is my recognition of being a nag about it in class!!!) The numerator of this is the number of pounds of BOD entering the aeration basins. The denominator is the number of pounds of MLVSS that are in the aeration basins. The F/M ratio value is 0.11. Of course this means that we want to have 0.11 pounds of BOD for every pound of MLVSS in the aeration basins. THAT IS THE KEY CONCEPT! If we indeed have 0.11 lbs BOD for every pound of MLVSS, then it stands to reason we can EXPAND on that ratio.

For now, lets place that into our formula on the right side:

It is not readily apparent HOW and WHY this works until I "un-stack" the units in the F/M ratio. (As I said, focus on the denominator.)So lets do that now. Our first step is to bring out from below the lb MLVSS. Just like we do with fractions, we "invert and multiply" to get:

Next step is to bring out "the next up," the 0.11 lb BOD/day, so we invert and multiply it to get:

We may now cancel out our units:

This is simplified to be:

This means then in text:

This is how we end up being able to waste pounds of MLVSS from what appears to be a "lb of BOD thing!"

Now back to our example with numbers! We will now insert the number of pounds of BOD entering the aeration basins, and the number of pounds we desire in the aeration basins to get:

We do the math to get:

We must pay attention here! We have just calculated the amount of MLVSS that we need to waste, but our work is not yet done. We must first convert these pounds into their MLSS counter-part, so we can then place the number of pounds of MLSS into a pounds formula and get the wasting flow rate. Here we go:

We now have the number of pounds of MLSS that we need to waste. We now go to the ole' pounds formula, and insert our information:

We now solve for the wasting rate in MGD units:

We now convert the MGD wasting rate into gal/min units:

Our answer is to waste at 50 gal/min. That is very close to the actual value we waste at. Remember, this math just puts you close to the target value. Your facility may need to do some adjusting for all of the variables that there are!

May I now offer my "in public thanks" to Anthony for his great question? Thanks Anthony!


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