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Convection Articles

turbine harnessing convection from skillet

Natural Convection Heat Transfer from an Isothermal Plate

Aubrey G. Jaffer


Using boundary-layer theory, natural convection heat transfer formulas which are accurate over a wide range of Rayleigh numbers (Ra) were developed in the 1970s and 1980s for vertical and downward-facing plates. A comprehensive formula for upward-facing plates remained unsolved because they do not form conventional boundary-layers.

From the thermodynamic constraints on heat-engine efficiency, the novel approach presented here derives formulas for natural convection heat transfer from isothermal plates. The union of four peer-reviewed data-sets spanning 1<Ra<1012 has 5.4% root-mean-squared relative error (RMSRE) from the new upward-facing heat transfer formula.

Applied to downward-facing plates, this novel approach outperforms the Schulenberg (1985) formula's 4.6% RMSRE with 3.8% on four peer-reviewed data-sets spanning 106<Ra<1012.

The introduction of the harmonic mean as the characteristic-length metric for vertical and downward-facing plates extends those rectangular plate formulas to other convex shapes, achieving 3.8% RMSRE on vertical disk convection from Hassani and Hollands (1987) and 3.2% from Kobus and Wedekind (1995).

Full Article

Thermo 2023 3(1): 148-175
arXiv 2022 2201.02612

Supplementary Data


Gray-code (self-similar) ramp-permutation roughness; L/ε≈13.86
Gray-code self-similar roughness

Skin-Friction and Forced Convection from Rough and Smooth Plates

Aubrey G. Jaffer


Since the 1930s, theories of skin-friction drag from plates with rough surfaces have been based by analogy to turbulent flow in pipes with rough interiors. Failure of this analogy at slow velocities has frustrated attempts to create a comprehensive theory.

Utilizing the concept of a self-similar roughness which disrupts the boundary layer at all scales, this investigation derives formulas for a rough or smooth plate's skin-friction coefficient and forced convection heat transfer given its characteristic length, root-mean-squared (RMS) height-of-roughness, isotropic spatial period, Reynolds number, and the fluid's Prandtl number.

This novel theory was tested with 456 heat transfer and friction measurements in 32 data-sets from one book, six peer-reviewed studies, and the present apparatus. Compared with the present theory, the RMS relative error (RMSRE) values of the 32 data-sets span 0.75% through 8.2%, with only four data-sets exceeding 6%. Prior work formulas have smaller RMSRE on only four of the data-sets.

Full Article

Skin-Friction and Forced Convection from Rough and Smooth Plates
arXiv 2023 1810.05743

Supplementary Data


Turbulent Mixed Convection from an Isothermal Plate

Aubrey G. Jaffer


While the heat transfer of laminar mixed convection from a vertical plate has been well studied, studies of heat transfer from other plate orientations and from turbulent mixed convection are rare.

The derivation by Jaffer (2023) of the heat transfer formula for natural convection from an external plate reveals a simple relation between heat transfer and fluid velocity. This relation leads to formulas for turbulent mixed convection from horizontal and vertical plates; these formulas are then combined to predict the mixed convection heat transfer at any plate inclination.

Heat transfer measurements of a 30.5 cm square plate with forced air velocities between 0.1 m/s and 2.5 m/s were made in each orthogonal combination of plate and airflow direction, and of a tilted downward-facing plate. Each data-set matches the present theory with root-mean-squared relative error between 1.4% and 4%.

Full Article

Turbulent Mixed Convection from an Isothermal Plate

Convection Machine electronics

Convection Measurement Apparatus and Methodology

Appendix A of Skin-Friction and Forced Convection from Rough and Smooth Plates details the apparatus and methodology with the plate facing down. STM32F303VCTx LQFP100 pinout

Technical details

Isothermal plate construction

Wind-tunnel construction

Apparatus electrical schematics (2 page pdf)

STM32F303VCTx configuration (pdf) (included in Firmware.zip)

"convect" source code Firmware.zip creates directory convect; the makefile is convect/Firmware.Makefile.

Apparatus operating instructions

Wind-Tunnel Calibrations

Supplementary Data

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.

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 Convection
natural.pdfNatural Convection at a dozen Rayleigh numbers in vertical and horizontal orientations.
angles.pdfNatural Convection at angles from −90 to +90
Mixed Convection
3mm/mixed-up.pdf3mm roughness face up; horizontal forced flow
1mm/mixed-up.pdf1mm roughness face up; horizontal forced flow
3mm/mixed-aid.pdf3mm roughness face vertical; upward forced flow
1mm/mixed-aid.pdf1mm roughness face vertical; upward forced flow
3mm/mixed-vt.pdf3mm roughness face vertical; horizontal forced flow
1mm/mixed-vt.pdf1mm roughness face vertical; horizontal forced flow
3mm/mixed-opp.pdf3mm roughness face vertical; downward forced flow
1mm/mixed-opp.pdf1mm roughness face vertical; downward forced flow
3mm/mixed-dn.pdf3mm roughness face down; horizontal forced flow
1mm/mixed-dn.pdf1mm roughness face down; horizontal forced flow
1mm/mixed-aid+82.pdf1mm roughness face down inclined aiding +82°
1mm/mixed-opp+82.pdf1mm roughness face down inclined opposing +82°
Zip of Supplementary Files
supplementary.zipZip Archive of All Files

datafiles.zip has the dimensionless measurements used to generate the summary graphs in each pdf above. The columns are:

  1. Re - Reynolds number
  2. Ra - Rayleigh number
  3. RSS combined measurement uncertainty %
  4. θ - surface angle from vertical degrees (−90 is facing up)
  5. ψ - flow angle from vertical degrees (0 is up)
  6. Pr - Prandtl number
  7. Nu - Nusselt number

Characteristic-length L = 0.305 m

Copyright © 2016, 2017, 2018, 2019, 2020, 2021, 2022, 2023 Aubrey Jaffer

I am a guest and not a member of the MIT Computer Science and Artificial Intelligence Laboratory.  My actions and comments do not reflect in any way on MIT.
agj @ alum.mit.edu
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