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Degrees Plato are used in the brewing industry to express the concentration of extract (dissolved solids, mostly sugars) in a wort or beer as a percentage by weight. Thus 100 grams of a 12 degree Plato (abbreviated 12 °P) wort contains 12 grams of extract.

[edit] Plato Table; Calculation of °P from Specific Gravity

In 1900 the Kaiserliche Normal-Eichungskomission, under the leadership of Fritz Plato, measured the specific gravity, to six decimal places, of pure sucrose solutions of known strength by weight and published tables of these values. These tables are more accurate than earlier, similar tables published first by Bohemian scientist Karl Balling (specific gravities determined to 3 decimal places) and later by German Adolf Brix. The original Plato table lists the density of wort at 20 °C normalized by the density of water at 4°C and the entries thus represent true specific gravities. In a modern table, exemplified by the one published by the ASBC, the data from the Plato table have been mathematically adjusted for the effects of air (buoyancy) on the instruments used to measure specific gravity (hydrometers and pycnometers) and normalized to the density of water at 20 °C. Tabulated specific gravities are, thus, apparent specific gravities and usually written with the annotation (20°C/20°C) to indicate that both sample and water reference temperatures are 20 °C.^[1]. Similar tables are in use by the EBC in Europe. As the Balling, Brix and Plato tables differ but slightly in the values of specific gravity reported for a given solution strength (grams sucrose per 100 grams solution) they are more or less interchangeable and Brix units are, for example, the standard in the fruit juice, soft drink, sugar and wine making industries.

To determine the extract content of wort a brewer measures the (20°C/20°C) apparent specific gravity of the wort using a hydrometer, pycnometer or oscillating U-tube electronic meter and then enters the ASBC or EBC tables with the value determined taking out the corresponding °P. The ASBC table can be fit by the third order polynomial

P = - 616.868 + 1111.14(S G) - 630.272(S G) 2 + 135.997(S G) 3

and as this is published by ASBC in their Methods of Analysis^[2], carries the same authority as the table itself. Note that electronic instruments will have this polynomial built in so that they read directly in °P.

Many brewers, particularly home brewers, like to think in terms of "points" defined as

$p_t = 1000 \times (SG-1)$

so that, for example, an SG of 1.040 would be equivalent to 40 points. Substituting $S G = (p t / 1000) + 1$ into the ASBC formula, expanding the binomial terms and summing the coefficients of like powers of $p t$ gives the equivalent of the ASBC polynomial in terms of points

$P = -0.003 + 0.258587p_t - 0.00022281p_t^2 + 0.000000135997p_t^3$

This is entirely equivalent to the ASBC formula and will compute exactly the same answer for a given specific gravity for those who prefer to work in points. Of equal significance is that there is apparently a near linear relationship between points and degrees Plato with the proportionality factor being close to 1/4 for the range of specific gravities of most interest. To obtain Plato from points, therefore, most home brewers simply divide points by 4. For SG 1.040 this yields 10 °P whereas use of the full formula or the ASBC table would give 9.993 °P. For SG 1.080 this very simple conversion yields 20 °P whereas the full formula gives 19.328°P.

A second order fit to the ASBC tabulated data yields, in factored form, the simpler (relative to the ASBC polynomial) formula (referred to as the Lincoln Equation^[3]):

P = (463 - 205 * S G) * (S G - 1)

At SG 1.040 this formula yields 9.992 °P and at 1.080 it gives 19.328 °P so it is clearly adequate for most purposes.

In calculating °P from specific gravity measurement the brewer is actually determining the amount of pure sucrose which would be found in an aqueous solution of the same specific gravity as his sample. While the typical wort contains only a small percentage of sucrose other sugars (maltose, fructose, glucose etc.) behave remarkably similarly to sucrose with respect to the specific gravity of their aqueous solutions so the extract found by this method is an accurate estimate of the extract in the wort except in cases where the brewing liquor is high in dissolved minerals.

Because of the 1 to 1 correspondence of specific gravity and extract content a conventional hydrometer can be marked in units of °P (or Brix or Balling). As the corrections elucidated by the Eichungskommission were in the fifth and sixth decimal places with respect to the Brix measurements such markings on a typical immersion hydrometer could represent either Brix or Plato. Many hydrometers sold today are marked in Brix/Plato and this leads to the common misconception that °P is another "scale" for specific gravity. It is not. It is representative of the mass of extract in a given mass of solution.

Sucrose solution strength (w/w) can also be measured by a suitably calibrated refractometer. As these are primarily used in the fruit juice, wine, sugar and similar industries they are usually calibrated in Brix. These instruments are sometimes used by brewers. The Brix reading can be considered to be the same as Plato readings which is a good assumption for all but the most precise work. To make calculations such as the ones in several of the examples which follow in the Applications section one must obtain the specific gravity number corresponding to the measured Brix value. This can be done by using the ASBC (or EBC) tables or by using one of the conversion methods under Calculation of Specific Gravity Equivalent to °P.

Consumers on the European continent are more familiar with Degrees Plato than in the US. They are frequently listed on German beer labels as the Stammwürze[1]. Slovak and Czech breweries also label their beers with the original gravity in °P though they will often print simply "12%" for a 12°P beer.

[edit] Calculation of Specific Gravity Equivalent to °P

Calculation of °P from specific gravity was described above. Going in the other direction is more difficult if highest accuracy is sought because the proper root of a third order polynomial must be found. There are closed form solutions. See the article cubic function where the proper root is the one designated $x 1$ , but, as is plain from this article, the closed form is so cumbersome that iterative root finding techniques (e.g. root bisection, Newton's method) are often simpler to code. For most purposes an approximate inverse^[4] of the Lincoln Equation is adequate

$SG = 1 + {P \over (258.6 - 0.8796P)}$

Note that this is not an exact inverse of the forward Lincoln Equation. The exact inverse is:

$SG = {668 - \sqrt{668^2 - 820(463+P)} \over 410}$

For approximate conversion multiply °P by 4, divide by 1000 and add 1. For example, 10°P yields SG 1.040.

[edit] Applications

Degrees Plato are of value wherever the actual mass of extract is of interest. Several examples follow.

[edit] Relationship between Plato and Extract Mass per Volume of Wort

Given that the mass of extract in a volume of wort is $p = V \times \rho_{wort} \times P_{wort}/100 = V \times SG_{wort} \times \rho_{water} \times P_{wort}/100$ the amount of extract per volume of wort is

E k g / h L = S G w o r t ρ w a t e r P w o r t

which is the relationship between Plato (extract concentration on a mass basis) and the extract concentration on a volume basis.

If $E k g / h L$ be computed from this formula for each tablulated value in the ASBC table (SG vs. °P ) and the resulting data fit against °P a second order polynomial which agrees with the exact data to better than 5 mg/hL, which is more than adequate for any practical purpose, is obtained:

E k g / h L = 0.0061289 + 0.99464 P + 0.0042888 P 2

In English units

E l b / g a l = 0.00051148 + 0.083001 P + 0.00035792 P 2

These are, of course, easily invertible.

[edit] Efficiency Calculations

A comparison between the mass of extract produced and the maximum mass of extract the malt is able to produce is indicative of the efficiency of the malting, milling, mashing and lautering processes. To illustrate suppose that a brewer has a hectoliter (100L) of wort cooled to 20°C for which he has measured a specific gravity of 1.040. This volume of wort will weigh $\rho_{water} \times 100L \times 1.040 = 103.813$ kg where $ρ w a t e r = 0.998203$ kg/L is the density of water at 20 °C on the ITS-90 temperature scale^[5]. Use of the ASBC tables indicates that wort of specific gravity 1.040 contains 9.993 grams of extract per 100 grams of wort (9.993% w/w) so this wort contains $\rho_{water} \times 100L \times 1.040 \times 0.09993 = 10.37$ kg extract. If the wort were prepared with 15 kg of malt the brewer would calculate his overall system efficiency as $100 \times 10.37/15 = 69.1%$ . Brewers often compare the performance of their brewhouses to the maximum that malt from a particular lot should be able to yield as reported in the producer's (or other party's) laboratory evaluation of the malt. For purposes of efficiency calculation the maximum extract is presumed to be the Dry Basis Fine Grind (DBFG) extract attained when the malt is finely ground and subjected to a mashing procedure as specified by the ASBC^[6]. An example number (for Pilsner malt) would be 81%^[7] of the weight of the grain on a dry basis i.e. adjusted downward by the moisture content percentage which is also reported by the maltster or determined by the breweries lab. Assuming a DBFG rating of 81% and a moisture content of 3% a brewer would calculate his brewhouse efficiency as $100 \times 10.37 /(0.81\times (1.00 - 0.03) \times 15) = 87.9%$ on a dry basis. The "as is" brewhouse efficiency, $100 \times 10.37 /(0.81\times 15) = 85.3%$ , does not consider the moisture content. Note that in order to make these calculations the brewer must have both the SG and Plato numbers.

Home brewers use a simpler approach. They assume that a particular malt has a maximum yield of a certain number of points when one pound of that malt is mashed in 1 gallon of water under optimum conditions. If one pound of sucrose were added to enough water to make up to 1 gallon of wort that wort would have an extract of 11.474 °P (found by setting $E l b / g a l = 1$ and finding the corresponding °P). The associated specific gravity is (from the ASBC table) 1.04617. If one pound (dry basis i.e. corrected for moisture content) of the Pilsner malt used above as an example were mashed under the same conditions it would be expected to produce 0.81 lb of extract which, in a wort volume of 1 gallon, would give 9.374 °P corresponding to specific gravity 1.03744. "As is" (not corrected for moisture content) a pound of the example malt at 3% moisture content mashed to produce 1 gallon of wort would be expected to produce 3% less extract or 0.786 pounds. This much extract gives wort of 9.106 °P with specific gravity 1.03634. Home brewers would say that sucrose has a potential extract of 46.17 points per pound per gallon (ppppg) whereas the Pilsner malt has a potential extract of 37.44 ppppg on dry basis and 36.34 ppppg as is (at 3% moisture content). Rather than do calculations based on the malt analysis or worry about moisture content most home brewers would consult malt tables in a home brewing text and find, for example, that Pilsner malt yields 37 ppppg.^[8]

Home brewers base their calculations on the assumption that points represent a measure of extract (which they do fairly well - as noted above the relationship between points and extract is nearly linear) and so a home brewer who had mashed 15 kg (33.1 lb) of the example Pilsner malt and obtained 100 L (26.4 gal) of 1.040 wort would expect to obtain $37 \times 33.1 /26.4 = 46.4$ points under ideal conditions. Given that he obtained 40 (SG 1.040) he would consider his efficiency to be $100 \times 40/46.4 = 86.2%$ . Discrepancies between efficiency values computed this way and extract based efficiencies are small and come from the facts that tabulated numbers can only be representative of the actual malts being used (both with respect to DBFG yield and moisture content) and that the relationship between points and extract is not exactly linear.

[edit] Estimation of Alcohol Content

Another common application is where it is desired to have an estimate of the alcohol content of a fermented beer and the instruments to measure it directly are not at hand. If the analyst knows the true extract of the beer (the number of grams of extract in 100 grams of beer) and the original extract of the wort which was fermented to produce it he can calculate the weight of alcohol in 100 grams of the beer by solving Balling's equation^[9] for $A w$ ,the alcohol by weight (grams alcohol per 100 grams beer) :

$A_w = {(P - n) \over (2.0665 - 1.0665P/100)}$

Here $P$ is the °P determined for the wort prior to fermentation (original extract) and $n$ the true extract of the beer. Note that true extract is the extract remaining in the beer after fermentation and cannot be determined from a specific gravity measurement of the beer because the presence of alcohol (which is appreciably less dense than water) lowers the specific gravity of the beer. In order to obtain true extract the analyst measures out a volume, $V$ of beer and evaporates it to about 1/3 its original volume. This is often done as part of a distillation^[10] and is sufficient to drive off nearly all the alcohol. The residue is made back up to the original volume with pure water and the specific gravity of the reconstituted (with respect to the extract) beer measured. This value is put into the ASBC table and the corresponding °P value taken out. This is the number of grams of extract in 100 grams of the reconstituted beer. The mass of the reconstituted beer is $V\times \rho_{water} \times SG_{recon}$ with $ρ w a t e r$ representing the density of water and $S G r e c o n$ the specific gravity of the reconstituted beer. The mass of extract in the reconstituted beer is thus $P_{recon} \times V\times \rho_{water} \times SG_{recon}/100$ i.e the mass of extract per 100 grams of reconstituted beer ( $P r e c o n$ ) times the number of units of 100 grams of reconstituted beer. The mass of the beer in volume $V$ is $V\times \rho_{water} \times SG_{beer}$ . Dividing the mass of extract in the reconstituted beer by the mass of the original beer (the factors of 100 and the density of water cancel) gives the grams of extract per 100 grams of beer. This is the true extract

$n = P_{recon}{SG_{recon} \over SG_{beer}}$

which, inserted in the previous formula give the alcohol by weight (ABW) or grams of alcohol per 100 grams of beer. It is clear that values for

$f_{pn} = {1 \over (2.0665 - 1.0665P/100)}$

can be precomputed. De Clerck^[11] gives tabulated values determined by Balling. The tabulated values are very closely approximated by

f p n = 0.48394 + 0.0024688 P + 0.00001561 P 2

Thus

$A w = f p n (P - n)$

and this form has intuitive appeal in that it indicates that the amount of alcohol produced is proportional to the amount of extract consumed. For 12 °P wort $f p n = 0.5158$ .

The volume of $A$ grams of ethanol is $A/(\rho_{water}\times SG_{EtOH})$ where $ρ E t O H = 0.790661$ is the specific gravity (20/20) of pure ethanol. The volume of 100 grams of beer is $100/(\rho_{water}\times SG_{beer})$ so that the ratio of the volume of alcohol to the volume of the beer, expressed as a percentage (the Alcohol by Volume - ABV) is

$A_v = A_w{SG_{beer} \over 0.79661}$

$A w$ and $A v$ both derive from conservation of mass but depend on Balling's determination (made in 1843) that 2.0665 grams of extract ferment to produce 1 gram of ethanol, 0.9565 grams of carbon dioxide and 0.11 grams of yeast biomass. If these assumptions are not true then Ballings formula produces inaccurate results. Despite the fact that modern brewing practices tend to produce less than 0.11 grams of yeast for each 2.0665 grams of extract Ballings formula is still in nearly universal use today.

[edit] Degree of Fermentation

A third application for mass based data is in monitoring the progress of a fermentation in particular the Real Degree of Fermentation (RDF), expressed as a percentage, is given by

$RDF = 100({P-n\over P})$

Similarly, the Apparent Degree of Fermentation (ADF) again as a percentage is

$ADF = 100({P-m\over P})$

where $m$ is the Plato equivalent to the specific gravity of the beer.

Because there is a near linear relationship between °P and "points", as defined above (and thus $S G - 1$ ) it is possible to use $S G - 1$ or points in the ADF and RDF formulas and even to put them into the alcohol formula though that formula must be scaled as 1 °P ~ 4 points. Most alcohol formulas in the popular home brewing literature are based on apparent extract expressed as points calculated from the specific gravity of the beer as most homebrewers do not wish to deal with the evaporation process required to measure true extract.

[edit] Dilution/Concentration Calculations

The mass of extract can be used to calculate the effect of dilution and concentration of wort. A wort of volume $V o$ with specific gravity $S G o$ and corresponding strength $P o$ (°P) weighs $\rho_{water} \times V_o \times SG_o$ , contains $P_o \times \rho_{water} \times V_o \times SG_o/100$ kg extract and $(100 - P_o) \times \rho_{water} \times V_o \times SG_o/100$ kg water. If the volume is changed by adding (dilution) or subtracting (evaporation) a volume of water $V δ$ , (positive in the case of addition, negative for subtraction) which weighs $\rho_{water}\times V_{\delta}$ the concentration of the new solution, by weight, as a percentage, is

$P_f = {P_oV_oSG_o \over{V_oSG_o + V_{\delta}}}$

also in °P. As was the case with the previous examples one may obtain an approximate result by using "points" for $P o$ in this formula in which case the result, $P f$ , will also be in points.

[edit] References

^ American Society of Brewing Chemists, Inc., ASBC Methods of Analysis, St. Paul. MN 2009, Preface to Table 1: Extract in Wort and Beer
^ ASBC Methods of Analysis: Wort-3
^ Lincoln, R.H. "Computer compatible parametric equations for basic brewing computations", MBAA TEch. Q., 24:129-132
^ Weissler, Harald E., "Brewing Calculations" in "Handbook of Brewing", Hardwick, W.A. Ed. Marcel Dekker, New York, 1994 p645 et. seq.
^ Bettin,H, Spieweck,F "Die Dichte des Wassers als Funktion der Temperatur nach Einfürung der Internationalen Temperaturskala von 1990" PTB-Mit. 100(1990). pg 195-196
^ ASBC Methods of Analysis Malt-4
^ Weyerman Specialty Malting Company, Bamberg, Germany. Product specification sheet for Bohemian Pilsner Malt
^ Palmer, J. How to Brew; Brewers Publications, Boulder 2006;Table 27 p193
^ De Clerck,J. A textbook of Brewing,Translated by Kathleen Barton-Wright, Chapman & Hall,, London 1958 Vol. II p427
^ ASBC Methods of Analysis, op. cit. Beer-4A
^ De Clerck, op. cit. Vol. II p 428

[0] American Society of Brewing Chemists, Inc., ASBC Methods of Analysis, St. Paul. MN 2009, Preface to Table 1: Extract in Wort and Beer

[1] ASBC Methods of Analysis: Wort-3

[2] Lincoln, R.H. "Computer compatible parametric equations for basic brewing computations", MBAA TEch. Q., 24:129-132

[3] Weissler, Harald E., "Brewing Calculations" in "Handbook of Brewing", Hardwick, W.A. Ed. Marcel Dekker, New York, 1994 p645 et. seq.

[4] Bettin,H, Spieweck,F "Die Dichte des Wassers als Funktion der Temperatur nach Einfürung der Internationalen Temperaturskala von 1990" PTB-Mit. 100(1990). pg 195-196

[5] ASBC Methods of Analysis Malt-4

[6] Weyerman Specialty Malting Company, Bamberg, Germany. Product specification sheet for Bohemian Pilsner Malt

[7] Palmer, J. How to Brew; Brewers Publications, Boulder 2006;Table 27 p193

[8] De Clerck,J. A textbook of Brewing,Translated by Kathleen Barton-Wright, Chapman & Hall,, London 1958 Vol. II p427

[9] ASBC Methods of Analysis, op. cit. Beer-4A

[10] De Clerck, op. cit. Vol. II p 428

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