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Heat transfer solutions in laminar co-current flow of immiscible liquids

Lösungen für die Wärmeübertragung von unvermischbaren Flüssigkeiten in einer laminaren Gleichströmung

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Abstract

Thermal entry region solutions are analytically determined for horizontal, co-current laminar flow of immiscible liquids in direct contact, inside circular tubes and parallel plate channels. The related eigenvalue problem for such a composite media is readily solved by extending the ideas in the recently advanced sign-count method. It is assumed a core-annular flow configuration for circular tubes and sheat-core flow for the parallel plates channel, without consideration of interface instabilities and stratified flow. First, the velocity problem is solved for fully developed flow and pumping power expressions established for different operating conditions. Then, the temperature problem is analytically handled to yield expressions for quantities of practical interest such as total heat exchange rates, along the duct length and, again, for different flow rates and pressure drop requirements. The analysis is illustrated through consideration of an application dealing with pumping of a very viscous oil with the addition of an external thin layer of a less viscous fluid (water). Pumping power and total heat exchange are then evaluated for both geometries and critically compared to the single fluid flow problem.

Zusammenfassung

Hier sind Lösungen für die thermische Eintritts-strecke von horizontalen, laminaren Gleichströmungen in unvermischbaren Flüssigkeiten, die in direktem Kontakt untereinander sind, analytisch bestimmt worden. Die Lösungen gelten für Gleichströmungen in Röhren und in parallelen flachen Kanälen. Das betreffende Eigenwertproblem für solch ein zusammengesetztes Medium ist vollkommen mit dem Gedanken des kürzlich weiterentwickelten Zeichenzählverfahrens gelöst worden. Für die Rohre sind kreisring- und kernförmige Strömungen und für die parallelen Plattenkanäle Schichtkernströmungen angenommen worden. Hierbei ist die Grenzflächeninstabilität nicht in Betracht gezogen worden. Als erstes ist das Geschwindigkeitsproblem für eine vollkommen entwickelte Strömung gelöst und es sind Ausdrücke für die Pumpleistung für verschiedene Betriebsbedingungen ermittelt worden. Dann ist das Temperaturproblem analytisch behandelt worden, um Ausdrücke für die Größen von praktischem Interesse zu erzielen, wie die gesamte Wärmeaustauschrate über die Kanallänge für verschiedene Strömungsgeschwindigkeiten und Druckverlustanforderungen. Die Berechnung ist durch die Betrachtung eines Anwendungsbeispiels veranschaulicht worden, bei dem sehr zähflüssiges Öl mit einer zusätzlichen äußeren, dünnen Schicht, die weniger zähflüssig ist, gepumpt wurde. Die Pumpleistung und der gesamte Wärmeaustausch sind für beide Geometrien ausgewertet und kritisch mit dem einfachen Strömungsproblem von Fluiden verglichen worden.

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Abbreviations

k m :

thermal conductivities (m=1 or 2)

\(\hat k = \frac{{k_1 }}{{k_2 }}\)=k 1/k 2 :

thermal conductivities ratio

\(\bar K = - \frac{{dp}}{{dz}}\) :

axial pressure gradient term

\(\dot m\) :

total flow rate

\(\dot m_m \) :

flow rates for each stream (m=1 or 2)

L :

duct's length

r 1 :

dimensional interface position

r 2 :

tube radius or half the distance between parallel-plates

T 0m :

inlet fluid temperatures (m=1 or 2)

T w :

duct wall temperature

\(\bar u\) :

average flow velocity

u m(r):

velocity fields (m=1 or 2)

\(\bar u_m \) :

average flow velocities for each stream (m=1 or 2)

\(\bar \alpha \) :

as defined in Eq. (8e)

\(\hat \alpha = \frac{{\alpha _1 }}{{\alpha _2 }}\) :

thermal diffusivities ratio

α m :

thermal diffusivities (m=1 or 2)

δ :

dimensionless interface position

λ i :

eigenvalues of problem (10)

ψ mi(R):

eigenfunctions of problem (10), (m=1 or 2)

\(\hat \mu = \frac{{\mu _2 }}{{\mu _1 }}\) :

viscosities ratio

μ m :

viscosities (m=1 or 2)

m=1 or 2:

internal and external fluid quantity, respectively

i :

order of eigenvalue and related quantities

*:

quantities for single fluid (internal) flow situation

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Authors and Affiliations

  1. Programa de Engenhaira Mecânica, COPPE/UFRJ, Cidade Universitária, Cx. Postal 68503, 21945, Rio de Janeiro - RJ, Brasil

    E. Nogueira & R. M. Cotta

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  1. E. Nogueira
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  2. R. M. Cotta
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Nogueira, E., Cotta, R.M. Heat transfer solutions in laminar co-current flow of immiscible liquids. Wäarme- und Stoffübertragung 25, 361–367 (1990). https://doi.org/10.1007/BF01811560

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