Aubrey G. Jaffer
The combination of a heated horizontal plate with a turbine above it constitutes a non-reversible heat-engine, converting the flow of heat into mechanical work. Thermodynamic analysis of this hypothetical apparatus finds that its maximum thermodynamic efficiency is 0.5 ΔT/T, which is half of the limit for reversible heat-engines. Furthermore, when measured at the plate, both the convection and upward fluid flow will be the maximum allowed by this thermodynamic efficiency. Fluid-dynamics analysis results in a comprehensive correlation for natural convection from an upward-facing horizontal plate:
Nu = (0.671 + 0.370 Ra1/6)2
This new correlation covers laminar and turbulent convection, and averages 9% higher than measurement-fitted correlations by Lloyd and Moran for 26000<Ra<1.5×109; and 4% higher than measurement-fitted correlations by Goldstein, Sparrow, and Jones for 1<Ra<104.
The same approach produces an upper bound for natural convection from a vertical plate:
Nu < (0.826 + 0.387 Ra1/6)2
For large Prandtl numbers, this matches the semi-empirical correlation reported by Churchill and Chu over its full range.
Thermodynamic Basis for Natural Convection from an Isothermal Plate
Aubrey G. Jaffer
Presented are new correlations for skin-friction and forced convection from an isothermal plate having root-mean-squared height-of-roughness εq. Corresponding to Prandtl and Schlichting's "fully-developed roughness" region, the skin-friction correlation is independent of Reynolds number; the convection correlation is a linear function of Reynolds number (versus an exponent of 4/5 for turbulent convection from a smooth plate).
fc = 1
Nu = Re Pr1/3
Measurements taken of a plate having precisely 3 mm of roughness match the convection correlation within 3% at Reynolds numbers from 5000 to 50000. A formula addressing the transition from fully-rough turbulent to smooth turbulent convection for this bi-level test plate matches those measurements within 2% from Re=5000 to 80000.
Convection measurements of a bi-level plate having 1.03 mm of roughness are within the expected measurement uncertainties of the transition formula.
An analysis of sand-roughness finds that its relation to root-mean-squared height-of-roughness is non-linear. The resulting scale error explains discrepancies between the Prandtl and Schlichting formula, two recent papers, and the results presented here.
These new correlations and measurements imply a threefold tighter upper bound for the height of admissible-roughness than is given in Schlichting's book Boundary-layer theory.
Skin-Friction and Forced Convection from an Isothermal Rough Plate
Aubrey G. Jaffer
Presented are new correlations for turbulent mixed convection from an isothermal rectangular surface having at least one horizontal edge and flow parallel to an edge of that surface.
When mixed with natural convection, laminar and turbulent forced flows behave quite differently. For turbulent forced flow, mixed convection is successfully modeled as a function of only the natural and forced convective surface conductances and plate and flow orientations. Furthermore, in each orientation turbulent mixed convection is bounded by the L4-norm and L2-norm of the forced and natural convective surface conductances.
Natural convection and horizontal forced flow mix as the L2-norm for vertical and upward-facing plates and the L4-norm for downward-facing plates. With vertical flow by a vertical plate, mixing transitions between the L4-norm and the L2-norm for both aiding and opposing flows.
Systematic measurements at Reynolds numbers from 2500 to 20000 of the 10 combinations of horizontal or vertical orientation and laminar natural flow mixed with turbulent and rough-turbulent forced flows were made in air using 0.305 m square heated plates having 3.0 mm and 1.03 mm RMS height-of-roughness. The formulas presented here match nearly all of these measurements within the apparatus' expected uncertainties.
Building on Fujii and Imura's approach to natural convection from an inclined plate, this mixed convection model is extended to any inclination of a rectangular plate having at least one horizontal edge and flow parallel to an edge.
Measurements of buoyancy-aided and buoyancy-opposed mixed flows made at a plate angle of 84.5° (nearly face down) also match this generalized formula within the expected uncertainties.
Turbulent Mixed Convection from an Isothermal Plate
In the temperature versus time graphs in the supplementary files, the green, blue, and black traces are the plate, (insulated) back, and ambient temperatures respectively. The upper red trace is a simulation of the plate temperature with the back and average ambient temperatures as inputs. The middle red trace is a simulation of the back temperature with the plate and ambient temperatures as inputs. The lower red line is the ambient temperature averaged over the measurement period.
The diamonds on the plate temperature trace mark the beginning and end of the measurement period, the ending temperature difference with ambient being at most half of the peak temperature difference with ambient. The simulated plate temperature is reset to the real plate temperature at the first diamond.
Underneath each temperature graph is a graph of the air velocity versus time for mixed convection runs and Rayleigh number versus time for natural convection runs.
|natural.pdf||Natural Convection at a dozen Rayleigh numbers in vertical and horizontal orientations.|
|angles.pdf||Natural Convection at angles from −90 to +90|
|mixed-up.pdf||3mm roughness face up; horizontal forced flow|
|mixed-up.pdf||1mm roughness face up; horizontal forced flow|
|mixed-aid.pdf||3mm roughness face vertical; upward forced flow|
|mixed-aid.pdf||1mm roughness face vertical; upward forced flow|
|mixed-vt.pdf||3mm roughness face vertical; horizontal forced flow|
|mixed-vt.pdf||1mm roughness face vertical; horizontal forced flow|
|mixed-opp.pdf||3mm roughness face vertical; downward forced flow|
|mixed-opp.pdf||1mm roughness face vertical; downward forced flow|
|mixed-dn.pdf||3mm roughness face down; horizontal forced flow|
|mixed-dn.pdf||1mm roughness face down; horizontal forced flow|
|mixed-aid+84.pdf||1mm roughness face down inclined aiding +84.5°|
|mixed-opp+85.pdf||1mm roughness face down inclined opposing +84.5°|
|Zip of Supplementary Files|
|supplementary.zip||Zip Archive of All Files|
Aubrey G. Jaffer
Presented are the design and operating methodology of an apparatus constructed to make accurate measurements of mixed convection at all horizontal and vertical orientations of an isothermal plate with forced airflow in the plane of the plate.
The measurements from this Convection Machine drove the development and validation of a comprehensive theory of turbulent mixed convection from a rectangular plate having at least one horizontal edge.
Convection Measurement Apparatus and Methodology
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|agj @ alum.mit.edu||Go Figure!|