# Mixing Water: Some Vital Math to Assist You Combine Waters of Completely different TDS

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### Some Vital Math to Assist You Combine Waters of Completely different TDS

Tasting Room Director Sandra Elisa Loofbourow and I just lately ran right into a troublesome downside to unravel on the Crown, and wished to share among the work it took to determine how one can combine two completely different waters to realize a desired focus and quantity.

Our downside began with needing a selected water formulation for an occasion.

We have been utilizing Third Wave Water which produces roughly 150ppm hardness when used with distilled water, and our faucet water usually falls proper round 60ppm. We wished to keep away from losing cash and plastic by buying distilled water, but nonetheless have *roughly* calibrated H2O to the supposed goal TDS and get a complete quantity of 5 gallons.

Vital Caveat: This won’t deal with the precise mineral content material. That matter is roofed intimately on this two half collection from 2018: Treading Water: A Sensible H2O Information for Espresso Professionals Half I Chemistry & Conditioning & Half II Sensory Knowledge. We’re not going to talk to that at present, we’re solely involved with the full mineral focus.

With some assist from Kathy Kornman, a physics and math trainer who can be my mom (Hello Mother!), we settled on a set of formulation to make a quantity of water at a desired focus. • g = Objective Water
• h = Excessive TDS water
• l = Low TDS water
• V = Quantity. Let’s use gallons at present.
• C = Focus. We’re utilizing elements per million (ppm) at present, however you should utilize a proportion or one other focus unit

Now, Let’s Focus

There are two methods to method this downside, and for each you’ll must know upfront the concentrations of your excessive and low water, in addition to your goal TDS (which should be in between the 2).

You’ll additionally need to know both a quantity purpose (Technique 1) or a quantity of a minimum of one of many elements (Technique 2). There’s a fast further formulation in Technique 2 that permits so that you can scale up your proportions simply.

## Technique 1 – Making a Desired Quantity & Focus

If you know the way a lot whole water you want, you may create it utilizing two related equations.

Low TDS water quantity will be discovered utilizing this formulation:
lV = gv * [(hC – gC ) ⁄ (hC – lC)]

Excessive TDS water quantity will be discovered utilizing this formulation:
hV = gv * [(gC – lC ) ⁄ (hC – lC)]

In our downside on the Crown, we knew we needed to attain 150 TDS (our gC) at 5 gallons (gv). We had 60 TDS faucet water (lC) and we knew we may make Third Wave Water with our faucet water: 150 TDS + 60 TDS in 1 gallon increments = 210 TDS (hC).

Low TDS quantity = 5 gallons * [(210 ppm -150 ppm)] / (210ppm – 60ppm)]
lV = 5 * (60/150)
lV = 2 gallons

Excessive TDS quantity = 5 gallons * [(150ppm – 60ppm) / (210ppm – 60ppm)]
hV = 5 * (90/150)
hV = Three gallons

So, to get 5 gallons of water at 150 TDS we mix 2 gallons of 60 TDS water with Three gallons of 210 TDS water. Or extra merely, use Three packets of Third Wave Water in 5 gallons of 60 TDS water.

## Technique 2 – Making a Desired Focus Utilizing a Identified Element Quantity

Approaching this downside from a unique angle, we are able to nonetheless remedy for an unknown element quantity.

The factor about these Third Wave Water particular person packets is that they’re supposed for 1 gallon of distilled water. If we’d blended 1 gallon (our hV) of our 60 TDS faucet water with 1 Third Wave Water Packet upfront, then we’d must dilute it utilizing an unknown amount of that very same faucet water. We have to remedy for lV.

This formulation is written within the type of an equation, the place each side are equal to one another. On the left facet, we’ll add the product of our two waters’ quantity and focus. On the appropriate, we’ll multiply the sum volumes of the 2 waters by our goal TDS, our gC.

hV * hC + lV * lC = gC(hV + lV)

Basically, what this says is that the volumes and concentrations of the 2 elements on the left, our elements, will equal the ultimate quantity and focus on the appropriate, our goal. This equation is definitely the start line for Technique 1, however we’ll deal with that later. Proper now, let’s plug in our numbers for Technique 2.

Simply to recap, we now have 1 gallon (hV) of 210 ppm (hC) water. We have to make 150 ppm (gC) water utilizing an unknown quantity (lV) of 60 ppm water.

hV gallons * hC ppm + lV gallons * lC ppm = gC ppm (hV gallons + lV gallons)
1 gallon * 210ppm + lV gallons * 60ppm = 150ppm * (1gallon + lV gallons)

The one unknown is the amount of low focus water. It’s our x. Let’s remedy for it. I’m going to take away the models for now, however keep in mind that we’re fixing for quantity in gallons.

Utilizing the distributive property, we all know that a*( b + c ) = (a * b) + ( a * c), so we are able to multiply that on the appropriate facet of the equation.

1 * 210 + lV * 60 = 150 * (1 + lV)
210 + 60x = 150 + 150lV

Now we simply must isolate lV.

210-150 = 150lV – 60lV
60 = 90lV
0.667 gallons = lV

So for each gallon of 200ppm water, we are able to add two thirds of a gallon of 60ppm to dilute it to 150ppm.

## Scaling Up Technique 2 to a Desired Quantity

For this explicit state of affairs, we wanted 5 whole gallons. To develop our fraction into the complete quantity, we’d like just a little extra math. We all know our whole yield with this formulation is 1.667 gallons (the sum of the volumes of 1 gallon of excessive and 0.667 gallons of low water). Dividing our goal quantity 5 gallons by our yield of 1.667 gallons = 3 (basically, what number of 1.667 gallon models match into 5 gallons).

So our models (u) will be discovered by dividing our purpose quantity by the sum of our excessive and low volumes.

gV / (hv + lV) = u
5 / (1 + 0.667) = 3

Now we take our models (3) and simply multiply them by the excessive and low volumes.

u * hv + u * lV = gV
3 * 1 gallon of excessive + 3 * 0.667 gallons of low water = 5 gallons at 150ppm.
Three gallons + 2 gallons = 5 gallons at 150ppm. Only for kicks, let’s strive a unique one as an example how having distilled water readily available could make this equation enterprise quite a bit easier.

Let’s say we now have tremendous onerous water, 666ppm (yikes!). We need to get again down to shut to SCA specs, at 150, and we’d like 7 gallons whole quantity. How a lot distilled water (Zero ppm) ought to we add?

### Technique 1

Low TDS water quantity:
lV = gv * [(hC – gC ) ⁄ (hC – lC)]
lV = 7 gallons * [(666 – 150)/(666-0)]
lV = 7 * (516/666)
lV = 5.423 gallons of distilled water

Excessive TDS water quantity:
hV = gv * [(gC – lC ) ⁄ (hC – lC)]
hV = 7 gallons * (150 – 0)/(666-0)
hV = 7 * (150)/(666)
hV = 1.576 gallons of water at 666 ppm

### Technique 2

hV * hC + lV * lC = gC(hV + lV )
666ppm * 1 gallon + 0pmm * ?gallons = 150 ppm * (1gallon + ?gallons)

See the place that is going? That 0ppm is performing some heavy lifting on the left facet, eliminating one among our x components. Let’s math this out actual fast.

666 = 150 + 150lV
516 = 150lV
3.44 = lV

So, 3.44 gallons of distilled water added to 1 gallon of our excessive TDS faucet will give us our desired focus. Now what number of models will get us to 7 gallons?

gV / (hv + lV) = u
7 / (1+3.44) = u
1.576 = u

So we’ll want 1.576 gallons at excessive TDS plus (3.44 * 1.567) = 5.423 gallons of distilled water to get 7 whole gallons. Should you’re something like me, you’ll need to understand how this explicit sausage is being made.

The important thing to unlocking these formulation is the substitution precept, which permits us to usefully change an unknown variable with the perform of a recognized one. Once more, I’m grateful to Kathy Kornman right here for breaking the steps into digestible parts for me.

The start line is an easy equation primarily based on what we all know.

Chemistry tells us that the focus of an answer is its solute’s mass divided by the liquid quantity: C=m/V. Subsequently a solute’s mass is the same as its quantity multiplied by its focus. m=C*V

We additionally know that the mass of our two elements, when added collectively, equals the ultimate mass.
hm + lm = gm

Now, measuring the mass of our solute (little ions of onerous minerals like calcium and magnesium) might be not probably the most environment friendly strategy to dilute an answer. So we substitute. If m=C*V, then we simply drop in our focus and quantity figures and

hm + lm = gm
turns into
(hc * hv) + (lc * lv) = gc * gv

Focus and quantity — these are numbers we are able to work with. Focus in elements per million and quantity in gallons, or no matter models you like.

Technique two is straightforward to get to from right here, utilizing substitution. We all know that our element volumes, when added collectively, equal our goal quantity. hv + lv = gv

If we don’t want to fret about last quantity, simply focus, we are able to drop this in as an alternative, and

(hc * hv) + (lc * lv) = gc * gv
turns into Technique 2:
hV * hC + lV * lC = gC * (hV + lV)

It takes just a little extra creativity to get to Technique 1, however the precept is similar. If we all know every part besides our two element volumes, we are able to nonetheless work them out individually as a result of hv + lv = gv

Subsequently
hv = gv – lv
And
lv = gv – hv

Because of this, we are able to substitute one or the opposite to isolate a single variable and remedy for it. Such that, if we’re on the lookout for lv

(hc * hv) + (lc * lv) = gc * gv
Turns into
(hc * (gv – lv)) + (lc * lv) = gc * gv
Or if on the lookout for hv it turns into
(hc * hv) + (lc * (gv – hv)) = gc * gv

If you work these two formulation right down to isolate the high and low water volumes, that’s Technique 1. I’ve the proof, as so many good ones are, scribbled on the again of serviette.

I received’t bore you with it. Should you made it this far, you’ve in all probability already bitten off greater than you bargained for! As Sandra requested me the opposite day, “Did you ever suppose you’d use a lot math in your espresso profession?”

No, I definitely didn’t.