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Experimental evaluation and transient simulation of detergent transport in household vertical axis washing machines
Accepted Manuscript Title: Experimental evaluation and transient simulation of detergent transport in household vertical axis washing machines Author: Luiz G.C. Campos Christian J.L. Hermes PII: DOI: Reference:
S0263-8762(16)30026-0 http://dx.doi.org/doi:10.1016/j.cherd.2016.03.021 CHERD 2233
To appear in: Received date: Revised date: Accepted date:
26-10-2015 13-2-2016 18-3-2016
Please cite this article as: Campos, L.G.C., Hermes, C.J.L.,Experimental evaluation and transient simulation of detergent transport in household vertical axis washing machines, Chemical Engineering Research and Design (2016), http://dx.doi.org/10.1016/j.cherd.2016.03.021 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Highlights
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Detergent transport in vertical axis washing machines is studied
A purpose-built test rig was used to gather data following a 24 factorial design
Agitation level and detergent insertion showed to be the most influencing factors
A semi-empirical model was developed and validated against experimental data
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Experimental evaluation and transient simulation of detergent
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transport in household vertical axis washing machines
Luiz G. C. Campos 1,2, Christian J. L. Hermes 2,*
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1
Advanced Product Development Fabric Care, Electrolux Major Appliances Latin America
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81520900 Curitiba, PR, Brazil, +55 41 3371 6109, [email protected] 2
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Laboratory of Thermodynamics and Thermophysics, Federal University of Paraná 81531990 Curitiba, PR, Brazil, +55 41 3361 3239, [email protected]
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* Corresponding
author
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ABSTRACT
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Detergent transport between the compartments and through the garments during the washing
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process in household top-load washing machines is investigated. An experimental methodology
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was put forward for evaluating the time evolution of the detergent concentration in each
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compartment, and also in a probe enveloped by a cotton-linen that emulates the garments. The
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experiments were carried out following a full factorial experiment, pointing out the detergent
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insertion location and the agitation speed as the rate-determining factors. A lumped transient
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model was put forward for the convective transport of detergent. The set of ODEs was solved
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numerically through an adaptive time-step algorithm. The closing parameters required by the
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model were reduced from the experimental data, thus providing the model with a semi-
37
empirical character. The model predictions for the time evolution of the detergent concentration
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and the experimental data agreed to within the range of the measurement uncertainties (~15%).
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Keywords: mass transfer; experimental analysis; transient simulation; detergent transport;
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washing machine
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Abbreviated
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title:
Detergent
transport
in
top-load
washing
machines
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NOMENCLATURE
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Roman
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A
agitation level
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Af
face area, m2
48
C
detergent concentration, kg m-3
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D
diffusivity of detergent in water, m2 s-1
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Deff
effective diffusivity of detergent, m2 s-1
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F
independent variables of the factorial experiment
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I
detergent insertion
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l
length, m
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S
source term, kg m-3 s-1
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t
time, s
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U
velocity, m s-1
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V
volume, m3
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X
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Y
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z
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Ac ce p
flow rate, m3 s-1
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63
Greek
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α
initial mass fraction of detergent in the clearance dependent variables of the factorial experiment mass transfer direction
volumetric fraction of detergent to the probe
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Subscripts
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0
load (but the probe)
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1
first control volume of the probe
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bb
between bowls
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cl
clearance
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cv
control volume
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dn
control volume at downstream
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i
i-th control volume of the probe
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N
last control volume of the probe
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nb
neighbour (upstream, downstream)
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ob
outer bowl
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pb
probe
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sb
spin bowl
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up
control volume at upstream
cr us
d
Superscripts
te
dimensionless variable of the factorial experiment (see equations 1 and 2)
Ac ce p
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INTRODUCTION
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A vertical axis (also known as top-load) washing machine is comprised of a water reservoir (the
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outer bowl), in which is placed a perforated drum (the spin bowl) that holds the garments which
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are moved together with the washing fluid (detergent-water solution) by an electric motor-
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driven rotating blade named agitator, as depicted in Fig. 1. The outer bowl is supplied with water
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by means of a feeding system comprised of an inlet valve, and a water-level sensor. The washing
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process in top-load washing machines involves mechanical and chemical phenomena. The
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former is regarded with fluid flow on and within the garments, thus promoting the stresses that
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remove the dirt. The chemical action, on the other hand, depends on the transport of detergent
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dissolved in water from one compartment (i.e., outer bowl) to the other (i.e., spin bowl), and
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from the latter to the garments. Once in the textile, the detergent helps to loosening the dirt,
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which can be dragged off by the mechanical action.
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Therefore, the washing process depends on a great variety of parameters, including the
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geometries of the spin bowl and the agitator, the washing stroke (i.e., the time evolution of the
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motor torque, angular speed, and angular direction), the water level and temperature, the
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amount of detergent, to cite just a few. There are different standards used around the globe for
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evaluating the washing efficiency (Bansal et al., 2011), most of them accounting for the water
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and energy consumption, the washing effectiveness, and the wear of the textile (e.g., AHAM
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HLW-1-2010, AS/NZS 2040.1, and IEC 60456:2010). The high number of geometric and
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operational parameters that interact non-linearly to each other together with the measurement
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uncertainties and the degree of subjectivity of the standardized processes turn the product
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design decision-making into a complex process, sometimes depending more on heuristics than
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on engineering calculations and on experimental observations (Ward, 2003). To aggravate, the
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open literature in this field is still scarce.
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In a pioneering work, van den Brekel (1987) modelled the detergent transport in horizontal axis
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(drum type) household washing machines considering a multi-reactor lumped-approach.
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Experiments were carried out using KCl as the tracer in place of detergent not only to gather the
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closing parameter required by the model, but also to be used in the model validation exercise. A
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decade later, van der Donck (1997) simulated the mechanical washing in drum type washing
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machines considering the compression of wet textiles. The model was validated against
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experimental data obtained for pure water. Charrette et al. (2001) investigated the mass transfer
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in cotton-linens, coming up with a model for the washing process in an industrial facility
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comprised of a series of water tanks. The model was compared with experiments in which the
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detergent was replaced by NaCl as the tracer. Warmoeskerken et al. (2002) assessed the
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capabilities of ultrasonic technologies to boost the washing process, and found out that
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ultrasound might promote a higher rate of NaCl removal from the inner portions of the garments
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in comparison to regular washing processes. It is worth of note that, in all above referenced
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studies, a salt (KCl, NaCl) was used as the tracer instead of detergent in the mass transport
124
analysis, which is not a realistic operating condition. More recently, Janacova et al. (2011)
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proposed an analytical model to optimise the washing process in a vertical axis washing machine
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aiming at reducing the consumption of washing fluid and energy. The detergent type was not
127
specified and no model validation exercise was reported. In addition, the model did not account
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for the convective transport of detergent, which turns out to play an important role in the
129
washing.
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To the authors’ best knowledge, studies of the washing process using real detergents or even
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standardized detergent blends, particularly in top-load washing machines, have not been found
133
in the open literature. Therefore, the present study is aimed at investigating the transient
134
transport of detergent in vertical axis washing machines. For doing so, a purpose-built washing
135
machine, with strict control of the operating conditions, such as the water volume (V), the
136
detergent concentration (C), the detergent insertion location (I), and the agitation speed (A) was
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carefully instrumented to gather data of detergent concentration over time. The experiments
138
were designed according to a factorial approach, in such a way that the effects of the washing
139
parameters (V, C, I and A) and their interactions on different response variables could be figured
140
out. A simulation model was additionally put forward based on detergent mass balances
141
between the compartments of the washing machine and within the garments. The closing
142
parameters required by the model were obtained from the experiments, providing it with a
143
semi-empirical character.
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EXPERIMENTAL WORK
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Testing Facility
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The experimental facility emulates a household top-load washing machine, as depicted in Fig.
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2.a. Differently from the washing machines commercially available on the market, the rig was
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designed and constructed to allow strict control of the working (e.g., agitator speed, spin
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direction and angular range, water volume, detergent concentration and insertion loci) and
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geometric (e.g., spin bowl and agitator geometries) conditions, as illustrated in Fig. 2.b. In
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addition, hoses were fixed on both the outer bowl (water reservoir) and the spin bowl (rotating
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drum), with the intake section placed ~50 mm from the bottom of the spin bowl, as can be seen
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in Fig. 2.c, in order to collect samples of the washing fluid, i.e., the detergent-water solution.
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An additional probe was manufactured using a cotton-linen with 800 mm2 (i.e., a typical pillow
159
case of the IEC 60456:2010 standard), folded four times to emulate a piece of garment being
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washed. Again, a hose was installed with one end inside the envelope to collect samples of the
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washing fluid, as shown in Figure 3. The edges of the probe were sealed with hot-melting
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adhesive (HMA), whereas the hoses are made of thermoplastic material (TPU).
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The concentration was measured using a UV-spectrophotometer to detect the absorbance of the
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washing fluid (which is related to the concentration due to the Lambert-Beer law), following the
166
procedure recommended by the AHAM HLW-1-2010 standard. Accordingly, the detergent blend
167
used in the experiments is also the one specified by the AHAM HLW-1-2010 standard, whose
168
main components are: sodium aluminosilicate solids (36.3%), sodium carbonate (14.9%), linear
169
alkylbenzene sulfonate (11.3%), nonionic surfactant (6%) and silicone (5%). The water supplied
170
to the facility was also controlled, with ~45 ppm of CaCO3 and temperature of ~20°C since the
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water hardness might reduce the cleaning potential of the detergent as calcium and magnesium
172
ions create bonds with surfactants. In addition to the probe (i.e., the folded cotton-linen,
173
hereafter named just probe), an extra load of 4 kg of 800 mm²-pillowcases was adopted. In all
174
cases, the washing fluid was collected manually by means of 60-ml syringes.
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The test itself, i.e. agitation and the measurements, were initiated 15 s after the detergent
177
insertion. Samples were acquired from the outer bowl, the spin bowl, and the probe by three
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different operators. In order to capture the start-up transients properly, the sampling interval
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was 15 s from the test initiation until the concentration equilibrium was reached between the
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outer and the spin bowl. From this point on, larger sampling intervals were adopted. The
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measurement errors associated with the instruments are lower than 1%. However, the
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experimental uncertainties are strongly affected by the manual sampling, which have been
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estimated to be ±15% of the measured value based on test repetition analyses. Significant foam
184
formation was not observed. More details can be found in Campos (2015).
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d
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Test Plan
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The experiments were carried out following a 24 (2-level, 4-factor) full factorial design. Such an
189
approach was employed to figure out the effects of the washing parameters (agitation speed, A;
190
detergent concentration, C; detergent insertion locus, I; and water volume, V) and their first and
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higher order interactions on two response variables: (i) the time to reach equilibrium between
192
outer and spin bowls; and (ii) the relative difference between the detergent concentration in the
193
probe and the average between outer and spin bowls after 30 minutes. Table 1 summarizes the
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lower (-) and upper (+) levels of the factorial experiment, whereas Table 2 details the test
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conditions for all 16 test runs.
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The regression model adopted in this study is as follows (Box et al., 1978):
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where
is the dimensionless response variable in the interval [-1,1],
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calculated from the best fit method, and
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1,1], calculated as follows:
are the coefficients
are the dimensionless factors also in the interval [-
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(1)
(2)
204 205
It is worth of note that each -coefficient provide the sensitivity of the response variable to its
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respective factor.
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SEMI-EMPIRICAL MODEL
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The mathematical model was based on the work of van den Brekel (1987), where all mass
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transfer processes that take place in the washing machine were treated as one-dimensional, as
212
follows:
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(3)
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where C is the detergent concentration [kg/m3], z the mass transfer direction [m], S is a term
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that can be a source of a sink [kg/m3s], D is the diffusion of detergent in water [m2/s], U the flow
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velocity in the z direction [m/s],
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[kg/m3s], and
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of C in the control surfaces [kg/m2s]. By means of a finite-volume approach (Patankar, 1980),
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equation (3) can be integrated in each control volume of the domain, yielding
te
is the transient variation of C in each control volume
and UC stand respectively for the diffusive and the convective transport
Ac ce p
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d
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(4)
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where the index cv refers to the control volume under analysis, whereas the indexes up and dn
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are related to the control volumes upstream and downstream of cv, respectively. Also,
and
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are the volume and the characteristic length of the control volume, whilst
is the flow rate
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of washing fluid from (to) the control volume to (from) its neighbours, denoted with index nb.
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Figure 4 illustrates the detergent flowsheeting. Washing fluid is transferred from the outer bowl
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to the spin bowl when the agitator is off, but follows the other way round when the agitator is on.
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The same behaviour is expected for the transport between the spin bowl and the garments,
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divided into two control volumes, namely the probe (where detergent diffusion takes place) and
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the load. Detergent is also transferred from (to) the outer bowl to (from) the clearance formed
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between the outer and spin bowls, where decanted detergent may accumulate.
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The model is aimed at predicting the time evolution of the detergent concentration in each of
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these control volumes (all but the probe treated as even lumps) in order to come out with the
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time to reach steady-state conditions, which must be as low as possible for the sake of washing
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performance. In addition, the transport coefficients (e.g., flow rates, and effective diffusivity in
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the textile) were not modelled through a first-principles approach, but best fitted to the
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experimental data thus providing the model with a semi-empirical character.
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Bearing in mind that the convective transport overrules the diffusive effects due to high Péclet
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numbers, equation (4), when applied to the outer bowl, the clearance, the spin bowl, and the
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load, yields
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(5)
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Page 11 of 36
ip t
(6)
(8)
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(7)
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where the indexes ob, sb, cl and 0 stand for the outer bowl, spin bowl, clearance and load,
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respectively,
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outer bowl and the clearance,
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load), being
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flow rate from the inner part of the probe to the spin bowl (due to leakage, e.g. the fluid taken
251
away by sampling). Also,
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the last (inner) layer of the probe, were diffusion takes place.
Ac ce p
is the flow rate between bowls (outer and spin),
is the flow rate from the spin bowl to the garments (probe and
the fraction directed to the load and
and
is the flow rate between the
to the probe (see Fig. 4), and
is the
stand for the detergent concentration in the first (outer) and
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The probe was then divided into N control volumes from the outer layer (1) to the inner layer
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(N). Figure 5 presents a schematic representation of the finite-volume formulation for the probe,
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where diffusion takes place from layers 1 to N, and convection is observed in the outer layer
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(from spin bowl to the probe) and in the inner layer (from the probe back to the spin bowl), as - 12 -
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illustrated in Fig. 4. The mass transport equations for the outer layer (1), the inner layer (N), and
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any layer in between (i) were obtained from equation (4), yielding
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(10)
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(11)
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(9)
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where the diffusivity of detergent in water D was replaced by the effective diffusivity of
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detergent in the textile, Deff. Also, Af refers to the face area of the probe, whereas l is the length of
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the i-th control volume. It is worth of mention that mesh independent results were achieved for
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N≥16 control volumes.
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Equations (5) to (11) provide a set of N+6 ordinary differential equations (ODEs) that must be
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integrated over time to come out with the time evolution of the detergent concentration in the
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probe, and in the bowls. In this work, the ODEs were solved by the Bogacki and Shampine’s
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(1999) so-called ODE23 method, whose main feature is an adaptive time-step calculation. - 13 -
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One should also note that the geometry (volumes, areas, lengths) must be provided to the model,
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whereas the transport parameters
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model to the experimental data. Table 3 summarizes the figures achieved for each test condition.
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The values of and Vo are 3.510-3 and 1.510-6 m3s-1, respectively. An additional empirical
276
information required by the model is the fraction of detergent that remained decanted before
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the agitation initiation, denoted as X. Hence, a mass fraction X must be artificially added to the
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clearance at the test beginning to ensure mass conservation. It is worth of note that X stands for
279
the mass fraction of detergent which must be added in the dead volume to ensure a proper mass
280
conservation, thus representing the amount of detergent decanted after insertion.
,
,
and
were obtained by best fitting the
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,
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The model therefore relies on seven empirical parameters that must be reduced from the
283
experimental data.
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RESULTS AND DISCUSSION
te
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Typical experimental results for the time evolution of the detergent concentration are illustrated
288
in Figures 6 and 7. One can see in Figure 6 the influence of the initial condition (i.e., detergent
289
insertion either in the outer, Fig. 6.a, or in the spin bowl, Fig. 6.b) on the detergent concentration
290
over time. When detergent is inserted in the outer bowl (run#3 in Tab. 2), the detergent
291
concentration in both bowls decrease until reaching an inflection point, as illustrated in Fig. 6.a.
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Such a behaviour is attributed to the detergent decantation in the initial stages that accumulate
293
in the clearance. From ~30 s on, the convective transport from the clearance to the outer bowl,
294
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, and from the latter to the spin bowl,
, make the concentration in both bowls to increase
295
exponentially until steady-state conditions are achieved after ~6 minutes. Figure 6.b shows the
296
results obtained in the case where detergent is inserted in the spin bowl (run#4 in Tab. 2). One
297
can see that no inflection (i.e., decantation) was observed: the concentration in the spin bowl - 14 -
Page 14 of 36
decreases while the concentration in the outer bowl increases until steady-state conditions are
299
reached after ~1 minute. This is due to the flow patterns observed in the spin bowl, where flow
300
agitation takes place. It is noteworthy that, in the very beginning of the experiment, a low
301
concentration was observed in the spin bowl, which is due to the distance between the insertion
302
locus (at the top) and the sampling intake (~50 mm from the bottom).
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Figure 7 shows the effects of the agitation profile (smooth and strong) on the time evolution of
305
the detergent concentration. When detergent is inserted in the outer bowl, decantation takes
306
place independently of the agitation profile. Looking at the detergent concentration in the outer
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bowl (see Fig. 7.a), one can see that the test condition under strong agitation (run#11 in Tab. 2)
308
reached the steady-state after 8 minutes, whereas the test conditions under smooth agitation
309
(run #9 in Tab. 2) took ~25 minutes. Figure 7.b depicts the time evolution of the concentration
310
in the probe for the same testing conditions. In both cases, the concentration increases as time
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goes, which is ruled by the diffusive transport within the probe and the convective transport
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from the spin bowl to the probe. It can be seen that the strong agitation enhances the detergent
313
transport to the probe, resulting in higher mass transfer rate. In both cases, the initial
314
concentration (1.25 g/L) was not reached, thus suggesting that part of the detergent mass
315
remained decanted in the clearance, which was confirmed by visual inspection after the tests.
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cr
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In the cases where the detergent is inserted in the spin bowl, as illustrated in Fig. 7.c, advection
318
rules from the very beginning. On the one hand, for a strong agitation (run#8 in Tab. 2), the
319
concentration measured at the bottom of the spin bowl is initially low due to the distance
320
between the insertion and the measurement loci, but higher concentrations are observed from
321
15 s on due to the flow patterns promoted by the agitator inside the spin bowl. The
322
concentration thus decreases until steady-state conditions are achieved after 2 minutes. On the
323
other hand, for a smooth agitation (run#6 in Tab.2) the transport of detergent from the top
324
(where it is inserted) to the bottom (measurement intake) slowly increases the concentration in
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Page 15 of 36
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the spin bowl, where the load is placed, since mass diffusion overrules the advection at low
326
speeds. The concentration thus increases at low rates until a peak and then decreases until
327
steady-state is reached after 4 minutes.
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Figure 8 illustrates the results of the factorial experiment in the form of the coefficients of
330
equation (1). The factors are: (A) agitation speed, (C) initial detergent concentration, (I)
331
insertion locus, and (V) water volume. The response variables are: (i) time to reach equilibrium
332
between bowls (white bars), and (ii) relative difference between the detergent concentration in
333
the probe and the average concentration between bowls after 30 minutes (black bars). It can be
334
observed that insertion, volume and agitation have a strong influence on the time to reach the
335
equilibrium. Both insertion and volume have positive effects, meaning that a higher water
336
volume and the insertion in the spin bowl tend both to increase the mixing time, which is an
337
undesired effect. Agitation, on the other hand, promotes a negative effect: a higher agitation
338
decreases the time to equilibrium, which is expected due to the augmented advection between
339
bowls and to the probe. High order interactions plays a minor role, except the pair agitation-
340
volume, which is due to the flow velocities achieved for the higher agitation level. One can also
341
note in Fig. 8 that insertion and agitation play dominant negative roles on the relative difference
342
after 30 minutes, i.e., both strong agitation and insertion in the spin bowl tend to reduce the
343
concentration difference.
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ip t
329
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The model parameters (
,
,
,
,
and X) were best fitted to the experimental data for
346
each of the running conditions of Table 2, whereas the parameter is fixed and known from the
347
experiments. On the one hand,
348
as the diffusive transport of detergent through the probe layers overrules the convective
349
transport to the outer layer. On the other hand, the other parameters varied substantially from
350
one test condition to another and, therefore, were fitted to the washing conditions used the same
351
procedure adopted for reducing the factorial experimental data (equation 1). The fitting
did not experience any practical change for all test conditions,
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Page 16 of 36
352
coefficients, which describe the sensitivity of the model parameter to the washing parameter (A,
353
C, I and V) are shown in Fig. 9. One can see that the flow between bowls,
354
insertion and agitation, the latter acting positively and the former negatively, indicating that
355
strong agitation and insertion in the outer bowl both increase the flow rate between bowls. The
356
flow rate from the probe to the spin bowl,
357
effect, followed by the insertion and the volume, both with negative effects. The effective
358
diffusivity of detergent in the textile is driven by all washing parameters, where agitation and
359
concentration showed positive effects, whereas volume and insertion affected the diffusivity
360
negatively. High order interactions involving all these parameters are of the same order of
361
magnitude, revealing that none factor dominates. A similar behaviour was observed for the
362
fraction of mass remained in the clearance. Finally, the flow rate between the outer bowl and the
363
clearance,
364
former negatively, thus indicating that a higher flow rate is achieved for a lower level of water.
, is mostly ruled by
an
us
cr
ip t
, is governed by the agitation speed, with a positive
M
, is affected mostly by agitation and water level, the latter acting positively and the
d
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Figure 10 compares the model predictions to the experimental data. The bars refer to the
367
measurement uncertainties. Figure 10.a shows the results for test condition #7 (see Tab. 2),
368
where the detergent was inserted in the outer bowl. One can see that the model predicts the
369
experimental trends for the bowls and the probe quite satisfactorily, with the simulation results
370
falling within the experimental uncertainty span. In this case, the adoption of the parameter X
371
was required. However, it is worth of note that the model was not able to predict the rapid
372
changes experienced by the detergent concentration in the spin bowl during the initial stages
373
(before 1 minute). Figure 10.b shows the results for the conditions of run #8 (see Tab. 2), where
374
the detergent was inserted in the spin bowl. Again, the experimental data and the model
375
predictions agreed to within the measurement uncertainty span. Figure 10.c shows the results
376
for the conditions of run #6 (see Tab. 2), where the model predictions for the spin bowl and the
377
experimental trends diverge quite substantially for the initial stages (until 90 s). A similar
378
behaviour was observed for conditions #2, #10 and #14, all reported in Table 2, in which
Ac ce p
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366
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Page 17 of 36
smooth agitation was adopted together with detergent insertion in the spin bowl. An explanation
380
relies on the fact that the model is supplied with a homogeneous initial condition (mass of
381
detergent inserted in the spin bowl divided by the volume of water in such compartment) which
382
is not realistic. Since the detergent is supplied at the top of the spin bowl, a richer solution in
383
detergent is found at the upper levels, whereas a poorer solution is observed at the bottom,
384
where the measurement is taken. Therefore, there does exist a delay until detergent reaches the
385
lower levels of water, which is aggravated by the presence of 4 kg of load. From 2 minutes on,
386
the model is able to predict the experimental trends for the bowls and the probe satisfactorily.
us
cr
ip t
379
387
Figure 11 assesses the model capabilities and its sensitivity to the washing parameters such as
389
agitation speed and detergent insertion. Figure 11.a compares the time evolution of detergent
390
concentration within the probe for different agitation speeds (from 70 to 100 rpm, 10 rpm step)
391
in the case when the detergent is inserted in the outer bowl. In addition, 80 litres of water and 1
392
g/L of detergent are assumed as working conditions. One can see in Fig. 11.a that higher
393
agitation speeds enhance the mass transfer rates to the probe, whose concentration varies
394
rapidly over time. A non-linear influence of the agitation speed can be observed as the difference
395
from 70 to 80 rpm is more perceptible than the difference from 90 to 100 rpm. Figure 11.b
396
explores the time evolution of detergent concentration in the spin bowl and within the probe for
397
different detergent insertion locations. Again, the water volume and the initial concentration are
398
held fixed at 80 litres and 1 g/L, whilst an agitation speed of 85 rpm was adopted. Firstly, the
399
concentration profile observed for the spin bowl may differ significantly depending on the
400
detergent insertion loci, thus affecting the behaviour of the probe. In the case where detergent is
401
inserted in the outer bowl, detergent must go first to the spin bowl and then be transferred to
402
the probe, whose concentration increase at lower rates. On the other hand, when detergent is
403
inserted in the spin bowl, it goes to the probe and to the outer bowl simultaneously, making the
404
probe concentration to increase at higher rates.
Ac ce p
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388
405
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Page 18 of 36
406
SUMMARY AND CONCLUSIONS
407
This study came out with an experimental methodology to evaluate the transient behavior of
409
detergent transport in the compartments and through the garments during the washing process
410
in a household top-load washing machines. In addition, a semi-empirical model based on the
411
transport equation of chemical species was put forward to predict the time evolution of the
412
detergent concentration in the outer and spin bowls, and also inside the garments. The closing
413
parameters required by the model were reduced from the experimental data. The model
414
predictions were compared with the experimental counterparts, when satisfactory agreements
415
(i.e., within the experimental uncertainty span) were observed for most conditions. The key
416
conclusions of this study are summarized as follows:
an
us
cr
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408
M
417
The factorial experiment pointed out the detergent insertion and the agitation level as
419
the most influencing washing parameters when one aims at obtaining a fast detergent
420
homogenization among the compartments, which can be achieved using a strong
421
agitation level and inserting detergent in the spin bowl.
Ac ce p
te
d
418
422
Detergent decantation took place every time the detergent was inserted in the outer
423
bowl. This was not observed when detergent was inserted in the spin bowl, which is
424 425 426
partly because the flow patterns in the outer bowl are not as turbulent as in the spin bowl, where the agitator works, and partly because there is a clearance between the outer and the spin bowl where detergent accumulates due to gravity and low velocities.
427
The detergent sampling should be carried out in such a way that the non-homogeneous
428
detergent distribution can be captured. When a single intake is placed at the bottom of
429
the spin bowl and the detergent is inserted at its top, there is a time lag in the detergent
430
transport that is not accounted for by the model, which turns out to be fed with a
431
homogeneous boundary condition. As a consequence, the model presents poor
- 19 -
Page 19 of 36
432
predictions for the detergent concentration in the spin bowl for the early stages after the
433
test initiation in the cases where detergent is inserted in this compartment.
434
ACKNOWLEDGMENTS
ip t
435 436
The authors are thankful to Electrolux do Brasil S/A for sponsoring this research project,
438
particularly to Messrs. Cirilo Cavalli and Cesar F. Silva. Financial support from the Brazilian
439
Government funding agency CNPq (Grant No. 470986/2012-3) is also duly acknowledged.
us
cr
437
441
an
440
REFERENCES
444
AHAM (2010) Performance Evaluation Procedures for Household Clothes Washers, Association of Home Appliance Manufacturers, AHAM HLW-1-2010, Washington-DC, USA
d
443
M
442
AS/NZS (2005) Performance of household electrical appliances – Clothes washing machines, Part 1:
446
Methods for measuring performance, energy and water consumption, Australia and New Zealand
447
Standard AS/NZS 2040.1, Wellington, New Zealand
449 450 451 452 453
Ac ce p
448
te
445
Bansal PK, Vineyard E, Abdelaziz O (2011) Advances in household appliances – A review, Applied Thermal Engineering 31 3748-3760 Bogaki P, Shampine LF (1999) A 3(2) pair of Runge–Kutta formulas, Applied Mathematics Letters 2 321-325
Box GEP, Hunter WG, Hunter JS (1978) Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building, Wiley, New York, USA
454
Campos LGC (2015) Uma metodologia para ensaio e simulação dos fenâmenos de transporte de
455
detergente em lavadoras domésticas de eixo vertical, MEng thesis, Federal University of
456
Paraná,
457
www.pgmec.ufpr.br/dissertacoes/dissertacao_171_luiz_guilherme_costa_campos.pdf
Curitiba-PR,
Brazil.
Available
at:
- 20 -
Page 20 of 36
459 460 461 462 463
Charrette H, Houzelot JL, Ventenant V (2001) Modelling and Experimental Studies of Mass Transfer in Cotton Fabrics, Textile Research Journal 71 954-959 IEC (2010) Clothes washing machines for household use - Methods for measuring the performance, IEC 60456:2010, International Electrotechnical Commission, Geneva, Switzerland Janacova D, Charvatova H, Kilomaznik K, Vasek V, Mokrejs P, Drga R (2011) Computer simulation
ip t
458
of washing processes, Int. J. Math. Models and Methods in App. Sc. 5 1094-1101
Patankar SV (1980) Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington-DC, USA
465
van den Brekel LDM (1987) Hydrodynamics and mass transfer in domestic drum-type fabric
us
466
cr
464
washing machines, PhD thesis, TR diss 1533, TU Delft, The Netherlands
van der Donck JCJ (1997) Compression of wet textile, Tenside Surfactants Detergents 34 322-326
468
Ward D (2003) A novel remote measurement and monitoring system for the measurement of
469
critical washing parameters inside a domestic washing machine, Measurement 34 193-205
470
Warmeskerken MMCG, van der Vlist P, Moholkar VS, Nierstrasz VA (2002) Laundry Process
M
Intensification by Ultrasound, Colloids and Surfaces A 210 277-285
d
471
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467
474
FIGURE CAPTIONS
Ac ce p
473
te
472
475
Figure 1. Schematic representation of a household top-load washing machine with its key
476
compartments and components
477
Figure 2. Photography of (a) the test facility, (b) the modular agitator, and (c) the washing fluid
478
sampling intake at the spin bowl
479
Figure 3. Probe enveloped by a cotton linen folded four times. The borders have been selead
480
with hot-melt adhesive (HMA) material
481
Figure 4. Flowsheeting of the convective transport of detergent between the compartments of a
482
household top-load washing machine
483
Figure 5. Schematic representation of the transport of detergent on and within the probe
- 21 -
Page 21 of 36
Figure 6. Time evolution of detergent concentration in the spin and outer bowls: (a) insertion in
485
the outer bowl; (b) insertion in the spin bowl. The other test conditions are: strong agitation, 70
486
litres of water, concentration of 0.75 g/L
487
Figure 7. Effect of agitation level on the time evolution of detergent concentration: (a) in the
488
outer bowl (insertion in the outer bowl, 70 litres, steady-state concentration 1.25 g/L); (b) in the
489
probe (insertion in the outer bowl, 70 litres, steady-state concentration 1.25 g/L); and (c) in the
490
spin bowl (insertion in the spin bowl, 90 litres, steady-state concentration 0.75 g/L)
491
Figure 8. Sensitivity coefficients from the factorial experiment. The response variables are: the
492
time to equilibrium (white bars) and the relative difference between the concentration in the
493
probe and in the spin bowl (black bars)
494
Figure 9. Sensitivity of the model parameters to the washing parameters: effective diffusivity
495
(white bar); flow rate between probe and spin bowl (light grey bar); flow rate between outer
496
and spin bowls (dashed bar); flow rate to the clearance (dark grey bar); and mass fraction in the
497
clearance (black bar)
498
Figure 10. Comparison between model predictions and experimental data: (a) insertion in the
499
outer bowl, strong agitation, 90 litres, concentration of 0.75 g/L; (b) insertion in the spin bowl,
500
strong agitation, 90 litres, concentration of 0.75 g/L; and (c) insertion in the spin bowl, smooth
501
agitation, 90 litres, concentration of 0.75 g/L
502
Figure 11. Model capabilities: time evolution of detergent concentration (a) for various agitation
503
speeds (smooth=70 rpm, strong=100 rpm, insertion in the outer bowl); and (b) for different
504
insertion locations (85 rpm) (common running conditions: 80 litres, 1 g/L)
Ac ce p
te
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M
an
us
cr
ip t
484
- 22 -
Page 22 of 36
505
Control Panel
Lid
Inlet Valve
cr
ip t
Detergent Dispenser
Agitator
us
Spin Bowl
Outer Bowl
an
Load
Clearance
M
Electric Motor
Cabinet
509
te
508
Figure 1
Ac ce p
507
d
506
- 23 -
Page 23 of 36
509
modular blades
spin bowl
50 mm
us
cr
outer bowl
ip t
hose
an
rotating base
(a)
(b)
M
510
(c)
Figure 2
511
Ac ce p
te
d
512
- 24 -
Page 24 of 36
512 hose
ip t
HMA seal
cr
folded cotton linen
514
Figure 3
an
515
us
513
Ac ce p
te
d
M
516
- 25 -
Page 25 of 36
516
Clearance
Outer Bowl
(decanted detergent)
Spin Bowl
Load
ip t
Probe
cr
517 518
Figure 4
us
519
Ac ce p
te
d
M
an
520
- 26 -
Page 26 of 36
520
1 control volumes
ip t
i N
cr
521
Figure 5
522
Ac ce p
te
d
M
an
us
523
- 27 -
Page 27 of 36
523
(a)
1.2
spin bowl
ip t
0.9
cr
0.6
0.3
0.0
(b)
2
4
6
Time [min]
1.6
10
12
spin bowl outer bowl
M
1.2
te
d
0.8
0.4
Ac ce p
Concentration [g/L]
8
an
0
us
Concentration [g/L]
outer bowl
0.0
0
524 525
0.5
1
1.5
Time [min]
2
2.5
3
Figure 6
- 28 -
Page 28 of 36
526 (a)
2.5
100 rpm 70 rpm
Concentration [g/L]
2.0
ip t
1.5
cr
1.0
0.0
(b)
5
10
15
Time [min]
25
30
M
1.2
Concentration [g/L]
20
an
0
us
0.5
d
0.8
Ac ce p
te
0.4
100 rpm 70 rpm
0.0
0
10
15
Time [min]
20
25
1.2
30
100 rpm 70 rpm
0.9
Concentration [g/L]
(c)
5
0.6
0.3
0.0 0
2
4
6
8
10
12
Time [min]
- 29 -
Page 29 of 36
Figure 7
Ac ce p
te
d
M
an
us
cr
ip t
527
- 30 -
Page 30 of 36
528
I*A*V*C time to equilibrium difference after 30 min
A*V*C I*V*C
ip t
I*A*C
I*A*V
cr
A*C
Washing Parameter
V*C
A*V
-0.5
-0.3
-0.1
M
an
us
I*C
0.1
0.3
I*V I*A C V A I
0.5
Sensitivity 529
d te
531
Figure 8
Ac ce p
530
- 31 -
Page 31 of 36
531
ip t
I*A*V*C A*V*C
cr
I*V*C
I*A*C I*A*V
flow from probe to spin bowl flow between bowls
A*C A*V
an
mass fraction in the clearance
V*C
I*C I*V
M
flow to the clearance
I*A C V
533 534
te
d 532
-0.3
A I
-0.1
Ac ce p
-0.5
Washing Parameter
us
effective diffusivity
0.1
0.3
0.5
Sensitivity Figure 9
- 32 -
Page 32 of 36
534
(a)
1.00
ip t
Concentration [g/L]
0.75
0.50
Outer bowl - predicted outer bowl - measured spin bowl - predicted spin bowl - measured probe - predicted probe - measured
0.00 0
600
900
Time [s]
1.50
Concentration [g/L]
1.25
M
1.00 0.75
1500
1800
outer bowl - predicted outer bowl - measured spin bowl - predicted spin bowl - measured probe - predicted probe - measured
0.25 0.00
300
Ac ce p
0
te
d
0.50
(c)
1200
an
(b)
300
us
cr
0.25
600
900
Time [s]
1200
1.50
1500
1800
outer bowl - predicted outer bowl - measured spin bowl - predicted spin bowl - measured probe - predicted probe - measured
Concentration [g/L]
1.25 1.00 0.75 0.50 0.25 0.00
0
300
600
900
Time [s]
1200
1500
1800
535
- 33 -
Page 33 of 36
Figure 10
536
(a)
1
ip t
0.6
cr
0.4
100 rpm (strong agitation) 90 rpm 80 rpm 70 rpm (smooth agitation)
0.2 0
(b)
300
600
900
Time [s]
1.8 1.6
1500
1800
M
1.4 1.2
d
1
spin bowl - inserted in the spin bowl probe - inserted in the spin bowl spin bowl - inserted in the outer bowl probe - inserted in the outer bowl
0.8 0.6
te
Concentration [g/L]
1200
an
0
us
Concentration [g/L]
0.8
0.4
Ac ce p
0.2
0
0
537 538 539
300
600
900
Time [s]
1200
1500
1800
Figure 11
- 34 -
Page 34 of 36
539
Table 1. Factors and levels of the factorial experiment
540
Level
Agitation speed [rpm] 70 (smooth) 100 (strong)
Volume [L] 70 90
Concentration [g/L] 0.75 1.25
ip t
(-) (+)
Insertion location outer bowl spin bowl
541 542
Insertion location
Agitation profile
Volume [L]
Concentration [g/L]
Time to equilibrium [min]
Difference after 30 min [%]
1
outer bowl
smooth
70
0.75
1.25
30
2
spin bowl
smooth
70
0.75
2.00
34
3
outer bowl
strong
70
0.75
0.50
21
4
spin bowl
strong
70
0.75
1.00
8
5
outer bowl
smooth
90
0.75
2.50
6
spin bowl
smooth
90
0.75
3.00
32
7
outer bowl
strong
90
0.75
0.75
18
70
10
spin bowl
smooth
70
11
outer bowl
strong
70
12
spin bowl
strong
70
13
outer bowl
smooth
14
spin bowl
smooth
15
outer bowl
strong
spin bowl
strong
an
90
smooth
0.75
1.00
26
1.25
1.00
49
1.25
2.00
17
1.25
0.75 0.75
14
1.25
M
strong
outer bowl
47
d
spin bowl
9
te
8
us
Run #
12
90
1.25
2.00
44
90
1.25
2.50
24
90
1.25
0.75
26
90
1.25
1.25
15
Ac ce p
16 544
cr
Table 2. Summary of the running conditions and the results of the factorial experiment
543
- 35 -
Page 35 of 36
544 545
[m3/s]
[m3/s]
[m2/s]
[m3/s]
X [-]
1
1.5010-3
1.5010-7
1.5010-8
3.0010-5
0.75
2
5.0010-4
1.1510-7
6.5010-9
3
1.5010-3
2.7510-7
1.5010-8
7.5010-5
0.50
4
1.0010-3
1.5010-7
7.5010-9
5
7.0010-4
8.510-8
4.5010-9
7.0010-6
0.50
6
4.0010-4
6.2510-8
7.5010-10
7
2.5010-3
2.2510-7
7.5010-9
4.2510-5
8
1.2510-3
1.2510-7
3.7510-9
9
1.7510-3
8.5010-8
1.0010-8
10
5.0010-4
1.5010-7
8.5010-9
11 12
2.0010-3
5.0010-7
5.0010-8
1.2510-3
1.5010-7
6.2510-9
13
8.2510-4
7.5010-8
1.0010-8
14
5.5010-4
6.2510-8
8.5010-10
15
1.7510-3
1.2510-7
5.0010-9
16
1.2510-3
1.5010-7
us
3.0010-5
0.70
1.2510-4
0.70
an
M
0.65
1.2510-5
0.65
5.0010-5
0.60
2.0010-8
Ac ce p
te
d
546
ip t
Run #
cr
Table 3. Summary of the empirical parameters of the model
- 36 -
Page 36 of 36
S0263-8762(16)30026-0 http://dx.doi.org/doi:10.1016/j.cherd.2016.03.021 CHERD 2233
To appear in: Received date: Revised date: Accepted date:
26-10-2015 13-2-2016 18-3-2016
Please cite this article as: Campos, L.G.C., Hermes, C.J.L.,Experimental evaluation and transient simulation of detergent transport in household vertical axis washing machines, Chemical Engineering Research and Design (2016), http://dx.doi.org/10.1016/j.cherd.2016.03.021 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
1 2
Highlights
3 4 5
Detergent transport in vertical axis washing machines is studied
A purpose-built test rig was used to gather data following a 24 factorial design
Agitation level and detergent insertion showed to be the most influencing factors
A semi-empirical model was developed and validated against experimental data
7
ip t
6
9
cr
8
11
us
10
12 13
Ac ce p
te
d
M
an
14
-1-
Page 1 of 36
14
Experimental evaluation and transient simulation of detergent
15
transport in household vertical axis washing machines
Luiz G. C. Campos 1,2, Christian J. L. Hermes 2,*
17 18
20
1
Advanced Product Development Fabric Care, Electrolux Major Appliances Latin America
cr
19
81520900 Curitiba, PR, Brazil, +55 41 3371 6109, [email protected] 2
us
21 22
ip t
16
Laboratory of Thermodynamics and Thermophysics, Federal University of Paraná 81531990 Curitiba, PR, Brazil, +55 41 3361 3239, [email protected]
23
* Corresponding
author
an
24 25
ABSTRACT
M
26 27
Detergent transport between the compartments and through the garments during the washing
29
process in household top-load washing machines is investigated. An experimental methodology
30
was put forward for evaluating the time evolution of the detergent concentration in each
31
compartment, and also in a probe enveloped by a cotton-linen that emulates the garments. The
32
experiments were carried out following a full factorial experiment, pointing out the detergent
33
insertion location and the agitation speed as the rate-determining factors. A lumped transient
34
model was put forward for the convective transport of detergent. The set of ODEs was solved
35
numerically through an adaptive time-step algorithm. The closing parameters required by the
36
model were reduced from the experimental data, thus providing the model with a semi-
37
empirical character. The model predictions for the time evolution of the detergent concentration
38
and the experimental data agreed to within the range of the measurement uncertainties (~15%).
39
Keywords: mass transfer; experimental analysis; transient simulation; detergent transport;
40
washing machine
41
Abbreviated
Ac ce p
te
d
28
title:
Detergent
transport
in
top-load
washing
machines
-2-
Page 2 of 36
42 43
NOMENCLATURE
44
Roman
46
A
agitation level
47
Af
face area, m2
48
C
detergent concentration, kg m-3
49
D
diffusivity of detergent in water, m2 s-1
50
Deff
effective diffusivity of detergent, m2 s-1
51
F
independent variables of the factorial experiment
52
I
detergent insertion
53
l
length, m
54
S
source term, kg m-3 s-1
55
t
time, s
56
U
velocity, m s-1
57
V
volume, m3
te
d
M
an
us
cr
ip t
45
59
X
60
Y
61
z
62
Ac ce p
flow rate, m3 s-1
58
63
Greek
64
α
initial mass fraction of detergent in the clearance dependent variables of the factorial experiment mass transfer direction
volumetric fraction of detergent to the probe
65 66
Subscripts
67
0
load (but the probe)
-3-
Page 3 of 36
1
first control volume of the probe
69
bb
between bowls
70
cl
clearance
71
cv
control volume
72
dn
control volume at downstream
73
i
i-th control volume of the probe
74
N
last control volume of the probe
75
nb
neighbour (upstream, downstream)
76
ob
outer bowl
77
pb
probe
78
sb
spin bowl
79
up
control volume at upstream
cr us
d
Superscripts
te
dimensionless variable of the factorial experiment (see equations 1 and 2)
Ac ce p
82
an
M
80 81
ip t
68
-4-
Page 4 of 36
83 84
INTRODUCTION
85
A vertical axis (also known as top-load) washing machine is comprised of a water reservoir (the
87
outer bowl), in which is placed a perforated drum (the spin bowl) that holds the garments which
88
are moved together with the washing fluid (detergent-water solution) by an electric motor-
89
driven rotating blade named agitator, as depicted in Fig. 1. The outer bowl is supplied with water
90
by means of a feeding system comprised of an inlet valve, and a water-level sensor. The washing
91
process in top-load washing machines involves mechanical and chemical phenomena. The
92
former is regarded with fluid flow on and within the garments, thus promoting the stresses that
93
remove the dirt. The chemical action, on the other hand, depends on the transport of detergent
94
dissolved in water from one compartment (i.e., outer bowl) to the other (i.e., spin bowl), and
95
from the latter to the garments. Once in the textile, the detergent helps to loosening the dirt,
96
which can be dragged off by the mechanical action.
cr
us
an
M
d
te
97
ip t
86
Therefore, the washing process depends on a great variety of parameters, including the
99
geometries of the spin bowl and the agitator, the washing stroke (i.e., the time evolution of the
100
motor torque, angular speed, and angular direction), the water level and temperature, the
101
amount of detergent, to cite just a few. There are different standards used around the globe for
102
evaluating the washing efficiency (Bansal et al., 2011), most of them accounting for the water
103
and energy consumption, the washing effectiveness, and the wear of the textile (e.g., AHAM
104
HLW-1-2010, AS/NZS 2040.1, and IEC 60456:2010). The high number of geometric and
105
operational parameters that interact non-linearly to each other together with the measurement
106
uncertainties and the degree of subjectivity of the standardized processes turn the product
107
design decision-making into a complex process, sometimes depending more on heuristics than
108
on engineering calculations and on experimental observations (Ward, 2003). To aggravate, the
109
open literature in this field is still scarce.
Ac ce p
98
-5-
Page 5 of 36
In a pioneering work, van den Brekel (1987) modelled the detergent transport in horizontal axis
111
(drum type) household washing machines considering a multi-reactor lumped-approach.
112
Experiments were carried out using KCl as the tracer in place of detergent not only to gather the
113
closing parameter required by the model, but also to be used in the model validation exercise. A
114
decade later, van der Donck (1997) simulated the mechanical washing in drum type washing
115
machines considering the compression of wet textiles. The model was validated against
116
experimental data obtained for pure water. Charrette et al. (2001) investigated the mass transfer
117
in cotton-linens, coming up with a model for the washing process in an industrial facility
118
comprised of a series of water tanks. The model was compared with experiments in which the
119
detergent was replaced by NaCl as the tracer. Warmoeskerken et al. (2002) assessed the
120
capabilities of ultrasonic technologies to boost the washing process, and found out that
121
ultrasound might promote a higher rate of NaCl removal from the inner portions of the garments
122
in comparison to regular washing processes. It is worth of note that, in all above referenced
123
studies, a salt (KCl, NaCl) was used as the tracer instead of detergent in the mass transport
124
analysis, which is not a realistic operating condition. More recently, Janacova et al. (2011)
125
proposed an analytical model to optimise the washing process in a vertical axis washing machine
126
aiming at reducing the consumption of washing fluid and energy. The detergent type was not
127
specified and no model validation exercise was reported. In addition, the model did not account
128
for the convective transport of detergent, which turns out to play an important role in the
129
washing.
cr
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d
te
Ac ce p
130
ip t
110
131
To the authors’ best knowledge, studies of the washing process using real detergents or even
132
standardized detergent blends, particularly in top-load washing machines, have not been found
133
in the open literature. Therefore, the present study is aimed at investigating the transient
134
transport of detergent in vertical axis washing machines. For doing so, a purpose-built washing
135
machine, with strict control of the operating conditions, such as the water volume (V), the
136
detergent concentration (C), the detergent insertion location (I), and the agitation speed (A) was
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Page 6 of 36
carefully instrumented to gather data of detergent concentration over time. The experiments
138
were designed according to a factorial approach, in such a way that the effects of the washing
139
parameters (V, C, I and A) and their interactions on different response variables could be figured
140
out. A simulation model was additionally put forward based on detergent mass balances
141
between the compartments of the washing machine and within the garments. The closing
142
parameters required by the model were obtained from the experiments, providing it with a
143
semi-empirical character.
cr
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137
145
us
144
EXPERIMENTAL WORK
147
an
146
Testing Facility
M
148
The experimental facility emulates a household top-load washing machine, as depicted in Fig.
150
2.a. Differently from the washing machines commercially available on the market, the rig was
151
designed and constructed to allow strict control of the working (e.g., agitator speed, spin
152
direction and angular range, water volume, detergent concentration and insertion loci) and
153
geometric (e.g., spin bowl and agitator geometries) conditions, as illustrated in Fig. 2.b. In
154
addition, hoses were fixed on both the outer bowl (water reservoir) and the spin bowl (rotating
155
drum), with the intake section placed ~50 mm from the bottom of the spin bowl, as can be seen
156
in Fig. 2.c, in order to collect samples of the washing fluid, i.e., the detergent-water solution.
te
Ac ce p
157
d
149
158
An additional probe was manufactured using a cotton-linen with 800 mm2 (i.e., a typical pillow
159
case of the IEC 60456:2010 standard), folded four times to emulate a piece of garment being
160
washed. Again, a hose was installed with one end inside the envelope to collect samples of the
161
washing fluid, as shown in Figure 3. The edges of the probe were sealed with hot-melting
162
adhesive (HMA), whereas the hoses are made of thermoplastic material (TPU).
163 -7-
Page 7 of 36
The concentration was measured using a UV-spectrophotometer to detect the absorbance of the
165
washing fluid (which is related to the concentration due to the Lambert-Beer law), following the
166
procedure recommended by the AHAM HLW-1-2010 standard. Accordingly, the detergent blend
167
used in the experiments is also the one specified by the AHAM HLW-1-2010 standard, whose
168
main components are: sodium aluminosilicate solids (36.3%), sodium carbonate (14.9%), linear
169
alkylbenzene sulfonate (11.3%), nonionic surfactant (6%) and silicone (5%). The water supplied
170
to the facility was also controlled, with ~45 ppm of CaCO3 and temperature of ~20°C since the
171
water hardness might reduce the cleaning potential of the detergent as calcium and magnesium
172
ions create bonds with surfactants. In addition to the probe (i.e., the folded cotton-linen,
173
hereafter named just probe), an extra load of 4 kg of 800 mm²-pillowcases was adopted. In all
174
cases, the washing fluid was collected manually by means of 60-ml syringes.
an
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M
175
The test itself, i.e. agitation and the measurements, were initiated 15 s after the detergent
177
insertion. Samples were acquired from the outer bowl, the spin bowl, and the probe by three
178
different operators. In order to capture the start-up transients properly, the sampling interval
179
was 15 s from the test initiation until the concentration equilibrium was reached between the
180
outer and the spin bowl. From this point on, larger sampling intervals were adopted. The
181
measurement errors associated with the instruments are lower than 1%. However, the
182
experimental uncertainties are strongly affected by the manual sampling, which have been
183
estimated to be ±15% of the measured value based on test repetition analyses. Significant foam
184
formation was not observed. More details can be found in Campos (2015).
186
te
Ac ce p
185
d
176
Test Plan
187 188
The experiments were carried out following a 24 (2-level, 4-factor) full factorial design. Such an
189
approach was employed to figure out the effects of the washing parameters (agitation speed, A;
190
detergent concentration, C; detergent insertion locus, I; and water volume, V) and their first and
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Page 8 of 36
higher order interactions on two response variables: (i) the time to reach equilibrium between
192
outer and spin bowls; and (ii) the relative difference between the detergent concentration in the
193
probe and the average between outer and spin bowls after 30 minutes. Table 1 summarizes the
194
lower (-) and upper (+) levels of the factorial experiment, whereas Table 2 details the test
195
conditions for all 16 test runs.
196
The regression model adopted in this study is as follows (Box et al., 1978):
cr
197
ip t
191
an
us
198
M
199
where
is the dimensionless response variable in the interval [-1,1],
201
calculated from the best fit method, and
202
1,1], calculated as follows:
are the coefficients
are the dimensionless factors also in the interval [-
Ac ce p
te
d
200
203
(1)
(2)
204 205
It is worth of note that each -coefficient provide the sensitivity of the response variable to its
206
respective factor.
207
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Page 9 of 36
208
SEMI-EMPIRICAL MODEL
209
The mathematical model was based on the work of van den Brekel (1987), where all mass
211
transfer processes that take place in the washing machine were treated as one-dimensional, as
212
follows:
ip t
210
(3)
an
us
cr
213
M
214
where C is the detergent concentration [kg/m3], z the mass transfer direction [m], S is a term
216
that can be a source of a sink [kg/m3s], D is the diffusion of detergent in water [m2/s], U the flow
217
velocity in the z direction [m/s],
218
[kg/m3s], and
219
of C in the control surfaces [kg/m2s]. By means of a finite-volume approach (Patankar, 1980),
220
equation (3) can be integrated in each control volume of the domain, yielding
te
is the transient variation of C in each control volume
and UC stand respectively for the diffusive and the convective transport
Ac ce p
221
d
215
(4)
222 223
where the index cv refers to the control volume under analysis, whereas the indexes up and dn
224
are related to the control volumes upstream and downstream of cv, respectively. Also,
and
- 10 -
Page 10 of 36
225
are the volume and the characteristic length of the control volume, whilst
is the flow rate
226
of washing fluid from (to) the control volume to (from) its neighbours, denoted with index nb.
227
Figure 4 illustrates the detergent flowsheeting. Washing fluid is transferred from the outer bowl
229
to the spin bowl when the agitator is off, but follows the other way round when the agitator is on.
230
The same behaviour is expected for the transport between the spin bowl and the garments,
231
divided into two control volumes, namely the probe (where detergent diffusion takes place) and
232
the load. Detergent is also transferred from (to) the outer bowl to (from) the clearance formed
233
between the outer and spin bowls, where decanted detergent may accumulate.
us
cr
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228
an
234
The model is aimed at predicting the time evolution of the detergent concentration in each of
236
these control volumes (all but the probe treated as even lumps) in order to come out with the
237
time to reach steady-state conditions, which must be as low as possible for the sake of washing
238
performance. In addition, the transport coefficients (e.g., flow rates, and effective diffusivity in
239
the textile) were not modelled through a first-principles approach, but best fitted to the
240
experimental data thus providing the model with a semi-empirical character.
d
te
Ac ce p
241
M
235
242
Bearing in mind that the convective transport overrules the diffusive effects due to high Péclet
243
numbers, equation (4), when applied to the outer bowl, the clearance, the spin bowl, and the
244
load, yields
245
(5)
- 11 -
Page 11 of 36
ip t
(6)
(8)
te
d
M
an
us
cr
(7)
246
where the indexes ob, sb, cl and 0 stand for the outer bowl, spin bowl, clearance and load,
247
respectively,
248
outer bowl and the clearance,
249
load), being
250
flow rate from the inner part of the probe to the spin bowl (due to leakage, e.g. the fluid taken
251
away by sampling). Also,
252
the last (inner) layer of the probe, were diffusion takes place.
Ac ce p
is the flow rate between bowls (outer and spin),
is the flow rate from the spin bowl to the garments (probe and
the fraction directed to the load and
and
is the flow rate between the
to the probe (see Fig. 4), and
is the
stand for the detergent concentration in the first (outer) and
253 254
The probe was then divided into N control volumes from the outer layer (1) to the inner layer
255
(N). Figure 5 presents a schematic representation of the finite-volume formulation for the probe,
256
where diffusion takes place from layers 1 to N, and convection is observed in the outer layer
257
(from spin bowl to the probe) and in the inner layer (from the probe back to the spin bowl), as - 12 -
Page 12 of 36
illustrated in Fig. 4. The mass transport equations for the outer layer (1), the inner layer (N), and
259
any layer in between (i) were obtained from equation (4), yielding
ip t
258
(10)
261
(11)
Ac ce p
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an
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cr
(9)
262
where the diffusivity of detergent in water D was replaced by the effective diffusivity of
263
detergent in the textile, Deff. Also, Af refers to the face area of the probe, whereas l is the length of
264
the i-th control volume. It is worth of mention that mesh independent results were achieved for
265
N≥16 control volumes.
266 267
Equations (5) to (11) provide a set of N+6 ordinary differential equations (ODEs) that must be
268
integrated over time to come out with the time evolution of the detergent concentration in the
269
probe, and in the bowls. In this work, the ODEs were solved by the Bogacki and Shampine’s
270
(1999) so-called ODE23 method, whose main feature is an adaptive time-step calculation. - 13 -
Page 13 of 36
271 272
One should also note that the geometry (volumes, areas, lengths) must be provided to the model,
273
whereas the transport parameters
274
model to the experimental data. Table 3 summarizes the figures achieved for each test condition.
275
The values of and Vo are 3.510-3 and 1.510-6 m3s-1, respectively. An additional empirical
276
information required by the model is the fraction of detergent that remained decanted before
277
the agitation initiation, denoted as X. Hence, a mass fraction X must be artificially added to the
278
clearance at the test beginning to ensure mass conservation. It is worth of note that X stands for
279
the mass fraction of detergent which must be added in the dead volume to ensure a proper mass
280
conservation, thus representing the amount of detergent decanted after insertion.
,
,
and
were obtained by best fitting the
an
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ip t
,
281
The model therefore relies on seven empirical parameters that must be reduced from the
283
experimental data.
M
282
286
RESULTS AND DISCUSSION
te
285
d
284
Typical experimental results for the time evolution of the detergent concentration are illustrated
288
in Figures 6 and 7. One can see in Figure 6 the influence of the initial condition (i.e., detergent
289
insertion either in the outer, Fig. 6.a, or in the spin bowl, Fig. 6.b) on the detergent concentration
290
over time. When detergent is inserted in the outer bowl (run#3 in Tab. 2), the detergent
291
concentration in both bowls decrease until reaching an inflection point, as illustrated in Fig. 6.a.
292
Such a behaviour is attributed to the detergent decantation in the initial stages that accumulate
293
in the clearance. From ~30 s on, the convective transport from the clearance to the outer bowl,
294
Ac ce p
287
, and from the latter to the spin bowl,
, make the concentration in both bowls to increase
295
exponentially until steady-state conditions are achieved after ~6 minutes. Figure 6.b shows the
296
results obtained in the case where detergent is inserted in the spin bowl (run#4 in Tab. 2). One
297
can see that no inflection (i.e., decantation) was observed: the concentration in the spin bowl - 14 -
Page 14 of 36
decreases while the concentration in the outer bowl increases until steady-state conditions are
299
reached after ~1 minute. This is due to the flow patterns observed in the spin bowl, where flow
300
agitation takes place. It is noteworthy that, in the very beginning of the experiment, a low
301
concentration was observed in the spin bowl, which is due to the distance between the insertion
302
locus (at the top) and the sampling intake (~50 mm from the bottom).
303
ip t
298
Figure 7 shows the effects of the agitation profile (smooth and strong) on the time evolution of
305
the detergent concentration. When detergent is inserted in the outer bowl, decantation takes
306
place independently of the agitation profile. Looking at the detergent concentration in the outer
307
bowl (see Fig. 7.a), one can see that the test condition under strong agitation (run#11 in Tab. 2)
308
reached the steady-state after 8 minutes, whereas the test conditions under smooth agitation
309
(run #9 in Tab. 2) took ~25 minutes. Figure 7.b depicts the time evolution of the concentration
310
in the probe for the same testing conditions. In both cases, the concentration increases as time
311
goes, which is ruled by the diffusive transport within the probe and the convective transport
312
from the spin bowl to the probe. It can be seen that the strong agitation enhances the detergent
313
transport to the probe, resulting in higher mass transfer rate. In both cases, the initial
314
concentration (1.25 g/L) was not reached, thus suggesting that part of the detergent mass
315
remained decanted in the clearance, which was confirmed by visual inspection after the tests.
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Ac ce p
316
cr
304
317
In the cases where the detergent is inserted in the spin bowl, as illustrated in Fig. 7.c, advection
318
rules from the very beginning. On the one hand, for a strong agitation (run#8 in Tab. 2), the
319
concentration measured at the bottom of the spin bowl is initially low due to the distance
320
between the insertion and the measurement loci, but higher concentrations are observed from
321
15 s on due to the flow patterns promoted by the agitator inside the spin bowl. The
322
concentration thus decreases until steady-state conditions are achieved after 2 minutes. On the
323
other hand, for a smooth agitation (run#6 in Tab.2) the transport of detergent from the top
324
(where it is inserted) to the bottom (measurement intake) slowly increases the concentration in
- 15 -
Page 15 of 36
325
the spin bowl, where the load is placed, since mass diffusion overrules the advection at low
326
speeds. The concentration thus increases at low rates until a peak and then decreases until
327
steady-state is reached after 4 minutes.
328
Figure 8 illustrates the results of the factorial experiment in the form of the coefficients of
330
equation (1). The factors are: (A) agitation speed, (C) initial detergent concentration, (I)
331
insertion locus, and (V) water volume. The response variables are: (i) time to reach equilibrium
332
between bowls (white bars), and (ii) relative difference between the detergent concentration in
333
the probe and the average concentration between bowls after 30 minutes (black bars). It can be
334
observed that insertion, volume and agitation have a strong influence on the time to reach the
335
equilibrium. Both insertion and volume have positive effects, meaning that a higher water
336
volume and the insertion in the spin bowl tend both to increase the mixing time, which is an
337
undesired effect. Agitation, on the other hand, promotes a negative effect: a higher agitation
338
decreases the time to equilibrium, which is expected due to the augmented advection between
339
bowls and to the probe. High order interactions plays a minor role, except the pair agitation-
340
volume, which is due to the flow velocities achieved for the higher agitation level. One can also
341
note in Fig. 8 that insertion and agitation play dominant negative roles on the relative difference
342
after 30 minutes, i.e., both strong agitation and insertion in the spin bowl tend to reduce the
343
concentration difference.
cr
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te
Ac ce p
344
ip t
329
345
The model parameters (
,
,
,
,
and X) were best fitted to the experimental data for
346
each of the running conditions of Table 2, whereas the parameter is fixed and known from the
347
experiments. On the one hand,
348
as the diffusive transport of detergent through the probe layers overrules the convective
349
transport to the outer layer. On the other hand, the other parameters varied substantially from
350
one test condition to another and, therefore, were fitted to the washing conditions used the same
351
procedure adopted for reducing the factorial experimental data (equation 1). The fitting
did not experience any practical change for all test conditions,
- 16 -
Page 16 of 36
352
coefficients, which describe the sensitivity of the model parameter to the washing parameter (A,
353
C, I and V) are shown in Fig. 9. One can see that the flow between bowls,
354
insertion and agitation, the latter acting positively and the former negatively, indicating that
355
strong agitation and insertion in the outer bowl both increase the flow rate between bowls. The
356
flow rate from the probe to the spin bowl,
357
effect, followed by the insertion and the volume, both with negative effects. The effective
358
diffusivity of detergent in the textile is driven by all washing parameters, where agitation and
359
concentration showed positive effects, whereas volume and insertion affected the diffusivity
360
negatively. High order interactions involving all these parameters are of the same order of
361
magnitude, revealing that none factor dominates. A similar behaviour was observed for the
362
fraction of mass remained in the clearance. Finally, the flow rate between the outer bowl and the
363
clearance,
364
former negatively, thus indicating that a higher flow rate is achieved for a lower level of water.
, is mostly ruled by
an
us
cr
ip t
, is governed by the agitation speed, with a positive
M
, is affected mostly by agitation and water level, the latter acting positively and the
d
365
Figure 10 compares the model predictions to the experimental data. The bars refer to the
367
measurement uncertainties. Figure 10.a shows the results for test condition #7 (see Tab. 2),
368
where the detergent was inserted in the outer bowl. One can see that the model predicts the
369
experimental trends for the bowls and the probe quite satisfactorily, with the simulation results
370
falling within the experimental uncertainty span. In this case, the adoption of the parameter X
371
was required. However, it is worth of note that the model was not able to predict the rapid
372
changes experienced by the detergent concentration in the spin bowl during the initial stages
373
(before 1 minute). Figure 10.b shows the results for the conditions of run #8 (see Tab. 2), where
374
the detergent was inserted in the spin bowl. Again, the experimental data and the model
375
predictions agreed to within the measurement uncertainty span. Figure 10.c shows the results
376
for the conditions of run #6 (see Tab. 2), where the model predictions for the spin bowl and the
377
experimental trends diverge quite substantially for the initial stages (until 90 s). A similar
378
behaviour was observed for conditions #2, #10 and #14, all reported in Table 2, in which
Ac ce p
te
366
- 17 -
Page 17 of 36
smooth agitation was adopted together with detergent insertion in the spin bowl. An explanation
380
relies on the fact that the model is supplied with a homogeneous initial condition (mass of
381
detergent inserted in the spin bowl divided by the volume of water in such compartment) which
382
is not realistic. Since the detergent is supplied at the top of the spin bowl, a richer solution in
383
detergent is found at the upper levels, whereas a poorer solution is observed at the bottom,
384
where the measurement is taken. Therefore, there does exist a delay until detergent reaches the
385
lower levels of water, which is aggravated by the presence of 4 kg of load. From 2 minutes on,
386
the model is able to predict the experimental trends for the bowls and the probe satisfactorily.
us
cr
ip t
379
387
Figure 11 assesses the model capabilities and its sensitivity to the washing parameters such as
389
agitation speed and detergent insertion. Figure 11.a compares the time evolution of detergent
390
concentration within the probe for different agitation speeds (from 70 to 100 rpm, 10 rpm step)
391
in the case when the detergent is inserted in the outer bowl. In addition, 80 litres of water and 1
392
g/L of detergent are assumed as working conditions. One can see in Fig. 11.a that higher
393
agitation speeds enhance the mass transfer rates to the probe, whose concentration varies
394
rapidly over time. A non-linear influence of the agitation speed can be observed as the difference
395
from 70 to 80 rpm is more perceptible than the difference from 90 to 100 rpm. Figure 11.b
396
explores the time evolution of detergent concentration in the spin bowl and within the probe for
397
different detergent insertion locations. Again, the water volume and the initial concentration are
398
held fixed at 80 litres and 1 g/L, whilst an agitation speed of 85 rpm was adopted. Firstly, the
399
concentration profile observed for the spin bowl may differ significantly depending on the
400
detergent insertion loci, thus affecting the behaviour of the probe. In the case where detergent is
401
inserted in the outer bowl, detergent must go first to the spin bowl and then be transferred to
402
the probe, whose concentration increase at lower rates. On the other hand, when detergent is
403
inserted in the spin bowl, it goes to the probe and to the outer bowl simultaneously, making the
404
probe concentration to increase at higher rates.
Ac ce p
te
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388
405
- 18 -
Page 18 of 36
406
SUMMARY AND CONCLUSIONS
407
This study came out with an experimental methodology to evaluate the transient behavior of
409
detergent transport in the compartments and through the garments during the washing process
410
in a household top-load washing machines. In addition, a semi-empirical model based on the
411
transport equation of chemical species was put forward to predict the time evolution of the
412
detergent concentration in the outer and spin bowls, and also inside the garments. The closing
413
parameters required by the model were reduced from the experimental data. The model
414
predictions were compared with the experimental counterparts, when satisfactory agreements
415
(i.e., within the experimental uncertainty span) were observed for most conditions. The key
416
conclusions of this study are summarized as follows:
an
us
cr
ip t
408
M
417
The factorial experiment pointed out the detergent insertion and the agitation level as
419
the most influencing washing parameters when one aims at obtaining a fast detergent
420
homogenization among the compartments, which can be achieved using a strong
421
agitation level and inserting detergent in the spin bowl.
Ac ce p
te
d
418
422
Detergent decantation took place every time the detergent was inserted in the outer
423
bowl. This was not observed when detergent was inserted in the spin bowl, which is
424 425 426
partly because the flow patterns in the outer bowl are not as turbulent as in the spin bowl, where the agitator works, and partly because there is a clearance between the outer and the spin bowl where detergent accumulates due to gravity and low velocities.
427
The detergent sampling should be carried out in such a way that the non-homogeneous
428
detergent distribution can be captured. When a single intake is placed at the bottom of
429
the spin bowl and the detergent is inserted at its top, there is a time lag in the detergent
430
transport that is not accounted for by the model, which turns out to be fed with a
431
homogeneous boundary condition. As a consequence, the model presents poor
- 19 -
Page 19 of 36
432
predictions for the detergent concentration in the spin bowl for the early stages after the
433
test initiation in the cases where detergent is inserted in this compartment.
434
ACKNOWLEDGMENTS
ip t
435 436
The authors are thankful to Electrolux do Brasil S/A for sponsoring this research project,
438
particularly to Messrs. Cirilo Cavalli and Cesar F. Silva. Financial support from the Brazilian
439
Government funding agency CNPq (Grant No. 470986/2012-3) is also duly acknowledged.
us
cr
437
441
an
440
REFERENCES
444
AHAM (2010) Performance Evaluation Procedures for Household Clothes Washers, Association of Home Appliance Manufacturers, AHAM HLW-1-2010, Washington-DC, USA
d
443
M
442
AS/NZS (2005) Performance of household electrical appliances – Clothes washing machines, Part 1:
446
Methods for measuring performance, energy and water consumption, Australia and New Zealand
447
Standard AS/NZS 2040.1, Wellington, New Zealand
449 450 451 452 453
Ac ce p
448
te
445
Bansal PK, Vineyard E, Abdelaziz O (2011) Advances in household appliances – A review, Applied Thermal Engineering 31 3748-3760 Bogaki P, Shampine LF (1999) A 3(2) pair of Runge–Kutta formulas, Applied Mathematics Letters 2 321-325
Box GEP, Hunter WG, Hunter JS (1978) Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building, Wiley, New York, USA
454
Campos LGC (2015) Uma metodologia para ensaio e simulação dos fenâmenos de transporte de
455
detergente em lavadoras domésticas de eixo vertical, MEng thesis, Federal University of
456
Paraná,
457
www.pgmec.ufpr.br/dissertacoes/dissertacao_171_luiz_guilherme_costa_campos.pdf
Curitiba-PR,
Brazil.
Available
at:
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459 460 461 462 463
Charrette H, Houzelot JL, Ventenant V (2001) Modelling and Experimental Studies of Mass Transfer in Cotton Fabrics, Textile Research Journal 71 954-959 IEC (2010) Clothes washing machines for household use - Methods for measuring the performance, IEC 60456:2010, International Electrotechnical Commission, Geneva, Switzerland Janacova D, Charvatova H, Kilomaznik K, Vasek V, Mokrejs P, Drga R (2011) Computer simulation
ip t
458
of washing processes, Int. J. Math. Models and Methods in App. Sc. 5 1094-1101
Patankar SV (1980) Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington-DC, USA
465
van den Brekel LDM (1987) Hydrodynamics and mass transfer in domestic drum-type fabric
us
466
cr
464
washing machines, PhD thesis, TR diss 1533, TU Delft, The Netherlands
van der Donck JCJ (1997) Compression of wet textile, Tenside Surfactants Detergents 34 322-326
468
Ward D (2003) A novel remote measurement and monitoring system for the measurement of
469
critical washing parameters inside a domestic washing machine, Measurement 34 193-205
470
Warmeskerken MMCG, van der Vlist P, Moholkar VS, Nierstrasz VA (2002) Laundry Process
M
Intensification by Ultrasound, Colloids and Surfaces A 210 277-285
d
471
an
467
474
FIGURE CAPTIONS
Ac ce p
473
te
472
475
Figure 1. Schematic representation of a household top-load washing machine with its key
476
compartments and components
477
Figure 2. Photography of (a) the test facility, (b) the modular agitator, and (c) the washing fluid
478
sampling intake at the spin bowl
479
Figure 3. Probe enveloped by a cotton linen folded four times. The borders have been selead
480
with hot-melt adhesive (HMA) material
481
Figure 4. Flowsheeting of the convective transport of detergent between the compartments of a
482
household top-load washing machine
483
Figure 5. Schematic representation of the transport of detergent on and within the probe
- 21 -
Page 21 of 36
Figure 6. Time evolution of detergent concentration in the spin and outer bowls: (a) insertion in
485
the outer bowl; (b) insertion in the spin bowl. The other test conditions are: strong agitation, 70
486
litres of water, concentration of 0.75 g/L
487
Figure 7. Effect of agitation level on the time evolution of detergent concentration: (a) in the
488
outer bowl (insertion in the outer bowl, 70 litres, steady-state concentration 1.25 g/L); (b) in the
489
probe (insertion in the outer bowl, 70 litres, steady-state concentration 1.25 g/L); and (c) in the
490
spin bowl (insertion in the spin bowl, 90 litres, steady-state concentration 0.75 g/L)
491
Figure 8. Sensitivity coefficients from the factorial experiment. The response variables are: the
492
time to equilibrium (white bars) and the relative difference between the concentration in the
493
probe and in the spin bowl (black bars)
494
Figure 9. Sensitivity of the model parameters to the washing parameters: effective diffusivity
495
(white bar); flow rate between probe and spin bowl (light grey bar); flow rate between outer
496
and spin bowls (dashed bar); flow rate to the clearance (dark grey bar); and mass fraction in the
497
clearance (black bar)
498
Figure 10. Comparison between model predictions and experimental data: (a) insertion in the
499
outer bowl, strong agitation, 90 litres, concentration of 0.75 g/L; (b) insertion in the spin bowl,
500
strong agitation, 90 litres, concentration of 0.75 g/L; and (c) insertion in the spin bowl, smooth
501
agitation, 90 litres, concentration of 0.75 g/L
502
Figure 11. Model capabilities: time evolution of detergent concentration (a) for various agitation
503
speeds (smooth=70 rpm, strong=100 rpm, insertion in the outer bowl); and (b) for different
504
insertion locations (85 rpm) (common running conditions: 80 litres, 1 g/L)
Ac ce p
te
d
M
an
us
cr
ip t
484
- 22 -
Page 22 of 36
505
Control Panel
Lid
Inlet Valve
cr
ip t
Detergent Dispenser
Agitator
us
Spin Bowl
Outer Bowl
an
Load
Clearance
M
Electric Motor
Cabinet
509
te
508
Figure 1
Ac ce p
507
d
506
- 23 -
Page 23 of 36
509
modular blades
spin bowl
50 mm
us
cr
outer bowl
ip t
hose
an
rotating base
(a)
(b)
M
510
(c)
Figure 2
511
Ac ce p
te
d
512
- 24 -
Page 24 of 36
512 hose
ip t
HMA seal
cr
folded cotton linen
514
Figure 3
an
515
us
513
Ac ce p
te
d
M
516
- 25 -
Page 25 of 36
516
Clearance
Outer Bowl
(decanted detergent)
Spin Bowl
Load
ip t
Probe
cr
517 518
Figure 4
us
519
Ac ce p
te
d
M
an
520
- 26 -
Page 26 of 36
520
1 control volumes
ip t
i N
cr
521
Figure 5
522
Ac ce p
te
d
M
an
us
523
- 27 -
Page 27 of 36
523
(a)
1.2
spin bowl
ip t
0.9
cr
0.6
0.3
0.0
(b)
2
4
6
Time [min]
1.6
10
12
spin bowl outer bowl
M
1.2
te
d
0.8
0.4
Ac ce p
Concentration [g/L]
8
an
0
us
Concentration [g/L]
outer bowl
0.0
0
524 525
0.5
1
1.5
Time [min]
2
2.5
3
Figure 6
- 28 -
Page 28 of 36
526 (a)
2.5
100 rpm 70 rpm
Concentration [g/L]
2.0
ip t
1.5
cr
1.0
0.0
(b)
5
10
15
Time [min]
25
30
M
1.2
Concentration [g/L]
20
an
0
us
0.5
d
0.8
Ac ce p
te
0.4
100 rpm 70 rpm
0.0
0
10
15
Time [min]
20
25
1.2
30
100 rpm 70 rpm
0.9
Concentration [g/L]
(c)
5
0.6
0.3
0.0 0
2
4
6
8
10
12
Time [min]
- 29 -
Page 29 of 36
Figure 7
Ac ce p
te
d
M
an
us
cr
ip t
527
- 30 -
Page 30 of 36
528
I*A*V*C time to equilibrium difference after 30 min
A*V*C I*V*C
ip t
I*A*C
I*A*V
cr
A*C
Washing Parameter
V*C
A*V
-0.5
-0.3
-0.1
M
an
us
I*C
0.1
0.3
I*V I*A C V A I
0.5
Sensitivity 529
d te
531
Figure 8
Ac ce p
530
- 31 -
Page 31 of 36
531
ip t
I*A*V*C A*V*C
cr
I*V*C
I*A*C I*A*V
flow from probe to spin bowl flow between bowls
A*C A*V
an
mass fraction in the clearance
V*C
I*C I*V
M
flow to the clearance
I*A C V
533 534
te
d 532
-0.3
A I
-0.1
Ac ce p
-0.5
Washing Parameter
us
effective diffusivity
0.1
0.3
0.5
Sensitivity Figure 9
- 32 -
Page 32 of 36
534
(a)
1.00
ip t
Concentration [g/L]
0.75
0.50
Outer bowl - predicted outer bowl - measured spin bowl - predicted spin bowl - measured probe - predicted probe - measured
0.00 0
600
900
Time [s]
1.50
Concentration [g/L]
1.25
M
1.00 0.75
1500
1800
outer bowl - predicted outer bowl - measured spin bowl - predicted spin bowl - measured probe - predicted probe - measured
0.25 0.00
300
Ac ce p
0
te
d
0.50
(c)
1200
an
(b)
300
us
cr
0.25
600
900
Time [s]
1200
1.50
1500
1800
outer bowl - predicted outer bowl - measured spin bowl - predicted spin bowl - measured probe - predicted probe - measured
Concentration [g/L]
1.25 1.00 0.75 0.50 0.25 0.00
0
300
600
900
Time [s]
1200
1500
1800
535
- 33 -
Page 33 of 36
Figure 10
536
(a)
1
ip t
0.6
cr
0.4
100 rpm (strong agitation) 90 rpm 80 rpm 70 rpm (smooth agitation)
0.2 0
(b)
300
600
900
Time [s]
1.8 1.6
1500
1800
M
1.4 1.2
d
1
spin bowl - inserted in the spin bowl probe - inserted in the spin bowl spin bowl - inserted in the outer bowl probe - inserted in the outer bowl
0.8 0.6
te
Concentration [g/L]
1200
an
0
us
Concentration [g/L]
0.8
0.4
Ac ce p
0.2
0
0
537 538 539
300
600
900
Time [s]
1200
1500
1800
Figure 11
- 34 -
Page 34 of 36
539
Table 1. Factors and levels of the factorial experiment
540
Level
Agitation speed [rpm] 70 (smooth) 100 (strong)
Volume [L] 70 90
Concentration [g/L] 0.75 1.25
ip t
(-) (+)
Insertion location outer bowl spin bowl
541 542
Insertion location
Agitation profile
Volume [L]
Concentration [g/L]
Time to equilibrium [min]
Difference after 30 min [%]
1
outer bowl
smooth
70
0.75
1.25
30
2
spin bowl
smooth
70
0.75
2.00
34
3
outer bowl
strong
70
0.75
0.50
21
4
spin bowl
strong
70
0.75
1.00
8
5
outer bowl
smooth
90
0.75
2.50
6
spin bowl
smooth
90
0.75
3.00
32
7
outer bowl
strong
90
0.75
0.75
18
70
10
spin bowl
smooth
70
11
outer bowl
strong
70
12
spin bowl
strong
70
13
outer bowl
smooth
14
spin bowl
smooth
15
outer bowl
strong
spin bowl
strong
an
90
smooth
0.75
1.00
26
1.25
1.00
49
1.25
2.00
17
1.25
0.75 0.75
14
1.25
M
strong
outer bowl
47
d
spin bowl
9
te
8
us
Run #
12
90
1.25
2.00
44
90
1.25
2.50
24
90
1.25
0.75
26
90
1.25
1.25
15
Ac ce p
16 544
cr
Table 2. Summary of the running conditions and the results of the factorial experiment
543
- 35 -
Page 35 of 36
544 545
[m3/s]
[m3/s]
[m2/s]
[m3/s]
X [-]
1
1.5010-3
1.5010-7
1.5010-8
3.0010-5
0.75
2
5.0010-4
1.1510-7
6.5010-9
3
1.5010-3
2.7510-7
1.5010-8
7.5010-5
0.50
4
1.0010-3
1.5010-7
7.5010-9
5
7.0010-4
8.510-8
4.5010-9
7.0010-6
0.50
6
4.0010-4
6.2510-8
7.5010-10
7
2.5010-3
2.2510-7
7.5010-9
4.2510-5
8
1.2510-3
1.2510-7
3.7510-9
9
1.7510-3
8.5010-8
1.0010-8
10
5.0010-4
1.5010-7
8.5010-9
11 12
2.0010-3
5.0010-7
5.0010-8
1.2510-3
1.5010-7
6.2510-9
13
8.2510-4
7.5010-8
1.0010-8
14
5.5010-4
6.2510-8
8.5010-10
15
1.7510-3
1.2510-7
5.0010-9
16
1.2510-3
1.5010-7
us
3.0010-5
0.70
1.2510-4
0.70
an
M
0.65
1.2510-5
0.65
5.0010-5
0.60
2.0010-8
Ac ce p
te
d
546
ip t
Run #
cr
Table 3. Summary of the empirical parameters of the model
- 36 -
Page 36 of 36