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.
For vertical plates, the laminar formula from Churchill and Chu's 1975 paper successfully fits experimental data over the full range of Rayleigh (Ra) and Prandtl (Pr) numbers, 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. Natural convection being a non-reversible heat-engine which converts the temperature difference between an object and fluid into fluid flow, the present work derives formulas for natural convection of isothermal flat plates from the thermodynamic constraints on heat-engine efficiency.
The resulting new 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 formula is consistent with measurements from Kobus and Wedekind (1995).
Thermodynamic Basis for Natural Convection from an Isothermal Plate
Aubrey G. Jaffer
Since the 1930s, models for skin-friction drag from plates with rough surfaces have been based on analogy to flow within pipes having rough interiors. However, this analogy fails at small Reynolds number (Re) flow rates. Significant discrepancies from the pipe analogy at small Re were found by the rough plate experiments of Hama (1954) and Pimenta, Moffat, and Kays (1975).
The present work derives exact formulas for a plate's skin-friction drag coefficient and turbulent forced convection given its root-mean-squared height-of-roughness, and Re bounds from the spatial frequency spectrum of its roughness. These formulas are in close agreement with the Mills-Hang (1983) formula, the Pimenta et al measurements, and experiments conducted by the present author.
The present work also derives an exact formula for skin-friction coefficient of a smooth plate; this formula is in very close agreement with measurements from Smith and Walker (1959) and Spalding and Chi (1964) spanning 4 decades of Re. Its new formula for turbulent forced convection is in agreement with Lienhard (2020), while expanding the range to all fluid Prandtl numbers.
Skin-Friction and Forced Convection from Rough and Smooth Plates
Aubrey G. Jaffer
When forced flow over an isothermal plate is turbulent, its total mixed convection can be computed as an algebraic function of only the forced and natural convections and the orientation of that surface.
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.
Also presented are series of total convection measurements at Reynolds numbers from 2500 to 25000 of the five combinations of horizontal and vertical plate orientation with turbulent horizontal and vertical flow, as well as at some intermediate angles.
Turbulent Mixed Convection from an Isothermal Plate
Aubrey G. Jaffer
Presented are the design and operating methodology of an apparatus constructed to make measurements of forced convection from an isothermal plate with a precisely rough surface. Measurements with a 2.5% root-sum-squared measurement uncertainty were achieved.
Mixed convection measurements at various plate orientations were also made, driving the development of a theory of turbulent mixed convection from any rectangular plate having at least one horizontal edge.
Convection Measurement Apparatus and Methodology
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-aid2.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-dnnt.pdf||1mm roughness face down; horizontal forced flow|
|mixed-dnnt2.pdf||1mm roughness face down; horizontal forced flow|
|mixed-dnrs.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|
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|agj @ alum.mit.edu||Go Figure!|