Wärme - und Stoffübertragung

, Volume 8, Issue 2, pp 87–100 | Cite as

Extensions of Merk's method to the calculation of laminar boundary layer heat transfer from a non-isothermal surface

  • Takao Sano
Article

Abstract

The Merk's method for calculating non-similar boundary layer heat transfer from an isothermal surface is extended to the case with non-isothermal surface and the reliability of the method is checked in detail. It is shown that except for some small ranges of β (pressure gradient parameter) and γ (wall temperature distribution parameter), the present method yields accurate heat transfer results. Examples of application of the present method are given for a flow around a circular cylinder and for that past a flat plate. The calculations are performed in the limits Pr→∞ and Pr→0, and the results are compared with those derived from the exact asymptotic theories for these limits.

Keywords

Heat Transfer Circular Cylinder Wall Temperature Flat Plate Asymptotic Theory 

Nomenclature

a

radius of the circular cylinder

c1(β),c2(β)

integration constants

C1(β), C2(β)

integration constants

f

non-dimensional stream function defined by (8)

f0,f1, ...

non-dimensional stream function defined by (20)

h

local heat transfer coefficient

j

variable defined to be zero for plane flow and one for axisymmetric flow

k

thermal conductivity

L

characteristic length of the body

Nu

Nusselt number

Pr

Prandtl number

Re

Reynolds number

r′

body transverse radius of curvature

r

non-dimensional body transverse radius of curvature

t′

temperature of the fluid

t

non-dimensional temperature of the fluid

tp

temperature at the edge of the boundary layer

tw

temperature at the surface of the body

u′

velocity component in x′ direction

u

non-dimensional velocity component in x direction

up

velocity of the inviscid flow on the surface

Ur

characteristic stream velocity

Up

non-dimensional velocity of the inviscid flow on the surface

U

free stream velocity

v′

velocity component in y′ direction

v

non-dimensional velocity component in y direction

x′

coordinate along the body

x

non-dimensional coordinate along the body

y′

coordinate normal to the body

y

non-dimensional coordinate normal to the body

Y

parameter defined by (84)

α

thermal diffusivity

β

pressure gradient parameter defined by (15)

β1234

constants defined by (51), (69), (70) and (71)

βs

the value of β corresponding to seperation point

γ

wall temperature distribution parameter defined by (16)

γ1

constant defined by (52)

ɛ(β)

perturbation parameter defined by (19)

ɛ1(γ)

perturbation parameter defined by (27)

ζ

non-dimensional variable defined by (35)

η

non-dimensional variable defined by (7)

θ

non-dimensional temperature defined by (8)

θ01,...

non-dimensional temperature defined by (21) or (28)

θ0(0)0(1)..., θ1(0)1(1)...

non-dimensional temperature defined by (46) and (47)

ϑ01,...

non-dimensional temperature defined by (35) or (73)

ϑ0(0)0(1)..., ϑ1(0)1(1),...

non-dimensional temperature defined by (36) and (37)

Θ01

non-dimensional temperature defined by (48) or (79)

Θ0(0)0(1)1(0)1(1)

non-dimensional temperature defined by (55) and (56)

ν

kinematic viscosity

ξ

non-dimensional variable defined by (7)

τ

non-dimensional variable defined by (48)

Φ

angle measured from the front stagnation point of the circular cylinder

Ψ

non-dimensional stream function defined by (9)

Zusammenfassung

Die Methode von Merk zur Berechnung des Wärmeübergangs in nicht-ähnlicher Grenzschicht an isothermer Fläche ist auf den nicht-isothermen Fall ausgedehnt; die Anwendbarkeit wird im einzelnen geprüft. Außer für kleine Werte von Parametern des Druckgradienten und der Temperaturverteilung gibt diese Methode genaue Ergebnisse. Als Anwendungsbeispiele dienen die Umströmung von Kreiszylindern und die Plattenströmung im Bereich Pr→∞ bis Pr→0. Die Ergebnisse werden mit jenen der genauen asymptotischen Rechnungen verglichen.

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Copyright information

© Springer-Verlag 1975

Authors and Affiliations

  • Takao Sano
    • 1
  1. 1.Department of Mechanical EngineeringUniversity of Osaka PrefectureOsakaJapan

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