P - Technische Universiteit Eindhoven

7. Thermodynamic and poromechanic crystallization pressure of sodium sulfate heptahydrate 78
started, at any moment in time the actual concentration of salt solution, c [M], can be
found from:
c = n s n 0 s − n c s
=
m w M w (n 0 w − ν w n c s) , (7.7)
where n s and n 0 s are the actual and initial amount in moles of sodium sulfate in the
solution, m w is the actual mass of water in the salt solution, n 0 w is the initial amount in
moles of water in the solution, ν w is the hydration factor of the crystals, and M w is the
molar mass of water. From equation (7.7) the amount of precipitated salt, n c s, can be
found as:
n c s = n0 s − n 0 wM w c
1 − ν w M w c = n0 wM w (c 0 − c)
, (7.8)
1 − cν w M w
where c 0 is the initial concentration of the sodium sulfate solution (in our case 3.4 mole/kg).
The initial mass of the salt solution, m 0 sol, is the sum of the initial masses of water, m 0 w,
and sodium sulfate, m 0 s:
m sol
0 = m 0 w + m 0 s = n 0 wM w + n 0 sM s ⇒ n 0 wM w = m0 sol
1 + c 0 M s
. (7.9)
If we assume that initially the sample is completely saturated, the initial mass of the
sodium sulfate solution can be written as m 0 sol = ρ sol V 0 , where ρ sol is the density of the
solution. By substituting equations 7.8 and 7.9 in equation 7.6, the volume fraction can
be written as
ρ sol V m c − c 0
S c =
(1 + c 0 M Na2 SO 4
) (cν w M H2 O − 1) . (7.10)
Using the relations presented in this section, the poromechanic crystallization pressure can
be obtained from measurements of the the expansion of the sample and the concentration.
7.5 Thermodynamic vs. poromechanic crystallization pressure
In our case the expansion, ε, is a function of time and will depend on both the material
properties and the crystallization pressure P c . If a material has no viscoelastic properties,
i.e., if there is no delay between the exerted pressure and the mechanical reaction of the
material, according to Coussy [42] the poromechanical and thermomechanical approaches
become equivalent. Therefore, for linear elastic materials P m = P c .
For most materials the situation is more complicated. The poromechanical pressure,
P m , represents the reaction of the system on the crystallization pressure. In general, it
will increase with the crystallization pressure, and when crystallization pressure decreases,
the poromechanical pressure will decrease too. However, the crystallization is a dynamic
process, slowly filling up the pores in a material. Moreover, as a result of the crystallization
in the pores the material of the pore matrix can be compressed and viscoelastic behavior
may influence the expansion. Hence, in general P m < P c . In the final situation, if there are
no irreversible changes in the material, e.g., cracks, and an equilibrium has been reached,
it is expected that P m = P c .