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Applied Scientific Research

, Volume 52, Issue 1, pp 67–79 | Cite as

Optimal convective heating of a hollow cylinder with temperature dependent thermal conductivity

  • Chung J. Chen
  • M. N. Ozisik
Article

Abstract

An infinitely long hollow cylinder with temperature dependent thermal conductivity is heated optimally by controlling the ambient temperature,Ta(t), such that at the final specified timetf, the final temperature distributionTf(r) of the hollow cylinder approximates a desired uniform temperatureTd as closely as possible with a minimum amount of energy spent for the heating.

Keywords

Thermal Conductivity Ambient Temperature Minimum Amount Final Temperature Hollow Cylinder 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

Bi

Biot number

cp

specific heat

g

gradient vector defined by equation (9)

H

Hamilton defined by equation (5d)

h

heat transfer coefficient

J

performance index defined by equation (4)

Jc

augmented performance index defined by equation (5a)

k0

reference thermal conductivity

k(T)

thermal conductivity defined by equation (1e)

T(r, t)

temperature

Tr

reference temperature

T0

initial temperature

Td

desired temperature

Ta(t)

ambient temperature

t, tf

time and final time, respectively

r

spatial coordinate

ri

inner radius

r0

outer radius

S(m)

direction of descent defined by equation (11)

u(ξ)=Ta(t)/Tr

dimensionless ambient temperature

α

weighting coefficient in equation (4)

β

nonlinearity coefficient of thermal conductivity

ε=βTr

dimensionless nonlinearity coefficient

η

dimensionless spatial coordinate

ηi

dimensionless inner radius

ρ

density

ξ=(k0t)/(ρcpr02)

dimensionless time

ξf

dimensionless final time

θ(η, ξ)=T(r, t)/Tr

dimensionless temperature

θd

dimensionless desired temperature

θ0

dimensionless initial temperature

Φ

function defined by equation (5b)

Ψ

function defined by equation (5c)

λ(η, ξ)

Lagrange multiplier

μ(m)

conjugate gradient parameter defined by equation (10)

Superscript

(m)

mth iteration

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References

  1. Butkovskii, A.G. and Lerner, A.Y., The optimal control of systems with distributed parameters'.Automn Remote Control 21 (1960) 472–477.Google Scholar
  2. Cavin, R.K. III and Tandon, S.C., Distributed parameter system optimum control design via finite element discretization.Automatica 13 (1977) 611–614.Google Scholar
  3. Lasdon, L.S., Mitter, S.K. and Waren, A.D., The conjugate gradient method for optimal control problems.IEEE Transactions on Automatic Control Vol. AC-12 (1967) pp. 132–138.Google Scholar
  4. Meric, R.A., Finite element analysis of optimal heating of a slab with temperature dependent thermal conductivity.International Journal of Heat and Mass Transfer 22 (1979) 1347–1353.Google Scholar
  5. Meric, R.A., Finite element and conjugate gradient methods for a nonlinear optimal heat transfer control problem.International Journal for Numerical Methods in Engineering 14 (1979) 1851–1863.Google Scholar
  6. Sage, A.P.,Optimum Systems Control. Englewood Cliffs, NJ: Prentice-Hall (1968).Google Scholar
  7. Sakawa, Y., Solution of an optimal control problem in a distributed-parameter system.IEEE Transactions on Automatic Control Vol. AC-9 (1964) pp. 420–426.Google Scholar
  8. Sakawa Y., Optimal control of a certain type of linear distributed-parameter system.IEEE Transactions on Automatic Control Vol. AC-11 (1966) pp. 35–41.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Chung J. Chen
    • 1
  • M. N. Ozisik
    • 1
  1. 1.Mechanical and Aerospace Engineering DepartmentNorth Carolina State UniversityRaleighUSA

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