Theoretical Foundations of Chemical Engineering

, Volume 40, Issue 5, pp 454–464 | Cite as

Mathematical modeling of turbulent heat and mass transfer with chemical conversions

  • L. P. Kholpanov
  • Yu. S. Polyakov
Article

Abstract

A transient model of heat and mass transfer with nonlinear sources (sinks) caused by first-and second-order chemical reactions is developed. The model uses a matching condition (equal temperature and local flux values) at the reaction zone-coolant interface. A finite-difference numerical solution to the problem is obtained using the alternating direction method. The model is tested by application to fast polymerization processes. The effect of the coolant velocity, reactor radius, and coolant temperature at the reactor inlet on the polymerization efficiency is studied.

Notation

c

concentration, mol/m3

cp

specific heat capacity, J/(kg K)

\(\tilde D\)

diffusion tensor

DT

averaged coefficient of turbulent diffusion, m2/s

Ea

effective activation energy for the reaction of monomer with active site (chain growth), J

Eb

effective activation energy for the chain termination reaction, J

F

vector of body forces

f1, f2, f3

parameters of hyperbolic equation, s

hi

longitudinal (axial) discretization step

hj

radial discretization step

hτ

discretization step in time

Ka

specific rate for the reaction of monomer with active site (chain growth), m3/(mol s)

Kb

specific rate for the chain termination reaction, 1/s

k

parameter accounting for the separate supply of monomer and catalyst

ka

preexponential factor for the rate of reaction between monomer and active site (chain growth), m3/(mol s)

kb

preexponential factor for the chain termination reaction, 1/s

L

reactor length, m

P

tensor of surface forces

Qa

molar heat for the reaction between monomer and active site, J/mol

R

universal gas constant, J/(mol K)

R2

tubular reactor radius, m

R3

cooling zone “radius,” m

r2

radial coordinate in zone II, m

r3

radial coordinate in zone III, m

r20

conversion factor for writing r2 in dimensionless form, m

r30

conversion factor for writing r3 in dimensionless form, m

\(\bar r_2 \)

dimensionless radial coordinate in zone II

\(\bar r_3 \)

dimensionless radial coordinate in zone III

T

temperature, K

T*

characteristic temperature in the modified Arrhenius equation, K

t

dimensionless time

u

velocity vector

u

averaged value of velocity, m/s

x

longitudinal coordinate, m

\(\bar x\)

dimensionless longitudinal coordinate

β = RT*/Ea

dimensionless parameter

ηa

characteristic time for the reaction between monomer and active site, s

ηb

characteristic time for the chain termination reaction, s

ϑ

dimensionless temperature

\(\tilde \lambda \)

thermal conductivity tensor

λ

thermal conductivity, J/(m s K)

λ*

effective thermal diffusivity, m2/s

λT

averaged thermal conductivity, J/(m s K)

ρ

density, kg/m3

τ

time, s.

Subscripts and Superscripts

a

monomer

b

catalyst (active site)

0

initial value

1

value at reactor inlet

2

zone II

3

zone III

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Kholpanov, L.P. and Shkadov, V.Ya., Gidrodinamika i teplomassoobmen s poverkhnost’yu razdela (Hydrodynamics and Heat Mass Transfer with Interface), Moscow: Nauka, 1990.Google Scholar
  2. 2.
    Frank-Kamenetskii, D.A., Diffuziya i teploperedacha v khimicheskoi kinetike (Diffusion and Heat Transfer in Chemical Kinetics), Moscow: Nauka, 1987.Google Scholar
  3. 3.
    Vol’pert A.I. and Khudyaev S.I., Analiz v klassakh razryvnykh funktsii i uravneniya matematicheskoi fiziki (Analysis in Classes of Discontinuity Functions and Equations of Mathematical Physics), Moscow: Nauka, 1975.Google Scholar
  4. 4.
    Trudy Vtoroi rossiiskoi natsional’noi konferentsii po teploobmeny: Tom 3. Svobodnaya konvektsiya. Teploobmen pri khimicheskikh prevrashcheniyakh (Proceeding of the Second Russian Conference on Heat Transfer: Vol. 3. Free Convection: Heat Transfer in Chemical Conversions), Moscow: Mosk. Energet. Inst., 1998.Google Scholar
  5. 5.
    Minsker, K.S. and Berlin, A.A., Fast Polymerization Processes, Amsterdam: Gordon & Breach, 1996.Google Scholar
  6. 6.
    Matkovskiy, P.E., Aldoshin, S.M., Troitskii, V.N., et al., Razrabotka i promyshlennaya realizatsiya protsessa polucheniya sinteticheskikh oligodetsenovykh masel (Development and Industrial Implementation of the Process of Producing Synthetic Oligodecene Oils), Chernogolovka: Inst. Problem Khim. Fiz., Russ. Akad. Nauk, 2004.Google Scholar
  7. 7.
    Kutateladze, S.S., Izbrannye trudy (Selected Works), Novosibirsk: Nauka, 1989.Google Scholar
  8. 8.
    Samarskii, A.A. and Vabishchevich, P.N., Vychislitel’naya teploperedacha (Computational Heat Transfer), Moscow: Editorial URSS, 2003.Google Scholar
  9. 9.
    Berlin, A.A., Minsker, K.S., Sangalov, Yu.A., et al., Calculation and Simulation of the Polymerization of Isobutylene As a Fast Reaction, Polymer Sci. USSR, 1981, vol. 22, no. 3, pp. 625–634.CrossRefGoogle Scholar
  10. 10.
    Berlin, A.A., Minsker, K.S., Prochukhan, Yu.A., and Enikolopyan, N.S., Macroscopic Kinetics of Fast Polymerization Processes, Vysokomol. Soedin., Ser. A, 1989, vol. 31, no. 9, pp. 1779–1798.Google Scholar
  11. 11.
    Minsker, K.S., Zakharov, V.P., and Berlin, A.A., Plug-Flow Tubular Turbulent Reactors: A New Type of Industrial Apparatus, Teor. Osn. Khim. Tekhnol., 2001, vol. 35, no. 2, pp. 172–177 [Theor. Found. Chem. Eng. (Engl. Transl), vol. 35, no. 2, pp. 162–167].Google Scholar
  12. 12.
    Minsker, K.S., Berlin, A.A., Zakharov, V.P., D’yakonov, G.S., Mukhametzyanova, A.G., and Zaikov, G.E., Fast Reactions in Polymer Syntheses, Zh. Prikl. Khim., 2003, vol. 76, no. 2, pp. 272–278 [Russ. J. Appl. Chem. (Engl. Transl), vol. 76, no. 2, pp. 264–270].Google Scholar
  13. 13.
    Oseen, C.W., Über die Stokes’sche Formel, und über eine verwandte Aufgabe in der Hydrodynamik, Ark. Math. Astron. Fys., 1910, vol. 6, no. 29.Google Scholar
  14. 14.
    Levich, V.G., Fiziko-khimicheskaya gidrodinamika (Physicochemical Hydrodynamics), Moscow: Fizmatgiz, 1959.Google Scholar
  15. 15.
    Roache, P.J., Computational Fluid Dynamics, Albuquerque: Hermosa, 1976. Translated under the title Vychislitel’naya gidrodinamika, Moscow: Mir, 1980.Google Scholar
  16. 16.
    Kennedy, J.P., Shinkawa, A., and Williams, F.J., Fundamental Studies on Cationic Polymerizations: Molecular Weights and Molecular Weight Distributions of Polyisobutylenes Produced by γ-Irradiation (Free Ions) and Chemical Catalysis (Ion Pairs), J. Polym. Sci. A-1, 1971, vol. 9, no. 6, pp. 1551–1561.CrossRefGoogle Scholar
  17. 17.
    Taylor, R.B. and Williams, F., Kinetics of Ionic Processes in the Radiolysis of Liquids: V. Cationic Polymerization of Isobutylene under Anhydrous Conditions, J. Am. Chem. Soc., 1969, vol. 91, no. 14, pp. 3728–3732.CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Inc. 2006

Authors and Affiliations

  • L. P. Kholpanov
    • 1
  • Yu. S. Polyakov
    • 2
  1. 1.Institute of Problems of Chemical PhysicsRussian Academy of SciencesChernogolovka Moscow oblastRussia
  2. 2.USPolyResearchAshlandUSA

Personalised recommendations