Liquid Blending Proportions - Problem Setup

[Vinocalc]

Hello MHB Forum,

Glad I found this forum, have spent quite some time getting reacquainted with past math concepts. I recently came across a simple algebraic problem in my home wine making, but would like some assistance creating the problem setup for this particular problem.

I want to add distilled spirits to a liquid to increase the alcohol %, but must keep the quantity exactly 59 gallons. Using the Pearson Square method I can determine that I need 2.134 gallons to fortify to my desired level.

The ultimate question (which seems really simple, but I seem rather dense here): If 2.134 gallons is needed to fortify 59 gallons, what quantity is needed to reach the desired level without going over 59 gallons (the exact size of a storage barrel)? If I add 2.134 gallons to 59 gallons then I am at 61.134 gallons going over the 59 gallon maximum capacity. Am I thinking of this correctly?

Known Factors:
Liquid #1: 74% of solution - 17.23% Alc.
Liquid #2: 26% of solution - 16.14% Alc.
180 Proof (90% Alc.) needed: 2.134 gallons/59 gallons
Desired Final Ethanol: 19.5%

Any assistance would be awesome. Thanks again!

 

[Opalg]

Hello MHB Forum,

Glad I found this forum, have spent quite some time getting reacquainted with past math concepts. I recently came across a simple algebraic problem in my home wine making, but would like some assistance creating the problem setup for this particular problem.

I want to add distilled spirits to a liquid to increase the alcohol %, but must keep the quantity exactly 59 gallons. Using the Pearson Square method I can determine that I need 2.134 gallons to fortify to my desired level.

The ultimate question (which seems really simple, but I seem rather dense here): If 2.134 gallons is needed to fortify 59 gallons, what quantity is needed to reach the desired level without going over 59 gallons (the exact size of a storage barrel)? If I add 2.134 gallons to 59 gallons then I am at 61.134 gallons going over the 59 gallon maximum capacity. Am I thinking of this correctly?

Known Factors:
Liquid #1: 74% of solution - 17.23% Alc.
Liquid #2: 26% of solution - 16.14% Alc.
180 Proof (90% Alc.) needed: 2.134 gallons/59 gallons
Desired Final Ethanol: 19.5%

Any assistance would be awesome. Thanks again!

59 gallons??! You must drink quite a lot of wine!

Hi Vinocalc, and welcome to MHB.

To reduce a quantity from 61.134 to 59, you have to multiply it by $\dfrac{59}{61.134}$. So the quantity of fortifier needed is $\dfrac{59}{61.134}\times 2.134 \approx 2.0595$ gallons.

 

[MarkFL]

I would let $x$ be the amount of wine in the final fortified solution and we want the total amount of alcohol to be \(\displaystyle 0.195\cdot59=11.505\). Thus, we may state:

\(\displaystyle 0.74x\cdot0.1723+0.26x\cdot0.1614+(59-x)0.9=11.505\)

Solving this for $x$, we find:

\(\displaystyle x=56.9378\)

So, what you need are:

42.133972 gal. of the 17.23% wine

14.803828 gal. of the 16.14% wine

2.0622 gal. of the 90% Ethanol

I'm not sure why this differs from what Opalg posted unless perhaps some rounding was done to get 61.134. :D

 

[Opalg]

I'm not sure why this differs from what Opalg posted unless perhaps some rounding was done to get 61.134. :D

As far as I can see, the discrepancy arises because the figures given by Vinocalc do not give precisely 19.5% for the final alcoholic content. In fact, his quantities are 74% of 59, or 43.66, gallons of liquid #1; and 26% of 59, or 15.34, gallons of liquid #2. Along with 2.134 gallons of ethanol, that gives the total alcohol content $$43.66\times0.1723 + 15.34\times0.1614 + 2.134\times0.9 \approx 11.919$$ gallons, out of an overall amount 61.134. The alcohol percentage is therefore $$\frac{100\times11.919}{61.134} \approx 19.49667%$$ (very close to 19.5% but not quite exact). My simple calculation preserved this percentage. But Mark's more accurate method restored a true 19.5 percentage.It looks as though Mark's method is superior to the Pearson Square method! (which I had not heard of – an internet search shows that it seems to be mainly used by dairy farmers).

 

[Vinocalc]

Wow! Thanks much MarkFL and Opalg.

It looks as though Mark's method is superior to the Pearson Square method! (which I had not heard of – an internet search shows that it seems to be mainly used by dairy farmers).

Most documentation available to basic winemakers references Pearson's for fortification, but indeed it is used in several agriculture disciplines. But yes, will absolutely use the methodology presented by you both in place of that from now on.

Solving this for x, we find:

x=56.9378

So, what you need are:

42.133972 gal. of the 17.23% wine

14.803828 gal. of the 16.14% wine

2.0622 gal. of the 90% Ethanol

The accuracy is amazing, now the challenge is measuring a millionth of a gallon :p

Again thanks, this method will certainly be used again in the future with much better and reliable results!