EMC time constant, tau

With EMC, equilibrium moisture content, we are required to leave the grain for some time before the air’s humidity and temperature come into equilibrium with the grain.  In the literature, the authors of the EMC experiments and resulting equations suggested anywhere from three to seven hours.  But indeed, even with this amount of time, equilibrium is never reached.  The temperature and humidity approach that of EMC of the grain by following an exponential path and then slowly, ever so slowly approaching, but never quite reaching equilibrium.  It approaches asymptotically.

In engineering and science there are many process that react in this manner:  a capacitor charging up to a battery voltage through a resistor, or a small tank being filled with water, through a narrow pipe from a large tank.  At first there is a large flow with a huge pressure on the pipe when the small tank is empty.  But as the tank fills, the water level difference decreases, the pressure decreases and the flow decreases.  The flow becomes slower and slower, and the level in the small tank approaches that of the level of the big tank, but it never gets there.

Why is it exponential and exactly what is this path?  Let’s walk through the math using the Capacitor with capacitance C, connected in series with a resistor, R, and battery with a voltage Ve.  The voltage (or pressure) across the capacitor is Vc and is determined by the amount of charge, Q on the capacitor.  C = Q/Vc  or Vc = Q/C. The voltage across the resistor depends on the flow of charge: Vr = dQ/dt.  Now according to Kirchoff’s, the series voltage, going all the way around the loop will be zero; or  Vc + Vr  – Ve = 0,    Q/C + R dQ/dt  – Ve = 0  and multiplying each term by C:  Q +  RC dQ/dt = C Ve – Q  and rearranging and separating the derivatives give:  dQ / (CVe – Q)  = dt / RC  taking the indefinite integral on both sides:  – ln(CVe-)/CVe = t/RC and raising both sides to an exponent:

Vc = Q/C = Ve(1 – e^-t/RC)   When the time, t is equal to RC such that the exponent of e is  -1, this is the time of one time constant, usually expressed with the stylized t or tau.   It is the time for Vc to become 0.632 that of the battery voltage.  It is the time it takes for the capacitor to charge about two thirds of the way to the battery voltage.  In this example with the capacitor charging through the resistor, one time constant is equal to RC.  Note that it only depends on the value of R and C and is totally independent of the battery voltage.  This is why time constant is useful.  It does not depending on the driving voltage or pressure, and it can be thought of as the time to cover two thirds of the way to the driving force.

In the case of grain and EMC, the air holding the water is the capacitor, the water is the charge, the resistance of kernels skin is the resistance, and the water in the seed or MC provides the pressure, like the battery voltage.  Using the EMC equations, we can plug in a MC and T, and get the relative humidity of what the grain will seek equilibrium.  Call this RHemc.   We can now use the RHemc and Temperature, plugged into the psychrometric equation to determine the absolute humidity the grain would like the surrounding air to be at.  Call this Hemc.  But the air around the grain does not contain this amount of water, it has an absolute humidity of Hair.  Hair will be working itself to that of Hemc, but just how long does it take?   What is the time constant?  How long does it take for the Hair to get two thirds of the way to Hemc.  Note we are not concerned about the specific value of Hair or Hemc, only the distance or difference between the two values.  For example, let’s say that Hemc is 20 gr. per cubic meter, and Hair is 10 gr.  There is 10 gr. difference and want to the time it takes for H air to become 16.32 gr. ( 2/3 of the way).  This amount of time is one time constant.

We can determine the EMC time constant experimentally.  By knowing the size of the bin in bushels, and CFM we can determine the amount of time to do one bin air exchange.  This is the amount of time that the air is in contact with the grain; let’s call it t.  We can also calculate Hemc and Hair and also the amount of water in the air leaving or being discharged from the bin, Hdis.  So air entering the bin with Hair, passes through the grain for a time t, and picks up water from the grain and gets closer to Hemc.  The percent distance to Hemc is what I will call %,   For one of our bins the transit or contact time worked out to 0.45 minutes.  If we measured % to be 0.632, the Hair would have got .632 closer to Hemc, this would be one time constant, and one time constant would be 0.45 minutes.  But we are never that lucky in getting 0.632, it will be something else.  However we can adjust for it:

one time constant =  – (contact time) / ln (% – 1)

Todate, our determination of the EMC time constant is about one minute.  If the grain is in contact with the air for one minute, it will get about two thirds of the way to EMC equilibrium.  This actually is much faster than I first would have thought.