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

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

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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.

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.

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Abbreviations

c p :

specific heat at constant pressure

f ap :

apparent friction coefficient, Eq. (23)

Gr :

Grashof number,gβ(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 2m

Pe :

Peclet number, 2U m R/α

Pr :

Prandtl number,μc p /k

Q T :

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

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)

b :

mean bulk

c :

center

i :

line

m:

mean

0:

entrance

w :

wall

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  1. Departamento de Termodinámica, Universidad Simón Bolívar, Caracas, Venezuela

    A. Campo

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Campo, A. Numerical prediction of combined laminar convection in isothermal vertical tubes via the method of lines. Wärme- und Stoffübertragung 23, 239–247 (1988). https://doi.org/10.1007/BF01807327

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  • DOI: https://doi.org/10.1007/BF01807327

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