Chemical Engineering Research and Design 1 5 3 ( 2 0 2 0 ) 507–516

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Chemical Engineering Research and Design

journal homepage: www.elsevier.com/locate/cherd

Negative temperature effect of complex reaction

kinetics system of Fe and Al mineral impurities in
liquid–solid process

Daijun Liu ∗ , Chengfa Jiang

Chemical Engineering institute, Sichuan University, Chengdu, Sichuan, 610065, China

a r t i c l e i n f o a b s t r a c t

Article history: Depletion of phosphate ore is one of the important resource crises in the world. The explo-
Received 20 April 2019 ration of the transfer regularity of mineral impurities to liquid in the digestion process of
Received in revised form 7 October phosphate rock is of great significance to the production of agricultural fertilizer. In this
2019 paper, the reaction kinetics experiments of iron and aluminum compounds in phosphate
Accepted 13 October 2019 ore particles were carried out in the range of temperature 50–70 ◦ and phosphoric acid con-
centration 15–25%. Under the combined temperature and concentration conditions, a large
number of experiments were carried out to determine the conversion fraction of impu-
Keywords: rities, and the relationship between conversion fraction and time was obtained. A novel
Negative temperature effect phenomenon of liquid-solid two-phase reaction kinetics was discovered. That is the neg-
Liquid–solid reaction kinetics ative temperature effect. When acid concentration is higher than the critical point, the
Reaction of Fe and Al mineral reaction conversion fraction increases with increasing temperature, but it decreases with
impurities increasing temperature below the critical value. It has been found that this phenomenon is
Coupled process caused by the mutual coupling of the phase transition, mass transfer and chemical reaction
Phosphate rock in the micro-pores structure, as well as specific solubility characteristics of iron-aluminum
Fractal percolation phosphate. According to the mechanism analysis, the kinetic model was established, which
is in good agreement with the experimental data. It can be used to predict the reaction
behavior of iron and aluminum mineral impurities of phosphate rock in phosphoric acid.
And the kinetics parameters such as Dpd , ␥, Dcop were thus obtained.
© 2019 Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

1. Introduction solid-liquid multiphase system, it may cause some abnormalities or

new phenomena, some of which may change the content distribution
Due to the dilution and depletion of phosphate rock resources, the in the reaction products.
producers of wet phosphoric acid in most regions of the world are Generally, the rate or yield of most chemical reactions in nature
increasingly facing the decline of phosphate rock quality in recent always increases with increasing temperature. Just only a few instances
years. Because a large amount of impurities such as Al3+ , Fe3+ , Mg2+ are reversed, such as ClO + NO → Cl + NO2 . Due to negative activation
in the low-grade phosphate ore entering the phosphoric acid solu- energy in the system, its value is −2460.9 J mol−1 . That is, the reaction
tion, it has a great influence on the reaction and crystallization process rate or conversion fraction decreases as temperature rise, because the
(El-Bayaa et al., 2011; Awwad et al., 2013). This is the key factor to deter- reaction rate constant becomes decrease with temperature rise accord-
mine the overall efficiency of the wet phosphoric acid process. However, ing to Arrhenius equation (Menon and Sathyamurthy, 1981). However
there is a lack of literature on the effect of specific impurities on the wet in some complex multiphase systems, even if the activation energy of
phosphoric acid process. Controlling the dissolution of impurities dur- the intrinsic chemical reaction is positive, macroscopic negative tem-
ing acid attack of phosphate rock is an important challenge. In complex perature effects may still exist.

∗
Corresponding author.
E-mail address: liudj@scu.edu.cn (D. Liu).
https://doi.org/10.1016/j.cherd.2019.10.024
0263-8762/© 2019 Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
508 Chemical Engineering Research and Design 1 5 3 ( 2 0 2 0 ) 507–516

approach was illustrated with numerical simulations and an experi-

mental case. It was successfully applied in leaching of zinc sulphide
Nomenclature
with ferric ions.
A ion reaction affinity potential, J mol−1 The interaction between the porous medium and the fluid, and the
As surface area, m2 coupling between transfer and chemical reaction process make the
kinetics very complicated. This kind of exploration also promotes the
C actual ionic concentrations of acid anion,
deepening of people’s understanding. Geetha and Surender (2000) stud-
mol m−3
ied the aeration leaching process for the removal of metallic iron from
CB actual ionic concentrations of metal cation,
reduced ilmenite for synthesizing rutile. This process, carried out in
mol m−3 a mechanically agitated reactor was a multi-phase system involving a
C* ionic concentrations of acid anion in an equi- complex interplay between mass transfer and chemical reactions. The
librium state, mol m−3 model in the paper especially accounted for the influence of the pre-
CB * ionic concentrations of metal cation in an equi- cipitated iron oxide of inert micro-particles on mass transfer rates, as
librium state, mol m−3 well as on the overall rate of iron removal. The results indicate that the
D fractal dimension overall iron removal process kinetics can be considered as constituted
Dco synergistic coefficient by an external mass transfer-controlled period followed by an internal
Dcop combining synergistic coefficient diffusion-controlled regime. As application of a new mathematic stool,
Molga et al. (2000) applied the neural networks to model the conversion
Dp diffusion coefficient, m2 s−1
rates of a heterogeneous oxidation reaction of oxidation of 2-octanol
Dpd combining diffusion coefficient, m2 s−1
with nitric acid. He found that the application of it was an efficient
H enthalpy of system, J mol−1
and accurate tool to solve modelling problems due to a more complex
K solubility product and unknown kinetics of the investigated reaction. But the general-
km constant of integral mean value theorem isation of the neural network approach to all series of experiments
M molecular weight, g mol −1 was impossible. Heydarpoura et al. (2011) studied the leaching process
N the amount of mass of the solid globe, mol to reduce carbonate content in calcareous phosphate rock using lac-
n mass flux, mol tic acid. The optimal conditions in laboratory scale were found. The
R gas constent, J mol−1 K−1 experimental data were tested through graphic and statistical meth-
r radius of a small globe, m ods and a shrinking core kinetics model was presented to describe

rp the dissolution and to analyze the data. Chen et al. (2015) developed a
radius of a capillary tuber, m
pore-scale two-phase multi-mixture model based on the lattice Boltz-
rs outside boundary radius of a small globe, m
mann method (LBM) for the reactive complex transport processes with
T temperature, K
dissolution–precipitation. The model is then applied to a physicochem-
t time, s
ical system encountered in shale gas/oil industry involving multiphase
Us unidirectional rate of mass transportation from flow. Mgaidi and Mokni (2018) developed a mathematical model to
solid to liquid phase, mol m−3 s−1 describe the behavior of phosphate particles into acidic solution under
x fraction of reactive substance disappearing or certain conditions. Results showed that the model is predictive in the
conversion case of a solid dissolution without ash film. Hu et al. (2019) investi-
gated the adsorption kinetics at the solid/solution interface by using
Greek Letters the fractal-like approach so as to revise the classical reaction kinet-
 scale index ics. This model could better describe the selected fixed-bed adsorption
ı conversion coefficient systems, and provide useful information for the design.
The percolation in complex systems and the state transitions in
 distribution fraction of holes in unit volume
the critical point are novel phenomena that arise in a class of kinetics
L density in the local transient region at the sur-
system. Avramov and Tonchev (2016) studied the percolation kinetics
face, kg m−3
of deposition sites of square lattice using Monte Carlo (MC) simulation
 surface free energy, J m−2 governed by dynamic rules. They found that the random probability
of connectivity can be p > 0 while the achievable concentration of the
sites was Cmax = 0.36 as a critical value. Lin et al. (2019) studied the per-
This phenomenon involves the chemical reaction kinetics system colation properties of 2D/3D porous media composed of homogeneous
actually. In recent years, there has been great interest in the fluid- solid matrix and overlapping pores of concave-shaped geometries. The
solid or liquid–solid process kinetics, as well as novel phenomenon corresponding percolation threshold ˚c was derived, and the relation
in kinetic system. Ramachandran and Smith (1977) proposed a single- between the characteristic of concave-shaped pores and the percola-
model for gas–solid non-catalytic reaction to predict the behavior in tion threshold was quantified by a numerical approximation of Monte
a porous pellet of reactant, and used it to illustrate the reduction of Carlo simulation.
nickel oxide pellets with carbon monoxide and the reaction of sulfur Theories and methods related to kinetics can describe and pre-
dioxide with calcium carbonate. Doroahkin (1996) studied the dissolu- dict changes in various processes, and provide an important basis for
tion kinetics and mechanism of single crystals of natural fluorapatite at the development of new process systems and industrial scaling up.
micro- and nano-levels. And he proposed new system of five chemical Today, its application covers almost most of the field. Nishida and
equations for the acidic dissolution of fluorapatite. The solid digestion Itoh (2012) developed a multiphase field method to simulate com-
in liquid phase system has been applied in metallurgical processes for plex microstructure formations in low-carbon steels, such as ␥-to-␣
many years. For example, metals are extracted from ores by hydromet- transformation. One was the general type derived rigorously from
allurgical processes to achieve selective dissolution (Habashi, 2005). a free-energy formulation, and another was the antisymmetric type
The kinetics of leaching selenium from Ni–Mo ore smelter dust in where only dual interface interaction was taken into account. The
H2 SO4 –HCl–H2 O system was investigated (Hou et al., 2010). The results difference between them reflects the higher-order corrections of the
indicated that the leaching of selenium increased sharply with the interface kinetics. Another interesting instance was microchannel pro-
increase of temperature. The leaching process was controlled by the cess of biomass production. Saccharomyces cerevisiae was cultivated
surface chemical reaction. The shrinking core model was expressed as aerobically in continuous chemostat mode using a multiphase microre-
1− (1−r) 1/3 = 185.4 exp(−44400/RT) t. Salmi et al. (2013) considered the actor (Krull and Peterat, 2016). The reaction kinetic parameters and
influence of solid particle morphology and developed a new shrink- the kinetic analysis were determined from the stationary concentra-
ing particle kinetics models for arbitrary geometries. The modelling tions of the biomass, glucose and ethanol. Maximal specific growth
 Chemical Engineering Research and Design 1 5 3 ( 2 0 2 0 ) 507–516 509

rate and Monod constant were determined using different lineariza-

tion methods. A review report of gas–liquid–liquid multiphase flow in
microfluidic system was presented (Chen et al., 2019). It involved differ-
ent type of the multiphase micro-flow, and was also extended to scaling
laws and numerical simulations used to predict the size of microbub-
bles or droplets. On the other hand, Schrimpf et al. (2019) introduced
gas–liquid–liquid process performed by homogeneous catalysts. This
critical review covers the basic aspects of gas-liquid-liquid mass trans-
fer and discusses a number of reactions in this type of environment. Ou
et al. (2019) studied gasification of coal in integrated supercritical water
reactor. An exploratory numerical model incorporating particle-fluid
flow dynamics, multispecies transport and thermal coupling between
endothermic coal gasification and exothermic product oxidation was
established to simulate the reacting multiphase flow process of coal
conversion. A kinetic model was proposed for the prediction of chem-
Fig. 1 – Instrument set.
ical reactions. The results showed that oxidation of gas products as
1 super thermostat. 2 data acquisition system. 3 ion
inner heat source could promote the gasification reaction under some
selective electrode. 4 sampling hole. 5 agitator. 6
suitable conditions. The experiments of a multiphase batch reaction of
catalyzed nitrobenzene hydrogenation were performed by Frikha et al. thermometer. 7 lid. 8 reactor. 9 temperature controller.
(2006). A simplified model associating the hydrodynamic gas–liquid
mass transfer parameters and the chemical kinetics processes was also
15%, 20%, 25% P2 O5 respectively. The particles were obtained
developed. It was showed that the model allows the precise simulation after pulverization and screening. The average size was
of the extent of reaction and the temperature evolution. 0.30 mm (0.28–0.315 mm).
In addition to the above, the kinetics of the decomposition of iron- The schematic diagram of the equipment is showed in
aluminum mineral impurities in phosphate rock has not been reported Fig. 1. The basic unit was a stirred tank under precise constant
so far, let alone some abnormal phenomena in this process. This paper temperature control, which was made of glass with 1000 ml
attempts to study the kinetic behavior of the decomposition of iron- volume. The lid was made in half for ease of loading. Four
aluminum mineral impurities in phosphoric acid through experiments, baffles were fixed on the reactor wall. A small sampling hole
and to analyze and explore the mechanism and rule of this phe-
was opened on the lid, and a syringe was used for sampling.
nomenon from the perspective of kinetic theory, so as provide reference
The inclined blade agitator was installed in the center of the
for industrial applications.
reactor with a diameter of about 48 mm. The ion selective elec-
trode was fixed on the lid and can be moved up and down.
2. The basic research methods Photoelectric speedometer was located near the top of agita-
tor shaft. The chemical compositions of impurities in liquid
The basic research methods include experiment and theo- phase and solid particle were measured by plasma emission
retical study. Firstly, through kinetic experiment, the rule of spectrometry (ICP). The chemical composition of phosphorus
reaction change was investigated and its mechanism was was determined by chemical analysis.
analyzed. Then the mathematical model of kinetics was devel- The decomposition rate or conversion fraction of the
oped to further explain and reproduce the reaction process, chemical reaction of related impurities can be calculated by
and to obtain some meaningful kinetic parameters for further measuring the impurity content in the solution
in-depth research and industrial application.
X = Wil ml /(Wis ms )

2.1. The experimental method where, X is the conversion fraction of the impurity. Wil is the
weight fraction of the impurity contained in the liquid phase,
The raw materials of the experiment were mainly powder of and ml is the total weight of the liquid phase. Wis is the weight
phosphate rock and phosphoric acid. The phosphoric acid was fraction of the impurity in the solid ore sample, and ms is the
prepared by using thermo-process phosphoric acid. The com- weight of the solid ore.
position of phosphate rock is more complicated. The main According to sample analysis at regular intervals, chem-
chemical components of phosphate rock are basically simi- ical compositions in the leaching solution can be obtained,
lar, but there are some differences due to the different genesis and a series of conversion results were thus calculated. The
and strata of the deposit. A medium-grade phosphate rock was impurity conversion fraction X indicates the ratio of the
used in the experiments. The ore came from JH phosphate amount of impurities entering the liquid phase over the origi-
mine in southwest China. The chemical components of the nal solid phase impurity mass. The change of iron/aluminum
ore are as follows — containing impurity conversion fraction with reaction
P2 O5 26.69%, CaO 36.55%, Fe2 O3 2.29%, Al2 O3 2.47%, MgO time/temperature in acid solution can be obtained.
1.568%, F2.72%, H2 O 0.205%, Acid insoluble matter 19.53%,
Burn loss 6.98%. 2.2. Kinetics theory exploration ideas
Detected by X-ray diffraction analysis (XRD), the pres-
ence of the solid phase in the mineral is mainly apatite Some interesting cases have been obtained in the experiment.
(Ca5(PO4 )3 F), dolomite (CaMg(CO3 )2 ), pyrite (FeS2 ), sulphur- Generally speaking, the conversion fraction of iron-aluminum
phosphorus aluminum strontium ore (SrAl3 [(OH)6 /SO4 PO4 ]), mineral impurities increases with temperature rise, but under
kaolinite (Al4 [(OH)8 /Si4 O10 ]), clinochlore (Mg6 (Mg, certain conditions in the experiment, the conversion fraction
Al)6 [(OH)16 /Al2 Si6 O2 ]), iron dolomite (Ca(Fe, Mg)(CO3 )2 ), decreases with temperature rise. This appears to be abnormal.
hematite (Fe2 O3 ), etc. While phosphoric acid concentration was as lower as
The experiments were carried out at temperature of 50◦ , 15% P2 O5 , it can be seen that conversion fraction of
60 , 70◦ respectively. And phosphoric acid concentration was
◦ iron-aluminum mineral impurities became lower with the
510 Chemical Engineering Research and Design 1 5 3 ( 2 0 2 0 ) 507–516

Fig. 3 – Profile of deposition layers on the reactive point.

and the solubility changes positively with temperature. That

is, the solubility increases with temperature rise.
Fig. 2 – Substance covering layer like snow heaps. It may be inferred from the situation that the solubility
of neutral iron or aluminum phosphate decreases with the
increase of temperature, and rapidly reaches or approaches
temperature increase. The higher the temperature in the range saturation in the localized region. The driving force of disso-
of 50 ◦ C–60 ◦ C, the lower the conversion fraction. In other lution is thus weakened. At the same time, the precipitated
words, the effect of temperature on the reactive situation has neutral salt adheres to the surface of the active point, form-
a negative effect. ing a local membrane or clogging, which also reduces the
As above description, this phenomenon occurs in the macroscopic reaction rate. From the kinetic process, though
liquid–solid reaction system. In the actual ore particles, a temperature rise speeds up the intrinsic chemical reaction
large number of micropores and microtubules are formed in rate while acid concentration is lower (<15%), the passiva-
the long-term geological action. The size and shape of these tion film is caused quickly and macro reaction rate is thus
micropores have fractal features to some extent. In localized decreased obviously as the reactive surface is masked. This
regions, crystallization may be induced owing to the local microscopic view is sketched in Fig. 3. However, when acid
supersaturation caused by chemical reaction. This conjecture concentration increases to 20% P2 O5 , dissolution rate of neu-
was verified by the test of SEM. In Fig. 2, we can see the newly tral phosphate salts are accelerated, and generated acidic
formed substances like snow heaps covered on the surface of salts are easy to dissolve in the liquid phase. In this case,
the reaction sites. Their phase contents were determined by X the macroscopic reaction rate of the impurities will increase
ray diffraction analysis and X ray photo electron spectroscopy, with temperature rise for the isolation membrane is reduced
which were mainly composed of some neutral phosphates or eliminated. From the above analysis it can be seen, in
such as FePO4 and AlPO4 . order to reduce the amount of iron and aluminum contain-
The electron probe was used to test a micro-region which ing impurities entering the liquid phase while treating with
is distributed with disordered substance layers. After mathe- low grade phosphate rock in the wet process, it is expected to
matical treatment with the result of X-ray diffraction, the main adopt suitable reaction conditions in the process. For exam-
solid phase was gave as follows. ple, if adopting higher reaction temperature while using lower
Neutral iron phosphate, FePO4 , 3.48% concentration of phosphoric acid, it is capable to inhibit the
Neutral aluminum phosphate, AlPO4 , 21.26% transfer of impurities from solid to liquid phase.
Orthoclase, K[AlSi3 O8 ], 20.87% Based on the above analysis, a kinetic model can be derived.
Clinochlore, Mg6 (Mg,Al)6 [(OH)16 /Al2 Si6 O20 ], 2.15% Not only the complex characteristics of microporou change,
Kaolinite, Al4 [(OH)8 /Si4 O10 ], 6.28% but also the effects of reaction and crystallization on the dif-
Quartz, SiO2 , 4.5% fusion process should be considered.
Apatite, Ca5 [F/(PO4 )3 ], 41.3%
It is a multiphase micro zone obviously. (Cameron and Bell,
1907; Carter and Hartshorne, 1923; Wu, 1991) Cammero et al 3. Mathematical model of the kinetics
investigated the solubility of aluminum phosphate and iron system
phosphate in phosphoric acid and found that their solubility in
low concentration phosphoric acid (<15% P2 O5 ) is very small, To predict their reaction kinetic behavior, a mathemati-
only a few thousandths or less. The solid phases formed are cal model can be built. Considering the above mentioned
mainly neutral phosphates such as compounds of FePO4 ·xH2 O complicated situation, the model for the conversion-time rela-
and AlPO4 ·xH2 O etc, and there are some amorphous sub- tionship of the liquid–solid reaction is suggested based on a
stances. Their solubility varies negatively with temperature. porous medium on account of thermodynamics and kinetic
That is, solubility decreases with temperature rise. When the influence.
concentration of phosphoric acid is higher than a certain When a type of salt is dissolved in the acid solvent, it will
threshold value, initial neutral salts dissolve, and acidic phos- ionize to the cations of the metal and the anions of the acid in a
phates such as Fe(H2 PO4 )3 and Al(H2 PO4 )3 etc will be formed, degree. The crystallization and dissociation of ions exist in the
 Chemical Engineering Research and Design 1 5 3 ( 2 0 2 0 ) 507–516 511

dynamic state. Here, Phosphate as a special case is expressed

by the following equation

[Fe, Al]PO4 ↔ [Fe, Al]3+ + PO3− 4

If the actual cations are represented by the general symbol

B,

BPO4 ↔ B3+ + PO3− 4

For a saturated solution of a poorly soluble electrolyte, the

dissolved state is often expressed in terms of solubility prod-
uct, K.

K(T) = C∗B C∗

here, CB * and C*, the ionic concentrations (mol L−1 ) in equilib-

rium, are the metal and acid ion concentration respectively.
According to the idea of nonequilibrium thermodynamics, the
ion reaction affinity potential can be defined as (Prigogine, Fig. 4 – Globe shell of particle.
1955)
K(T) temperature rise. Endotherm helps crystal growth and hinders
A = RT ln
CB C liquid passing.
here, CB and C are the actual ion concentrations in their non- Usually, if the micro pores are too small, it is always easy
equilibrium thermodynamics state, respectively. The rate of to induce crystallization in the local area, whether from the
total solid phase formation can be thus expressed as view point of thermodynamics or kinetics.
  A  The above analysis is based on the microstructure. If
Uc = Us 1 − exp − expanding an angle of the view to a small sphere and making
RT
a thin spherical shell in it (Fig. 4), a differential equation can
where, Us is unidirectional rate of liquid to solid phase trans- be proposed. Eq. (1) represents the percolation process, which
fer, expressed as mol m−3 s−1 . Since the clusters are formed in contains two factors of diffusion flow and resistance caused
the fractal channels, considering the effect of the capillary on by crystal deposition.
the liquid saturation, a minus sign is added in the exponent
brackets. After arrangement we have ∂n ∂C
= Dp 4 r2 − Dco U (C)(rs − r).4 r2 (1)
  ∂t ∂r
 CB C
 CB C
Uc = Us 1 − = Us 1 − =U here, Dp and Dco are diffusion and synergistic coefficient,
K(T) C∗B C∗
   respectively. This  is the distribution fraction of pores in the
2M C unit volume, which has fractal characteristic. And n is the total
s 1 − exp −
L RTrp C∗ mass flux. The intrinsic chemical reaction may be neglected
for its rate is too fast in the situation. If zero order steady state
here  is the average surface free energy of the local transition is taken,
zone near the wall, which is related to both the temperature
and concentration of the liquid phase. dn/dt = 0
Generally, if the system absorbs heat while dissolving, the
solution heat is positive. So the temperature rise is beneficial Dp dC
= (rs − r)dr
to dissolve the matter and restricts crystal formation. It is pos- Dco U (C)
sible to bring about a special flow in the porous region near the
interface while dissolving, which is called as the percolation As a boundary condition, r = rs , C = C0 , where C0 is the con-
layer or semi-flow layer. It can be supposed that the semi-flow centration of bulk liquid. If first order steady state is set, there
layer may be formed as a transitional zone between two phase is
when mass transfers from surface of the solid to bulk liquid.
dn/dt = const = n’
The ␴ means the deviation of the free energy between the
transitional zone and the solid surface phase. According to So, we have
thermodynamics principle (Birdi, 1997; Zhu and Zhao, 1996).   
 dH  n Dp dC 2M C
= − Dco Us 1 − exp − (rs − r)
 = 4 r2 dr L RTrp C∗
dAs S,P,ni

The enthalpy change between semi-flow phase and solid

phase, H, may cause the free energy variation between n Dp dC
= − G (1 − Ke C) (rs − r)
these two phase in the reversible process. While dissolution 4 r2 dr
occurs, the area of the micro region always increases due to here, G = Dco Us . Supposing that bulk liquid concentration is
pore emerging. That is dAs > 0. If dissolution heat is positive, invariable and C* = C0 , we have
dH > 0, ␴ > 0, the both of values of H and ␴ increase with 
temperature rise. Endotherm is favorable to dissolving and liq- 1 2M
Ke = exp −
uid flowing. On the contrary, If dissolution heat is negative, C0 L RTrp
dH < 0, ␴ < 0, the both of value of H and ␴ decrease with
512 Chemical Engineering Research and Design 1 5 3 ( 2 0 2 0 ) 507–516

Based on zero order steady state, a solution is obtained. The last form is available
Here is set 1-Ke C = E, which seems to be a relative driving force
of interface mass transfer. C0 Mt r2  2 1  
= s 1 − (1 − x)1/3 − 1 − (1 − x)1/3
ıkm 2Dp GKe
 
GKe 2
E = Q exp (rs − r) (2)
2Dp
C0 Mt r2  2
= s 1 − (1 − x)1/3 − (1−
If adopting the variable coefficient method for the inte- ıkm 2Dp
gral constant Q, the solution of first order steady state can be 
1/3
 C0 2M
obtained. The integral variable is (1 − x) exp −
  Dco Us km L RTrp

expr GKe
− 2D (rs − r)
2
Ke n p
Q(r) = dr
4 Dp r2 It can be simplified into the following form by combin-
rs
ing coefficients. Considering that the particle structure has
Since the variation of the definite integral is continuous fractal characteristics, the surface is not dense and smooth,
in the boundary range, the mean value theorem is applied to and many breakpoints are formed, so it has fractal dimension
above equation. Eq. (2) becomes scale. The power exponent is revised to scale index.

1   
Ke n km 1 GKe 2 C0 Mt r2     C0
E − E0 ≈ − exp (rs − r) (3) = s 1 − (1 − x)1/3 − 1 − (1 − x)1/3 exp
4 Dp r rs 2Dp  2Dpd Dcop

At the boundary, r → rs , E → E0 , and E-E0 = Ke (C0 –C). 2M
− (8)
Eq. (3) is a relationship between liquid concentration and L RTrp
the radius of the pellet under the first order steady state. Of
course, this is only a quasi steady state in a short time. The
variation with time can be seen in the next step. The substance
variation of the solid pellet may be expressed below. 4. Results of experiment and discussion
4 3
N= r The basic regularity of the transfer of iron-aluminum impuri-
3 M
ties from phosphate rock to phosphoric acid liquid phase were
studied from both experiments and theoretical models.
dN dN dr 4  2 dr
= = r (4) A series of conditions involved in experiment include tem-
dt dr dt M dt
peratures (50 ◦ , 60 ◦ , 70 ◦ ) and phosphoric acid concentrations
Considering that transfer flux passing the interface equals (15%, 20%, 25%P2 O5 ). The reaction time was 60–90 min. Under
almost the decreased amount of the pellet. the combined temperature and concentration conditions,
more than 200 experiments were carried out to determine
dN the conversion fraction of impurities, and the relationship
n = −ı (5)
dt between conversion fraction and time was obtained.
We consider the variation range of conversion fraction of
The Eq. (5) can be substituted into the appropriate position
aluminum ions within the reaction time limit of 60 min. If the
of the Eq. (3). And the interface concentration C is small and
acid concentration was 15%, the conversion fraction can reach
negligible. At the same time, the bulk liquid concentration Co
to 0.25 and 0.2 at 60 ◦ C and 70 ◦ C, respectively. If the concen-
can be regarded as the constant due to the large amount of
tration was 20%, the conversion fraction can reach to 0.12 and
the liquid phase.
0.21 at 50 ◦ C and 70 ◦ C,. respectively. If the concentration was

t
r   25%, the conversion fraction can reach to 0.14, 0.20 and 0.40 at
ıkm r GKe 2 50 ◦ C, 60 ◦ C and 70 ◦ C, respectively.
C0 dt = − (rs − r) exp (rs − r) dr (6)
MDp rs 2Dp Similarly, the reaction time is limited to a range of 60 min
0 rs considering the conversion of iron ions. If the concentration
was 15%, the conversion fraction can reach to 0.20 and 0.045

rs   at 60 ◦ C and 70 ◦ C,. respectively. If the concentration was 20%,
ıkm r GKe 2 the conversion fraction can reach to 0.09 and 0.12 at 50 ◦ C and
C0 t = d exp (rs − r)
MGKe rs 2Dp 70 ◦ C,. respectively. If the concentration was 25%, the conver-
r
sion fraction can reach to 0.05, 0.10 and 0.20 at 50 ◦ C, 60 ◦ C and
In the above partial integration, only the first term is 70 ◦ C, respectively.
retained due to the similarity of the items, and the rest is For the JH phosphate rock used, it can be seen that under
ignored. the same reaction time, phosphoric acid concentration and
   temperature conditions, aluminum-containing impurity is
ıkm r GKe 2 more likely to enter into phosphoric acid. When the concen-
C0 t = exp (rs − r) − 1 (7)
MGKe rs 2Dp tration is 25% and the temperature is 70 ◦ , the conversion
fraction of aluminum-containing impurity reaches 0.4, while
After simplifying exponent, the relationship between the that of iron is 0.2. This is because the iron-containing impurity
conversion and radius is assumed to be of the JH phosphate rock is mainly in the form of pyrite FeS2 ,
r = rs (1 − x)1/3 which is less soluble in phosphoric acid than other common
iron-containing impurity like hematite.
 Chemical Engineering Research and Design 1 5 3 ( 2 0 2 0 ) 507–516 513

and concentration. Where, the values of the ␴ can be estimated

Table 1 – Basic parameter values for numerical fit
calculations. simply by the following equation.

Parameter Value Parameter Value

␴ = ␴’
(9)
FePO4 2870 kg/m3 MFe2O3 159.70 kg/kmol
AlPO4 2566 kg m3 MAl2O3 101.96 kg/kmol
T
C 
0
liquid 1200 kg/m3 R 8314.3 J/kmol K ς =1+ −1 (10)
273.15 Cr
particle 2900 kg/m3 rs 0.0003 m
 FePO4 0.03957 J/m2 Cr 0.17
 AlPO4 0.03186 J/m2 C0 0.15, 0.20, 0.25 The ␴ in the formula is only a macroscopic parameter. In
rp 1.09 × 10−6 m T 323.15, 333.15, 343.15K order to simplify the calculation, the constant ␴’ is merged into
Dcopw which becomes a comprehensive parameter reflecting
synergy.
Within 60 min of reaction time, the conversion fraction of The conversion-time curves and data points of impuri-
phosphorus can reach 0.95 in the range of 15%–25% and tem- ties containing iron and aluminum are depicted in Fig. 5–6.
perature 50–70 ◦ C. Both in the situation of impurities containing aluminum and
The change of conversion fraction versus time was iron, the conversion fractions exhibit a sudden change near
measured under each combination condition. It has been the critical point. Below the critical concentration of phos-
previously known that, above and below the critical concentra- phoric acid, the conversion or reactive fraction decrease with
tion, the conversion trend of the impurities with temperature temperature rising. If above it, the conversion fraction tends
rising is opposite, so there is a mathematical condition at the towards increasing along with temperature rising. The critical
critical point, . Since Eq. (8) is an implicit function, concentration estimated by experiment data in the system is
and the constants Dpd , Dcop , , etc. are also related to con- about 17% P2 O5 . It was discovered by XDR and SEM that there
centration and temperature, it is difficult and complicated to were dense layers of neutral phosphate salts FePO4 or AlPO4
derive the analytical formula of critical concentration. The below the critical point. Because the solubility of these neu-
more appropriate method is to define the critical concentra- tral phosphates decreases with temperature rising, it is able
tion based on the experimental conditions, so setting Cr = 17%. to accelerate crystallites forming, which covers the surface
The basic information for calculation is listed in Table 1. thereby impeding the diffusion. But while the concentration
The combining method of nonlinear and linear was of phosphate acid is higher than this critical point, the state of
adopted to calculate mathematical model fitting experimental the system goes into another thermodynamics phase region,
data. The calculation was performed by the Matlab pro- acidic salts will form, and its solubility will increase with
gram. The fitting of Eq. (8) with experimental data shows temperature rising. The covering layer becomes much thin-
that the mathematical model is satisfactory for the fitting of ner than the former, and is even easy to dissolve again. The
experimental data. The relevant parameters obtained can be transfer process can thus be enhanced.
expressed in Tables 2 and Table 3. In addition to obtaining Regarding the change of Dpd , the variation range of Dpd
the relevant kinetic parameters, the curve graphs can also be in iron-containing impurity reaction is larger than that in
directly obtained, such as Fig. 5, Fig. 6 and Fig. 7. the case of aluminum-containing impurity. Taking the vari-
Owing to influence of negative temperature effect, param- ation of Dpd in the reaction of aluminum-containing impurity
eters such as Dpd , Dcop , , ␴ in Eq.(8) are related to temperature as an example, some interesting phenomena can be found.

Table 2 – Reaction parameters of aluminum-containing impurity obtained by fitting the experimental data.
Parameters ConcentrationP2 O5 %w 50 ◦ 60 ◦ 70 ◦

Dpd /m2 min−1 15 2.5198 × 10−11 1.5616 × 10−11

20 1.134 × 10−11 5.524 × 10−11
25 1.782 × 10−11 7.0719 × 10−11 5.4870 × 10−10

␥ 15 2.791 2.726
20 2.253 2.088
25 2.194 1.842 1.470
3
Dcop /kmol/min m 15 1.9103 0.78951
20 0.1743 0.3685
25 0.3845 0.2553 0.4714

Table 3 – Reaction parameters of iron-containing impurity obtained by fitting the experimental data.
Parameters ConcentrationP2 O5 %w 50 ◦ 60 ◦ 70 ◦

Dpd /m2 min−1 15 5.510473 × 10−14 0.40237 × 10−15

20 1.314 × 10−16 4.35368 × 10−15
25 1.969292 × 10−13 3.985096 × 10−13 7.25022 × 10−13

␥ 15 4.588 4.498
20 5.318 4.527
25 2.629 2.782 3.331

Dcop /kmol/min m3 15 1.247 1.1358

20 0.8500 0.42964
25 1.2720 0.5987 0.3343
514 Chemical Engineering Research and Design 1 5 3 ( 2 0 2 0 ) 507–516

Fig. 5 – Digestion kinetics curves of aluminum-containing impurity in phosphoric acid.

(a) 15%P2 O5 ; (b) 20% P2 O5 ; (c) 25% P2 O5 .

Fig. 6 – Digestion kinetics curve of iron-containing impurities in phosphoric. acid (a) 15%P2 O5 ; (b) 20% P2 O5 ; (c) 25% P2 O5.

Under the different phosphoric acid concentration, the vari-

ation of diffusion coefficient with temperature is different. In
fact, the negative temperature effect can be described by per-
colation flow theory. The liquid penetrates a solid membrane
as if it diffuses through a lattice made of crystal cluster. The
substance can coagulate or dissolve. Phase transition, liquid
flow and chemical reaction are interrelated. If below the criti-
cal concentration, the diffusion coefficient decreases with the
increase of temperature. This indicates that the diffusion pro-
cess tends to stop when salting-out forms a dense layer at
the reaction interface. But above this critical point, the diffu-
sion channel gradually opens as the temperature increases.
This gradual process is not exactly like a normal percolation
characterized by a sudden opening and closing. Fig. 7 indicates
the variation of diffusion coefficients of aluminum-containing
impurities associated with temperature. When the phosphoric
acid concentration is relatively low, such as 15%, the diffu-
sion coefficient decreases with increasing temperature, which
indicates that the diffusion resistance in the membrane of
micropores increases with temperature rising. In this case,
a new insoluble substance in the pore blocks the channel. Fig. 7 – Diffusion coefficients of Al -containing impurity.
Under the condition of higher phosphoric acid concentra-
the particle, the generation of crystal clusters and the large
tion, the diffusion coefficient increases when the temperature
number of bifurcation and pore channel blockage caused
rises, indicating that the blocked substances in the pores
by the initial structure, the fractal dimension of radius
has been dissolved, which makes the mass transportation
length or interface is always different from the topological
smoother.
dimension.
It can be assumed that the radius change of the gran-
The gamma coefficient ␥ reflects the degree of spatial seg-
ule ball is uniformly reduced. However, due to breakage of
mentation. In fact, the change of radius in one dimension is
 Chemical Engineering Research and Design 1 5 3 ( 2 0 2 0 ) 507–516 515

Table 4 – Change of surface energy of impurity containing Al and Fe.

Concentration ␴ value/J m−2
Sort of impurity
P2 O5 %w
50 ◦ 60 ◦ 70 ◦

Al salt 15 0.02729 0.02715

20 0.03851 0.03892
25 0.04960 0.05015 0.05070

Fe salt 15 0.03389 0.03372

20 0.04783 0.04834
25 0.06160 0.06228 0.06296

not continuous. Considering the integral over the radius, if phase of phosphoric acid are studied from the aspects of
the fractal dimension on the radius is suggested as D, there experiments and theoretical model. The range of temperature
is D + 1 = ␥. As can be seen about the aluminum-containing and phosphoric acid concentration was 50–70 ◦ and 15–25%,
impurity situation from Table 2, when the phosphoric acid respectively. Under the combined temperature and concentra-
concentration is 15%, ␥ = 2.79–2.73, the fractal dimension tion conditions, a large number of experiments were carried
D = 1.79–1.73 on the radius can be obtained. It is shown that out to determine the conversion fraction of impurities for the
a large number of non-shaped whiskers are generated in selected JH phosphate ore. The relationships between the con-
the radial direction at this time, resulting in space fold- version fraction of iron and aluminum mineral impurities in
ing, so the fractal dimension is larger than one. When the digestion process versus reactive time were obtained through
phosphoric acid concentration is 25%, ␥ = 2.19 1.47, and the the experiment. It was noticed that the fraction of transfer of
fractal dimension on the radius is D = 1.19 0.47. The fractal aluminum impurities to liquid within 60 min could reach 0.4,
dimension is able to be lower than 1 when the tempera- while that of iron impurities was only 0.2. Moreover, a novel
ture is higher. This indicates that there are voids or special phenomenon was found that the conversion fraction of the
breaking points along the radial direction. However, in the reaction did not increase with temperature as it usually does,
reaction of iron-containing impurity, a large number of three- showing a special rule.
dimensional structures are generated due to the rapid initial In the two-phase reaction complex kinetics system of phos-
reaction rate of iron-containing impurity. The reaction curve phate rock leaching in phosphoric acid, there exists a novel
is also steep, and the scale factor is also high, which may be macroscopic reaction phenomenon, namely negative temper-
called “hyperspace”. When the concentration increases, the ature effect. Under the experimental conditions, when the
voids increase and folds decrease, so the fractal dimension phosphoric acid concentration is higher than the critical point
decreases (Mandelbrot and Benoit, 1983). of 17% P2 O5 , the conversion fraction of the iron-aluminum
In general, as the fractal dimension increases, the space mineral impurity transferred into the liquid phase increases
fold increases, thereby increasing the fine passage, increas- with increasing temperature. When it is below the critical
ing the surface area, and complicating the structure. If the point, it decreases with increasing temperature. The trend of
fractal dimension is reduced, the voids and breakpoints may the kinetic curve is similar synchronously.
increase, and the surface area will decrease. However, the The negative temperature effect is present in the kinetic
situation of macroscopic response needs to be determined system of complex reactions of some solid–liquid two phases,
according to the correlation between the transfer process and caused by coupling multiple influences. The solid phase
the internal structure. medium contains a large number of micropores. Due to
In addition, the significance of ␴ value is the macroscopic the reaction of iron-aluminum mineral impurities at dif-
surface energy of the interfacial transition phase, which can ferent phosphoric acid concentrations, two kinds of salts,
indicate whether the mass transport from solid to liquid is neutral or acidic salts, are formed. Their solubility varies
smooth or not. The evaluated values in Table 4 roughly show differently with temperature. When the phosphoric acid con-
the change trend of the surface energy. When the concen- centration is below a critical value, the state of the system
tration of liquid is higher than the critical point, the ␴ value is in the crystalline region of neutral iron phosphate or alu-
increases with the temperature rise, and the dissolution heat minum phosphate, and a large amount of salt precipitation
also increases. The temperature rise facilitates dissolution, will form. Furthermore, the precipitate covers the surface
thereby promoting the mass transfer from solid to liquid of the active reaction site, hindering the transfer of mass.
(Table 4). If the concentration is lower than the critical value, The overall macroscopic result is that the acid decomposi-
both of the ␴ value and the dissolution heat decrease when the tion rate of iron or aluminum mineral impurities decreases
temperature is raised, which are favorable for the precipitation with increasing temperature, although the chemical reac-
of the substance and blocking the micropores. Therefore, the tion may be enhanced by temperature rise. On the other
␴ value is a significant parameter in the transport of interfa- hand, if the phosphoric acid concentration is above the crit-
cial substance, which has an important effect on the diffusion ical point, the system enters the phase region formed by the
process. acidic salt, and their solubility will increase as the temper-
ature increases. The acidic salt deposition is less likely to
occur at this time. The original crystalline material on the sur-
5. Conclusion
face changes to other forms or disappears, and the diffusion
resistance decreases. Under such conditions, the total decom-
The basic rules of the transfer of iron and aluminum min-
position rate of these impurities increases as the temperature
eral impurities from phosphate rock particles to the liquid
516 Chemical Engineering Research and Design 1 5 3 ( 2 0 2 0 ) 507–516

increases, and the substance easily diffuses to the liquid Frikha, Nader, Schaer, Eric, Houzelot, Jean-L.éon, 2006.
bulk. Methodology of multiphase reaction kinetics and
The kinetic model established according to this mechanism hydrodynamics identification: Application to catalyzed
nitrobenzene hydrogenation. Chem. Eng. J. 124, 19–28.
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Geetha, K.S., Surender, G.D., 2000. Experimental and modelling
the experimental data satisfactorily. The kinetics parameters studies on the aeration leaching process for metallic iron
were thus obtained such as Dpd , ␥, Dcop . When above the crit- removal in the manufacture of synthetic rutile.
ical point, the kinetic curve of 70 ◦ C lies above that of 50 ◦ C. Hydrometallurgy 56, 41–62.
When it is below the critical point, the position of the kinetics Habashi, F., 2005. A short history of hydrometallurgy.
curve is reversed, and the 70 ◦ C curve is below 50 ◦ C. The vari- Hydrometallurgy 79, 15–22.
ation of the diffusion coefficient Dpd is also enough to show Heydarpoura, T., Rezaia, B., Gharabaghi, M., 2011. A kinetics study
of the leaching of a calcareous phosphate rock by lactic acid.
this trend.
Chem. Eng. Res. Des. 89, 2153–2158.
Negative temperature effects are useful for limiting the Hou, Xiaochuan, Xiao, Liansheng, Gao, Congjie, Zhang, Qixiu,
transport of some substances from the solid phase to the liq- Zeng, Li, 2010. Kinetics of leaching from Ni–Mo ore smelter
uid phase. For example, it can be used to achieve selective dust using sodium chlorate in a mixture of hydrochloric and
leaching in the wet process of phosphate rock and rare earth sulfuric acids. Hydrometallurgy 104, 76–80.
enrichment. Hu, Qili, Xie, Yanhua, Feng, Chuanping, Zhang, Zhenya, 2019.
Fractal-like kinetics of adsorption on heterogeneous surfaces
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Declaration of interests Krull, Rainer, Peterat, Gena, 2016. Analysis of reaction kinetics
during chemostat cultivation of Saccharomyces cerevisiae
The authors declare that they have no known competing using a multiphase microreactor. Biochem. Eng. J. 105,
financial interests or personal relationships that could have 220–229.
appeared to influence the work reported in this paper. Lin, Jianjun, Zhang, Wulong, Chen, Huisu, Zhang, Rongling, Liu,
Lin, 2019. Effect of pore characteristic on the percolation
The authors declare the following financial inter-
threshold and diffusivity of porous media comprising
ests/personal relationships which may be considered as
overlapping concave-shaped pores. Int. J. Heat Mass Transf.
potential competing interests: 138, 1333–1345.
Mandelbrot, Benoit B., 1983. Chen, S.J., Ling, F.H. (Translators),
Acknowledgment 1998. The Fractal Geometry of Nature (Updated and
Augmented), The Far East Press of Shanghai.
This work is supported by the National Natural Science Foun- Menon, A., Sathyamurthy, N., 1981. Negative activation energy for
the Cl(Br)O + NO → Cl(Br) + NO2 reactions. J. Phys. Chem. 85,
dation of China [Grant No. 20046001]
1021–1023 (SICHUAN UNIV on September 19, 2018 at 14:23:15
UTC) https://pubs.acs.org/.
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