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Effects of pressure work and viscous dissipation on Graetz problem for gas flows in parallel-plate channels

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Abstract

The analytical solutions are obtained for the Graetz problem with pressure work and viscous dissipation in the thermal entrance region of the parallel-plate channels for two basic boundary conditions of uniform wall temperature and uniform wall heat flux involving fully developed laminar gas flows. The asymptotic Nusselt number is found to be zero instead of the conventionally accepted value of 7.54 for the uniform wall temperature case and (140/17)/ [1+(27/17) PrEc] for uniform wall heat flux case. The effects of pressure work and viscous dissipation contribute significantly to the asymptotic results for heat transfer and cannot be neglected under any circumstances in the case of uniform wall temperature. Sample results are presented to illustrate the effects of pressure work and viscous dissipation on heat transfer characteristics in the thermal entrance region.

Zusammenfassung

Für den thermischen Einlauf im ebenen Spalt bei voll ausgebildeter Laminarströmung (Graetz-Problem) wird eine analytische Lösung unter Berücksichtigung der Druckarbeit und der Dissipation durch die Zähigkeit sowohl für konstante Wandtemperatur als auch für konstante Wärmestromdichte an der Wand angegeben. Die asymptotischen Nusselt-Zahlen werden Null für konstante Wandtemperatur (anstatt des üblichen Wertes 7,54) und (140/17)/[1+(27/17) PrEc] für konstante Wärmestromdichte. Druckarbeit und Dissipation wirken stark auf den Wärmeübergang ein, insbesondere bei konstanter Wandtemperatur. Beispiele illustrieren diese Einflüsse.

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Abbreviations

b:

one-half of the channel height

Cn, Dn, Kn :

coefficients in the series expansion ofφ, see Eq. (12)

Cn′, Kn′:

coefficients in the series expansions of Eq. (18)

Ec:

Eckert number, u′2 m p (t0−tw) for tw == constant or u′m 2/cp (qwb/k) for qw=constant

cp :

specific heat at constant pressure

h:

local heat transfer coefficient

k:

thermal conductivity

Nu:

local Nusselt number, (4 b h/k)

Pr:

Prandtl number, cp μ/k

p:

axial pressure

qw :

uniform wall heat flux per unit area

Re:

Reynolds number, 4 b u′m/ν

t, tb, t0, tw :

gas temperature, bulk temperature, uniform entrance temperature and constant wall temperature, respectively

u:

dimensionless axial velocity, u′/(3/2) u′m

u′, u′m :

gas velocity and mean velocity in axial direction

x, y:

dimensionless axial and transverse coordinates, x′/(3/8) b Re Pr and y′/b, respectively

x′, y′:

axial and transverse coordinates

Yn, Zn :

eigenfunctions of Eqs. (11) and (19), respectively

β n, λn :

eigenvalues of Eqs. (19) and (11), respectively

θ :

dimensionless temperature difference, (t−tw)/(t0−tw) for tw=constant or (t−t0)/(qwb/k) for qw=constant

θ b,θ w :

dimensionless bulk and wall temperatures

μ :

absolute viscosity

ν :

kinematic viscosity

ρ :

gas density

φ :

dimensionless temperature difference defined in Eq. (7)

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Alberta, Edmonton 7, Alberta, Canada

    Jenn-Wuu Ou &  K. C. Cheng

Authors
  1. Jenn-Wuu Ou
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  2. K. C. Cheng
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Ou, JW., Cheng, K.C. Effects of pressure work and viscous dissipation on Graetz problem for gas flows in parallel-plate channels. Warme- und Stoffubertragung 6, 191–198 (1973). https://doi.org/10.1007/BF02575264

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  • DOI: https://doi.org/10.1007/BF02575264

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