Aubrey G. Jaffer
There are successful theoretical models for laminar natural convection from downward-facing and vertical plates, but not for upward-facing plates. Moreover, there are no successful theoretical models for turbulent natural convection in any plate orientation.
But for vertical plates, the total laminar formula from Churchill and Chu's 1975 paper successfully fits experimental data over the full range of Prandtl (Pr) and Rayleigh numbers (Ra), notably better than their turbulent formula in the turbulent regime.
This suggests that there may be a principle uniting laminar and turbulent natural convection. The fundamental laws of thermodynamics make no distinction between laminar and turbulent flows, but are applicable only to a system as a whole. The present work derives formulas for total natural convection of isothermal flat plates from the thermodynamic constraints on heat-engine efficiency.
The resulting upward-facing formula is consistent with measurements from Fujii and Imura (1972), Goldstein, Sparrow, and Jones (1973), and Lloyd and Moran (1974) over nearly 11 decades of Ra. The vertical and downward-facing formulas are within 1.2% of the Churchill and Chu (1975) and the Schulenberg (1985) formulas in fluids with high Pr, and within 6% overall.
This heat-engine approach is a new paradigm for natural convection modeling with uniform treatment of laminar and turbulent flows. Its power is further demonstrated by the coefficients and exponents in the final formulas being exact algebraic expressions.
Thermodynamic Basis for Natural Convection from an Isothermal Plate
Aubrey G. Jaffer
The current models for skin-friction drag from a plate with a rough surface are derived by analogy to flow within pipes having rough interiors. But this analogy fails at low Reynolds number (Re) flow rates because the boundary-layer must compress into the center of the pipe as it grows with shrinking Re, while the plate boundary-layer is not similarly constrained. A significant discrepancy from the pipe analogy at low Re was found by the rough plate experiments of Pimenta, Moffat, and Kays (1975) and by experiments conducted by the author.
An additional problem is that the roughness parameter in pipe analogy theories is tied to the drag measurements of flows inside Nikuradse's assortment of sand-roughened pipes (1933). Prandtl and Schlichting (1934) caution that their theory applies only to sand-roughness. It would be a great advance to develop a theory based on direct measurements of roughness profiles.
The present work derives a formula for a plate's skin-friction drag coefficient given its root-mean-squared height-of-roughness and spatial frequency spectrum.
This new theory is in close agreement with the Mills-Hang (1983) theory, the Pimenta, Moffat, and Kays measurements, and the experiments conducted by the author over their respective Re ranges.
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!|