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A laminar boundary layer model of heat transfer due to a nonuniform planar jet impinging on a moving plate

Laminares Grenzschichtmodell des Wärmeübergangs von einem ungleichmäßigen ebenen auf eine bewegte Platte auftreffenden Strahl

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Abstract

A theoretical model is developed of planar jet impingement heat transfer on a moving plate. Boundary layer equations in their integral forms are used in order to include, both near and away from the stagnation line, the effects of surface motion directed perpendicular to the jet plane, an arbitrary surface temperature variation, and nonuniform jet discharge velocity profiles. The validity and accuracy of the approximate solution is assessed by comparison to an exact solution restricted to regions near the stagnation line. Results indicate that the influence of a nonuniform jet discharge velocity decreases with distance from the stagnation line. Surface motion affects heat transfer at regions away from the stagnation line, but has little influence near the stagnation line when the surface temperature is constant.

Zusammenfassung

Ein theoretisches Modell für die Wärmeübertragung beim Auftreffen eines ebenen Strahls auf eine bewegte Platte wird hier entwickelt. Es werden Grenzschichtgleichungen in Integralform benutzt um, sowohl in der Nähe als auch in einiger Entfernung der Staulinie, die Effekte der Oberflächenbewegung senkrecht zum Strahl, bei willkürlichen Veränderungen der Oberflächentemperatur und ungleichmäßigen Strahlgeschwindigkeitsprofilen zu beschreiben. Die Gültigkeit und Genauigkeit der approximativen Lösung wird beurteilt in dem sie mit der exakten Lösung für den Bereich in der Nähe der Staulinie verglichen wird. Die Ergebnisse zeigen, daß der Einfluß einer ungleichmäßigen Strahlaufprallgeschwindigkeit mit zunehmenden Abstand von der Staulinie abnimmt. Bewegungen der Oberfläche beeinflussen den Wärmeübergang in Bereichen außerhalb der Staulinie, aber haben einen kleinen Einfluß in der Nähe der Staulinie wenn die Oberflächentemperatur konstant gehalten wird.

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Abbreviations

c p :

specific heat at constant pressure

C :

freestream velocity gradient near stagnation line

\(\bar C\) :

dimensionless velocity gradient =w C/v j

h :

heat transfer coefficient

H :

vertical distance above plate to nozzle opening

k :

thermal conductivity

Nu w :

jet Nusselt number =h w/k

Nu * :

ratio of local Nusselt number for moving plate to local Nusselt number for stationary plate

P :

static pressure

\(\bar P\) :

dimensionless pressure=(P−P ∞)/(1/2)ϱ v 2 j

Pr :

Prandtl number =μ c p /k

q w :

wall heat flux

Re w :

Reynolds number=v j w/v

T :

temperature

T p :

local surface temperature of plate

T ∞ :

freestream temperature (jet temperature)

u :

x-component of velocity

u ∞ :

x-component of velocity beyond velocity boundary layer

\(\bar u_\infty\) :

velocity ratio=u ∞/v j (Eq. (20))

v j :

jet impingement velocity along jet centerline

v p :

velocity of plate

\(\bar v_p\) :

velocity ratio=v p /v j

w :

jet width

x * :

value for\(\bar x\) where\(\bar P = 0\) (Eq. (19))

x :

horizontal distance from stagnation line (Fig. 2)

\(\bar x\) :

dimensionless position =x/w

y :

vertical position above plate (Fig. 2)

γ :

boundary layer thickness ratio=δ/Δ

Γ :

boundary layer thickness ratio=Δ/δ

δ :

velocity boundary layer thickness

\(\bar \delta\) :

dimensionless velocity boundary layer thickness=δ/tw

Δ :

thermal boundary layer thickness

\(\bar \Delta\) :

dimensionless thermal boundary layer thickness=Δ/tw

τ xy :

shear stress =μ ∂u/∂y

θ :

dimensionless surface temperature of plate=(T p −T ∞)/(T p0 −T ∞)

μ :

dynamic viscosity

ν :

kinematic viscosity=μ/ϱ

ϱ :

mass density

ψ :

coefficient related to boundary layer growth (Eqs. (10), (11), (14)–(17))

0:

pertaining to stagnation line

p :

plate

∞:

jet free stream

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Authors and Affiliations

  1. Thermal/Fluids Laboratory Department of Mechanical Engineering, Clemson University, 29634, Clemson, SC, USA

    D. A. Zumbrunnen

  2. Heat Transfer Laboratory School of Mechanical Engineering, Purdue University, 47907, West Lafayette, IN, USA

    F. P. Incropera & R. Viskanta

Authors
  1. D. A. Zumbrunnen
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  2. F. P. Incropera
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  3. R. Viskanta
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Zumbrunnen, D.A., Incropera, F.P. & Viskanta, R. A laminar boundary layer model of heat transfer due to a nonuniform planar jet impinging on a moving plate. Wärme- und Stoffübertragung 27, 311–319 (1992). https://doi.org/10.1007/BF01589969

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  • DOI: https://doi.org/10.1007/BF01589969

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