BSysE 595 Biosystems Engineering for Fuels and Chemicals
Spring 2013
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
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.
Filip Stankovikj
4/18/2012
Washington State University 2 of 17

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).
Filip Stankovikj
4/18/2012
Washington State University
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
Representative
Component BP [K] BP [C]
Mol weight
[g/mol]
4
5
6
7
1
2
3
8
9
10
11
Formaldehyde
Ethanol
Formic acid
Acetic acid
Propionic acid n-butric acid p-cresol
Eugenol
Hydroquinone
Pyrolytic lignin
Extractives
253.9 -19.1
351.6 78.6
373.9 100.9
391.0 118.0
414.2 141.2
436.0 163.0
475.0 202.0
526.0 253.0
558.0 285.0 p*=0 p*=0
285.0
285.0
31.4
48.4
61.6
60
71
105.8
125.3
150.9
160
450
460
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4/18/2012
Washington State University 4 of 17

12 Non-volatiles (unknown) p*=0 285.0 1050
13 Water 373 100 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
Sulfuric Acid
Water
First
Condenser
150 – 200 ⁰C
Second
Condenser
80 ⁰C
Third
Condenser
20 ⁰C
Gases
Lignin oligomers and
Sugars
Pyrolytic Lignin
Precipitation
Pyrolytic
Lignin
Mono-phenols and Furans
Esterification
Esters
C1-C4
Oxygenates
C4-Olefins
Esters
Ethanol &
Butanol
Sugars Fermentation
Figure 1. Proposed bio-refienery concept to be modeled
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
Filip Stankovikj
4/18/2012
Washington State University 5 of 17

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.
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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Washington State University 6 of 17

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 , 𝐶 𝑖
(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:
𝐶 𝑖
(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 k i
. 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 𝑑𝑘
= 𝑑𝑖 𝑑𝜃
∙ 𝑑𝜃 𝑑𝑘
(4)
For hydrocracking, the dependence k vs. 𝜃 is assumed to obey a power law ( 𝛼 is a model parameter): 𝑘
= 𝜃 1/𝛼
(5) 𝑘 𝑚𝑎𝑥
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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Washington State University 7 of 17

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)
𝐷(𝑘) = 𝑁 ∙ 𝑘 𝛼 𝛼 𝑚𝑎𝑥
∙ 𝑘 𝛼−1
(6)
When you integrate the function above in the limits from 𝑘 = 0 to 𝑘 𝑚𝑎𝑥 that she species-type distribution function satisfies the following criteria:
, you will obtain
1 𝑘 𝑚𝑎𝑥
∙ ∫ 𝐷(𝑘) ∙ 𝑑𝑘 = 1
𝑁
0
Material balance for the species with reactivity k can be defined with:
(7) 𝑑𝑐(𝑘, 𝑡) 𝑘 𝑚𝑎𝑥
= −𝑘 ∙ 𝑐(𝑘, 𝑡) + ∫ 𝑝(𝑘, 𝐾) ∙ 𝐾 ∙ 𝑐(𝐾, 𝑡) ∙ 𝐷(𝑘) ∙ 𝑑𝐾 (8) 𝑑𝑡 𝑘 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
∙
[ e
−[
{( 𝑘
𝐾
) 𝑎0 𝑎
1
−0.5}
]
2
Where
𝐴 = 𝑒
−(
0.5
𝑎
1
)
2 is obtained when 𝑝(𝑘, 𝐾) = 0 at 𝑘 = 𝐾 .
𝐵 = 𝛿 ∙ (1 − 𝑘
𝐾
)
When 𝑘 = 0 𝑝(𝑘, 𝐾) = 𝛿
𝑆 ∙ √2 ∙ 𝜋
− 𝐴 + 𝐵
]
(9)
(10)
(11)
(12)
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4/18/2012
Washington State University 8 of 17

Parameters in this model 𝑎
0
, 𝑎
1
, 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.
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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4/18/2012
Washington State University 9 of 17

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 ): 𝜃 𝑖
=
𝑇𝐵𝑃 𝑖
− 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 𝜃 𝑖 for the feed and product for those N points. Calculate the values for 𝑤𝑡 𝑖
% 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 , 𝑎
0
, 𝑎
1
, 𝛿 , and 𝑘 𝑚𝑎𝑥
.
Next step, use equation (5) to obtain the rate of reaction for each of the N points (k vs. 𝜃 plot): 𝑘 𝑖
= 𝑘 𝑚𝑎𝑥
∙ 𝜃 𝑖
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 𝑘 𝑖
≤ 𝑘 ≤ 𝑘
Lagrange polynomial: 𝑖+1
and for any residence time t, and represent it with a first degree 𝑘 − 𝑘 𝑐(𝑘, 𝑡) = ( 𝑘 𝑖+1 𝑖
− 𝑘 𝑖
) ∙ 𝑐(𝑘 𝑖+1 𝑘 − 𝑘
, 𝑡) + ( 𝑘 𝑖
− 𝑘 𝑖+1 𝑖+1
) ∙ 𝑐(𝑘 𝑖
, 𝑡)
For the initial time t=0 i.e. for the bio-oil feed we get: 𝑘 − 𝑘 𝑐(𝑘, 0) = ( 𝑘 𝑖+1 𝑖
− 𝑘 𝑖
) ∙ 𝑐(𝑘 𝑖+1 𝑘 − 𝑘
, 0) + ( 𝑘 𝑖
− 𝑘 𝑖+1 𝑖+1
) ∙ 𝑐(𝑘 𝑖
, 0)
(16)
(17)
Filip Stankovikj
4/18/2012
Washington State University 10 of 17

The weight fraction of any species with reactivities between 𝑘 𝑖 calculated: 𝑤𝑡 𝑖 𝑘 𝑖+1
= ∫ 𝑐(𝑘, 0) ∙ 𝐷(𝑘) ∙ 𝑑𝑘 𝑘 𝑖
If we substitute (17) in (18):
≤ 𝑘 ≤ 𝑘 𝑖+1
(18)
can be 𝑤𝑡 𝑖
Or abbreviated:
= 𝑐(𝑘 𝑖+1 𝑘 𝑖+1 𝑘 − 𝑘
, 0) ∫ ( 𝑘 𝑖 𝑘 𝑖+1 𝑘 𝑖+1 𝑖
− 𝑘 𝑘 − 𝑘
∙ ∫ ( 𝑘 𝑖 𝑘 𝑖
− 𝑘 𝑖+1 𝑖+1 𝑖
) ∙ 𝐷(𝑘) ∙ 𝑑𝑘
) ∙ 𝐷(𝑘) ∙ 𝑑𝑘 ∙
+ 𝑐(𝑘 𝑖
, 0) 𝑤𝑡 𝑖
= 𝑐(𝑘 𝑖
, 0) ∙ 𝑎 𝑖1
+ 𝑐(𝑘 𝑖+1
, 0) ∙ 𝑎 𝑖2 𝑎 𝑖1
= 𝑘 𝑖
(19)
(20)
1
− 𝑘 𝑖+1
𝑁 ∙ 𝛼
∙ 𝑘 𝛼 𝑚𝑎𝑥
[( 𝑘 𝛼+1 𝑖+1 𝛼 + 1
− 𝑘 𝛼+1 𝑖+1
) − ( 𝛼 𝑘 𝑖 𝛼+1 𝛼 + 1
− 𝑘 𝑖+1 𝑘 𝑖 𝛼
)] (21) 𝛼 𝑎 𝑖2
= 𝑘 𝑖+1
1
− 𝑘 𝑖
𝑁 ∙ 𝛼
∙ 𝑘 𝛼 𝑚𝑎𝑥
[( 𝑘 𝛼+1 𝑖+1 𝛼 + 1
− 𝑘 𝑖 𝑘 𝛼 𝑖+1
) − ( 𝛼 𝑘 𝑖 𝛼+1 𝛼 + 1
− 𝑘 𝑖 𝛼+1
)] (22) 𝛼
At t=0 equation (20) becomes: 𝑤𝑡 𝑖
(0) = 𝐴(𝑘) ∙ 𝑐(𝑘 𝑖
, 0) 𝑤𝑡 𝑖
(0) = [𝑤
1
𝑤
2
𝑤
3
… 𝑤 𝑛
] 𝑇
(23)
(24)
Filip Stankovikj
4/18/2012
Washington State University 11 of 17

(25) 𝑐(𝑘 𝑖
, 0) = (𝑐(𝑘
1
, 0) 𝑐(𝑘
2
, 0) 𝑐(𝑘
3
, 0) … 𝑐(𝑘 𝑛+1
, 0))
𝑇 𝑤𝑡 𝑖
(0) can be calculated from the feed TBP as:
(26) 𝑤𝑡 𝑖
(0) = 𝑤𝑡| 𝜃+∆𝜃
− 𝑤𝑡| 𝜃
(27) 𝑐(𝑘 𝑖
, 0) is obtained by solving the equation (23) with Matlab’s minimization function:
𝑁
𝑀𝑖𝑛 𝐽(𝑐(𝑘, 0)) = ∑[𝑐(𝑘 𝑖
, 0) − 𝑐(𝑘 𝑖+1
, 0)] 2
(28) 𝑖=1
Now we go onto solving the mass balance, substituting equation (16) in (8) and discretizing: 𝑑𝑐(𝑘 𝑖
, 𝑡) 𝑑𝑡
= −𝑘 𝑖
∙ 𝑐(𝑘 𝑖
, 𝑡) 𝑛 𝑘 𝑖+1
+ ∑ ∫ 𝑝(𝑘 𝑖 𝑘 𝑖 𝑖=1 𝑘 − 𝑘 𝑖
∙ [( 𝑘 𝑖+1
− 𝑘
∙ 𝐷(𝑘) ∙ 𝑑𝐾 𝑖
, 𝐾) ∙ 𝐾 ∙ 𝑐(𝐾, 𝑡)
) ∙ 𝑐(𝑘 𝑖+1 𝑘 − 𝑘
, 𝑡) + ( 𝑘 𝑖
− 𝑘 𝑖+1 𝑖+1
)] ∙ 𝑐(𝑘 𝑖
, 𝑡)
(29)
Filip Stankovikj
4/18/2012
Washington State University 12 of 17

𝑑𝑐(𝑘 𝑖
, 𝑡) 𝑑𝑡
= 𝑐(𝑘 𝑖
, 𝑡) [−𝑘 𝑖 𝑘 𝑖+1
+ ∫ 𝑝(𝑘 𝑖 𝑘 𝑖 𝑛+1 𝑘 𝑗
𝐾 − 𝑘
, 𝐾) ∙ 𝐾 ∙ ( 𝑘 𝑖
− 𝑘 𝑖+1 𝑖+1
) ∙ 𝐷(𝐾) ∙ 𝑑𝐾 ]
+ ∑ ∫ 𝑗=𝑖+1 𝑘 𝑗−1 𝑝(𝑘 𝑖
𝐾 − 𝑘 𝑗−1
, 𝐾) ∙ 𝐾 ∙ ( 𝑘 𝑗
− 𝑘 𝑗−1
) 𝑐(𝑘 𝑗
, 𝑡)
∙ 𝐷(𝐾) ∙ 𝑑𝐾 𝑛 𝑘 𝑗+1
+ ∑ ∫ 𝑝(𝑘 𝑖 𝑗=2 𝑘 𝑗
𝐾 − 𝑘
, 𝐾) ∙ 𝐾 ∙ ( 𝑘 𝑗
− 𝑘 𝑗+1 𝑗+1
) 𝑐(𝑘 𝑗
, 𝑡)
∙ 𝐷(𝐾) ∙ 𝑑𝐾
Or abbreviated: 𝑛+1 𝑑𝑐(𝑘 𝑖
, 𝑡 𝑟
) 𝑑𝑡
= 𝑐(𝑘 𝑖
, 𝑡 𝑟
)[−𝑘 𝑖
+ 𝐼
1𝑖
] + ∑ 𝑐(𝑘 𝑗 𝑗=𝑖+1
, 𝑡 𝑟
) ∙ 𝐼
2𝑗 𝑛
+ ∑ 𝑐(𝑘 𝑗
, 𝑡 𝑟
) ∙ 𝐼
3𝑗 𝑗=𝑖+1 𝑖 = 1,2, … , 𝑛 + 1 0 < 𝑡 𝑟
< 𝑡 𝑚𝑎𝑥
𝐼
1𝑖 𝑘 𝑖+1
= ∫ 𝑝(𝑘 𝑖 𝑘 𝑖
𝐾 − 𝑘
, 𝐾) ∙ 𝐾 ∙ ( 𝑘 𝑖 𝑖+1
− 𝑘 𝑖+1
) ∙ 𝐷(𝐾) ∙ 𝑑𝐾
𝐼
2𝑗 𝑘 𝑗
= ∫ 𝑝(𝑘 𝑖 𝑘 𝑗−1
𝐾 − 𝑘
, 𝐾) ∙ 𝐾 ∙ ( 𝑘 𝑗
− 𝑘 𝑗−1 𝑗−1
) ∙ 𝐷(𝐾) ∙ 𝑑𝐾
𝐼
3𝑗 𝑘 𝑗+1
= ∫ 𝑝(𝑘 𝑖 𝑘 𝑗
𝐾 − 𝑘 𝑗+1
, 𝐾) ∙ 𝐾 ∙ ( 𝑘 𝑗
− 𝑘 𝑗+1
) ∙ 𝐷(𝐾) ∙ 𝑑𝐾
The functions in (30) can be expressed in matrix form:
(30)
(31)
(32)
(33)
(34)
Filip Stankovikj
4/18/2012
Washington State University
(35)
13 of 17

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 𝑘 𝑛+1
and then
(31) for 𝑘 𝑛+1
will be: 𝑑 𝑑𝑡 𝑐(𝑘 𝑛+1
, 𝑡 𝑟
) = −𝑘 𝑛+1
∙ 𝑐(𝑘 𝑛+1
, 𝑡 𝑟
) (36) 𝑐(𝑘 𝑛+1
, 𝑡 𝑟
) = 𝑐(𝑘 𝑛+1
, 𝑡 𝑟−1
) ∙ exp(−𝑘 𝑛+1
∙ 𝑡 𝑟
) (37)
0 ≤ 𝑡 𝑟−1
< 𝑡 𝑟
≤ 𝑡 𝑚𝑎𝑥
For the first calculation 𝑡 𝑟−1
= 0 : 𝑐(𝑘 𝑛+1
, 𝑡 𝑟
) = 𝑐(𝑘 𝑛+1
, 0)
The compound with lower reactivity 𝑘 𝑛
is solved as: 𝑑 𝑑𝑡 𝑐(𝑘 𝑛
, 𝑡 𝑟
) = [−𝑘 𝑛+1
+ 𝐼
1𝑛
] ∙ 𝑐(𝑘 𝑛
, 𝑡 𝑟
) + 𝐼
2𝑛
∙ 𝑐(𝑘 𝑛+1
, 𝑡 𝑟
)
(38)
(39)
Again this equation can be solved like (36) as initial value problem: 𝑐(𝑘 𝑛
, 𝑡 𝑟
) = 𝑐(𝑘 𝑛
, 𝑡 𝑟−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 . 𝑤𝑡 𝑖
(𝑡) = 𝐴(𝑘) ∙ 𝑐(𝑘 𝑖
, 𝑡) (41)
𝑁
𝐽(𝑤𝑡 𝑖
(𝜃)) = ∑[𝑤𝑡 𝑖
𝐸𝑥𝑝
| 𝑡 𝑖=1
− 𝑤𝑡 𝑖
𝑃𝑟𝑒𝑑
| 𝑡
]
2
(42)
𝑁
∑ 𝑤𝑡 𝑖=1 𝑖
= 1 (43) 𝑐(𝑘 𝑖
, 𝑡) − 𝑐(𝑘 𝑖
, 𝑡 − 𝛿𝑡) 𝛿𝑡
𝑁−1
= 𝑐(𝑘 𝑖
, 𝑡)[−𝑘 𝑖
+ 𝐼
1
(𝑘 𝑖
, 𝑘 𝑖+1
)] +
𝑁−1
∑ 𝑐(𝑘 𝑗
, 𝑡) ∙ 𝐼
1
(𝑘 𝑗
, 𝑘 𝑗+1
) + ∑ 𝑐(𝑘 𝑗
, 𝑡) ∙ 𝐼
2
(𝑘 𝑗
, 𝑘 𝑗+1
) 𝑗=𝑖+1 𝑗=𝑖+1
(44)
Filip Stankovikj
4/18/2012
Washington State University 14 of 17

𝑘 𝑗+1
𝐼
1
(𝑘 𝑗
, 𝑘 𝑗+1
) = ∫ 𝑝(𝑘 𝑖 𝑘 𝑗
, 𝐾) ∙
𝐾 − 𝑘 𝑘 𝑖
− 𝑘 𝑗+1 𝑗+1
∙ 𝐷(𝐾) ∙ 𝑑𝐾
𝐼
2
(𝑘 𝑗
, 𝑘 𝑗+1 𝑘 𝑗+1
) = ∫ 𝑝(𝑘 𝑖 𝑘 𝑗
, 𝐾) ∙
𝐾 − 𝑘 𝑘 𝑗+1 𝑗
− 𝑘 𝑖
∙ 𝐷(𝐾) ∙ 𝑑𝐾 𝑘
2
𝐶
1,2
(𝑡) = ∫ 𝑐(𝑘, 𝑡) ∙ 𝐷(𝑘) ∙ 𝑑𝑘 𝑘
1
(45)
(46)
(47)
Filip Stankovikj
4/18/2012
Washington State University 15 of 17

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Ignacio Elizalde, Jorge Ancheyta, “Modeling the Simultaneous Hydrodesulfurization and
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Filip Stankovikj
4/18/2012
Washington State University 16 of 17

Results:
Filip Stankovikj
4/18/2012
Washington State University 17 of 17