Abstract:
Opportunities for process intensification and increased productivity
have made the field of Continuous Biomanufacturing an area of high
interest and active research. Within the purification train of producing
biologics, Tangential Flow Filtration (TFF) is typically employed after
chromatographic separations, to increase drug substance concentration,
making the process more economical and further meeting dosage
specifications. In a batch operation, concentration occurs via
recirculation of the feed material where desired output concentration is
attained through multiple pump-passes over the TFF membrane, while
steadily excluding the buffer. Single-Pass Tangential Flow Filtration
(SPTFF) enables continuity of this process by achieving similar
concentration factors through a single – pass over these membranes
while operating at low feed flow rates.
Our work elucidates the development of a mechanistic process model to
predict SPTFF performance across a relatively wide design space using a
first principles approach. The developed model is found to be accurate
for a range of high feed flow rates but is inaccurate at flow rates
below 25 L/m2/hr. At very low flow rates, small
differences in the mass transfer coefficient have been observed to
significantly alter the prediction of the retentate concentration. We
thus describe the challenges in predictive process modeling of SPTFF in
antibody biomanufacturing.
Keywords: Antibody Concentration, Continuous Biomanufacturing,
SPTFF, Ultrafiltration, Membrane Filtration
Introduction
Tangential flow filtration (TFF) is widely performed in
biopharmaceutical downstream purification processes to concentrate
proteins and perform buffer exchange (Foley, 2013; H. Lutz, 2015; Teske
et al., 2010). An ultrafiltration membrane with a nominal molecular
weight cutoff (3-5x smaller than the protein of interest) is typically
employed to facilitate full retention of the protein, enabling buffer
permeation (H. Lutz, 2015; Millipore, 1999) . The retentate stream is
recirculated into the feed vessel during the startup phase - resulting
in multiple passes through the filter assembly, to achieve the desired
output concentration, making it an inherently batch process.
Alternatively, this can be performed with the retentate stream directed
into a separate collection vessel. This configuration, referred to as
Single Pass Tangential Flow Filtration (SPTFF), has the product pass
only once over the membranes in the unit process. To achieve target
concentrations without recirculation, SPTFF can be performed at much
lower flow rates than a traditional (recirculation - enabled) TFF,
resulting in increased residence time through the filter assembly and
thus higher conversions leading to higher product concentrations
(Dizon-Maspat et al., 2012; H. Lutz, 2015; Millipore, 1999; Teske et
al., 2010).
SPTFF has been utilized in a variety of downstream processing
applications. Volume reduction of in-process products can minimize
intermediate hold-tank requirements, prevent facility fit deliberations,
and decrease filter and column sizes (Dizon-Maspat et al., 2012; H.
Lutz, 2015; Miranda & Campos, 2002; Teske et al., 2010). In addition,
it is difficult to accommodate a batch TFF into a continuous process due
to retentate recirculation that may result in product accumulation
upstream of the unit operation and disrupt product flow to the
downstream operations. SPTFF is therefore an operation that can
facilitate end-to-end continuous processing. However, using SPTFF in a
truly continuous process to achieve a target product concentration is
challenging due to flow rate constraints required to ensure continuity
of the purification process. Therefore, development of an effective
concentration step requires an in-depth understanding of the effect of
filter configuration and input flow rate on output concentrations.
Research is being increasingly conducted to understand these effects
specifically (Thakur & Rathore, 2021). The case of SPTFF is
particularly challenging when compared to batch TFF, where targets can
be relatively easily reached by operating in recirculation until the
target concentration is reached.
In our studies, we develop and demonstrate the performance of a model
capable of predicting the output concentration of the SPTFF
ultrafiltration step at various operating parameters. Such an effort is
expected to facilitate proper membrane sizing and configuration for a
wide range of processes where input flow rate and concentration are
fixed, with minimal experimental work. To this end, flux excursion
experiments were performed to generate a data set to which the stagnant
film model was applied, and additional experiments were performed to
test the model and empirically fine - tune it. As described in the
forthcoming sections, we observed model inaccuracies at low feed fluxes
that led to the evaluation of maximum conversion across a single
membrane. The behavior observed could not be accurately predicted by the
stagnant film model, so the assumptions of the model in this low flux
regime were analyzed. Through the course of this paper, we provide an
overview of implementing the stagnant film model for SPTFF and highlight
the limitations of this analysis.
Materials and Methods
All the experiments described in this paper were performed using
88cm2 Millipore Ultracel Pellicon 3 C-Screen cassettes
with a nominal molecular weight cutoff of 30 kDa (Cat. No P3C030C00). A
monoclonal antibody (147.8 kDa) at a concentration of 8 mg/mL - prepared
by purifying harvested CHO cell culture through protein A
chromatography, low pH viral inactivation, anion exchange
chromatography, and viral nanofiltration, was used as the feed material.
Various membrane configurations were formulated using diverter plates
between individual membranes (Cat. No. XXSPTFF01). The orientation of
the diverter plates dictated the series / parallel nature of the
membrane configuration, as described in Millipore’s TFF Operating Guide
(Millipore, 1999).
Membranes were assembled in a Millipore Pellicon Mini Cassette Holder
(Cat. No. XX42PMINI) and sanitized according to the manufacturer’s
guidelines. Briefly, sanitization was conducted with a flush volume of
20 L/m2 0.5M NaOH at 4 L/m2/min. The
system was held in 0.5M NaOH for an hour and flushed using 0.1M NaOH
before storage. The system was restarted by re-equilibrating the system
with appropriate buffers.
Pressure independent regions for operation were deduced through flux
excursion studies using one 88cm2 membrane. The design
space used for conducting these experiments is listed in Table 1.
Pressure was monitored at the feed inlet and retentate and permeate
outlets using PressureMAT™ sensors (PendoTECH, PMAT4R).
Steady state was assumed to be achieved when the retentate concentration
stabilized, which was measured using the SoloVPE instrument (C
Technologies, Inc.). Once steady, the output concentration (CR) and
permeate flux (J) were recorded. TMP was controlled manually using a
needle valve (Swagelok) at the retentate outlet. A predictive model was
built using this flux excursion data according to the stagnant film
model theory described in section 3.
Two SPTFF experiments were performed to validate the model developed
with a single membrane. The first experiment involved a three-membrane
configuration with alternating diverter plates separating them into
three stages of equal area. The TMP was controlled only across the final
stage using an automated pinch valve (PendoTECH Throttle Valve™
PDKT-PVT-P). The inlet pressure to the first membrane was used as the
feed pressure and the retentate pressure was measured at the final
outlet of the filter assembly. Protein was fed into the system at 68 and
116 LMH respectively. Retentate concentration at the outlet of the third
stage was recorded as the steady state value. Retentate concentration
was also measured at the first and second stages to collect additional
data for understanding the system operation. Finally, the measurement of
maximum conversion was performed by concentrating a feed at 5.4 g/L
using a single 88cm2 membrane at various feed fluxes
ranging from 5 to 60 LMH.
Theory
Tangential flow filtration is governed by stagnant film theory, which
states that permeate flux is directly proportional to the concentration
gradient. At a steady state, mass transfer of solute to the membrane
surface is equal to diffusion away from the membrane, leading to a
constant fluid boundary gel layer of a specific thickness and
concentration. Mass transfer is assumed to occur across this fluid
boundary layer (Zydney, 1997). Mass balance of this layer can be written
as:
\(J=\ -D\ \frac{\text{dC}}{\text{dz}}\) (3.1)
Integrating equation (3.1) and applying boundary conditions, we obtain
\(J\ =\ \frac{D}{\delta}\text{\ ln\ }\frac{Cw-Cp}{Cf-Cp}\text{\ \ \ }\)(3.2)
The term \(\frac{D}{\delta}\) is the mass transfer coefficient k. For an
intact, retaining membrane, Cp is zero, therefore
simplifying equation 3.2 to:
\(J\ =\ k\ ln\ \frac{\text{Cw}}{\text{Cf}}\) (3.3)
Permeate flux is also governed by the average differential pressure
across the membrane, also known as the transmembrane pressure (TMP).
\(T{MP=\frac{P_{\text{Feed}}+P_{\text{Retentate}}}{2}-P_{\text{Permeate}}}\ \)(3.4)
Flux across the membrane can be increased by increasing the TMP until a
critical pressure where mass transfer limits flux across the membrane.
This region is known as the pressure independent region. Such limiting
flux behavior is typically explained by two mechanisms: (1) an increase
in resistance to flow and (2) an increase in osmotic pressure, both due
to the tightly packed protein gel layer at the membrane surface. A
fundamental limitation to flux in this regime can be understood as the
mass transfer of protein away from the membrane, rather than pressure.
To operate in such a range, the region of pressure independence is
characterized through flux excursion experiments carried out at various
feed flow rates and concentrations.
A plot of J vs ln(Cb), where Cb is the
bulk feed concentration, in the pressure independent region at a
particular flow rate results in a straight line with slope of –k and
x-intercept of ln (Cw). Data from each flow rate
produces a line with a different slope, or mass transfer coefficient.
The effect of feed flow rate on the mass transfer coefficient k is then
characterized by using the Sherwood number.
\(Sh=\ \frac{\text{k\ L}}{D}=\ \beta.\text{Re}^{a}\text{.\ }\text{Sc}^{b}\text{.\ }{(\frac{L}{D})}^{c}.({\frac{\mu_{b}}{\mu_{w}})}^{d}\)(3.5)
A common assumption to determine the mass transfer coefficient from the
above equation is that the fluid properties at the membrane wall and in
the bulk stay constant across flow rates (Teske et al., 2010).
Therefore, all the terms in equation 3.5 except the Reynolds number are
constant and the mass transfer coefficient at different flow rates can
be expressed as:
\(\frac{k_{1}}{k_{0}}=({\frac{Q_{1}}{Q_{0}})}^{a}\) (3.6)
The value of ‘a’ can be empirically determined resulting in a
relationship between k and feed flow rate, which can be used to predict
the conversion of a process at various bulk concentrations. The value of
Cw is calculated by taking the exponential of the
x-intercept in the J vs ln (Cb) plot. The conversion for
a membrane separation process is defined as the ratio of the permeate
flux to the feed flux.
\(\varnothing=\ \frac{\text{J.A}}{q_{f}}=\ \frac{J}{Q}=\ \frac{k}{Q}\text{ln\ }\left(\frac{C_{w}}{C_{b}}\right)\ \)(3.7)
The equation for the final retentate concentration is derived using mass
balance:
\(C_{R}=\ \frac{C_{f}}{1-\ \varnothing}\) (3.8)
Results and Discussion
Flux Excursions
Flux excursion experiments were performed across a single
88cm2 cassette by varying feed flow rate, TMP, and
feed concentration, as described in Figure 1. These flux excursion
experiments for developing SPTFF are identical to those required for
batch TFF, except they are conducted at much lower feed fluxes. We
observed that the permeate flux, J, was independent of TMP above 15 psi
for all conditions. Due to the pressure drop across membrane cassettes,
the TMP on the last membrane was slightly lower than the preceding
membranes. For a multi-stage setup, the TMP across the final stage was
controlled by an automated back pressure valve at the outlet of the
final stage. Therefore, to ensure all stages operate in the pressure
independent region, the TMP of the third stage was set to 15 psi.
Operating at the edge of this pressure independent region maximized our
permeate flux and the concentration factor while minimizing membrane
fouling (van Reis et al., 1997; Zydney, 1997). Additionally, it
implemented an ease of operation and process development, as well as
reduced sensitivity to pressure fluctuations.
Stagnant Film Modeling – First Principles Approach
Predictive Model Development
To build the stagnant film model, the measured permeate flux at 15 psi
for each condition was plotted against the natural log of feed
concentration at each flow rate tested (Figure 2). The slope and
intercepts were used to determine the mass transfer coefficient and wall
concentration respectively at each feed flux. The Sherwood number
approximation (Equation 3.5) was then used to make appropriate
predictions of the mass transfer coefficient k, using the stagnant film
theory, simplified as equation 3.6, assuming constant bulk fluid
properties. The values of k evaluated using equation 3.6 have been shown
in Figure 3A. The initial properties (Q0 and
k0) were chosen from the data points generated from our
experiments. For the analyses, we set Q0 to 82 LMH and
k0 to its corresponding mass transfer coefficient. The
exponent ‘a’ was determined empirically from the least-squared best-fit
analytical value of the remaining flux excursion data (Teske et al.,
2010).
This approach and model required approximating k and wall concentration
(Cw) to predict the overall conversion of the process
(Equation 3.7). The x-intercept from the J vs ln(Cw)
plot was used as the value of the wall concentration. Based on the
stagnant film theory, Cw was assumed constant across all
flow rates (Teske et al., 2010; Zydney, 1997). However, it was then
observed that Cw had an inverse linear relationship with
feed flow rate, as shown in Figure 3B. One of the possible hypotheses
for this observation is the fact that the SPTFF is operated at lower
cross flow rates than batch TFF. This reduced shear appears to lead to
higher deposition of concentrated material on the membrane surface.
Equation 4.1 is the complete stagnant film model for permeate flux
across a single membrane at a given input concentration and flow rate.
\(J=k_{o}\left(\frac{Q}{Q_{o}}\right)^{a}\text{ln\ }\left(\frac{mQ+b}{C_{b}}\right)\ \)(4.1)
Where, mQ+b is the linear fit of the Cw vs. Q graph in
Fig. 3B
This model, combined with equation 3.6 and 3.7, was then extended to a
multi-stage SPTFF by using the output concentration and flow rate of one
stage as the input to the next. This led to the construction of a
predictive model that could calculate output concentration based on
input flow rate, feed concentration, number of stages, and relative
surface area of those stages. Figure 4 shows the predicted output
concentration curves for a configuration with stages of the same
relative area and a feed concentration of 8.2 mg/mL.
To validate the model predictions, a 3-stage SPTFF with
88cm2 membranes in series was first conducted with a
feed concentration of 8.2 mg/mL at feed fluxes of 116 LMH and 68 LMH.
The output concentration measured from each stage is shown in overlay
with the model predictions in Figure 4. The stagnant film model with the
simplified expression for k(Q) derived from the Sherwood number was
observed to work well within the range of feed fluxes that the model was
built on, i.e., 34 - 136 LMH. The model could also be used to accurately
predict the output concentration of different membrane configurations at
different feed flow rates and concentrations within this range.
Model Prediction at Low Flow Rates
Accuracy of the model at lower feed fluxes relevant to a continuous
process was then investigated. As shown in Figure 4, an exponential
increase was observed in the model output concentration at the first
stage, around 25 LMH. This inaccuracy was observed to prevail at low
feed fluxes and high conversions. This phenomenon is further
demonstrated in Figure 5A, where predicted conversion and concentration
factors at low flow rates are compared. High conversions of about 0.85
and above occurred at low flux and feed concentrations
(Cf) into a single stage. These conditions led to a high
degree of membrane polarization (Cw/Cb),
which in turn increased the predicted permeate flux (Equation. 3.7) to
be greater than the feed flux (Q) - which is physically erroneous due to
conversion values > 1 (Equation 3.7).
Figure 5B shows the predicted conversion as a function of feed flux at
different input concentrations. The conversion was observed to be
greater than 1 at a flux of 20 LMH at a Cf of 4 mg/mL,
whereas at Cf of 15 mg/mL it did not exceed the same
until it reached 5 LMH. This observation clarified the fact that low
feed concentrations led to a higher polarization term. However, the
effect of the polarization on the permeate flux calculation is
diminished by the natural log operation. Therefore, it is likely that
this sharp increase in conversion at low feed fluxes is due to an
overestimation of the mass transfer coefficient. However, these
predictions were extrapolated outside the experimental conditions used
to build the model, as the lowest feed flux tested was only 34 LMH.
Therefore, additional flux excursion experiments were performed at 7.2
and 18.7 LMH to account for behavior at lower feed fluxes (Figure 6).
Upon incorporation of additional low flow rate data into the model, the
prediction of the mass transfer coefficient was not significantly
affected (Fig. 6B). However, large increases in Cw were
seen at low flow rates thereby further increasing the polarization term
and decreasing the accuracy of the model (Fig. 6C). This data confirms
that the original model was not inaccurate due to a lack of data points
within the design space but that it is only accurate for moderate to
small values of Cw/Cb. This confirms the
constraint on polarization concentration values for the stagnant film
model as previously reported (Zydney, 1997). At these low flow rates,
the estimation of k and Cw as a function of flow rate
alone is no longer appropriate.
To corroborate this notion, a series of experiments were conducted at
decreasing feed fluxes to determine the maximum possible conversion for
a single feed concentration. A feed flux range of 5 to 60 LMH was
investigated across one 88cm2 membrane with a feed
concentration of 5.45 mg/mL. These results are shown in Figure 7. The
model fits the experimental data accurately above 20 LMH. Below this
point, the modeled conversion continued to exponentially increase while
the experimental conversion plateaued. However, the experimental output
concentration continued to increase significantly, approaching the
theoretical maximum Cw for this feed concentration. This
was possibly due to small increases in conversion resulting in large
increases in output concentration at low fluxes. For example, at these
experimental conditions, the difference between a conversion of 0.96 and
0.97 is an output concentration of 135 and 180 mg/mL respectively. Thus,
output concentration for low feed concentration systems was seen to be
very sensitive to flow rates below 20 LMH. Therefore, any small errors
in predicted conversion led to vast changes in predicted concentration.
The conversion plateau seen in Fig 7A could not be predicted by the
stagnant film model. Furthermore, the nature of the plateau would likely
vary with feed concentration. For example, at a feed concentration of 15
mg/mL in this experiment, the maximum output concentration would be
around 390 mg/mL, which is not achievable in practice. To accurately
characterize the plateau behavior and improve the accuracy of the model
at low feed fluxes, this experiment would need to be repeated with a
range of feed concentrations. This data could be used in combination
with stagnant film modeling at higher feed fluxes to extend the range of
prediction.
Limitations of Stagnant Film Model
Applying certain corrective measures to the inaccuracies of k and
Cw is challenging due to the interdependency of both
these parameters. Firstly, Cw cannot be easily or
reliably measured directly, and our data suggests that it varies with
the feed flux and is capable of increasing the error margins of model
predictions. Secondly, deriving k from Sherwood number (Equation 3.5)
encompasses Cw information and also includes terms that
are difficult to measure accurately, such as diffusivity.
As an exercise, it was assumed that k was accurately deduced from the
slope of the J vs ln(Cb) graph. Cw was
then calculated by rearranging Equation 3.3 into Equation 4.2 and
solving for J from the experiment in Figure 7. This approach was
inspired by van Reis et al. who combined the stagnant film and osmotic
pressure models to create a constant Cw ultrafiltration
process control (van Reis et al., 1997).
\(C_{w}=C_{b}*exp(\frac{J_{\exp}}{k_{\text{model}}})\) (4.2)
Results of this analysis are shown in Figure 8A. According to this
calculation, the Cw at 5 LMH was 59 mg/mL. This could
not have been the case since the output concentration at this flux was
measured to be 112 mg/mL (Figure 7B).
Conversely, the measured wall concentrations from the flux excursion
experiments were assumed to be true. In other words, it was assumed that
Cw could be accurately derived from the x-intercepts of
the J vs ln(Cb) graph (Figure 6). k was then calculated
by rearranging equation 3.3 into equation 4.3 using the flux values
determined in Figure 7.
\(k=\frac{J_{\exp}}{ln(\frac{C_{w,\ model}}{C_{b}})}\) (4.3)
The results of this analysis are shown in Figure 8B. This calculation
supports the idea that the method for calculating k with the simplified
model (equation 3.6) overpredicts k at low flow rates. These exercises
reinstated the higher accuracy of the stagnant film model at high
fluxes, which decreased as the low flow rate regime was entered.
Sherwood Number Approximation
Thus far, we worked with a simplified equation for k based on equation
3.6, allowing us to neglect the effects of concentration-dependent
terms, such as viscosity and diffusivity. This approach only granted our
model accuracy down to about 25 LMH. With the large range of calculated
wall concentrations, it is not valid to assume that fluid properties
remain the same. Specific to our operating regime, Cwlikely has a more complex dependency on other operating parameters.
Therefore, at low flow rates, we hypothesize that k would be a function
of the bulk fluid properties as well.
To investigate this phenomenon, a viscosity curve for mAb A was
generated as a function of concentration up to 260 mg/mL (Figure 9). An
exponential increase in viscosity with increasing concentration was
observed as expected. At low feed fluxes, highly concentrated protein at
the membrane wall can be imagined to be very viscous compared to the
bulk solution. However, producing sufficient protein at high enough
concentrations to mimic wall concentration is nearly impossible. Even if
it could be done, then the accuracy of the viscosity measurements would
be the limiting factor. Additionally, mass transfer coefficient
calculations often leave out the dependence of the diffusion coefficient
on concentration (Foley, 2013). High concentration and viscosity at the
surface of the membrane ultimately affect the diffusivity of the
protein. Therefore, the diffusivity term used in the Sherwood number
calculation should be a function of concentration. However, measuring
diffusivity would have the same limitations as viscosity.
Furthermore, even if these terms could be accurately measured, they rely
on knowing the value of Cw at a given feed flux and
concentration, which is difficult to measure or estimate without already
knowing k. Calculating the mass transfer coefficient to develop a
predictive model using the complete Sherwood number can be tedious.
Alternatively, data-based empirical approaches can be adapted to predict
these concentrations with greater accuracy than first principles-based
approaches. However, such models would need to be restricted to similar
experimental conditions, setups, and process parameters to achieve a low
degree of error on the predictions.
Conclusions
The stagnant film model has been widely used to describe the behavior of
filtrate flux and mass transfer for the ultrafiltration of
biopharmaceuticals (Foley, 2013; Teske et al., 2010; Zydney, 1997). In
this work, we demonstrate that the model in description was observed to
be highly accurate down to a feed flux of about 25 LMH, below which it
could no longer accurately predict the system’s behavior. A potential
explanation for the lack of predictability of the model arises from the
use of a simplified version of the Sherwood number to estimate the mass
transfer coefficient as a function of feed flux. This common
simplification assumes constant fluid properties in the bulk and at the
membrane surface. Additionally, the mass transfer coefficient also
serves as a lumped parameter that captures the properties of the
membrane as well as the protein. However, since the wall concentration
and viscosity vary so widely across feed flux, this assumption could
lead to the discrepancies we see with the model predictions. Thus, the
model fails at low feed fluxes and feed concentrations due to high
membrane polarization and a lack of accurate understanding of the
effects of each of these variables on each other - and the system - at
these conditions.
In order to utilize the Sherwood number to accurately calculate the mass
transfer coefficient, viscosity and diffusivity must be determined as a
function of concentration. These values are difficult to accurately
measure and can introduce significant errors in the calculation. In
addition, Cw must be known in order to calculate
viscosity at the wall. Determining Cw using the stagnant
film model negates any effect of feed concentration. Furthermore,
estimating Cw by rearranging the stagnant film model
equation requires very accurate estimates for k. Thus, even if the
concentration-dependent terms could be measured accurately, the complete
Sherwood calculation depends on knowing Cw.
Another possible explanation could be the fact that the stagnant film
theory, equation 3.1, which is derived from Fick’s second law, neglects
the axial diffusion terms to derive a finite solution to the following
differential equation:
\(V_{x}\frac{\partial C}{\partial x}+V_{z}\frac{\partial C}{\partial z}=D\left[\frac{\partial^{2}C}{\partial x^{2}}+\frac{\partial^{2}C}{\partial z^{2}}\right]\)(5.1)
where x is the direction of axial diffusion and z is the direction of
convective transfer. To provide an analytical solution to this equation,
axial diffusion and the velocity distribution in the x direction are
neglected (Miranda & Campos, 2002). However, it also needs to be taken
into consideration that at high conversion observed at lower feed flow
rates the bulk concentration increases and the velocity decreases
significantly along the length of the membrane (H. Lutz, 2015). It can
be hypothesized that axial diffusion becomes significant and should not
be neglected in eq. 5.1. The contribution of both axial and convective
mass transfer can be a possible explanation to the plateauing of the
conversions at low fluxes, as their consideration would result in lower
concentration factors than predicted by the stagnant film model. Lastly,
throughout steady state, concentration gradients within the physical
system of operation, protein - protein interactions, buffer effects and
specific geometries of membranes on the concentration product may impact
the relationship between variables used in to build the model, which
have not been considered in our studies. Jabra et. al, considered such
variables to model and predict output concentrations within a different
set of inlet flow rates (Jabra et al., 2021). While accounting for
system variations at low flow rates such as the range in question within
this paper is possible, doing so would require a battery of additional
studies and severely increase the complexity of the model.
In summary, the stagnant film model can be used to accurately predict
the output concentration of SPTFF processes at reasonable feed fluxes,
which in this work were above 25 LMH. This facilitates proper membrane
sizing and configuration for a wide range of processes with minimal
additional experimental work to perform. This is particularly attractive
for SPTFF based ultrafiltration in continuous biomanufacturing where a
target product concentration must be achieved under feed flow rate and
concentration constraints resulting from upstream unit operations.
Therefore, this model can be used to design membrane configurations
capable of achieving a target concentration across a range of feed
concentrations and flow rates with minimal process development. However,
we show that the typical assumptions made to utilize this model fail at
combinations of low feed fluxes and concentrations. In this regime,
there appears to be a more complex relationship between the mass
transfer coefficient, wall concentration, feed concentration, and flow
rate, than the model can currently capture. At lower fluxes, an
empirical characterization of the design space or alternative modeling
approaches can be utilized to continue achieving the benefits of
model-based concentration prediction.
Acknowledgements
The authors would like to acknowledge
Megan Le for executing viscosity measurements, Bryan Chan, Nidhi Thite,
and Sushmitha Krishnan for SPTFF development efforts, and Antonio Ubiera
for providing a review of the manuscript.
Conflict of Interest
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