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 CCHHEEMMIICCAALL  EENNGGIINNEEEERRIINNGG  TTRRAANNSSAACCTTIIOONNSS  
 

VOL. 45, 2015 

A publication of 

 
The Italian Association 

of Chemical Engineering 

www.aidic.it/cet 
Guest Editors: Petar Sabev Varbanov, Jiří Jaromír Klemeš, Sharifah Rafidah Wan Alwi, Jun Yow Yong, Xia Liu  

Copyright © 2015, AIDIC Servizi S.r.l., 

ISBN 978-88-95608-36-5; ISSN 2283-9216 DOI: 10.3303/CET1545198 

 

Please cite this article as: Hernández-Vargas J., Martinez-Gomez J. , González-Campos J.B., Lara-Romero J., Ponce-

Ortega J.M., 2015, Optimal processing of carbon nanotubes, Chemical Engineering Transactions, 45, 1183-1188  

DOI:10.3303/CET1545198 

1183 

Optimal Processing of Carbon Nanotubes 

Julia Hernández-Vargas
a
, Juan Martinez-Gomez

a
, Janett Betzabe González-

Campos
b
, Javier Lara-Romero

a
, José María Ponce-Ortega

a,
* 

a
Chemical Engineering Department, Universidad Michoacana de San Nicolás dwie Hidalgo, Morelia, Michoacán México. 

b
Institute of Chemical and Biological Researches, Universidad Michoacana de San Nicolás de Hidalgo, Morelia 

 Michoacán, México. 

 jmponce@umich.mx  

This paper presents an optimisation approach for determining the best synthesis method and their 

corresponding operating conditions for synthesising carbon nanotubes for a desired application accounting 

for technical and economic issues as objective functions. Proper correlations for the interaction between 

the considered variables are proposed, and these correlations are based on experimental data obtained in 

the lab. The optimisation formulation is a mixed-integer nonlinear programming problem. A case study 

considering three synthesis methods of carbon nanotubes (arc discharge, laser ablation and chemical 

vapour deposition) is presented. High-temperature techniques such as electrical arc-discharge and laser 

ablation produce CNTs with the highest quality in terms of the graphitic structure; however these 

techniques are slow and expensive, while the CVD technique is able to produce extremely dense and pure 

materials, and properties such as crystal structure, surface morphology and orientation of the products can 

be manipulated and customised via the CVD’s process parameters. The results show that the proposed 

model can be useful to determine the best method and operating conditions with the maximum efficiency, 

quality and safety at the minimum cost. 

1. Introduction 

Carbon nanotubes (CNTs) are one of the most exciting discoveries in nanoscale sciences because they 

have superior properties with respect to other materials in several applications (Ng et al., 2013). Since the 

discovery of CNTs, a variety of techniques for their production have been developed, where numerous key 

parameters and operating conditions have to be manipulated to yield CNTs with the desired final 

properties (Mubarak et al., 2014). Among these techniques are the electric-arc discharge, laser ablation 

and chemical vapour decomposition (CVD), where different structures of CNTs can be produced in the 

form of vapour grown, carbon fibre and different types of carbon nanostructured materials (Liu et al., 

2014). 

Mubarak et al. (2014) carried out different experimental researches, where the simultaneous interactions 

between the manipulated variables have not been optimised to yield the desired properties, and specific 

targets such as the production of CNTs with the minimum cost, minimum environmental impact and 

minimum risk have not been considered. In this context, Hernández-Vargas et al. (2013) recently reported 

an optimisation formulation for the synthesis of nanofibers at the minimum cost, where the importance of 

taking into account these types of objective functions is highlighted. On the other hand, safety in chemical 

processes is another very important factor that is rarely taken into account as a key parameter to propose 

new strategies to obtain new materials (Schmidt, 2013). Therefore, in order to account for a proper 

methodology for identifying the optimal synthesis and operating conditions for yielding CNTs with desired 

properties, this paper proposes an optimisation formulation based on experimental correlations to 

determine the optimal synthesis method and operating conditions for producing CNTs accounting for 

technical, economic and safety issues. The optimisation formulation is based on a mathematical 

programming formulation that considers several variables and objectives as shown in Figure 1.  

 



 

 

1184 

 

Technology

CNT synthesis

Carbon source

Catalyst

Temperature

Time

Gas flow

Efficiency

Cost

Safety

Quality

Variable Inlet (V) Outlet
 

Figure 1: Interrelations for yielding CNTs with desired properties for specific applications  

1.1 Proposed optimisation formulation 
This paper proposes a disjunctive programming formulation to select the type of technology used (t) as 

well as the operating parameters (V) including the weight of carbon source, weight of catalyst, 

temperature, time and flow rate of the carrier gas for synthesizing carbon nanotubes determining the 

efficiency (Eff), cost of production (Cost), safety of the process (Saf) and quality of the obtained products 

(Qualy). The proposed disjunctive formulation is stated as follows: 

 

 

 

 
min max

, ,

t

Eff

t U

cost

p U

Saf

t U
t

Qual

t U

U t U U t U

Y

Eff f V

Cost f V

Saf f V

Qual f V

V V V U

 
 

 
 

 
 


 
 
 
    
 
  



 
In previous formulation, Yt is a Boolean variable that is true when the technology t is selected as the 

optimum one, and for this case the corresponding relationships for the efficiency, cost, safety and quality 

are applied. In the case when the Boolean variable Yt is false, the relationships are not considered. It 

should be noted that for each technology there are specific relationships for the diameter and cost as well 

as limits for the involved variables. In addition, only one technology must be selected as the optimal one. 

Then, previous disjunctive formulation must be reformulated as a set of algebraic relationships to be 

implemented as a formal mathematical programming formulation (Gutiérrez-Arriaga et al., 2013). This way, 

to state that only one technology can be selected (i.e. only one binary variable yt must be one) the 

following Eq(1) is used: 

1
t

T

y 
 

(1) 

Then, the relationships to determine the efficiency (Eff) for the process must be stated only for the selected 

technology t as follows: 

       Eff Eff -M 1 M 1 ,Eff Efft U t t U t Tf V y Eff f V y t       (2) 

where M
Cost

 is a big-M parameter used to activate the corresponding efficiency. When the technology t is 

selected, then the binary variable yt is one and the last terms of Eq(2) are zero. This means that the 

associated cost has to be calculated according to the relationship for the technology t (  Efft Uf V ). On the 

other hand, when the technology is not selected, the binary variable yt is zero and the last terms of Eq(2) 

are M
Eff

 and -M
Eff

. Notice that for this case these terms M
Eff

 and -M
Eff

 relax Eq(2); this means that the cost 

is not calculated using the corresponding function for this technology.  

Then, the relationships to determine the cost (Cost) for producing carbon nanotubes must be stated only 

for the technology t selected to be the optimum one as follows: 

       os Cost os Cost-M 1 M 1 ,C t C tt U t t U t Tf V y Cost f V y t       (3) 

The relationships to determine the safety of the process are stated as follows: 

       -M 1 M 1 ,Saff Saff Saff Safft U t t U t Tf V y Saff f V y t       (4) 

The quality of the obtained product is given by Eq(5): 



 

 

1185 

       -M 1 M 1 ,Qual Qual Qual Qualt U t t U t Tf V y Qual f V y t       (5) 

Notice that the limits for the variables involved depend on the type of selected technology, and because 

the type of technology is an optimisation variable, then this is modelled as follows: first there are upper 

(
up

U
V ) and lower (

lo

U
V ) limits for the involved variables U; however, these limits are optimisation variables 

that are determined depending on whether the technology is selected or not. This is stated as follows: 

,
lo up

U U U U
V V V U    (6) 

where these lower and upper bounds are determined as stated in Eq(7) and Eq(8):  

min

t,U
,

lo

U t Ut
V V y U    

max

t,U
,

up

U t Ut
V V y U    

(7) 

(8) 

In previous relationships, 
up

U
V  and 

lo

U
V  are lower and upper limits for the variable U associated to the 

technology t. Notice that the variables 
up

U
V  and 

lo

U
V are equal to the one of the selected technology 

because for this case the binary variable is activated. Table 1 shows the considered variables of the 

process. A set of correlations, based on experimental data (Liu et al., 2014) were obtained using the 

Statgraphics software. By a multiple regression analysis, a polynomial equation was found; it shows a 

relationship between the dependent variable (efficiency, cost and safety) and independent variables [initial 

weight of carbon source (g), weight catalyst (g), temperature (°C), gas flow rate (mL/min) and time (min)].  

1.2 Case study  

High-temperature techniques such as electrical arc-discharge and laser ablation produce CNTs with the 

highest quality in terms of the graphitic structure; however these techniques are slow and expensive 

(Prasek et al., 2011), while the CVD technique is able to produce extremely dense and pure materials, and 

properties such as crystal structure, surface morphology and orientation of the products can be 

manipulated and customised via the CVD’s process parameters, however, this process includes inherent 

chemical and safety hazards caused by the use of toxic, corrosive, flammable, and/or explosive precursor 

gases. Prasek et al. (2011) analysed the parameters that influence the growth mechanism of CNTs as the 

catalyst, carrier gas type and flow rate, substrate, synthesis temperature and reaction time. The authors 

determined that the most effective and widely used catalysts are the Fe/Co/Ni based because of the high 

solubility of carbon in these metals at high temperature and the high carbon diffusion rate in these metals. 

Glaser (2012) showed that typically nanometer-size metal particles are required to enable hydrocarbon 

decomposition and the growth rate of CNT depends on the catalyst particle size and the diffusion rate of 

carbon through the catalyst; the higher the catalyst particle size, the lower the growth rate, while the 

growth rate of CNTs is directly proportional to the diffusion rate of carbon through the catalyst. He 

demonstrated that the most used CNT precursors are methane, ethylene, acetylene, benzene, xylene, 

carbon monoxide and botanic source and the carrier gas also affects the growth of CNTs, and mainly H2 

and Ar are used. In this paper we studied three technologies for synthesising carbon nanotubes; these are 

arc discharge, laser ablation and chemical vapour deposition, where in the latter two carbon sources were 

analysed (camphor and turpentine). For each technology a correlation between cost and efficiency as 

function of initial weight of carbon source (g), weight catalyst (g), temperature (°C), gas flow rate (mL/min) 

and time (min) was obtained and it is shown in Table 1. Table 2 shows the carbon source, catalyst and 

flow gas for each technology analysed, and in the Table 3 the correlations of each technology studied in 

this paper are shown. 

Table 1: Variables for carbon nanotubes synthesis for the presented case study 

Independent  variable Process parameter 

v1 Weight of carbon source, g 

v2 Weight of catalyst, g 

v3 Temperature, °C 

v4 Gas flow rate, mL/min 

v5 Time, min 



 

 

1186 

 
Table 2: Carbon source, catalyst and flow gas for each technology 

Technology Carbon source,  

g 

Catalyst,  

g 

Flow gas, 

mL/min 

 

Arc discharge  3.75 2.755 200  

Laser ablation 0.67 0.001 400  

CVD camphor 0.10 0.035 40.0  

CVD turpentine 4.00 0.107 10.0  

Table 3: Correlations between cost and efficiency of each technology  

Technology Correlation for efficiency  Correlation for cost 

Arc discharge  2 3 4 5 2

2 4

 = 20.5048 +45.3291• + 0.0258•  0.0553• 0.2602• - 0.8614•

0.0154•  

1
Eff v v v v v v

v v

   

 

 
1 2 4 5

0.0042 1.106 0.032 0.002TotCost v v v v         

Laser ablation 
2 2 4 2 3

1 5

 = 32.16 856.77•  + 71369.4•  - 216.889•  - 40.2681•

-5.5714•

1
Eff v v v v v v

v v

  



 
1 2 4 5

0.0042 0.810 0.032 0.003TotCost v v v v         

CVD camphor 3 2
2 2 2 3

 = 63.9175 - 3.03707•  + 50439.6•  + 23303.8•  3056.96• +0.0572793•

0.513682• 2.32872 •

1

4 5

Eff

 

v v v v v

v v

 

 

 
1 2 4 5

0.023 0.273 0.032 0.0015TotCost v v v v         

CVD turpentine  = 116.273 - 61.6575•  + 1665.57•   + 0.053• - 1.102• + 4.66•1 2 3 4 5Eff  v v v v v
 

1 2 4 5
0.003 0.273 0.032 0.0015TotCost v v v v         

 

The presented model also takes into account the analysis of the safety of different technologies for the 

production of carbon nanotubes, it was determined according to the toxicity caused by exposure to harmful 

chemicals during each process. The analysis of the security in the synthesis of carbon nanotubes process 

is determined by Eq(9) - Eq(13), where the safety for each technology is a function of the variables of the 

process defined in the optimisation of the synthesis process. 

 Saft USaf f V  (9) 

Saff Carbonsource Catalyst gas

t
f R R R  

 
(10) 

MLD
mfR

CSseCarbonsour

seCarbonsour

50

5.0
))(

 

(11) 

MLD
mfR

CCatalyst

Catalyst

50

5.0
))(

 

(12) 

MLD
mfR

GIgas

Gas

50

5.0
))(

 
(13) 

The LD50 values were obtained from the database of the software SCRI (SCRI 2013) and the considered 

average weight of an adult (w) was 70 kg. 

Table 4 shows the correlation for carbon source and solvent as function of process parameters for each 

technology. 

Table 4: Toxicity Correlations for each technology  

Technology Correlation for carbon source Correlation for solvent 

Arc discharge  
1 2 3 4 5
 = 10.9 + 7.7• 0.008• 0.0015•  +0.103•v v v v v   2 1 3 4 5 = -1.42+ 0.115• +0.0099• 0.00089• 0.0082•v v v v v 

 

Laser ablation 
1 2 3 4 5
 = 0.32 + 5.06• 0.002• 0.032•  +0.24•v v v v v   2 1 3 4 5 = -0.304+ 0.027• +0.00027• 0.00055• +0.0024•v v v v v  

CVD camphor 
1 2 3 4 5
 = 4.42 2.351• 0.0029• 0.123•  +0.086•v v v v v    2 1 3 4 5 = 0.237 0.03• 0.00032• 0.0081• +0.005•v v v v v    

CVD turpentine 
1 2 3 4 5
 = -2.005 + 0.19• 0.000055• 0.04•  +0.29•v v v v v   2 1 3 4 5 = -0.239+ 0.0025• +0.00028• 0.0015• +0.0094•v v v v v  



 

 

1187 

2. Results and discussion 

2.1 Optimisation for efficiency and costs. 
The correlations obtained in Statgraphics and the equations for determining the cost were coded in the 

software GAMS (Brooke et al., 2015). The problem consists of 26 variables, 35 constraints and 3 binary 

variables and it is solved in 0.16 s of CPU time in a computer with an Intel processor at 2.4 GHz with 8 GB 

of RAM. The results are shown in Table 5, it should be noticed that the best economic solution involves the 

use of CVD using turpentine as carbon source; whereas the use of camphor as carbon source in CVD 

provides a solution with a moderate cost but with the minimum efficiency. Arc discharge and laser ablation 

yield the worst economic scenario with a moderate efficiency. For a further analysis, Figure 2 shows a 

comparison for the different technologies and the obtained costs, where the relationships between the cost 

and the efficiency for the technologies studied in this paper are shown. It should be noted that the 

minimum cost with the maximum efficiency is for the CVD with turpentine as carbon source. Notice in 

Figure 2 that for the arc discharge method the cost per gram of carbon nanotubes increases from zero with 

0 % efficiency to 18.645 USD at ~100 % efficiency. With respect to the laser ablation method, the cost 

increases from zero at 0 % of efficiency to 14.438 USD at ~100 % efficiency. While for the chemical 

vapour deposition method, the cost increases from zero to 3.419 at ~100 % efficiency when camphor is 

used as carbon source, and it increases from zero to 0.374 USD at ~100 % efficiency when the turpentine 

is used as carbon source. 

2.2 Toxicity determination.  
The toxicity results for the different optimal solutions are shown graphically in Figure 3. Notice that the 

damage for the arc discharge method caused by exposition to the carbon source is too small due to the 

occasioned damage associated to the catalyst is 0.002 % of fatalities for exposure to 0.050 g during 5 min. 

For the laser ablation method the used catalyst represents 0.225 % of fatalities caused for exposure to 

0.001 g during 1 min. With respect to the CVD method, using camphor as carbon source, it represents 

0.001656 % of fatality caused for the exposure to 0.004 g of catalyst during 12 min and 0.01945 % 

fatalities for exposure to 0.100 g of carbon source during 12 min. Finally, for the CVD method using 

turpentine as carbon source, a value of 0.00088 % fatalities for exposure to 0.1 g of catalyst during 8 min 

and 0.000006 % of fatalities for exposure to 4 g of carbon source during 8 min were observed. Therefore, 

from the safety point of view, the safest technology is the one associated to the CVD method using 

turpentine as carbon source and ferrocene as catalyst. Notice that for the arc discharge and laser ablation 

methods the exposure value of the carbon source is small and so it is represented as N/A. 

Table 5: Optimal results 

Technology Cost (USD/g) Efficiency (%) 

Arc discharge 15.6528 1 

Laser ablation 13.4163 12 

CVD camphor 

CVD turpentine 

1.3030 

0.3713 

1 

97 

 

 

Figure 2: Results for cost and efficiency for the evaluated technologies. 



 

 

1188 

 

 

Figure 3: Percentage of fatalities for each method of synthesis of CNTs. 

3. Conclusion 

This paper has proposed a set of relationships to determine the best technology for synthesising carbon 

nanotubes considering three options of synthesis: arc discharge, laser ablation and chemical vapour 

deposition as a function of some independent variables involved in the different processes (i.e. carbon 

source weight, catalyst weight, temperature, gas flow, time, production cost and safety of the process); 

these relationships are integrated into a disjunctive programming formulation to determine the optimal 

conditions to yield the desired efficiency and the safest process at the minimum cost. A case study is 

presented, where, the safest technology with the minimum cost and maximum efficiency was the CVD 

using turpentine as carbon source, followed by CVD using camphor as carbon source. The laser ablation 

method represents the worst scenario with respect to safety and the arc ablation laser method is the worst 

solution with respect to the cost and efficiency. The proposed model was applied to a case study where 

the advantages of the proposed approach are highlighted. This approach can be useful to determine the 

minimum cost, maximum safety and operating conditions to yield a desirable maximum efficiency. Finally, 

the proposed approach is general and it can be easily extended to analyse other technologies for the 

synthesis of carbon nanotubes and different parameters. 

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