Comparison of experiment and simulation: Influence of the aspect ratio101

this simple force balance). As expected,θ₀is low for low volume flows and large spacings but rapidly decreases with decreasing post-to-post spacing or increasing volume flow (Fig. 6.7 b).

6.2.6 Comparison of experiment and simulation: Influence

0.0 0.5 1.0 1.5 2.0 2.5 ( V)²dP²Q 1 1dhS¹ (V²RMSh m ²) ^×105 0

20 40 60 80 100

Separation efficiency (%) ^{hS = 130 µm}

dataset 1 dataset 2 dataset 3

Fig. 6.9: Separation efficiency η as a function of (almost) all parameters employed in this study (∆V)²d_P²Q⁻¹ε_1dh_S⁻¹ atconstant post spacingd =100 µm. Since almost all points collapse, the separation is only dependent on one single variable. To obtain a constantηthe parameters are interchangeably variable as long asd is kept constant and as long ash_Sis sufficiently larger thand. Three different datasets in which different parameters have been changed independently have been used to compile the figure (cf. Tbl. 6.1). The empty symbols denote h_S=130 µm (from dataset 3, green) which is too close tod =100 µm so thatηdeviates from the other values.

The black line shows a fit according to Eq. (6.6).

ciency was not predicted and is due to the small spacing. It vanishes for largerd³; this is what was expected from the force distribution presented in Chapter 4. Nevertheless, with increasing AR, the experimental results come closer towards the theoretical predictions. This holds espe-cially forQ =0.1 mL h⁻¹where the∆ηbetween simulation and experiment is almost 15 % at AR=1but only 5 % at AR=1.67. This is directly linked to the rapid drop inθ₀from AR 1 to 1.67. The drop inθ₀is more moderate for 0.2 mL h⁻¹and so is the decrease of∆ηwhich changes only from 20 % at AR=1to 15 % at AR=1.67. It is expected that at higher AR, the θ₀will also drop for the higher flow rate.

Albeit there might be other wonders at play, the results presented in Fig. 6.8 indicate that the deviation between simulation and experiment is caused by the unfavorable force balance at the surface of the post.

6.2.7 Influence of geometry and operational parameters on separation efficiency

As already shown in Fig. 5.5 a, it makes sense to incorporate the dependence of the separa-tion effieciency on all parameters in one single figure (unfortunately, two figures are necessary).

From Fig. 5.5 we know that the trapping effectiveness (H_crit/d_S) of a single post is proportional to E₀², d_P², 1/v_F, and d_S (which is equivalent to h_S in this study) as long as d_S > 10H_crit. This is for a single post but the same dependencies hold for the entire post array. This was

3This is not shown, but ford =160 µm a change of AR from 1 to 1.67 causes only an increase inηby less than 5 % compared to the 20 % increase atd =38 µm.

Tab. 6.1: List of the parameters employed in the three datasets that were used to compile Fig. 6.9.

dataset ∆V Q d_P h_S d

1 350–4000 V_RMS 0.05–1 mL h⁻¹ 1 µm 262 µm 100 µm

2 1400 V_RMS 0.05–0.1 mL h⁻¹ 0.2–5 µm 262 µm 100 µm

3 1050 V_RMS 0.05–0.4 mL h⁻¹ 1 µm 130–1200 µm 100 µm

tested using three different parameter sets in which several different parameters have been var-ied independently (cf. Table 6.1). The separation efficiency η is then plotted as a function of(∆V)²d_P²Q⁻¹ε_1dh_S⁻¹ (cf. Fig. 6.9). In Fig. 5.5 a, the variable defining the throughput isv_F. Here, instead, the volume flow is given. Whenh_Schanges, then the void fraction changes and thus also the velocity will change (at constantd). The velocity directly between the posts is pro-portional to the volume flow scaled by the one-dimensional “projected” porosityε_1d =d/h_S, which has been introduced to the term defining the separation efficiency. As expected, all datasets collapse on one line.

This means that the separation efficiency (theoretically) only depends on a single parameter, η=η (∆V)²d_P²Q⁻¹ε_1dh_S⁻¹

(and shows the same proportionality as H_crit/d_S). This is only valid if the post-to-post spacingd is kept constant (here,d =100 µm was chosen) and as long as the post is sufficiently larger than the post-to-post distance. Whenh_Sandd come close, the separation efficiency deviates towards lower values, as it can be inferred from the labeled points forh_S=130 µm, which is quite close tod =100 µm.

The black line shows a fit based on the equation η(x) =k_a

1−exp−x k_b

(6.6)

0 2 4 6

( V)²d_P²Q 1 1dh_S¹ (V²_RMSh m ²)×10⁴ 0

20 40 60 80 100

Separation efficiency, (%)

d a.

d = 33 µm d = 66 µm d = 99 µm d = 132 µm

20 40 60 80 100 120 140

Post-to-post spacing, d (µm) 0

1 2 3 4

kb (V2 h m2)

×10⁴

= 100%·exp(1 ( V)²d_P²Q 1 1dh_S¹k_b¹) b.

Fig. 6.10: Separation efficiency η as a function of all parameters employed in this study (∆V)²d_P²Q⁻¹ε1dh_S⁻¹at different post spacingd =33–130 µm (a). With increasing post spac-ing, the factork_b of Eq. (6.6) increases (b). This decreases the responsiveness ofηtowards a change in the parameters and thus makes the separation less efficient. The fit in (b) isk_b=ad² witha=2.134×10⁻¹²V_RMSh²m⁻⁴.

with the term x = (∆V)²d_P²Q⁻¹ε_1dh_S⁻¹ which incorporates all the parameters, k_a =100 % andk_b=18.93×10³V_RMS²h m⁻². The constantk_bdefines how sensitive the system reacts to a change of one of the four key parameters. A variation of the post-to-post distance changes the parameterk_b as indicated by Fig. 6.10. In Fig. 6.10 a the results of Fig. 6.5 ford =33 µm up to 132 µm are plotted versus x instead of d. An increase in d decreases theη when the other parameters are kept constant. This is becausek_bincreases withd (Fig. 6.10 b) and thusη becomes less responsive to a change of the parametersx. This means, whenk_bis low, the system reacts fast to a change in the parameters. With increasingk_bthe response becomes slower and the system becomes more difficult to adjust by changing the parameters.

The fit throughk_b(d)isk_b=ad²(a=2.134×10⁻¹²V_RMS²h m⁻⁴). Thus, with increasing d, the possibility to tune the separation efficiency by a change in the parameters decreases quadratically. This holds until d comes close to h_S at which point these predictions fail. To put in other words: For alld,ηis 0 when the parameter combination approaches 0 andηwill always reach 100 % when the parameters (x) are high enough. However, at intermediate values, ηwill decrease with increasingd.

When pluggingε_1dinto(∆V)²d_P²Q⁻¹ε_1dh_S⁻¹, it yields(∆V)²d_P²Q⁻¹d h_S⁻². This means, a decrease in the post size could actually compensate a decrease in particle size or applied voltage one to one. This also means, that the separation can be tuned by decreasingh_Suntilh_Sreaches d or by decreasingh_Sandd in concert (decrease in xand decrease ink_b).

These findings match the results of Čemažar et al. (2016) for a contactless DEP device that have been discussed in Sec. 2.14.3. There, the post size was reduced from 100 µm to 20 µm at constant d =60 µm while simultaneously the throughput was increased by a factor of 6.

Actually, from the model, a reduction ofh_S at constant d should allow an increase inQ by a factor 25. Note, however, that h_S =20 µm is reduced to a value three times smaller than d =60 µm, thus the fit from Eq. (6.6) will over-predict the actual separation efficiency.

In Sec. 5.3.4 it was discussed that a change inv_For system size alone is not enough to scale up a separation process, because a change in v_F will require a reciprocal change in post size to compensate and this would cause a reduction of the throughput. In that discussion it was assumed that the number of posts perpendicular to the flow direction is constant (that means, a reduction in post size would cause a reduction in channel size). In contrast, in this study it was assumed that the channel width is constant and that the number of posts adapts to the spacing and post diameter. Because of this it is possible in this study to increase the throughput by decreasing the system size (h_S²/d). For a separation process employing actual macroscopic porous media, this means, at constant filtration area, the separation efficiency can be increased by an increase in porosity at constant pore size (that is, increase the pores per inch, ppi, but keep thed_poreconstant).

6.2.8 Correction of the simulated separation efficiency

In general, Eq. (6.6) together with Fig. 6.10, is sufficient to (theoretically!) describe the separation dynamics in the microchannel. The actually employed proportionality factor a (Fig. 6.10 b) required for calculatingk_b in Eq. (6.6), k_b =ad², depends on the details of the channel (for example channel height and width).

0.0 0.5 1.0 1.5 2.0 2.5 3.0 tan( 0)

0.0 0.2 0.4 0.6 0.8

(simexp)/sim

0.34 tan( 0) 0.12 R²= 0.72

Calculated, Fit

Fig. 6.11: Strictly empirical correlation between the results obtained via experiments and simulations η_sim,η_expand the intersection angleθ₀that gives a rough quantification of the finite size effect.

The fit has been obtained with a trial-and-error approach.

To obtain the actual separation efficiency from the theoretical prediction a correction ofη in cases whereθ₀is large is required. The concept ofθ₀(Sec. 6.2.5) to quantify the influence of the finite size effect is rather crude. Nevertheless, there might be a correlation betweenθ₀and the difference between the simulated and experimental separation efficiency∆η=η_sim−η_exp, so thatf η_sim,η_exp

=g(θ₀). This would allow to entirely model the separation in the channel without the need to perform any experiments after the correlation has been obtained.

Several definitions for f andg have been tested (not shown) from which∆η/η_sim∝tanθ₀ appears to show the best correlation (cf. 6.11). This proportionality is entirely empirical and has been obtained manually (that means, by testing several possible combinations until a satisfac-tory fit was obtained). The data points presented in Fig. 6.11 are compiled from all experimental data points presented so far (in total 35 points). The linear fit through the points yields

η_sim−η_exp

η_sim =0.34 tanθ₀−0.12, (6.7) so that

η_sim,cor=η_sim(1.12−0.34 tanθ₀). (6.8) TheR²is rather low,R²=0.72.

Using the correlation, a simulated result can be corrected to account for the finite size effect.

The1.12in the correction term indicates that for very lowθ₀the simulation under-predicts the experimental results, indicating that the first guess of the Re”_∼

f_CM—

was too low. The correction according to Eq. (6.8) has been applied to the results of Figs. 6.4 and 6.5 which is shown in Fig. 6.12. As expected,η_sim,cormatchesη_expmuch better as the uncorrected values. The trend of the four curves as a function ofd in Fig. 6.12 b is different from what was expected. Only Q =0.1 mL h⁻¹ (blue) shows the expected maximum ofη(d). This might be due to the to smaller number of data points plotted in both directionsd andQ (spacing ind is 30 µm). It could also be that not all curves show a maximum due to the complex dependence ofθ₀on all parameters; more data points need to be evaluated to make better informed statements.