6.7 TRANSPORT OF SUBSTANCES
6.7.5
Osmosis: Soaking Dried Lentils
When dried lentils are soaked, they take in water and their volume grows. It turns out
that at least a part of the phenomenon can be understood in terms of chemical processes. However, a first model will have to be augmented by ideas that make use of our
understanding of the dynamics of fluids discussed in Chapter 1.
Experiment, observations, and basic idea. Dried brown lentils are put in a contain-
er with plenty of distilled water (see Fig. 6.34, top). In the course of time, their volume
grows to about 2.6 times the initial value (see the graph in Fig. 6.34).
The fundamental process must be intake of water. We assume the increase in volume
of the legumes to be due completely to the added amount of water. Therefore, we can
attempt to understand their swelling in terms of a single law of balance of volume applied to their water content:
Vlentils = IV ,water
(6.62)
(see Fig. 6.35). The important task is to model the flow of water into the lentils. The
phenomena discussed in before in Section 6.1 suggest that the flow may be due to osmosis: The pressure of the remaining water in the dried lentils (and therefore its chemical potential) is smaller than the pressure of the water outside (i.e., its chemical
potential). The chemical potential difference is the driving force for the flow of water.
As the water content in the lentils increases, the concentration of the solutes in the cells
decreases. We have seen that solutes reduce the chemical potential (pressure) of the
solvent (Section 6.6.3). When the concentration is reduced, the reduction of pressure
must be smaller. Therefore, in the course of time, the pressure difference between outside and inside will decrease, letting the current of water decrease.
Water pressure: Osmosis. As before, we should assume the flow of amount of water
to depend upon the chemical driving force:
I n,water = k ' A ( µoutside − µinside )
(6.63)
k’A is a chemical conductance. Alternatively, the expression can be put in hydraulic
form to fit the law of balance in Equ. 6.62:
IV ,water = kA ( poutside − pinside )
(6.64)
The water pressure inside dried lentils is smaller than on the outside since substances
are dissolved in the remaining water. According to Section 6.6.3, we can write:
pinside = 1 − psolute = 1 − a
nsolute
Vwater
(6.65)
We can leave the pressure values arbitrary by setting poutside = 1. ninside is the amount
of solutes inside the lentils. Remember that we set the pressure of pure water equal to
1. The product a·ninside is unknown and should be determined by comparing simulation results to data (Fig. 6.34).
The shape of the lentils will be taken to be a pillbox whose height does not change
much. This makes the surface area of the lentils is proportional to their volume. We
PART II
187
Volume / mL
300
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10000 20000
Time / s
30000
Figure 6.34: Volume of lentils
soaking in distilled water, as a
function of time. Data were taken
every 15 minutes by eye using
the markings on the container.
CHAPTER 6. TRANSPORT AND REACTION OF SUBSTANCES
do not have to worry about the value of the factor of proportionality—changing it will
change only the flow factor k in Equ. 6.64 which has to be found by simulation.
V lentils
Figure 6.35: Diagram of a first
V lentils init
system dynamics model of the
soaking of lentils. Note the balance of amount of water, and two
effects upon the flow: (1) Pressure difference because of solutes
in the lentils, and (2) change of
surface due to change of volume.
V water
Thickness
Surface
V water init
IV water
c solute
Flow factor
n solute
p outside
p inside
p solute
If we choose an amount of initial water in the dried lentils of 25 (25% of the 100 mL
of the initial volume of lentils), simulation of the model results in a function shown in
Fig. 6.36 (dashed line). It turns out that we can fit the initial rise of volume. However,
the model predicts that the lentils continue to grow, contrary to what is observed in the
experiment. What is the reason for this discrepancy?
Water pressure: Elastic walls. The chemical factor—solution of substances in the wa-
ter inside the lentils—is not the only one affecting the pressure of the liquid. In the lentils, the water is contained in cells which act as storage elements having elastic walls.
We know this condition as a capacitive effect (Chapter 1): increasing the amount of
water raises the pressure. Combining chemical and capacitive effects means that
pinside = 1 − psolute + pelastic
(6.66)
Let us assume that the capacitive component of pressure is zero when the lentils are
dried and rises steeply when the cell walls are stretched strongly. A nonlinear capacitive relation of the form
pelastic = b (Vwater − Vwater ,initial )
4
(6.67)
can accommodate these ideas and results in a simulation that follows our experimental
data fairly well (see the solid curve in Fig. 6.36).
400
Figure 6.36: Simulation results
Model 1
Volume / mL
for the first and second models,
and data (dots). A second effect
upon the pressure of the water inside the lentils has been taken
into account (pressure due to the
elasticity of the cell walls).
188
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Model 2
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100 [
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16000
24000
Time / s
THE DYNAMICS OF HEAT