Advertisement

Wärme - und Stoffübertragung

, Volume 23, Issue 4, pp 239–247 | Cite as

Numerical prediction of combined laminar convection in isothermal vertical tubes via the method of lines

  • A. Campo
Article

Abstract

Simultaneous hydrodynamic and thermal development of a laminar flow due to superimposed natural and forced convection in an isothermal vertical pipe is investigated numerically. The distinct feature of the computational procedure is that it uses the method of lines, wherein the Navier-Stokes equations accounting for buoyancy and constant properties are reduced to a system of first order ordinary differential equations. The latter is readily solved by a standard fourth-order Runge-Kutta technique. The paper includes a detailed discussion of the proposed methodology, and in addition to this, it provides the distributions of velocities, pressure and temperature. All calculations are based on a coarse grid with ten lines uniformly distributed in the cross-stream direction of the pipe when the velocity profile at the entrance is assumed parabolic. Computed results accounting for both upflow and downflow situations are in good agreement with other more elaborate numerical investigations and also with available experimental data employing air and water as working fluids.

Keywords

Convection Laminar Flow Coarse Grid Forced Convection Thermal Development 
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

cp

specific heat at constant pressure

fap

apparent friction coefficient, Eq. (23)

Gr

Grashof number,(Tw−To) R3/v2

g

acceleration of gravity

h

convection coefficient

k

thermal conductivity

l

tube length

L

parameter related to the sign of the buoyancy term, Eq. (3)

m

mass flow rate

Nu

local Nusselt number, 2hR/k

n

number of lines

p

pressure

P

dimensionless pressure, (p-p o -ϱgx L)/ϱ U m 2

Pe

Peclet number, 2U m R/α

Pr

Prandtl number,μc p /k

QT

total heat transfer

r

radial coordinate

R

radius

Re

Reynolds number,2U m R/v

rT

temperature

u

xial velocity

U

dimensionless axial velocity,u/U m

v

radial velocity

V

dimensionless radial velocity,v/U m

x

axial coordinate

Z

dimensionless axial coordinate,x/R

Greek symbols

α

thermal diffusivity

β

coefficient of thermal expansion, (l/ϱ)(∂)p

η

dimensionless radial coordinate,r/R

θ

dimensionless temperature, (T - Tw)/(To - Tw)

v

kinematic viscosity

Ω

dimensionless heat transfer, Eq. (25 a)

ϱ

density

ψ

generalized variables, Eq. (28)

Subscripts

b

mean bulk

c

center

i

line

m

mean

0

entrance

w

wall

Numerische Berechnung von kombinierter laminarer Konvektion in isothermen senkrechten Rohren nach der Method of lines

Zusammenfassung

Die simultane hydrodynamische und thermische Entwicklung einer laminaren Strömung infolge der Uberlagerung von freier und erzwungener Konvektion in einem isothermen senkrechten Rohr wird numerisch untersucht. Die besondere Eigenschaft der rechnerischen Prozedur ist, daß diese die Method of lines“ benutzt, worin die Navier-Stokes-Gleichungen für Antrieb bei konstanten Stoffwerten zu einem System einfacher Differentialgleichungen erster Ordnung reduziert werden. Das letztere ist ohne weiteres mit der Runge-Kutta-Technik 4. Ordnung zu lösen. Die Untersuchung beinhaltet eine detaillierte Diskussion der vorgeschlagenen Methodik und in Anfügung an diese liefert es die Verteilung von Geschwindigkeiten, Druck und Temperatur. Alle Kalkulationen basieren auf einem groben Gitter mit 10 Linien, die einheitlich in der Querstromrichtung des Rohres verteilt sind, wenn das Geschwindigkeitsprofil am Eingang des Rohres parabolisch angenommen wird. Die errechneten Ergebnisse, die sowohl für Aufwärts- als auch Abwärtsströmungen gelten, stimmen gut mit anderen aufwendigen Untersuchungen und ebenso mit verfügbaren experimentellen Daten, die Luft und Wasser als Arbeitsfluide gebrauchen, überein.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Martinelli, R. C.; Boelter, L. M. K.: An analytical prediction of superimposed free and forced viscous convection in a vertical pipe. Univ. Calif. Berkeley Publ. Eng. 24 (1942) 393–400Google Scholar
  2. 2.
    Martinelli, R. C.; Boelter, L. M. K.: Heat transfer and pressure drop for a fluid flowing in the viscous region through a vertical pipe. Trans. AIChE 38 (1942) 493–530Google Scholar
  3. 3.
    Pigford, R. L.: Nonisothermal flow and heat transfer inside vertical tubes. Chem. Eng. Progr. Symp. Ser. 51 (1955) 79–92Google Scholar
  4. 4.
    Rosen, E. M.; Hanratty, T. J.: Use of boundary-layer theory to predict the effect of heat transfer on the laminar flow field in a vertical tube with a constant temperature wall. AIChE J. 7 (1961) 112–123Google Scholar
  5. 5.
    Marner, W. J.: Combined free and forced laminar nonNewtonian convection in a vertical tube. Ph.D. Thesis Univ. of South Carolina 1969Google Scholar
  6. 6.
    Marner, W. J.; McMillan, H. R.: Combined free and forced lamiar convection in a vertical tube with constant wall temperature. J. Heat Tranfer (1970) 559–562Google Scholar
  7. 7.
    Zeldin, B.; Schmidt, F. W.: Developing flow with combined force-free convection in an isothermal vertical tube. J. Heat Transfer (1972) 211–223Google Scholar
  8. 8.
    Collins, M. W.: Finite-difference analysis for developing laminar flow in circular tubes applied to forced and combined convection. Int. J. Num. Meth. Eng. 15 (1980) 381–404Google Scholar
  9. 9.
    Suzuki, K.; Kieda, S.; Chichiki, T.; Sato, T.: Numerical study of combined convective heat transfer with variable fluid properties in the inlet region of a circular pipe. Int. Conf. of Numerical Methods in Thermal Problems, Venice, Italy, 1981Google Scholar
  10. 10.
    Shaddy, M. A.: Combined forced-free convection through vertical tubes at high Grashof numbers. Int. Heat Transfer Conference 3 (1986) 1433–1438, Munich, FRGGoogle Scholar
  11. 11.
    Liskovets, D. A.: The method of lines (review). Differential Equations 1 (1965) 1308–1323Google Scholar
  12. 12.
    Campo, A.; Lacoa, U.; Morales, J. C: Semi-analytical computation for laminar flow in the thermal entrance region of circular tubes. To appear in Int. J. Mech. Eng. Education 1987Google Scholar
  13. 13.
    Campo, A.; Schuler, C: Thermal radiation and laminar forced convection in a gas pipe flow. Wärme-Stoffübertrag. 22 (1988) 251–257Google Scholar
  14. 14.
    Campo, A.: El metodo de las lineas aplicado al desarrollo termohidráulico en tubos isotérmicos. To appear in Revista Internacional de Métodos Numéricos en Ingenieria, 1988Google Scholar
  15. 15.
    Marner, W. J.: Written discussion of Ref. [7]Google Scholar
  16. 16.
    Shah, R. K.: Thermal entry length solutions to the circular tube and parallel plates. National Heat Transfer Conference, Bombay, India, 1975Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • A. Campo
    • 1
  1. 1.Departamento de TermodinámicaUniversidad Simón BolívarCaracasVenezuela

Personalised recommendations