Explanation of DDT formula

This calculation is only used to get you in the ballpark of temperature, especially for the home baker. It is not a scientific formula by any means, because it leaves out so many of the variables. Even bakeries have proofing chambers so that they can adjust the dough temperature. 

That said, your reference to 10*C as a friction temperature seems awfully high to me. Most references for home bakers that I have seen say to ignore the friction temperature for hand kneading and use 1-3*F for home machines. That is an order of magnitude less than your reference. Again there are many variables including the type of flour, amount of dough, hydration, kind of mixing system, etc. that all affect the friction temperature and a professional bakery would probably work out a different number for each formula.  I have made batches up to 3 kg by hand and found that I could ignore it and still get pretty close. Even the few doughs I mix by machine don’t see a massive increase in temperature after mixing. 

-Brad

This formula works for open space low speed mixing, because of the room temperature. Even if friction heats the dough little by little, low room temperature and cold kneading surface temperature continuously bring the dough temp down. In the end you will have your 24C dough. If not, then your friction factor is higher than that which you inputed into formula.

High speed kneading in enclosed and warm space, as in bread machine, in thermomix or in food processor, or even inside mixer capable of high speed mixing is a bit different. There I use ice cold water or ice without exception and often have to chill the dough right after kneading down to DDT before in begins fermenting.

High friction factor is rather commonplace according to this baker: 

https://www.busbysbakery.com/desired-dough-temperature/

Gareth says that it depends on time and on the chosen kneading method, i.e. on kneading intensity

Light incorporation: 0C (32F)

Standard mix (8 mins): 4C(40F)

Long mix (14 mins): 7C (44F).

My research has indicated that:

Light incorporation: 1C (32F)

Standard mix (10 mins): 7C(40F)

Long mix (20 mins): 14C (44F)

But it depends heavily on how you knead your dough!

Part One

The Desired Dough Temperature Formula is basically a mathematical model. Modeling requires assumptions. The reason why the DDT formula is non-intuitive is because the assumptions are not usually stated. 

To see this, first rearrange the formula.

Usually, we see the DDT formula given as:

(water temp) = 3*(DDT) - (room temp) - (flour temp) - (friction factor)

or

(water temp) = 4*(DDT) - (room temp) - (flour temp) - (friction factor) - (preferment temp)

Let's collapse the two formulas into one by including a multiplication factor M.

(water temp) = M*(DDT) - (room temp) - (flour temp) - (friction factor) - (preferment temp)

In fact, the above should really be called the Water Temperature Formula. See Required Water Temperature formula from The Perfect Loaf.

Do some algebra to get:

Desired Dough Temperature = [(water temp) + (room temp) + (flour temp) + (friction factor) + (preferment temp)] / M

where M is some constant.

Does the above look more intuitive?

If M were the number of (non-zero) temperature variables, then what the above equation says is that the desired dough temperature will be an average of the variables. 

This formula is correct, assuming that each variable (water, flour, room etc.) contributes equally to the final dough temperature. Obviously the assumption is not exactly true. But since the DDT formula seems to work for a lot of people in various environments, it's likely that the approximation is good enough for the formula's intended purpose. 

But hold on! The multiplication factor isn't the number of variables! It's the number of variables minus 1.

King Arthur Flour's article on Desired Dough Temperature says that the multiplication factor is "the number of variable temperatures other than water temperature that affect dough temperature". 

But why water? Seems improbable. Bread dough is basically flour and water. The more likely candidate for the 'odd thing out' is the friction factor. In other words, we do not think that friction (expressed in degrees Celsius as a 'factor') contributes equally to the final dough temperature. 

Think about it like this: when you make the dough, the process of kneading raises the temperature of the remaining variables (flour, water, room, preferment) by some amount in comparison to the measured temperature. The friction factor accounts for that additional amount. In fact, you could write the DDT equation like this:

DDT = [(water temp + ff1) + (room temp + ff2) + (flour temp + ff3) + (preferment temp + ff4)] / M

where ff1, ff2, ff3, ff4 are the respective increases in temperature to each variable due to friction, and 

(friction factor) = ff1 + ff2 + ff3 + ff4

When you think that kneading won't raise the temperatures, then naturally

(friction factor) = 0.

Notice that this is different from when there is no preferment. The absent preferment obviously cannot contribute heat to the dough, so we set

(preferment temperature) = 0.

But in addition, the value of the constant M also must decrease by 1. In contrast, when friction is negligible M should remain the same.  

This would all be more transparent if we wrote the DDT formula like this:

DDT = [(water temp) + (room temp) + (flour temp) + (preferment temp) + (friction factor)] / [M - 1]

where M is the total number of variables.

Note: Remember that DDT isn't a variable. DDT is a constant which we can decide. Solve for (water temp).

In theory, we could also write an equation for brioche:

DDT = [(water/egg temp) + (room temp) + (flour temp) + (preferment temp) + (butter temp) + (friction factor)] / [M - 1]

In this case M = 6. 

Part Two

Rearrange the DDT formula again:

DDT = { [(water temp) + (room temp) + (flour temp) + (preferment temp)] / [M - 1] } + { (friction factor) / [M - 1] }

What the DDT formula above says is that the increase in final dough temperature due to kneading is the friction factor divided by [M - 1]. So to calculate the friction factor:

(friction factor) = [M - 1] * (dough temp increase from knead),

In your example, the dough rises 10 degrees in kneading, so the friction factor needs to be 30 degrees.

According to what I've read, the friction factor is usually either assumed via standard rules of thumb, or determined experimentally for a specific machine/method. But in any case, the key is that the friction factor (in degrees) does not represent the increase in final dough temperature due to kneading.

If you generalise the improved formula you offer above:

(water temp) = { [DDT - (friction factor)]*[1 + H] - (flour temp) } / H

where H is the hydration in decimals.

Rearrange to get:

DDT = [1 / (1+ H)]*(flour temp) + [H / (1+ H)]*(water temp) + (friction factor)

First of all, the reason I've put friction factor in italics is because the (friction factor) as expressed here is no longer the friction factor in the usual sense, but rather the contribution to the final dough temperature from kneading in degrees. 

However, the more interesting thing here is that the above formula essentially weights each temperature variable by a reasonable approximation of their relative contributions to the final dough temperature. You're taking a weighted average.

Rewrite as:

DDT = A*(flour temp) + B*(water temp) + C*(room temp) + (friction factor)

where A + B + C = 1.

In the conventional DDT formula, we've assumed that A = B = C. 

This is by no means the intuition of the DDT formula. I'm not familiar with its history (Does anyone know its history?) Nor do I really use the DDT formula myself, so this is all theoretical. Corrections welcome. 

It’s not particularly complicated. 

Determine the final dough temperature you want to achieve, say 75 F. 

Determine the temperatures of the room, the flour, the preferment, if using, and a friction factor—the amount of temperature increase caused by machine mixing (Hamelman has a method to do this; 25 degrees works for a Kitchen Aid mixer at home and most commercial mixers in bakeries).  If using a preferment, you have four temperature factors, if not, you have three. Multiply that number by desired DDT (A). Add up the various temperatures of the factors (B). Subtract B from A. The result is the required water temperature. 

Easy. Works every time. 

This link is probably the most comprehensive analysis of the many nuances that are involved solving the complexities of the thermodynamic  interaction of the ingredients of a dough mass.https://madebywindmill.com/rise/ddt/

In addition to using the DDT calculator, read through all parts of the study.

Anton