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Heat transfer from the heated concave wall of a return bend with rectangular cross section

Wärmeübertragung von einer beheizten konkaven Wand eines Umkehrkrümmers mit rechteckigem Querschnitt

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Abstract

The characteristics of the turbulent heat transfer along the heated concave walls of return bends which have rectangular cross sections with large aspect ratio have been examined for various clearances of the ducts in detail.

The experiments are carried out under the condition that the concave walls are heated at constant heat flux while the convex walls are insulated. Water as the working fluid is utilized. Using three kinds of clearance of 9, 34, and 55 mm, the Reynolds number in the turbulent range are varied from 5×103 to 8×104 with the Prandtl numbers ranging from 4 to 13.

As a result it is elucidated that both the mean and the local Nusselt numbers are always greater than those for the straight parallel plates or for the straight duct, respectively. This is attributed to Görtier vortices, which are visualized here. It is also found that the more the clearance increases, the more both the local and the mean Nusselt numbers increase.

Correlation equations for the mean and the local Nusselt numbers are determined in the range of parameters covered. Introducing the Richardson number, it appears that the local Nusselt number,Nu x , may be described as the following equation:Nu x =447.745 ·Re 1.497 x ·De −1.596 x ·F 0.960 ·Pr 0.412

Zusammenfassung

Es wird der turbulente Wärmeübergang längs der beheizten konkaven Wand eines Umkehrkrümmers mit rechteckigem Querschnitt und großem Verhältnis Breite zu Höhe bei verschiedenen Höhen untersucht. Die konkaven Wände werden mit konstanter Wärmestromdichte beheizt, die konvexen sind isoliert. Arbeitsfluid ist Wasser. Für die drei Kanalhöhen 9, 34 und 55 mm liegen die Reynolds-Zahlen zwischen 5·103 und 8·104, die Prandtl-Zahlen reichen von 4 bis 13. Es zeigt sich, daß sowohl die mittleren wie die lokalen Nusselt-Zahlen immer größer sind also die für gerade Platten oder gerade Kanäle. Das wird durch Görtier-Wirbel verursacht, die hier sichtbar gemacht werden. Mit steigender Kanalhöhe steigen auch die mittleren und die lokalen Nusselt-Zahlen.

Die lokalen Nusselt-Zahlen im untersuchten Bereich lassen sich durch die GleichungNu x =447.745 ·Re 1.497 x ·De −1.596 x ·F 0.960 ·Pr 0.412 wiedergeben mitDe als Dean-Zahl undF als Richardson-Zahl.

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Abbreviations

c :

clearance of duct

c p :

specific heat at constant pressure

De :

Dean number,\(\left[ {\bar u \cdot \left( {2c} \right)/v} \right]\sqrt {c/R} \)

De x :

local Dean number,\(\left[ {\bar u \cdot x/v} \right]\sqrt {c/R} \)

dA :

minute area of cross section of duct

d e :

hydraulic diameter of duct

F :

Richardson number, defined in Eq. (8)

f c :

centrifugal force per unit volume, ϱ(¯u 2 R)

G :

Görtler number, (¯u · δ **/v) √δ **/R 0

h m :

mean heat-transfer coefficient, defined in Eq. (3)

h x :

local heat-transfer coefficient atx=x

\(\overline {Nu} \) :

mean Nusselt number,h m · (2c)/λ

Nu x :

local Nusselt number, defined in Eq. (1)

\((Nu_x )_{d_e } \) :

local Nusselt number based on hydraulic diameter,h x ·d e λ

\((Nu_\infty )_{d_e } \) :

Nusselt number of hydrodynamically and thermally fully developed flow through a straight duct according to Deissler's equation based ond e

Pr :

Prandtl number,μ · c p λ

q :

constant heat flux per unit area from concave wall

R :

radius of curvature of center line of passage,R 0-c/2

Re :

Reynolds number,\(\bar u \cdot \left( {2c} \right)/v\)

Re d :

Reynolds number based ond e ,\(\bar u \cdot d_e /v\)

R 0 :

radius of curvature of concave wall

St :

Stanton number,h x /ϱc p \(\bar u\)

T :

local temperature on concave wall atx=x

T f :

temperature of fluid

T in :

bulk temperature in inlet of return bend, defined in Eq. (2)

T w :

general wall temperature

u :

fluid velocity

¯u :

fluid mean velocity

u + :

u/u *

u * :

friction velocity, √τ w /ϱ

W :

width of duct

x :

streamwise coordinate along concave wall with origin at beginning of heating

y :

coordinate perpendicular tox]

y + :

u * ·y/v

δ ** :

momentum thickness

θ :

angle of advance of concave wall taken from inlet

λ :

thermal conductivity of fluid

μ :

coefficient of viscosity of fluid

ν :

kinematic viscosity of fluid

ϱ :

density of fluid

τ w :

wall shear stress

Φ :

implicit function to determineh x

χ + :

Re /1.20 x ·De /−1.28 x ·F 0.77 ·Pr 0.333

ψ + :

Re 0.8 ·De 0.8 ·pr 0.75

d e :

condition based on hydraulic diameter

L :

condition in which Eq. (6) is realized

∞:

condition of hydrodynamically and thermally fully developed straight flow

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Authors and Affiliations

  1. Department of Mechanical Engineering Faculty of Engineering, Hokkaido University, 060, Sapporo, Japan

    N. Seki (Professor), S. Fukusako (Associate Professor) & M. Yoneta (Graduate Student)

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  1. N. Seki
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  2. S. Fukusako
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  3. M. Yoneta
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Seki, N., Fukusako, S. & Yoneta, M. Heat transfer from the heated concave wall of a return bend with rectangular cross section. Wärme- und Stoffübertragung 17, 17–26 (1982). https://doi.org/10.1007/BF01686961

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