BSysE 595 Biosystems Engineering for Fuels and Chemicals
Spring 2013
Title: Application of Continuous Kinetic Lumping Model for Description of
Pyrolysis Oil Hydrodeoxygenation
Spring 2013
Author: Filip Stankovikj
Department of Biological Systems Engineering, Washington State University,
Pullman WA 99164-6120
Date: April 8, 2013
Project Title: Application of Continuous Kinetic Lumping Model for Description of Pyrolysis
Oil Hydrodeoxygenation
Contents:
1.
Project Description.................................................................................................... 2
2.
Introduction and Objectives ...................................................................................... 3
3.
Historical perspective of lump modeling .................................................................. 5
4.
Continuous kinetic lumping model – mathematical description............................... 6
5.
Matlab implementation ............................................................................................. 9
6.
Results and discussion ............................................................................................ 15
7.
Summary ................................................................................................................. 15
8.
References ............................................................................................................... 16
9.
Appendix ................................................................................................................. 17
1. Project Description
Hydrodeoxygenation has been identified as an important and almost inevitable step in
pyrolysis upgrading. The pyrolysis oil is consisted of approximately 400 compounds, and the
reactions that are involved in the process of catalytic upgrading are numerous, including:
cracking, derabonylation, decarboxylation, hydrocracking, hydrodeoxygenation, and
hydrogenation (fig. below). Today, even with the most sophisticated analytical tools we are not
able to quantify and qualify this complex mixture. It is around 40 wt% unidentifiable compounds
that are present in the bio-oil. Furthermore, knowing the kinetic parameters of the reactions of
each of these 400 compounds would make the modeling effort of the hydrotreatment formidable
and impossible task. Modeling of the complex kinetics of hydrodeoxigenation, predicting the
yields of the process at different operating conditions in crucial for process development,
optimization, control, design and selection of catalysts, etc. In order to address similar complex
problem in the processes of hydrodesulfurization and hydrocracking, throughout many decades
scientist have developed so called continuous lumping models which uses the true boiling points
distribution of the crude oil mixture, obtained experimentally, to construct continuous
concentration-reactivity function. The model follows the changes of this concentration-reactivity
function throughout the hydrocracking process, and determines the fractional yield distribution
of the species based on just few parameters. This seems to be an elegant and powerful method
for modeling various processes in chemical engineering, however very few papers tried to use
this modeling approach to describe hydrodeoxgenation processes in bio-fuel.
In this class project I will investigate the applicability of a proposed methodology for
modeling VGO (vacuum gas oil) hydrocracking to model bio-oil hydrodeoxygenation.
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Verification of the model will be done against the data for hydrodeoxygenation and the true
boiling point distributions present in the literature. Moreover, I will propose a strategy to verify
the results of this model in our lab. A programing code will be developed in Matlab, and
integrated within the framework of the modeling efforts in my thesis. It is as well worth
investigating the ways to modify the distribution function of this model in order to get a more
precise model of hydrotreatment, while using not more than 13 representative component groups.
I have chosen this particular number of groups since one of my previous efforts in modeling a
fractional condensation system demonstrated that this number of groups is sufficient to
appropriately describe the system, and those may represent an input in the next step of the biorefining concept (refer to figure and table in the next section).
2. Introduction and Objectives
There is no doubt among the scientists today that the number of fossil crude oil sources is
diminishing, and that our only alternative for production of liquid transportation fuels and
chemicals is biomass utilization. However, using biomass to produce gasoline, diesel and jet fuel
is followed by much inefficiency and although many universities and national laboratories
managed to develop viable concepts, they are still not able to produce fuels with prices
competitive to the current fossil fuel prices. One of the most viable paths for bio-fuels production
consists of biomass harvesting, fast pyrolysis and stabilization, and then refining the crude biooil in larger refineries by upgrading and distillation in order to obtain final transportation fuel
cuts. Within this concept, developed as a design case by PNNL in 2009, diverse biomass sources
might be utilized, however the large consumption of hydrogen together with the high prices of
initial biomass were identified as main reasons that prevent widespread production of bio-fuels
with acceptable price tags. In this spirit, the main goal of my graduate research is to develop new
bio-refinery concept that will utilize the advantages of multi-step condensation systems,
hydrotreatment and esterification in order to reduce hydrogen consumption and to produce novel
partially deoxygenated transportation fuels. The expensive hydrogen is necessary for the
processes of hydrodeoxygenation and hydrocracking, where oxygenated molecules are reacted
with hydrogen on a catalyst surface in order to remove the oxygen in form of water and CO2 and
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produce stabilized oil. Rather than hydrotreating the whole bio-oil which consist of compounds
that after hydrotreatment do not lead to liquid-fuel precursors and in the same time consume a lot
of hydrogen, it is better to separate the oil in fractions with distinctly different composition and
characteristics and then hydrotreat only those fractions that will lead to drop-in fuels. This
separation in distinct fractions may be achieved by stepwise condensation of pyrolysis vapors. In
one of my previous efforts I modeled a single counter-flow spray condenser as an isothermal
equilibrium flash separator in order to obtain a relationship between the product distribution and
temperature at the outlet of the condenser. Naturally, the next step in modeling of the whole biorefining concept would be modeling of the hydrotreatment step. The assumption,
approximations, and methods used in this modeling project, as well as the results and the
discussion on further developed will be presented in the text below. But first I will introduce the
results from my previous modeling research and how this project can make use of these results in
order to build one comprehensive model of a bio-refinery.
The basic hypothesis behind my previous study of the multistep condensation is that by
controlling the temperature of each of the condensation steps, it will be possible to separate fast
pyrolysis products into fractions that are easier to refine [1]. Fast pyrolysis of biomass is one of
the most promising technologies to convert solid biomass into liquid bio-oil. Bio-oil yields as
high as 70 wt.% can be achieved at 450–500 C using high heating rates and with short residence
time of the bio-oil vapors in the hot zones. The composition of the bio-oil will be the input in the
mathematical model of the condenser, which on its own hand may be obtained by modeling the
actual fast pyrolysis. Since the pyrolysis vapors consist of almost 400 chemical compounds, it is
appropriate to make assumptions in order to simplify the condenser model, and that is without
significant accuracy losses. In my case, the bio-oil vapors will be represented as a mixture of 13
groups of organic compounds with boiling points ranging between 250 and 550 K. Every group
will be embodied in a single compound, which has a boiling point in the middle of the groups
boiling range. It is these 13 compounds for which we will say that approximately represent the
pyrolysis vapor and that have to be given in advance, together with the condenser pressure p, and
temperature T. The output of the condensation step and the condenser model will be the weight
fraction of each of these compounds in the liquid and vapor phases. The list of the compounds
with their boiling points and molecular weight is given in following table:
Group
1
2
3
4
5
6
7
8
9
10
11
Filip Stankovikj
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Representative
Component
Formaldehyde
Ethanol
Formic acid
Acetic acid
Propionic acid
n-butric acid
p-cresol
Eugenol
Hydroquinone
Pyrolytic lignin
Extractives
BP [K]
253.9
351.6
373.9
391.0
414.2
436.0
475.0
526.0
558.0
p*=0
p*=0
BP [C]
-19.1
78.6
100.9
118.0
141.2
163.0
202.0
253.0
285.0
285.0
285.0
Washington State University
Mol
weight
[g/mol]
31.4
48.4
61.6
60
71
105.8
125.3
150.9
160
450
460
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12
13
Non-volatiles (unknown)
Water
p*=0
373
285.0
100
1050
18
The mathematical model developed along the lines of this class project is intended to use
the output, i.e. the bio-oil compounds distribution from the condensers in the liquid phase. The
governing phenomena that determines the real composition of bio-oil out of a non-ideal
condenser are dependent on the mass and energy transfer limitations, on the extent of interaction
of the components in the mixture, which on the other hand depend on many measurable and
immeasurable factors such as the size of the sprayed liquid particles, actual pressure and
temperature in the condenser, its height, flow patterns, velocity of vapors and non-condensable
gases inside the condenser, etc. However the results of my previous model are good enough to
coarsely describe the bio oil.
A general form of the aforementioned model build on the previous assumptions will be
implemented in Matlab.
C1-C4 Oxygenates
Hybrid Poplar
Selective
Pyrolysis
First
Condenser
150 – 200 ⁰C
Second
Condenser
80 ⁰C
Lignin
oligomers and
Sugars
Sulfuric Acid
Water
Ethanol &
Butanol
Pyrolytic Lignin
Precipitation
Pyrolytic
Lignin
Third
Condenser
20 ⁰C
Mono-phenols
and Furans
Gases
C1-C4
Oxygenates
Esterification
Esters
C4-Olefins
Esters
Sugars Fermentation
Figure 1. Proposed bio-refienery concept to be modeled
3. Historical perspective of lump modeling
There is a scarce literature on kinetic lumping modeling of hydrodeoxygenation reactions
during pyrolysis oil upgrading. Most of the research in this area has been done on describing
hydrocracking kinetics of standard petroleum oils, since prediction of yields of desired and
undesired products leads to better optimization, control, and design of refining processes, as well
as selection and design of catalyst. In any case, the feedstock in either hydrocracking or
hydrotreatment is a complex mixture of hundreds of compounds. The actual processes of
hydrocracking and hydrodeoxigenation are composed of many different reactions. This makes it
difficult, even more impractical, to follow every single compound and reaction involved in these
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chemical processes. In the beginning, as for every new reaction, the researchers tested various
model compounds in hydrocracking conditions to obtain data for product yield distribution.
Based on this data different models for hydrocracking were proposed. Product distribution
patterns. The first models proposed were in the 70’s, various discrete lumping models, where the
complex feedstock was presented as a series of lumps (lumps for gasoline, diesel, LPG, tar) and
for each of them, parallel reaction models were developed. It is obvious that the precision of
these models depended on the number of lumps, and number of reactions describing the
processes. Since every reaction carries with itself optimization parameters, increasing the
number of lumps meant increasing the number of parameters to be found, which increased the
complexity of the models, and in some cases computer algorithms can fail to estimate large
number of parameters. Few other discrete models were develop, which lowered the number of
parameters one have to deal with, but either limited the overall reaction kinetics to first order
(Stangeland, 1974 – model using comminution of particles analogy for predicting the TBP
curve), or skewed the product yield at higher reaction severities (Krishna and Saxena, 1989).
However, many of these models are still used in describing hydrocracking reactions.
If discrete lumping approaches divided the feedstock and product mixtures into cuts
according to different ranges of boiling points, molecular weights, carbon number, etc., and
presented them as a sum of subgroups of pseudocomponents, the continuous lumping model sees
the reactive mixture as a continuous mixture i.e. continuous function of the same descriptors
(molecular weight, TBP, reactivity, etc.). In other words, this concept considers the reaction
mixture as composed of infinite number of components. This concept was first introduced by
DeDoner in 1931, and then this idea found application in describing distillation,
thermodynamics, polymerization, reactions in continuous mixtures, oligomerization reactions,
chromatographic separation, coke formation, and in many other fields. Weekman used this
approach first in 1979 to describe cracking of crude oil. Chou and Ho (1988) used this approach
to describe parallel reactions of n-th order, and they showed that continuum lumping accurately
estimates the overall order of reaction in complex mixtures, which is important prerequisite for
precise design of reactors. Cicarelli (1992) introduced fragmentation and Langimuir-Hinselwood
kinetics in lump modeling, where he assumed equal stoichiometric distribution of all compounds
in the reaction. Later, McCoy and Wang developed a method where different stoichiometric
distribution for the reactants can be used.
Mathematically the continuous lumping model is based on solving integrodifferential
equation that describes the mass balance equation in hydrocracking reactions within the reactors.
Its advantage is that one needs to optimize less parameters than the discrete models in order to
precisely describe the reactions. This will be shown in the section below.
4. Continuous kinetic lumping model – mathematical description
In this class project applicability of continuous kinetic lump modeling for hydrocracking
of petroleum oil on modeling of hydrodeoxygenation of pyrolysis oil will be assessed. Pyrolysis
mixture can be described by the true boiling point (TBP) and in this case we will accept first
order kinetics for individual components in the pyrolysis oil. Any other type of kinetics can be
easily adopted in the later stages of development of this model by simply changing the yield
distribution function p(k,K). TBP as characterization parameter of the bio-oil continuously
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changes over the reaction time in the hydrotreater. TBP is a distribution function that represents
the boiling temperature vs. cumulative weight fraction of the compounds in the mixture, and for
modeling purposes it is usually represented in as a function of dimensionless temperature 𝜃
defined as:
𝜃=
𝑇𝐵𝑃 − 𝑇𝐵𝑃(𝑙)
𝑇𝐵𝑃(ℎ) − 𝑇𝐵𝑃(𝑙)
(1)
TBP(h) and TBP(l) represent the boiling points of the heavies and lightest component in
the mixture. Of course with the modeling we would like to describe the concentration
distribution of the reaction mixture at any given residence time 𝑡. If we designate with 𝐶(𝜃, 𝑡)
the concentration of species with boiling point 𝜃 at time 𝑡, than 𝐶(𝜃, 𝑡) ∙ 𝑑𝜃 will represent the
concentration of a lump of species in the boiling range of 𝜃 + 𝑑𝜃. This equals to the product of
the concentration of a pseudo component i, 𝐶𝑖 (𝑡), and 𝑑𝑖, the infinitesimal range of that
component. The following equation gives the link between the continuous and discrete lumping,
and here it is assumed that i’s are equally spaced along the i-species axis:
𝐶𝑖 (𝑡) ∙ 𝑑𝑖 = 𝐶(𝜃, 𝑡) ∙ 𝑑𝜃
(2)
We can make one more change in coordinates which gives us significant advantage on
the further development of the hydrotreating model. Namely, every species in the mixture is
strictly defined (described) by its normalized temperature 𝜃, and to each of these species in the
mixture certain reaction rate constant can be attached ki. This means that now from the TBP
curve we can obtain a relation between the concentration of the species and their reactivity
𝑐(𝑘, 𝑡). This change from one coordinate system and the other based on 𝑐(𝑘, 𝑡) is give with the
equation below:
𝐶(𝜃, 𝑡) ∙ 𝑑𝜃 = 𝑐(𝑘, 𝑡) ∙ 𝐷(𝑘) ∙ 𝑑𝑘
(3)
𝐷(𝑘) is called species-type distribution function. 𝐷(𝑘) ∙ 𝑑𝑘 is the number of species with
reactivity between 𝑘 and 𝑘 + 𝑑𝑘. The distribution character of the right-hand side of the equation
above is transferred to 𝐷(𝑘), where 𝑐(𝑘, 𝑡) remains concentration of component i with reactivity
k. 𝐷(𝑘) can be viewed as Jacobian and it is defined as:
𝐷(𝑘) =
𝑑𝑖
𝑑𝑖 𝑑𝜃
=
∙
𝑑𝑘 𝑑𝜃 𝑑𝑘
(4)
For hydrocracking, the dependence k vs. 𝜃 is assumed to obey a power law (𝛼 is a model
parameter):
𝑘
𝑘𝑚𝑎𝑥
= 𝜃1/𝛼
(5)
However, for hydrodeoxygenation this function might have to get different shape since
the assumption that the reaction rate k=0 when 𝜃=0 does not hold i.e. the species with the lowest
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boiling point 𝜃=0 takes part in the hydrogenation reactions (k>0). We may use model compound
to obtain more precise relationship for the k vs. 𝜃 dependence.
If we differentiate equation (5) and supplement it in equation (4) we will obtain equation
(6). Have in mind that 𝑑𝑖/𝑑𝜃 = 𝑁 when 𝑁 → ∞ (N stands for number of species in the mixture)
𝛼
𝐷(𝑘) = 𝑁 ∙
𝛼
𝑘𝑚𝑎𝑥
∙ 𝑘 𝛼−1
(6)
When you integrate the function above in the limits from 𝑘 = 0 to 𝑘𝑚𝑎𝑥 , you will obtain
that she species-type distribution function satisfies the following criteria:
𝑘𝑚𝑎𝑥
1
∙∫
𝐷(𝑘) ∙ 𝑑𝑘 = 1
𝑁 0
(7)
Material balance for the species with reactivity k can be defined with:
𝑘𝑚𝑎𝑥
𝑑𝑐(𝑘, 𝑡)
= −𝑘 ∙ 𝑐(𝑘, 𝑡) + ∫
𝑝(𝑘, 𝐾) ∙ 𝐾 ∙ 𝑐(𝐾, 𝑡) ∙ 𝐷(𝑘) ∙ 𝑑𝐾
𝑑𝑡
𝑘
(8)
𝑝(𝑘, 𝐾) is yield distribution function and it represents the amount of formation of species
with reactivity k from species with reactivity K. This function for hydrocracking of petroleum
products was adopted to have skewed Gaussian distribution and is given by the equation (9). I
will check the compatibility of this function with modeling of hydrodeoxygenation of pyrolysis
oil.
𝑝(𝑘, 𝐾) =
1
𝑆 ∙ √2 ∙ 𝜋
𝑘 𝑎0
{( ) −0.5}
𝐾
−[
]
𝑎1
2
∙ e
(9)
−𝐴+𝐵
[
]
Where
𝐴=𝑒
is obtained when 𝑝(𝑘, 𝐾) = 0 at 𝑘 = 𝐾.
−(
0.5 2
)
𝑎1
𝑘
𝐵 = 𝛿 ∙ (1 − )
𝐾
(10)
(11)
When 𝑘 = 0
𝑝(𝑘, 𝐾) =
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𝛿
𝑆 ∙ √2 ∙ 𝜋
Washington State University
(12)
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Parameters in this model 𝑎0 , 𝑎1 , 𝛿 are tuning parameters for this model and depend on
the catalyst type, activity, and impurities present in the feed. The 𝑝(𝑘, 𝐾) function can be
obtained from experimental data and for hydrocracking it should have the following properties:
1) 𝑝(𝑘, 𝐾) = 0 for 𝑘 = 𝐾 since species with reactivity k cannot be transformed into the
“same” kind of species under hydrocracking conditions.
2) The 𝑝(𝑘, 𝐾) function should satisfy material balance:
𝐾
∫ 𝑝(𝑘, 𝐾) ∙ 𝐷(𝑘) ∙ 𝑑𝑘 = 1
(13)
0
3) 𝑝(𝑘, 𝐾) should be non-zero at k=0.
4) 𝑝(𝑘, 𝐾) should always have a positive value.
5. Matlab implementation
The process of solving the mass balance equation (8) is iterative and goes through couple
of loops. Final product of this algorithm are the optimal model parameters, and the starting point
are TBP for the feed and for the product from a batch hydrocracking reactor. TBP of the feed and
the product can be obtained by ASTM 5307. Fast and sufficiently precise determination of TBP
can be done by ASTM D2887. This method uses traditional GC/MS instrument with a specific
(shorter) column and few standard alkanes in order to obtain simulated distillation curve.
However, the algorithm is presented in the following figure and the explanation of the steps
follows:
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First step will be to read the TBP (wt.% vs. boiling temperature °C) of the bio-oil fed into
the reactor (at residence time t=0) and the TBP at after the reaction (at final residence time t max).
Next using equation (1) this curve is transformed in dimensionless units ((wt.% vs. boiling
temperature 𝜃):
𝜃𝑖 =
𝑇𝐵𝑃𝑖 − 𝑇𝐵𝑃(𝑙)
𝑇𝐵𝑃(ℎ) − 𝑇𝐵𝑃(𝑙)
(14)
A caution here: TBP(h) of the component with highest boiling point is much higher than
the limit of the ASTM D2887 method 538°C, so this temperature should be carefully
guessed/estimated in such a way that it is close to the final boiling point of the heaviest
compound in the bio-oil mixture, and it accurately reproduces the experimental distillation curve
obtained for the entire range of distillation. TBP(l) as well includes the gasses in the feed such as
CH4, C2H6, etc.
Next ,set the number of compounds for the calculation N>40, and obtain the values for 𝜃𝑖
for the feed and product for those N points. Calculate the values for 𝑤𝑡𝑖 % for all N points by
linear interpolation between the points obtained by simulated distillation. After we start the loops
for optimizing the parameters, we need to make a guess for the values of the five parameters: 𝛼,
𝑎0 , 𝑎1 , 𝛿, and 𝑘𝑚𝑎𝑥 .
Next step, use equation (5) to obtain the rate of reaction for each of the N points (k vs. 𝜃
plot):
𝑘𝑖 = 𝑘𝑚𝑎𝑥 ∙ 𝜃𝑖 1/𝛼
(15)
In order to solve integrodifferential equation (8), we have to calculate the concentration
of species at any time of reaction t, 𝑐(𝑘, 𝑡). If we suppose linear dependence, which is reasonable
if our domain is represented with many points N>40, than we can make a linear interpolation for
𝑐(𝑘, 𝑡) at 𝑘𝑖 ≤ 𝑘 ≤ 𝑘𝑖+1 and for any residence time t, and represent it with a first degree
Lagrange polynomial:
𝑘 − 𝑘𝑖
𝑘 − 𝑘𝑖+1
𝑐(𝑘, 𝑡) = (
) ∙ 𝑐(𝑘𝑖+1 , 𝑡) + (
) ∙ 𝑐(𝑘𝑖 , 𝑡)
𝑘𝑖+1 − 𝑘𝑖
𝑘𝑖 − 𝑘𝑖+1
(16)
For the initial time t=0 i.e. for the bio-oil feed we get:
𝑘 − 𝑘𝑖
𝑘 − 𝑘𝑖+1
𝑐(𝑘, 0) = (
) ∙ 𝑐(𝑘𝑖+1 , 0) + (
) ∙ 𝑐(𝑘𝑖 , 0)
𝑘𝑖+1 − 𝑘𝑖
𝑘𝑖 − 𝑘𝑖+1
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Washington State University
(17)
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The weight fraction of any species with reactivities between 𝑘𝑖 ≤ 𝑘 ≤ 𝑘𝑖+1 can be
calculated:
𝑘𝑖+1
𝑤𝑡𝑖 = ∫
𝑐(𝑘, 0) ∙ 𝐷(𝑘) ∙ 𝑑𝑘
(18)
𝑘𝑖
If we substitute (17) in (18):
𝑘𝑖+1
𝑘 − 𝑘𝑖
(
) ∙ 𝐷(𝑘) ∙ 𝑑𝑘 + 𝑐(𝑘𝑖 , 0)
𝑘𝑖+1 − 𝑘𝑖
𝑘𝑖
𝑘𝑖+1
𝑘 − 𝑘𝑖+1
∙∫
(
) ∙ 𝐷(𝑘) ∙ 𝑑𝑘 ∙
𝑘𝑖 − 𝑘𝑖+1
𝑘𝑖
𝑤𝑡𝑖 = 𝑐(𝑘𝑖+1 , 0) ∫
(19)
Or abbreviated:
𝑤𝑡𝑖 = 𝑐(𝑘𝑖 , 0) ∙ 𝑎𝑖1 + 𝑐(𝑘𝑖+1 , 0) ∙ 𝑎𝑖2
(20)
𝛼+1
𝛼+1
1
𝑁 ∙ 𝛼 𝑘𝑖+1
𝑘𝑖+1
𝑘𝑖𝛼+1
𝑘𝑖𝛼
𝑎𝑖1 =
∙ 𝛼 [(
−
)−(
− 𝑘𝑖+1 )]
𝑘𝑖 − 𝑘𝑖+1 𝑘𝑚𝑎𝑥
𝛼+1
𝛼
𝛼+1
𝛼
(21)
𝛼+1
𝛼
1
𝑁 ∙ 𝛼 𝑘𝑖+1
𝑘𝑖+1
𝑘𝑖𝛼+1 𝑘𝑖𝛼+1
𝑎𝑖2 =
∙ 𝛼 [(
− 𝑘𝑖
)−(
−
)]
𝑘𝑖+1 − 𝑘𝑖 𝑘𝑚𝑎𝑥
𝛼+1
𝛼
𝛼+1
𝛼
(22)
At t=0 equation (20) becomes:
𝑤𝑡𝑖 (0) = 𝐴(𝑘) ∙ 𝑐(𝑘𝑖 , 0)
𝑤𝑡𝑖 (0) = [𝑤1 𝑤2 𝑤3 …
Filip Stankovikj
4/18/2012
𝑤𝑛 ]𝑇
Washington State University
(23)
(24)
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(25)
𝑐(𝑘𝑖 , 0) = (𝑐(𝑘1 , 0) 𝑐(𝑘2 , 0) 𝑐(𝑘3 , 0) …
𝑐(𝑘𝑛+1 , 0))
𝑇
(26)
𝑤𝑡𝑖 (0) can be calculated from the feed TBP as:
𝑤𝑡𝑖 (0) = 𝑤𝑡|𝜃+∆𝜃 − 𝑤𝑡|𝜃
(27)
𝑐(𝑘𝑖 , 0) is obtained by solving the equation (23) with Matlab’s minimization function:
𝑁
𝑀𝑖𝑛 𝐽(𝑐(𝑘, 0)) = ∑[𝑐(𝑘𝑖 , 0) − 𝑐(𝑘𝑖+1 , 0)]2
(28)
𝑖=1
Now we go onto solving the mass balance, substituting equation (16) in (8) and
discretizing:
𝑑𝑐(𝑘𝑖 , 𝑡)
= −𝑘𝑖 ∙ 𝑐(𝑘𝑖 , 𝑡)
𝑑𝑡
𝑛
𝑘𝑖+1
+ ∑∫
𝑝(𝑘𝑖 , 𝐾) ∙ 𝐾 ∙ 𝑐(𝐾, 𝑡)
𝑖=1 𝑘𝑖
(29)
𝑘 − 𝑘𝑖
𝑘 − 𝑘𝑖+1
∙ [(
) ∙ 𝑐(𝑘𝑖+1 , 𝑡) + (
)] ∙ 𝑐(𝑘𝑖 , 𝑡)
𝑘𝑖+1 − 𝑘𝑖
𝑘𝑖 − 𝑘𝑖+1
∙ 𝐷(𝑘) ∙ 𝑑𝐾
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4/18/2012
Washington State University
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𝑑𝑐(𝑘𝑖 , 𝑡)
= 𝑐(𝑘𝑖 , 𝑡) [−𝑘𝑖
𝑑𝑡
𝑘𝑖+1
+∫
𝑝(𝑘𝑖 , 𝐾) ∙ 𝐾 ∙ (
𝑘𝑖
𝑛+1
𝐾 − 𝑘𝑖+1
) ∙ 𝐷(𝐾) ∙ 𝑑𝐾 ]
𝑘𝑖 − 𝑘𝑖+1
𝑘𝑗
𝐾 − 𝑘𝑗−1
𝑝(𝑘𝑖 , 𝐾) ∙ 𝐾 ∙ (
) 𝑐(𝑘𝑗 , 𝑡)
𝑘𝑗 − 𝑘𝑗−1
𝑘𝑗−1
+ ∑ ∫
𝑗=𝑖+1
(30)
∙ 𝐷(𝐾) ∙ 𝑑𝐾
𝑛
𝑘𝑗+1
+ ∑∫
𝑝(𝑘𝑖 , 𝐾) ∙ 𝐾 ∙ (
𝑗=2 𝑘𝑗
𝐾 − 𝑘𝑗+1
) 𝑐(𝑘𝑗 , 𝑡)
𝑘𝑗 − 𝑘𝑗+1
∙ 𝐷(𝐾) ∙ 𝑑𝐾
Or abbreviated:
𝑛+1
𝑑𝑐(𝑘𝑖 , 𝑡𝑟 )
= 𝑐(𝑘𝑖 , 𝑡𝑟 )[−𝑘𝑖 + 𝐼1𝑖 ] + ∑ 𝑐(𝑘𝑗 , 𝑡𝑟 ) ∙ 𝐼2𝑗
𝑑𝑡
𝑗=𝑖+1
𝑛
(31)
+ ∑ 𝑐(𝑘𝑗 , 𝑡𝑟 ) ∙ 𝐼3𝑗
𝑗=𝑖+1
𝑖 = 1,2, … , 𝑛 + 1
𝑘𝑖+1
𝐼1𝑖 = ∫
𝑘𝑖
𝐾 − 𝑘𝑖+1
𝑝(𝑘𝑖 , 𝐾) ∙ 𝐾 ∙ (
) ∙ 𝐷(𝐾) ∙ 𝑑𝐾
𝑘𝑖 − 𝑘𝑖+1
𝑘𝑗
𝐼2𝑗 = ∫
𝑘𝑗+1
𝐼3𝑗 = ∫
𝑘𝑗
(32)
𝐾 − 𝑘𝑗−1
) ∙ 𝐷(𝐾) ∙ 𝑑𝐾
𝑘𝑗 − 𝑘𝑗−1
(33)
𝐾 − 𝑘𝑗+1
𝑝(𝑘𝑖 , 𝐾) ∙ 𝐾 ∙ (
) ∙ 𝐷(𝐾) ∙ 𝑑𝐾
𝑘𝑗 − 𝑘𝑗+1
(34)
𝑝(𝑘𝑖 , 𝐾) ∙ 𝐾 ∙ (
𝑘𝑗−1
0 < 𝑡𝑟 < 𝑡𝑚𝑎𝑥
The functions in (30) can be expressed in matrix form:
(35)
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Washington State University
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Matrix 𝐵(𝑘) can be calculated by knowing the values of the assumed five parameters. In
order to solve (31) one should start from the compound with highest reactivity 𝑘𝑛+1 and then
(31) for 𝑘𝑛+1 will be:
𝑑
𝑐(𝑘𝑛+1 , 𝑡𝑟 ) = −𝑘𝑛+1 ∙ 𝑐(𝑘𝑛+1 , 𝑡𝑟 )
𝑑𝑡
(36)
𝑐(𝑘𝑛+1 , 𝑡𝑟 ) = 𝑐(𝑘𝑛+1 , 𝑡𝑟−1 ) ∙ exp(−𝑘𝑛+1 ∙ 𝑡𝑟 )
(37)
0 ≤ 𝑡𝑟−1 < 𝑡𝑟 ≤ 𝑡𝑚𝑎𝑥
For the first calculation 𝑡𝑟−1 = 0:
𝑐(𝑘𝑛+1 , 𝑡𝑟 ) = 𝑐(𝑘𝑛+1 , 0)
(38)
The compound with lower reactivity 𝑘𝑛 is solved as:
𝑑
𝑐(𝑘𝑛 , 𝑡𝑟 ) = [−𝑘𝑛+1 + 𝐼1𝑛 ] ∙ 𝑐(𝑘𝑛 , 𝑡𝑟 ) + 𝐼2𝑛 ∙ 𝑐(𝑘𝑛+1 , 𝑡𝑟 )
𝑑𝑡
(39)
Again this equation can be solved like (36) as initial value problem:
𝑐(𝑘𝑛 , 𝑡𝑟 ) = 𝑐(𝑘𝑛 , 𝑡𝑟−1 )
(40)
This procedure goes on until the lightest component in the mixture is obtained. After that
we will have the 𝑐(𝑘, 𝑡) curve at time t. This is the starting point for solving the same equations,
but now for higher residence times t.
𝑤𝑡𝑖 (𝑡) = 𝐴(𝑘) ∙ 𝑐(𝑘𝑖 , 𝑡)
(41)
𝑁
𝐽(𝑤𝑡𝑖 (𝜃)) = ∑[𝑤𝑡𝑖𝐸𝑥𝑝 |𝑡 − 𝑤𝑡𝑖𝑃𝑟𝑒𝑑 |𝑡 ]
2
(42)
𝑖=1
𝑁
∑ 𝑤𝑡𝑖 = 1
(43)
𝑖=1
𝑐(𝑘𝑖 , 𝑡) − 𝑐(𝑘𝑖 , 𝑡 − 𝛿𝑡)
= 𝑐(𝑘𝑖 , 𝑡)[−𝑘𝑖 + 𝐼1 (𝑘𝑖 , 𝑘𝑖+1 )] +
𝛿𝑡
𝑁−1
𝑁−1
(44)
∑ 𝑐(𝑘𝑗 , 𝑡) ∙ 𝐼1 (𝑘𝑗 , 𝑘𝑗+1 ) + ∑ 𝑐(𝑘𝑗 , 𝑡) ∙ 𝐼2 (𝑘𝑗 , 𝑘𝑗+1 )
𝑗=𝑖+1
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4/18/2012
𝑗=𝑖+1
Washington State University
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𝑘𝑗+1
𝐼1 (𝑘𝑗 , 𝑘𝑗+1 ) = ∫
𝑝(𝑘𝑖 , 𝐾) ∙
𝐾 − 𝑘𝑗+1
∙ 𝐷(𝐾) ∙ 𝑑𝐾
𝑘𝑖 − 𝑘𝑗+1
(45)
𝑝(𝑘𝑖 , 𝐾) ∙
𝐾 − 𝑘𝑗
∙ 𝐷(𝐾) ∙ 𝑑𝐾
𝑘𝑗+1 − 𝑘𝑖
(46)
𝑘𝑗
𝑘𝑗+1
𝐼2 (𝑘𝑗 , 𝑘𝑗+1 ) = ∫
𝑘𝑗
𝑘2
𝐶1,2 (𝑡) = ∫ 𝑐(𝑘, 𝑡) ∙ 𝐷(𝑘) ∙ 𝑑𝑘
(47)
𝑘1
6. Results and discussion
7. Summary
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Washington State University
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8. References
[1]
R. J. M. Westerhof et al., “Fractional Condensation of Biomass Pyrolysis Vapors,” Energy
& Fuels, vol. 25, no. 4, pp. 1817-1829, Apr. 2011.
[2]
M. Garcia-perez, A. Chaala, H. Pakdel, D. Kretschmer, and C. Roy, “Characterization of
bio-oils in chemical families National Bureau of Standards,” Biomass and Bioenergy, vol.
31, pp. 222-242, 2007.
[3]
Karl F. Knopf. “Modeling, analysis and optimization of process and energy systems.”
John Willey and Sons. 2012
[4]
Warren L. McCabe, Julian C. Smith, Peter Harriott. “Unit operations of chemical
engineering-7th ed.” McGraw-Hill. 2005
[5]
Ignacio Elizalde, Jorge Ancheyta. “On the detailed solution and application of the
continuous kinetic lumping modeling to hydrocracking of heavy oils.” Fuel 90, 35423550, 2007
[6]
C.S. Laxminarasimhan, R.P. Verma, “Continuous Lumping Model for Simulation of
Hydrocracking.” AIChE Journal, Vol.42, No.9, Sep.1996
[7]
J. Govindhakannan, J.B. Riggs, “On the Construction of a Continuous Concentration –
Reactivity Function for the Continuum Lumping Approach.” Ind. Eng. Chem. Res., 46,
1653-1656, 2007
[8]
Ignacio Elizalde, Jorge Ancheyta, “Modeling the Simultaneous Hydrodesulfurization and
Hydrocracking of Heavy Residue Oil by Using the Continuous Kinetic Lumping
Approach.”, Energy Fuels, 3, 1999-2004, 2012
[9]
M. Sau, K. Basak, U. Manna, M. Santra, R.P. Verma, “Effects of organic nitrogen
compounds on hydrotreating and hydrocracking reactions.”, Catalysis Today, 109, 112119, 2005
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Washington State University
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9. Appendix
Results:
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