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### Specific Gravities in Large Volumes

```Date: 04/09/2002 at 18:05:33
From: Rob
Subject: Specific gravities in large volumes

I am trying to create a chart that could be used to help determine
what weight of methanol/water to use to either raise or lower the
specific gravity of a volume of corrosion inhibitor in bulk mixing
tanks.

We mix 20,000 gallons of these fluids a day in each tank. We would
like a simple equation that could be used that utilizes the measured
specific gravity of the fluid in the mixing tank, the known current
volume of fluid in the mixing tank, and the ideal specific gravity of
the fluid in the mixing tank to determine what volume of a second
fluid with a known specific gravity should be added to the bulk mixing
tank to achieve the ideal specific gravity in the bulk mixing tank.

Currently we are doing it by trial and error. Although I have a
chemistry degree I am not sure where to go to get this equation. I
figured since there is only one unknown in this mystical equation
there must be one equation that will solve for it.

Thanks,

Rob Gustafson
```

```
Date: 04/09/2002 at 19:22:47
From: Doctor Rick
Subject: Re: Specific gravities in large volumes

Hi, Rob.

Sounds good, let's try it! I'll start by defining variables:

t = target specific gravity of fluid
a = actual specific gravity of fluid in tank
b = specific gravity of fluid to be added
v = volume of fluid in tank
x = (unknown) volume of second fluid to be added to reach target s. g.

Specific gravity is the density of a fluid divided by the density of
water. In SI units, 1 kg/l = 1 g/ml is the density of water, so
specific gravity is essentially the density in kg/l.

What is the specific gravity of a mixture of two fluids? The mass of
v liters of fluid a (specific gravity = a) is v*a; the mass of
x liters of fluid b (specific gravity = b) is x*b. The total mass of
the mixture is (va+xb); the total volume is (v+x). Therefore the
specific gravity of the mixture is the mass divided by the volume,
(va+xb)/(v+x).

You want the specific gravity of the mixture to be t. This gives us an
equation:

(va+xb)/(v+x) = t

Solve this for x:

va + xb = (v+x)t
va + xb = vt + xt
xb - xt = vt - va
x(b-t) = v(t-a)
x = v(t-a)/(b-t)

How's that for a formula? The desired volume of the second fluid is
the volume in the tank times the ratio of the deficiency of the
specific gravity in the tank to the excess of the specific gravity of
the second fluid.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 04/10/2002 at 12:52:05
From: Rob
Subject: Specific gravities in large volumes

Doctor Rick,

I sincerely appreciate the help you have provided and the formula for
our needs. I will have the mixing guys try this out and see if it
works for actual conditions. Looking at the equation it now seems so
intuitively obvious. I guess I did not think to set it up that way
and gave up.

Thanks again,
Rob
```
Associated Topics:
College Physics
High School Physics/Chemistry

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