Summary
This paper suggests the application of the minimum principle of Pontryagin to the solution of an optimal control problem for a porous packed bed cooled by a flow of incompressible fluid. The procedure for determination of the optimal initial temperature distribution in a one-dimensional packed bed is developed. The amount of heat transferred to the fluid phase is utilized as the optimization criterion. It is necessary to maximize this amount under the following constraints: (a) a given amount of heat is initially stored in the packed bed and (b) a given duration of the process. As the control the initial temperature of the packed bed is considered. Qualitative changes in the behavior of the optimal initial temperature distribution take place as the duration of the process is increased.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- a sf :
-
specific surface area common to solid and fluid phases, m2/m3
- c p :
-
specific heat at constant pressure, J kg−1 K−1
- h sf :
-
fluid-to-solid phase heat transfer coefficient, W m−2 K−1
- I v :
-
modified Bessel function of the orderv
- L :
-
dimensionless length of the porous slab
- L′:
-
length of the porous slab, m
- t :
-
dimensionless time
- t′:
-
time, s
- t f :
-
dimensionless duration of the process
- T :
-
temperature, K
- T 1 andT 2 :
-
reference temperatures, K
- u min :
-
the lower boundary for admissible controls
- u max :
-
the upper boundary for admissible controls
- v :
-
velocity of the fluid phase, m s−1
- z :
-
dimensionless Cartesian coordinate
- z′:
-
Cartesian coordinate, m
- ε:
-
porosity
- Φ:
-
dimensionless temperature of the fluid phase
- Φ in :
-
dimensionless inlet temperature of the fluid phase
- λ 1 :
-
the Lagrange multiplier
- θ:
-
dimensionless temperature of the solid phase
- θ 0 :
-
dimensionless initial temperature of the solid phase
- ϱ:
-
density, kg m−3
- f :
-
fluid
- s :
-
solid
References
Bejan, A.: The optimal spacing for cylinders in crossflow forced convection. ASME J. Heat Transfer117, 767–770 (1995).
Bejan, A., Morega, Al. M.: The optimal spacing of a stack of plates cooled by turbulent forced convection. Int. J. Heat Mass Transfer37, 1045–1048 (1994).
Bejan, A.: How to distribute a finite amount of insulation on a wall with nonuniform temperature. Int. J. Heat Mass Transfer36, 49–56 (1993).
Mereu, S., Sciubba, E., Bejan, A.: The optimal cooling of a stack of heat generating boards with fixed pressure drop, flowrate or pumping power. Int. J. Heat Mass Transfer36, 3677–3686 (1993).
Vafai, K., Sözen, M.: Analysis of energy and momentum transport for fluid flow through a porous bed. ASME J. Heat Transfer112, 690–699 (1990).
Vafai, K., Sözen, M.: An investigation of a latent heat storage porous bed and condensing flow through it. ASME J. Heat Transfer112, 1014–1022 (1990).
Sozen, M., Vafai, K.: Analysis of the non-thermal equilibrium condensing flow of a gas through a packed bed. Int. J. Heat Mass Transfer33, 1247–1261 (1990).
Amiri, A., Vafai, K.: Analysis of dispersion effects and non-thermal equilibrium, non-Darcian, variable porosity incompressible flow through porous media. Int. J. Heat Mass Transfer37, 939–954 (1994).
Sözen, M., Vafai, K., Kennedy, L. A.: Thermal charging and discharging of sensible and latent heat storage packed beds. J. Thermophys. Heat Transfer5, 623–630 (1991).
Kuznetsov, A. V.: An investigation of a wave of temperature difference between solid and fluid phases in a porous packed bed. Int. J. Heat Mass Transfer37, 3030–3033 (1994).
Schumann, T. E. W.: Heat transfer: liquid flowing through a porous prizm. J. Franklin Inst.208, 405–416 (1929).
Carslaw, H. S., Jaeger, J. C.: Conduction of heat in solids, 2nd ed., pp. 391–394, Oxford: University Press 1959.
Arpaci, V. S., Clark, J. A.: Dynamic response of fluid and wall temperatures during pressurized discharge for simultaneous time-dependent inlet gas temperature, ambient temperature, and/or ambient heat flux. Adv. Cryogen. Eng.7, 419–432 (1962).
Hung, F. T., Nevins, R. G.: Unsteady-state heat transfer with a flowing fluid through porous solids. ASME paper No. 65-HT-10 (1965).
Jang, W. J., Lee, C. P.: Dynamic response of solar heat storage systems. ASME Paper No. 74-WA/HT-22 (1974).
Burch, D. M., Allen, R. W., Peavy, B. A.: Transient temperature distributions within porous slabs subjected to sudden transpiration heating. ASME J. Heat Transfer98, 221–225 (1976).
Riaz, M.: Analytical solutions for single- and two-phase models of packed-bed thermal storage systems. ASME J. Heat Transfer99, 489–492 (1977).
White, H. C., Korpela, S. A.: On the calculation of the temperature distribution in a packed bed for solar energy applications. Solar Energy23, 141–144 (1979).
Spiga, G., Spiga, M.: A rigorous solution to a heat transfer two phase model in porous media and packed beds. Int. J. Heat Mass Transfer24, 355–364 (1981).
Pontryagin, L. S., Boltyanskii, V., Gamkrelidze, R., Mishchenko, E.: The mathematical theory of optimal processes. New York: Interscience 1962.
Athans, M., Falb, P. L., Optimal control: an introduction to the theory and its applications. New York: McGraw-Hill 1966.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kuznetsov, A.V. Determination of the optimal initial temperature distribution in a porous bed. Acta Mechanica 120, 61–69 (1997). https://doi.org/10.1007/BF01174316
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01174316