6.3 Model Parameterization
6.3.1 Dispersion Coefficient
The dispersion coefficient is determined from a comparison of conductivity tracer simu-lations to experiments, which were described for setup b in Section 4.3.1. Additionally, tracer experiments were conducted in setup a at 309 K and are now compared to simula-tions.
For the simulations, the total tube length has to be known. The tube length la of setup a, which was measured from the seed addition position to the outlet, was given in Table 6.1. In the tracer experiments, the tracer passes an additional tube section before the seed addition valve including the pump and, for downward experiments, including the
Table 6.2: Experimental parameters for the experiments in Sections 6.3.2 and 6.3.3. Su-persaturation calculated on the basis of Eq. (4.2) for the given initial masses in the continuous phase, which correspond to initial and feed saturation tem-peratures from T =313 K to 314 K.
Experiment σin σout mf,anh,in mf,w,in
Figure 6.7 -4 % 1 % 5.1×10−4kg 4.26×10−3kg Figure 6.8a,b -10 % 10 % 5.1×10−4kg 4.26×10−3kg Figure 6.8c,d -12 % 2 % 5.2×10−4kg 4.38×10−3kg Figure 6.9a -4 % 4 % 4.9×10−4kg 4.17×10−3kg Figure 6.9b -8 % 14 % 4.9×10−4kg 4.17×10−3kg Figure 6.9c -5 % 19 % 4.9×10−4kg 4.17×10−3kg
debubbler. The length of this section is called linlet and is determined in the following.
To calculate the tube length, the mass flow rate is required. The conductivity experi-ments were carried out at four different pump speeds. To measure the conductivity, the flow-through microscope was replaced by a probe. This may influence the mass flow rate.
For upward flow, the mass flow rate was determined once at the four different pump speeds by averaged weighing measurements. For downward flow, the mass flow rate was measured for each experiment by mass flow measurements with a Coriolis type mass flow sensor. Then, the mass flow rate was applied to calculate the average fluid velocity vf using the known tube diameter.
For each tracer experiment, the offset additional length linlet is the product of the average fluid velocity vf and the mean fluid RT τf,inlet of the tracer signal. τf,inlet was calculated from the conductivity signal of the experiments at the seed addition position, as depicted as “inlet” in Figure 4.6. For each experiment, the calculated vf and linlet are illustrated in Figure 6.3 by a marker. A simple linear regression model was fitted minimizing least-squares for the upward and downward tracer experiments at the inlet position. The resulting linear polynomials are also shown. For the experiments at the outlet position, linlet was calculated from these polynomials. The resultingltracer, which is the sum of linlet and la, is given in Table 6.3. For setup a, the additional tube lengthlinlet is set to 4 m. As explained in Section 6.3, Nj increases proportionally to the increase in tube length. For all tracer simulations, a fluid density ρf of water at room temperature of
0.2 0.25 0.3 0.35 vf in m/s
4 6 8 10
l inlet in m
upward downward
Figure 6.3: Tracer measurements at the seed-addition valve in the HCT setup in Fig-ure 4.4b (Wiedmeyer et al., 2017a) for upward (orange) and downward (black) flow.
Table 6.3: Values of dynamic viscosity, average fluid velocity, Reynolds number, tracer tube length, and inlet pulse duration in the tracer simulations.
Symbol and Unit Setup b Setup a
Figure 6.4a-d Figure 6.5a-b
Flow direction upward upward downward downward upward upward ηf×104 [kg m−1s−1] 8.992 8.992 8.693 8.891 7.058 7.058 vf [m s−1] 0.23058 0.32518 0.23649 0.34095 0.2564 0.2564
Re[−] 1534 2163 1627 2294 2173 2173
ltracer [m] 37.6 38.6 40.7 42.0 37 37
tin [s] 15 15 15 15 19 20
997 kg m−3 is assumed. Regarding the simulation, no crystals are added, fin is zero, and the PBE in Eq. (6.9) remains zero during the simulation. In the mass balance Eqs. (6.10) and (6.11), the same dispersion coefficient is assumed. Initially, the tube is filled with water and the initial condition in Eq. (6.16) for each control volume is
mf,anh,0 = 0
mf,w,0 =ρf∆zπR2 = 4.35×10−3kg
where ρf = 997 kg m−3 and ∆z = 0.15 m. For the tracer mass fraction, which is given in Section 4.3.1, the boundary condition at the inlet in Eq. (6.15) becomes
mf,anh,in = 1.6×10−4kg mf,w,in = 4.20×10−3kg
For a dispersion coefficient D = 0.015 m2s−1, a good agreement between experiments and simulations can be achieved as displayed in Figures 6.4 and 6.5. The dispersion co-efficient is seven to eight magnitudes larger than the diffusion coco-efficient of potash alum solutions (Mullin et al., 1965), hence, the diffusion coefficient is not further discussed. For setup b, the same tracer outlet signals are depicted in Figure 6.4 as in Figure 4.6 in the experimental section. In Section 4.3.1, only the tube section that is relevant for crystal growth, which starts at the seed-addition valve, was considered to calculate τf. The inlet signal, also, was measured at the seed addition position a few seconds after the start of an experiment. Here, the whole tube section is considered to calculateτf. The inlet signal illustrates the mass fraction of the tracer solution and the duration of tracer addition at the reservoir position. The axial dispersion coefficient influences the width of the tracer signal at the outlet. The width agrees well for experiments and simulations in Figure 6.4.
At downward flow, the simulations show slightly narrower curves, but because the peak height agrees well, the axial dispersion coefficient is not adjusted further. In Figure 6.5 for setup a, the simulations and experiments also agree well. The noise in the second exper-iment results from air bubbles that disturb the conductivity signal, but the overall peak height and width are in accordance. Hence, the axial dispersion coefficient was validated in setup a. In simulations, axial dispersion results from the dispersion term in Eqs. (6.10) and (6.11) as well as from numerical diffusion. Therefore, the axial dispersion coefficient, which is used for the simulations, is smaller than the physical dispersion coefficient. The extent of the numerical diffusion depends on the grid refinement of the FVM. For compa-rability of simulations of varying tube length, the number of finite volumes is scaled with the tube length, as noted in Table 6.1. The hydrodynamic axial dispersion coefficient can
a
0 100 200
in s 0
20 40 60
w in g hydrate per kg added H 2O
inlet outlet
f b
0 100 200
in s 0
20 40 60
w in g hydrate per kg added H 2O
inlet outlet
f
c
0 100 200
in s 0
20 40 60
w in g hydrate per kg added H 2O
inlet outlet
f d
0 100 200
in s 0
20 40 60
w in g hydrate per kg added H 2O
inlet outlet
f
Figure 6.4: Tracer measurements in the HCT setup in Figure 4.4b for (a,b) upward and (c,d) downward flow. (a,c) Low Reynolds numbers Reup/down = 1534/1627;
(b,d) high Reynolds numbersReup/down = 2163/2294. Average RT as deter-mined by the Coriolis-type mass flow sensor (dashed vertical lines). Exper-imental outlet signal as in Figure 4.6 (solid) and outlet signal determined from simulations (dotted).
Source: Adapted with permission from Wiedmeyer et al. (2017a). Copyright 2017 American Chemical Society.
a
0 100 200
in s 0
20 40 60
w in g hydrate per kg added H 2O
inlet outlet
f b
0 100 200
in s 0
20 40 60
w in g hydrate per kg added H 2O
inlet outlet
f
Figure 6.5: Tracer measurements indicating the fluid RT in the HCT setup in Figure 4.4a (Wiedmeyer et al., 2017b) for similar conditions atReup = 2173. Average RT as determined by the Coriolis-type mass flow sensor (dashed vertical lines).
Experimentally measured outlet signal (solid) and outlet signal determined from simulations (dotted).
be derived from the Bodenstein number (Klutz et al., 2015). The Bodenstein number is estimated from a minimization of least squares of the dimensionless tracer concentration, which is estimated from Figures 6.4 and 6.5. The resulting Bodenstein number is 330±92
and the hydrodynamic axial dispersion coefficient is (0.036±0.017) m2s−1.