Very interesting Lb...
You've put quite some effort into this, will wait with bated breath for the "continuation" findings.
Mal.
Linus School
A roast does not develop heat momentum, aka thermal momentum, because this doesn’t exist. As far as I can tell from what has been written about it by Rob Hoos and others (including some on this forum), the idea is that the rate of rise (RoR) of the bean mass has its own inertia that must be accounted for when making changes to roast conditions.
The purpose of this post is to apply some fundamental thermodynamics to this and hopefully make it disappear in a puff of logic.
RoR is the rate of change of bean mass temperature with time, in normal notation this is δT_{b }/ δt ; T is in Kelvins, t is in seconds. If you are not familiar with Kelvins, the more familiar Celsius scale is derived from the Kelvin scale by
^{ o}C = K – 273.1.
There is a good reason for using this unit which will become obvious later.
We know that δT_{b }/ δt = Q_{b} / C_{b}
Where Q_{b} is the net heat input to the bean mass and C_{b} is the heat capacity of the bean mass. C is in Joules per kg.
This is what’s known as a lumped parameter model: the parameter C_{b} incorporates such things as the heat required to vaporise the water content and the heat required to drive the reactions while the parameter Qb allows for heat losses outside the beans (eg to ambient). I will tease out some of these factors later but for now we’ll stick with lumped parameters, hence the partial differential indicated by the lower case deltas.
The fundamental rule of heat transfer is
Q = U . A . ΔT
Where U is the thermal transmittance*, A is the system boundary area and ΔT is the temperature difference across that boundary. U is in watts/ m^{2} /K, A is in m^{2.}. In this case the area with which we will be concerned is the net area of the air layer next to the skin of the beans, this will again become obvious later. ΔT = T_{g} – T_{b} where T_{g} is the temperature of the heating gas ( air) and T_{b} is the bean mass temperature as above, so
Q_{b} = U . A . (T_{g} – T_{b})
Putting this together, we have
δT_{b }/ δt = U . A . (T_{g} – T_{b}) / C_{b} or
δT_{b }/ δt = U . A . T_{g} / C_{b} - U . A . T_{b} / C_{b}
Letting U . A / Cb = k, this is
δT_{b }/ δt = k . T_{g} - k . T_{b}
For constant T_{g} this means the rate of change of T_{b} is proportional to T_{b}, so we have an exponential** decay curve. Over small time scales where k can be considered as a constant we can convert to a full differential equation
dT_{b }/ dt = k . T_{g} - k . T_{b} so we can solve to get
T_{b }(t)= (k . T_{g}).t - T_{b}(0)e^{-kt}
which has a characteristic time constant = 1/k which from the above = Cb / U . A
Since Cb and A depend on the coffee bean while U is a function of the roaster design, the value of this time constant varies with roaster and bean and can be between about 30 seconds and a few minutes. Here’s what a time constant of 60 seconds and an input temperature of 240 ^{o}C looks like:
Exponential decay
This time constant is important because it controls the rate of system response to an input: the larger the value of the time constant, the slower the system responds.
This result was derived with the assumption of the constituent factors being constant over small time scales. Over longer time scales the individual factors vary. The boundary area (A) increases as the beans expand during the roast, the thermal transmittance (U) decreases for more or less the same reasons. The heat capacity Cb changes as the roast develops as will be teased out later. Taken together these mean that the time constant reduces somewhat over the history of the roast but the changes are small enough that it is still a useful guide.
If you are measuring your roast profile with a temperature probe it will also have a finite lag and thus its own time constant. These two are additive in terms of system control.
The analysis so far assumes that the heating medium is entirely gas, as is the case for a fluidised bed roaster. If there are significant amounts of metal in contact with the gas and the beans, as is the case with a drum roaster, the analysis is more complex and includes another time constant for the response of the metal structures.
The important thing to note here is that the rate of rise is entirely passive, it is an output from the roasting system, not an input to it. It indicates the balance between the heat input into the roasting system and the ability of the bean mass to absorb that input. There is no need to invoke the mythical quantity of thermal momentum.
The existence of the thermal time constant means that when something changes the system will continue on its exising path for some time before the effect of the change become apparent: this can look like the rate of rise has a life of its own but that isn't the case.
Quote Rob Hoos (Modulating the Flavour Profile of Coffee p 16): “If the turn-around happens earlier than planned, you are entering the roast with more thermal momentum than you may have expected and should lower your heat application”.
This can be replaced by:
if the turnaround happens earlier than planned the total energy in the system is higher than expected so energy input can be reduced to compensate.
Quote Paul Songer Coffee Thermodynamics
"The change in the internal energy of the system at this stage is the sum of the heat entering the system from the environment and the heat being created by the sugar browning reactions. Like a snowball rolling downhill, the sugar browning reactions develop a momentum of their own and the increase in internal energy depends less and less on the heat in the environment as the roast progresses and more on the heat already taken on, the buildup of pressure within the bean, and the heat released by sugar browning."
He is wrong about almost every point here. The browning reactions are endothermic so there is no internal heat generation. The heat already taken on has increased the temperature of the bean mass so there’s no need to account for it again. The buildup of pressure within the bean probably actually decreases reaction rate by itself: the pressure is however a symptom of increased temperature which will increase reaction rate but again we’ve already accounted for that.
To be continued.
* U is the inverse of the R value used in insulation, so U * R = 1
** If you really care, visit Khan academy for tutelage
Last edited by Lyrebird; 16th October 2019 at 08:34 PM.
Very interesting Lb...
You've put quite some effort into this, will wait with bated breath for the "continuation" findings.
Mal.
Nah, this was the easy bit: my wife says I never met an equation I didn't like.
The hard work was trolling through several hundred pages of research on the reaction kinetics of coffee roasting and trying to make it all add up to a consistent theory. The stuff above came out of that.
BTW happy to see you are one of the few who can spell "bated" correctly.
Wow Lyrebird, impressive body of work! Genuinely looking forward to this and the debate around it.
Without sitting down and looking at everything for half an hour, I understood everything up to "It'll take him twelve years to unlearn everything Lucy's been teaching him" Any comments or debate will help my learning process. This may be the post in years to come people point to.
Would be interesting if thermal momentum doesn't exist, to find out what people are observing/experiencing which looks like that to them.
Buried beneath the equations in my post are a couple of answers to that. I think the time lag is the most important factor.
Here's a prediction, after all predictive power trumps explanatory power: if I am on the right track, the phenomenon formerly known as heat momentum will be much more commonly observed in drum roasting than fluidised bed. This is because there's a secondary heat store in a drum roaster which can be at a higher temperature than the bean mass so the time lag in system response can result in overshoot. The time constant for a drum roaster is also much longer.
I am assuming the drum roaster theory is because the metal of the drum roaster is more efficient at conducting or storing heat compared to the beans? (I would have guessed that recently, but after the explanation that 400 grams of water has the same thermal energy as 4.5kg of metal, everything is open to be more heavily thought on)
The full answer to that is long and complex and would keep me up all night but yes, the standard theory of drum roasting is that some of the heat transfer to the beans is from the metal of the drum and thermodynamics requires the drum to be hotter than the beans for this to occur.
OK, in the first post above I made some pretty broad claims which I will now attempt to justify.
Firstly I should acknowledge that one of the sentences in the post above doesn’t say what I wanted it to. Where I said
“Taken together these mean that the time constant reduces somewhat over the history of the roast but the changes are small enough that it is still a useful guide.”
I should have said
“Taken together these mean that the time constant changes somewhat over the history of the roast but the changes at the critical points are small enough that it is still a useful guide.”
I lumped the thermal transmission into one factor and said the area involved was that of the air film on the surface of the bean, where many people writing on the subject have assumed that intra-bean heat transfer is an important factor. Much of the relevant info here comes from a paper by Basile et al which goes into exhaustive detail building a model of heat transfer during roasting, I have had the temerity to vastly simplify it for this audience. Amongst the interesting factors in the Basile paper is the section on using Biot numbers to estimate the importance of gas to film heat transfer vs intra-bean heat transfer.
Basile states that the Biot numbers for different coffee roasting methodologies are all below 5 and thus the film heat transfer dominates over the intra-bean transfer, to the extent that the thermal gradient across the bean is less than 10% of the gradient across the film for the drum process and reduces from about 30% at T = 0 to around 5% at the later stages of roast for the fluidised bed process. Since my intention was to show where the time lag in roasting originates rather than building a precise analytical model, I felt justified in lumping the intra-bean transfer in with the film transfer as one generalised thermal transmittance value.
The data from Basile also show that the time constant for a drum roaster is much longer than that for a fluidised bed roaster, with a spouted bed somewhere in between (FWIW the “Corretto” process is pretty close to a spouted bed).
I acknowledged that the net surface area A of the film increases during the roast but how much? Given that we know that for Arabica beans the volume increase during the roast is of the order of 50% and for a consistent shape the surface area varies as the square of the cube root of volume, simple algebra gives us a surface area increase of 1.5^(2/3) - 1 or about 31% for the beans themselves. Since the surface film depth should stay constant, the percentage increase in film area is somewhat less than this, if the film is of the order of 0.5mm the surface area increase should be around 25%.
Then there’s the matter of the heat capacity of the beans. This has three major contributing factors: the heat capacity of the dry matter in the beans given by Strezov as approximately 1.5 kJ / kg / K but the total heat capacity reduces over the roast because of organic roast loss which is around 6 – 10 % of initial dry weight, depending on the depth of roast. In addition there is the enthalpy of vaporisation of the water present in the greens, almost all of which is lost during the roast (the water content however does not decline to zero as several of the roast reactions give off water vapour). The enthalpy of vaporisation of water is taken at 2260 kJ/ kg and the heat capacity at 4.18 kJ / kg / K.
Lastly there is the energy absorbed by the roasting reactions themselves, again according to the Strezoff paper these add up to 100 kJ / kg over the course of the roast to a normal end point but do so in a non-linear way, being much higher in the early part of the roast, peaking at about 2.5 kJ / kg /K at around 120 ^{o}C internal temperature.
BTW the data in the Strezoff paper also shows that the roast reactions are never exothermic in the normal roasting range. Basically if your roast is exothermic it’s because it’s on fire.
A suitable model for water loss is easily calculated using a simple model derived from the complex moving front model validated by Fadai. For our simplified version we will assume that the water is evenly distributed through the bean at the start of roast and that a drying front moves inwards as the roast progresses until the water produced by the roast reactions equals the rate of water loss and water content stabilises. We will assume that net water loss is proportional to the area of this drying front which is in turn proportional to the 2/3 power of the remaining water volume as outlined above. Water production from roast reactions accelerates after about 180 ^{o}C internal temperature and reaches equilibrium around first crack, as shown by Niya Wang.
BTW this puts paid once and for all to the notion that first crack is due to the sudden release of pent up steam, if it were the water content would go down not up.
The data in N. Wang also gives us a means of working out the proportionality constant for the water loss rate per unit area. Combining this with the total roast loss data from Xiujie Wang (same family name, same research group, different person) will give us a method for estimating the organic roast loss as mentioned above.
Putting all this together we can build a piecewise linear model of the energy absorbed by the various factors over the course of the roast, the difference between the energies at any two temperatures divided by the difference between temperatures is then our lumped parameter heat capacity Cb.
This is most easily presented in graphical format.
The first graph shows the contributions to the cumulative energy over the course of the roast.
Chart 1
The second graph shows the energy being used against time. I’m sorry for the wobbly lines, they are artefacts caused by noise in the temperature data used which in turn is due to the poor signal to noise ratio of the Heatsnob.
Chart 2
The temperature plot in the second graph is on a different axis (x10).
Note that the reaction energy never goes to zero eg it is always endothermic.
Stacking these on top of each other, the value for Cb starts and finishes around the value for inert coffee mass (1.5 kJ / kg /K) but peaks at a value of around 6 early in the roast due to the reaction energy and water vapour components.
Lastly, I am pleased to note that the model predicted 16.3% roast loss for this roast which is bang on.
Last edited by Lyrebird; 18th October 2019 at 09:49 AM.
Excellent stuff Lb...
Certainly clears up some interpretations I have been using over time, to explain the various milestones and outcomes of a roast batch progression. Any chance of transferring all of this to a single PDF document that could be stored here on CS for CSers' future reference?
Mal.
"Water production from roast reactions accelerates after about 180 ^{o}C internal temperature and reaches equilibrium around first crack, as shown by Niya Wang.
BTW this puts paid once and for all to the notion that first crack is due to the sudden release of pent up steam, if it were the water content would go down not up."
Hey Lyrebird, what does "reaches equilibrium" mean in the above statement, does it mean that the rate of water production from chemical reactions equals the rate of water loss of the bean?
If my understanding of what you're saying is correct, does it mean you've got a situation where the water content of the bean is decreasing as the beans are heated through the roast, up until around the 180c mark when increasingly water is produced by chemical reactions up to around first crack when water production and loss cancel each other out? If first crack isn't due to moisture loss, what's proposed as the reason for the beans cracking, could it be a chemical reaction only?
The coffee beans become more brittle as they cook, could it be possible that the "stabilization" of water is due a change in the structure of the surface of the bean at higher temperatures which may hinder water loss (resulting in stabilization), rather than water created by chemical reactions, causing pressure buildup resulting in first crack? If i'm understanding you though, you're saying this isn't the case?
Is it true that the water content of the beans drops following first crack?
In Niya Wang's thesis paper (page 60) they claim that first crack is due to pressure buildup (steam and gasses) and the resulting release/crack.
Just a thought and happy to be schooled (if i can understand the response )
Lyrebird I don't know if this is too trivial and lightweight to add to your well thought out hypothesis, but did a few roasts on the weekend in my home made corretto which is predominantly lightweight ss with a few heavy panels and insulation around. Not much lag in the system. One roast I raced up to first crack then just after removed the heat. Another roast removed the heat at second crack after a steady approach. I was hoping to induce some thermal runaway, like a lithium ion battery on heat. Unfortunately nothing happened, removed heat and ROR just dropped till after 30 seconds there was none, at which point total temp dropped. I might try burning a roast, racing straight up and past second crack and see if I can get the heat momentum.
That said, the beans still pop and crack when you put them in the cooler, certainly giving the impression something is happening. With no sensors I have no idea what.