Abstract
The conjugate heat transfer across a thin horizontal wall separating two fluids at different temperatures is investigated both numerically and asymptotically. The solution for large Rayleigh numbers is shown to depend on two nondimensional parameters;α/ε 2, withα being the ratio of the thermal resistance of the boundary layer in the hot medium to the thermal resistance of the wall andε the aspect ratio of the plate, andβ, the ratio of the thermal resistances of the boundary layers in the two media. The overall Nusselt number is an increasing function ofα/ε 2 taking a finite maximum value forα/ε 2 → ∞ and tending to zero forα/ε 2 → 0.
Zusammenfassung
Der Wärmeaustausch zwischen zwei, durch eine dünne horizontale Wand getrennten Fluiden unterschiedlicher Temperatur wird numerisch und asymptotisch untersucht. Die Lösung für große Rayleigh-Zahlen hängt von zwei dimensionlosen Parametern ab: (1)α/ε 2, wobeiα das Verhältnis des thermischen Widerstandes der Grenzschicht im heißeren Medium zu dem der Wand bedeutet undε das Seitenverhältnis der Platte. (2)β, das Verhältnis der thermischen Widerstände der Grenzschichten in den beiden Medien. Die gemittelte Nusselt-Zahl ist eine mitα/ε 2 anwachsende Funktion, welche fürα/ε 2 → ∞ einem endlichen Maximalwert zustrebt und fürα/ε 2 → 0 gegen Null geht.
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Abbreviations
- c i :
-
specific heat of fluidi
- f i :
-
nondimensional stream function for fluidi defined in Eq. (15)
- g :
-
acceleration of gravity
- h :
-
thickness of the strip
- L :
-
half-width of the strip
- Nu⋆:
-
reduced Nusselt number defined in Eq. (11)
- Nu :
-
Nusselt number defined in Eq. (10)
- \(\overline {Nu^ \star } \) :
-
overall reduced Nusselt number defined in Eq. (32)
- Pr i :
-
Prandtl number of fluidi (=μ i c i /λ i )
- Ra i :
-
Rayleigh number of mediumi (=gβ i ΔTPr i L 3/ν /3 i )
- T i∞ :
-
temperature of mediumi far from the plate
- x, y :
-
horizontal and vertical Cartesian coordinates
- z :
-
nondimensional transversal coordinate for the plate (=y/h)
- α :
-
heat conduction parameter defined in Eq. (3)
- β :
-
ratio of boundary layer thermal resistances defined in Eq. (4)
- β i :
-
thermal expansion coefficient of fluidi
- χ :
-
nondimensional longitudinal coordinate (=x/L)
- ΔT :
-
temperature difference, (=T 1∞−T 2∞)
- ε :
-
aspect ratio of the plate (=h/L)
- φ i :
-
nondimensional pressure in fluidi defined in Eq. (15)
- λ i :
-
thermal conductivity of mediumi
- λ w :
-
thermal conductivity of the solid
- μ i :
-
dynamic viscosity of fluidi
- v i :
-
kinematic viscosity for mediumi
- ψ i :
-
stream function for mediumi
- θ i :
-
nondimensional temperature of mediumi defined in Eq. (15)
- θ w :
-
nondimensional temperature of the plate defined in Eq. (15)
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This work has been supported by the research grant IN 107795, DEAPA at UNAM, Mexico.
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Luna, E., Treviño, C. & Higuera, F.J. Conjugate natural convection heat transfer between two fluids separated by a horizontal wall: steady-state analysis. Heat and Mass Transfer 31, 353–358 (1996). https://doi.org/10.1007/BF02184050
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DOI: https://doi.org/10.1007/BF02184050