Natural Convection Heat Transfer from an Inclined Cylinder

Abstract

Based on Jaffer’s (2023) heat engine analysis of natural convection, this investigation mathematically derives a novel, comprehensive formula predicting the natural convective heat transfer from an inclined cylinder given its length, diameter, angle, and Rayleigh number and the fluid’s Prandtl number and thermal conductivity. The present formula was tested with 93 inclined cylinder measurements having length-to-diameter ratios between 1.48 and 104 in nine data-sets from three peer-reviewed studies, yielding (data-set) root-mean-squared relative error values between 1.9% and 4.7%.

1. Introduction

Natural convection is the flow caused by nonuniform density in a fluid under the influence of gravity. Natural convection is a fundamental process with application from engineering to geophysics.

Changes in fluid density can be caused by changes in temperature or solute concentration. Under the influence of gravity, density changes cause fluid flow, which also transports heat or solute. Rates of transfer grow until reaching a plateau. This investigation seeks to predict the overall steady-state heat transfer rate from an external, cylindrical, isothermal surface inclined at any angle in a Newtonian fluid.

An “external” surface is one that fluid can flow around freely, especially horizontally. If enclosed, the enclosure must have dimensions much larger than the heated or cooled surface.

Unbounded vertical extent allows the imagining of a hypothetical apparatus to extract as much of the rising fluid’s mechanical energy as possible. This then is a (non-reversible) heat engine driven by the temperature difference between the heated object and the unheated fluid. The analysis in Jaffer [1] finds that the maximum efficiency, the fraction of the heat energy which can converted into mechanical work, is $|T-T_{\infty }|[2/T_{\infty }]$, which is 1/2 of the Carnot efficiency limit for reversible heat engines.

From this thermodynamic constraint on heat-engine efficiency, conservation laws, and flow topologies gleaned from streamline photographs, Jaffer [1] mathematically derives convection heat-transfer upper-bound formulas for vertical and horizontal flat surfaces. Actual convective heat-transfer is less than these upper bounds when fluid flow is (partially) obstructed.

Horizontal downward-facing and vertical surfaces are partially self-obstructing. The Jaffer [1] treatment of self-obstruction is similar to prior works, but unifies prior work self-obstruction factors into a single exact factor which is more plausible (measurements not significantly exceeding the upper bound) over the full range of Prandtl numbers (the fluid’s momentum diffusivity per thermal diffusivity ratio).

1.1. Characteristic Length

The characteristic length L is the length scale of a physical system. As with a vertical rectangular plate, a vertical cylinder’s characteristic length is its height.

The characteristic length of a level circular cylinder is its diameter d. Generalizing to convex cylinders is the hydraulic diameter, which is 4 times the area-to-perimeter ratio of the cylinder’s cross-section. Note that the diameter and hydraulic diameter are identical for a level circular cylinder.

1.2. Fluid Mechanics

In engineering, convection heat transfer rates are expressed using the average surface conductance $\overline{h}$ with units $\mathrm{W}/({\mathrm{m}}^2·\mathrm{K})$.

In fluid mechanics, the convective heat transfer rate is represented by the dimensionless average Nusselt number ($\overline{Nu}\equiv \overline{h}L/k$), where k is the fluid’s thermal conductivity with units $\mathrm{W}/(\mathrm{m}·\mathrm{K})$, and L is the system’s characteristic length (m).

The Rayleigh number $Ra$ is the impetus for fluid flow due to gravity acting on density differences caused by temperature or solute concentration. A fluid’s Prandtl number $Pr$ is its momentum diffusivity per thermal diffusivity ratio. For mass transfer, the fluid’s Schmidt number $Sc$ is analogous; $Pr$ will be used in formulas.

When rising convection-induced fluid flow must pass along or around the object’s surface, $Ra$ is scaled by a “self-obstruction” factor $1/\mathsf{\Xi }$, which depends only on $Pr$. This applies to both cylinder flow topologies.

The system’s characteristic length L scales $\overline{Nu}$, while $L^3$ scales $Ra$. Variables $\mathsf{\Xi }$, $\overline{h}$, $Pr$, and $Sc$ are independent of L.

1.3. Flow Topologies

Jaffer [1] derived a natural convection formula for external flat plates (with convex perimeter) in any orientation from its analyses of horizontal and vertical plates. Similarly, this investigation will derive its formula for an inclined cylinder from its analyses of horizontal and vertical cylinders.

There are two topologies of convective flow from external, convex cylinders. Figure 1 shows the induced fluid flows around heated vertical and horizontal cylindrical surfaces.

An important aspect of both flow topologies is that fluid is pulled horizontally before being heated by the cylinder. Pulling horizontally expends less energy than pulling vertically because the latter does work against the gravitational force. Inadequate horizontal clearance around a cylinder can obstruct flow and reduce convective heat transfer.

There is a symmetry in external natural convection; a cooled cylinder induces downward flow instead of upward flow. The rest of the present work assumes that the cylindrical exterior surface is warmer than the fluid.

1.4. Turbulence

Jaffer [1] derives the formula for an external flat surface’s total natural convective heat transfer from the thermodynamic constraints on heat-engine efficiency, conservation laws, and flow topology.

Thermodynamic and conservation laws make no distinction between laminar and turbulent flows. What about the fluid flow topology? If the transition to turbulence is far from a heated convex object, then it will not affect heat transfer from the object. Otherwise, the fluid near the object will form a turbulent boundary layer. This turbulent boundary layer will have a viscous sublayer between it and the object. Regarding forced flow-induced boundary layers, Lienhard and Lienhard [2] (p. 321) state

Because turbulent mixing is ineffective in the sublayer, the sublayer is responsible for a major fraction of the thermal resistance of a turbulent boundary layer.

The boundary layer thickness grows with decreasing forced velocity. Without forced flow, the transition to turbulence will be far from the object, increasing turbulent mixing in the boundary layer and reducing its thermal resistance. This makes the non-turbulent viscous sublayer responsible for most of the thermal resistance. If the distinction between laminar and turbulent natural convection effects heat transfer, its effect will be small.

Prior plate investigations assumed that natural convection heat transfer formulas would differ substantially when the convection was turbulent versus laminar. For their upward-facing plate, Lloyd and Moran [3] reported that the transition from laminar to turbulent flow occurred at $Ra\approx 8\times {10}^6$. The straight line segments they fitted to their data at greater and lesser $Ra$ were disjoint at $Ra=8\times {10}^6$. However, with their fit lines removed, if $Ra\approx 8\times {10}^6$ represents a discontinuity, then it is one of several, and subsumed within the scatter of their measurements (their data is plotted in Jaffer [1]).

About their measurements of vertical and downward tilted plates, Fujii and Imura [4] wrote

Though the boundary layer was not always laminar near the trailing edge for large $GrPr$ [$=Ra$] values, no influence of the flow regime on the data shown in [their] Fig. 6 is appreciable.

Churchill and Chu [5] conclude that one of their equations

...based on the model of Churchill and Usagi [6] provides a good representation for the mean heat transfer for free convection from an isothermal vertical plate over a complete range of Ra and Pr from 0 to ∞ even though it fails to indicate a discrete transition from laminar to turbulent flow.

Although there are substantial differences between laminar and turbulent fluid flow along plates, a single formula appears to govern both laminar and turbulent natural convective heat transfer in all orientations.

The flow topology from a vertical cylinder is quite similar to that from a vertical plate. The laminar–turbulent differences in flow topology from a level cylinder happen far from the cylinder. Hence, this investigation expects a single formula to govern both laminar and turbulent natural convection from cylinders.

1.5. Combining Transfer Processes

Formula (1) is an unnamed form for combining functions which appears frequently in heat or mass transfer formulas:

$$F^p=F_1^p+F_2^p$$

(1)

Churchill and Usagi [6] stated that such formulas are “remarkably successful in correlating rates of transfer for processes which vary uniformly between these limiting cases.” Convection transfers heat (or solute) between the cylinder and the fluid.

1.6. The ${\ell }^p$-Norm

When $F_1\ge 0$ and $F_2\ge 0$, taking the pth root of both sides of Equation (1) yields a vector-space functional form known as the ${\ell }^p$-norm, which is notated $\parallel F_1,F_2{\parallel }_p$ :

$${∥F_1,F_2∥}_p\equiv {\left[|F_1{|}^p+{\left|F_2\right|}^p\right]}^{1/p}$$

(2)

Norms generalize the notion of distance. Formally, a vector-space norm obeys the triangle inequality $\parallel F_1,F_2{\parallel }_p\le |F_1|+|F_2|$, which holds only for $p\ge 1$. However, $p<1$ is also useful.

When $p>1$, the processes modeled by $F_1$ and $F_2$ compete and $\parallel F_1,F_2{\parallel }_p\ge max\left(\right|F_1|,|F_2\left|\right)$; the most competitive case is $\parallel F_1,F_2{\parallel }_{+\infty }\equiv max\left(\right|F_1|,|F_2\left|\right)$.

The ${\ell }^1$-norm models independent processes; $\parallel F_1,F_2{\parallel }_1\equiv |F_1|+|F_2|$.

When $0<p<1$, the processes cooperate and $\parallel F_1,F_2{\parallel }_p\ge |F_1|+|F_2|$. Cooperation between conduction and flow-induced heat transfer can occur in natural convection systems.

2. Data-Sets and Evaluation

Heat transfer measurements were captured from graphs in the cited works by measuring the distance from each point to its graph’s axes, then scaling to the graph’s units using the “Engauge” software (version 12.1).

Churchill and Chu [7] collected level cylinder (angle $\vartheta =0^{\circ }$) heat and mass transfer measurements from eleven studies spanning more than 23 orders of magnitude of $Ra$. The Kutateladze [8] data-set (with the largest $Ra$ values) is treated separately in Table 1.

Al-Arabi and Khamis [9] measured natural convection heat transfer from a cylinder at six angles. They measured local temperatures along the cylinder, but incorrectly inferred the average heat transfer.

Popiel, Wojtkowiak, and Bober [10] measured natural convection heat transfer from four vertical cylinders with $1<H/d<59$. Unfortunately, they tested with the bottom of the cylinder resting directly on a flat platform, which would impede horizontal fluid flow at the bottom. Extra unheated walls were found to significantly affect convective heat transfer in Jaffer [1].

Goldstein, Khan, and Srinivasan [11] measured natural convection mass transfer from three cylinders at four inclinations.

Heo and Chung [12] measured natural convection mass transfer from five cylinders with $3.7<H/d\le 25$ at inclinations $0^{\circ }\le \vartheta \le {90}^{\circ }$.

These data files, used for generating the present work’s graphs and tables, along with digitization details and estimated digitization inaccuracies are collected in Supplementary Materials.

Table 1. Cylinder natural convection data-sets.

Source	Study		$\mathsf{\vartheta }$	${\mathit{Ra}}_{\mathit{d}}/{\mathsf{\Xi }}_{•}\ge$	${\mathit{Ra}}_{\mathit{d}}/{\mathsf{\Xi }}_{•}\le$	±	#
Churchill & Chu [7]	Kutateladze [8]		$0^{\circ }$	$4.3\times {10}^9$	$5.3\times {10}^{12}$		6
Churchill & Chu [7]	all		$0^{\circ }$	$7.5\times {10}^{-12}$	$5.3\times {10}^{12}$		63
Churchill & Chu [7]	10 others		$0^{\circ }$	$7.5\times {10}^{-12}$	$3.3\times {10}^9$		57
Source	$\mathit{Pr}$ or $\mathit{Sc}$	$\mathit{H}/\mathit{d}$	$\mathsf{\vartheta }$	$\mathit{Ra}/{\mathit{L}}^{\mathbf{3}}\ge$	$\mathit{Ra}/{\mathit{L}}^{\mathbf{3}}\le$	±	#
Goldstein et al. [11]	$Sc=2300$	0.63–2.34	$0^{\circ }$	$3.2\times {10}^{13}$	$1.6\times {10}^{14}$		4
Goldstein et al. [11]	$Sc=2300$	0.63–2.34	${30}^{\circ }$	$1.4\times {10}^{13}$	$3.2\times {10}^{14}$		11
Goldstein et al. [11]	$Sc=2300$	0.63–2.34	${60}^{\circ }$	$6.8\times {10}^{12}$	$4.4\times {10}^{14}$		11
Goldstein et al. [11]	$Sc=2300$	0.63–2.34	${90}^{\circ }$	$2.3\times {10}^{12}$	$6.1\times {10}^{14}$		16
Heo & Chung [12]	$Sc=2094$	25	$0^{\circ }$–${90}^{\circ }$	$1.3\times {10}^{14}$	$1.3\times {10}^{14}$	0.9%	13
Heo & Chung [12]	$Sc=2094$	$7.4$	$0^{\circ }$–${90}^{\circ }$	$1.7\times {10}^{14}$	$1.7\times {10}^{14}$	0.9%	13
Heo & Chung [12]	$Sc=2094$	$3.7$	$0^{\circ }$–${90}^{\circ }$	$1.7\times {10}^{14}$	$1.7\times {10}^{14}$	0.9%	13
Heo & Chung [12]	$Sc=2094$	13	$0^{\circ }$–${90}^{\circ }$	$1.7\times {10}^{14}$	$1.7\times {10}^{14}$	0.9%	9
Heo & Chung [12]	$Sc=2094$	$6.7$	$0^{\circ }$–${90}^{\circ }$	$1.7\times {10}^{14}$	$1.7\times {10}^{14}$	0.9%	9
AlArabi & Khamis [9]	$Pr=0.708$	15.5–104	$0^{\circ }$	$4.8\times {10}^9$	$4.8\times {10}^9$		7
AlArabi & Khamis [9]	$Pr=0.708$	15.5–104	${30}^{\circ }$	$4.8\times {10}^9$	$4.8\times {10}^9$		7
AlArabi & Khamis [9]	$Pr=0.708$	15.5–104	${45}^{\circ }$	$4.8\times {10}^9$	$4.8\times {10}^9$		7
AlArabi & Khamis [9]	$Pr=0.708$	15.5–104	${60}^{\circ }$	$4.8\times {10}^9$	$4.8\times {10}^9$		7
AlArabi & Khamis [9]	$Pr=0.708$	15.5–104	${75}^{\circ }$	$4.8\times {10}^9$	$4.8\times {10}^9$		7
AlArabi & Khamis [9]	$Pr=0.708$	15.5–104	${90}^{\circ }$	$4.8\times {10}^9$	$4.8\times {10}^9$		7

Table 1. Cylinder natural convection data-sets.

Source	Study		$\mathsf{\vartheta }$	${\mathit{Ra}}_{\mathit{d}}/{\mathsf{\Xi }}_{•}\ge$	${\mathit{Ra}}_{\mathit{d}}/{\mathsf{\Xi }}_{•}\le$	±	#
Churchill & Chu [7]	Kutateladze [8]		$0^{\circ }$	$4.3\times {10}^9$	$5.3\times {10}^{12}$		6
Churchill & Chu [7]	all		$0^{\circ }$	$7.5\times {10}^{-12}$	$5.3\times {10}^{12}$		63
Churchill & Chu [7]	10 others		$0^{\circ }$	$7.5\times {10}^{-12}$	$3.3\times {10}^9$		57
Source	$\mathit{Pr}$ or $\mathit{Sc}$	$\mathit{H}/\mathit{d}$	$\mathsf{\vartheta }$	$\mathit{Ra}/{\mathit{L}}^{\mathbf{3}}\ge$	$\mathit{Ra}/{\mathit{L}}^{\mathbf{3}}\le$	±	#
Goldstein et al. [11]	$Sc=2300$	0.63–2.34	$0^{\circ }$	$3.2\times {10}^{13}$	$1.6\times {10}^{14}$		4
Goldstein et al. [11]	$Sc=2300$	0.63–2.34	${30}^{\circ }$	$1.4\times {10}^{13}$	$3.2\times {10}^{14}$		11
Goldstein et al. [11]	$Sc=2300$	0.63–2.34	${60}^{\circ }$	$6.8\times {10}^{12}$	$4.4\times {10}^{14}$		11
Goldstein et al. [11]	$Sc=2300$	0.63–2.34	${90}^{\circ }$	$2.3\times {10}^{12}$	$6.1\times {10}^{14}$		16
Heo & Chung [12]	$Sc=2094$	25	$0^{\circ }$–${90}^{\circ }$	$1.3\times {10}^{14}$	$1.3\times {10}^{14}$	0.9%	13
Heo & Chung [12]	$Sc=2094$	$7.4$	$0^{\circ }$–${90}^{\circ }$	$1.7\times {10}^{14}$	$1.7\times {10}^{14}$	0.9%	13
Heo & Chung [12]	$Sc=2094$	$3.7$	$0^{\circ }$–${90}^{\circ }$	$1.7\times {10}^{14}$	$1.7\times {10}^{14}$	0.9%	13
Heo & Chung [12]	$Sc=2094$	13	$0^{\circ }$–${90}^{\circ }$	$1.7\times {10}^{14}$	$1.7\times {10}^{14}$	0.9%	9
Heo & Chung [12]	$Sc=2094$	$6.7$	$0^{\circ }$–${90}^{\circ }$	$1.7\times {10}^{14}$	$1.7\times {10}^{14}$	0.9%	9
AlArabi & Khamis [9]	$Pr=0.708$	15.5–104	$0^{\circ }$	$4.8\times {10}^9$	$4.8\times {10}^9$		7
AlArabi & Khamis [9]	$Pr=0.708$	15.5–104	${30}^{\circ }$	$4.8\times {10}^9$	$4.8\times {10}^9$		7
AlArabi & Khamis [9]	$Pr=0.708$	15.5–104	${45}^{\circ }$	$4.8\times {10}^9$	$4.8\times {10}^9$		7
AlArabi & Khamis [9]	$Pr=0.708$	15.5–104	${60}^{\circ }$	$4.8\times {10}^9$	$4.8\times {10}^9$		7
AlArabi & Khamis [9]	$Pr=0.708$	15.5–104	${75}^{\circ }$	$4.8\times {10}^9$	$4.8\times {10}^9$		7
AlArabi & Khamis [9]	$Pr=0.708$	15.5–104	${90}^{\circ }$	$4.8\times {10}^9$	$4.8\times {10}^9$		7

2.1. Not Empirical

Empirical theories derive their coefficients from measurements, inheriting the uncertainties from those measurements. Theories developed from first principles derive exact coefficients and exponents mathematically. For example, Incropera, DeWitt, Bergman, and Lavine [13] (p. 210) give the thermal conductance (units $\mathrm{W}/\mathrm{K}$) of a diameter d sphere ($L_s=d/2$) into an unbounded stationary, uniform medium having thermal conductivity k as

$$U_0=2\pi dk$$

(3)

The present theory predicting natural convective heat transfer from a round cylindrical surface derives from first principles; it is not empirical. Each formula is tied to aspects of the cylinder geometry and orientation, fluid, and flow.

2.2. RMS Relative Error

Root-mean-squared (RMS) relative error (RMSRE) provides an objective, quantitative evaluation of theory versus experimental data. It gauges the fit of measurements $g\left({Ra}_j\right)$ to function $f\left({Ra}_j\right)$, giving each of the n samples equal weight in Formula (4). Along with presenting RMSRE, charts in the present work split RMSRE into the bias and scatter components defined in Formula (5). The root-sum-squared of bias and scatter is RMSRE.

$$\begin{array}{cc}\hfill \mathrm{RMSRE}=& \sqrt{{\displaystyle \frac{1}{n}}\sum _{j=1}^n{\left|{\displaystyle \frac{g\left({Ra}_j\right)}{f\left({Ra}_j\right)}}-1\right|}^2}\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \mathrm{bias}={\displaystyle \frac{1}{n}}\sum _{j=1}^n\left\{{\displaystyle \frac{g\left({Ra}_j\right)}{f\left({Ra}_j\right)}}-1\right\}& \mathrm{scatter}=\sqrt{{\displaystyle \frac{1}{n}}\sum _{j=1}^n{\left|{\displaystyle \frac{g\left({Ra}_j\right)}{f\left({Ra}_j\right)}}-1-\mathrm{bias}\right|}^2}\hfill \end{array}$$

(5)

3. Theory from Prior Works

Subscripts and variable names are not uniform among prior works; they have been renamed consistently for inclusion in the present work. Where possible, the formulas are written using the ${\ell }^p$-norm.

3.1. Vertical Cylinder

Sparrow and Gregg [14] and Cebeci [15] created differential equations modeling the thermal boundary layer surrounding a vertical cylinder. Solved numerically, they created graphs and tables relating the cylinder’s local temperature profiles to a that of a vertical plate having the same height. Sparrow and Gregg’s graphs only address $Pr=0.71$ and $Pr=1$; Cebeci expanded the range to $0.01<Pr<100$.

Popiel et al. [10] fit a complicated formula to numerical solutions of the Cebeci [15] equations.

Of greater interest is a simple formula (similar to the vertical plate formula from Churchill and Chu [5]) that they attribute to S. M. Yang [16]:

$${∥0.36{\displaystyle \frac{H}{d}},0.150\sqrt[3]{{\displaystyle \frac{{Ra}_H}{{\parallel 1,0.492/Pr\parallel }_{9/16}}}}∥}_{1/2}$$

(6)

${\parallel 1,0.492/Pr\parallel }_{9/16}$ is the denominator used for a vertical plate in Churchill and Chu [5]. Jaffer [1] finds that $\mathsf{\Xi }={\parallel 1,0.5/Pr\parallel }_{\sqrt{1/3}}$ is more accurate. The self-obstruction factor ($1/\mathsf{\Xi }$) for a vertical cylinder and vertical plate should be the same; this investigation uses $\mathsf{\Xi }={\parallel 1,0.5/Pr\parallel }_{\sqrt{1/3}}$ for vertical cylinders.

3.2. Level Cylinder

Formula (7) is the ${\ell }^p$-norm form of what Churchill and Chu [7] propose as the natural convective ${\overline{Nu}}_d$ from a level diameter d isothermal cylinder (excluding end faces).

$${∥0.36,0.150\sqrt[3]{{\displaystyle \frac{{Ra}_d}{{∥1,0.559/Pr∥}_{9/16}}}}∥}_{1/2}$$

(7)

They also propose a laminar flow formula with an exponent of 1/4 instead of 1/3:

$$0.36+0.518\sqrt[4]{{\displaystyle \frac{{Ra}_d}{{∥1,0.559/Pr∥}_{9/16}}}}$$

(8)

The denominator ${\parallel 1,0.559/Pr\parallel }_{9/16}$ differs from ${\parallel 1,0.492/Pr\parallel }_{9/16}$ only in its coefficient, which is within 1% of 9/16. This investigation uses ${\mathsf{\Xi }}_{•}={\parallel 1,\sqrt{1/3}/Pr\parallel }_{\sqrt{1/3}}$ as a denominator for level cylinders.

3.3. Inclined Cylinder

Al-Arabi and Khamis [9] propose empirical power-law Formula (9) to match their laminar heat transfer measurements:

$$\begin{array}{cc}\hfill & {\overline{Nu}}_H=\left[2.9-2.32{cos}^{0.8}\vartheta \right]{Gr}_d^{1/12}{Ra}_H^{n}n={\displaystyle \frac{1}{4}}+{\displaystyle \frac{{cos}^{1.2}\vartheta }{12}}\hfill \\ \hfill & 1.08\times {10}^4<G{r}_d<6.9\times {10}^{5}9.88\times {10}^7<R{a}_H<4\times {10}^9\hfill \end{array}$$

(9)

and Formula (10) to match their turbulent measurements:

$$\begin{array}{cc}\hfill & {\overline{Nu}}_H=\left[0.47+0.11{cos}^{0.8}\vartheta \right]{Gr}_d^{-1/12}{Ra}_H^{1/3}\hfill \\ \hfill & 1.08\times {10}^4<G{r}_d<6.9\times {10}^{5}4\times {10}^9<R{a}_H<2.95\times {10}^{10}\hfill \end{array}$$

(10)

Al-Arabi and Khamis [9] do not report observing turbulence; they claim to infer its occurrence where the local h at the cylinder’s end is insensitive to the cylinder’s length.

Heo and Chung [12] also fit empirical power-law curves to their data:

$${\overline{Nu}}_d=0.3{Ra}_d^{0.25}[1+0.7cos\vartheta ]\mathrm{Laminar}$$

(11)

$${\overline{Nu}}_d=0.13{Ra}_d^{0.3}[1+0.6cos\vartheta ]\mathrm{Turbulent}$$

(12)

$${\overline{Nu}}_H=0.67{Ra}_H^{0.25}[1+1.44{Ra}_d^{-0.04}cos\vartheta ]\mathrm{Laminar}$$

(13)

$${\overline{Nu}}_H=0.26{Ra}_H^{0.28}[1+1.89{Ra}_d^{-0.044}cos\vartheta ]\mathrm{Turbulent}$$

(14)

$$\begin{array}{cc}\hfill 1.69\times {10}^8& <{Ra}_d<5.07\times {10}^{10}3.7<H/d<25\hfill \\ \hfill 2.64\times {10}^{12}& <{Ra}_H<1.54\times {10}^{13}Pr=2094\hfill \end{array}$$

(15)

The only mention of a turbulence threshold in Heo and Chung [12] is ${Gr}_H\approx {10}^9$, which they attribute to Al-Arabi and Salman [17]. By that criterion, most of the $d=0.01$ m cylinder would be laminar, and the others mostly turbulent.

Neither study’s formulas include a conduction (heat transfer into static fluid) term, so they do not apply to small $Ra$ values. Each study’s formulas apply to a single $Pr$ or $Sc$ value, while horizontal and vertical cylinder formulas scale $Ra$ by a function of Pr (or $Sc$). Although dimensionally consistent, both of these studies mix variables having different characteristic lengths, H and d, making it difficult to conjecture a corresponding physical interpretation.

Although lacking a conduction term, and proposed only for $Sc=2300$, Goldstein et al. [11] address the characteristic-length issue by proposing a formula using an angle-dependent characteristic length G:

$$\begin{array}{cc}\hfill & {\overline{Sh}}_G=0.712{Ra}_G^{1/4}G={\displaystyle \frac{Hsin\vartheta +dcos\vartheta }{{(sin\vartheta +cos\vartheta )}^{7/3}}}Sc=2300\hfill \\ \hfill & 0.63<H/d<2.342.0\times {10}^9<{Ra}_G<9.5\times {10}^{11}\hfill \end{array}$$

(16)

where $\overline{Sh}$ is the (average) Sherwood number, an analog of the (average) Nusselt number $\overline{Nu}$.

About Formula (16), Goldstein et al. [11] writes

The correlation suggests that the flow is in the laminar region for the most part. The horizontal cylinder data taken alone suggest the onset of turbulent flow; however, within the range of the Rayleigh numbers studied, the correlation above will give a Sherwood number that is reasonably accurate.

The approach of this investigation is to derive the general formulas for horizontal and vertical cylinders from thermodynamic constraints on heat-engine efficiency, conservation laws, and flow topology. Streamline photos of natural convection from a cylinder were not found. Experience developing the plate natural convection flow models led to the flows proposed in Figure 1a,b.

Horizontal and vertical flow modes compete in the case of an inclined plate. This investigation posits that the same is true for an inclined cylinder.

4. Natural Convection

First, a review of flat surface convection (which will also be used for cylinder end-caps):

From thermodynamic constraints and conservation laws, Jaffer [1] derives generalized natural convection Formula (17) with the parameters specified in

Table 2. Plate natural convection parameters.

Face	$\mathsf{\theta }$	L	$\overline{\mathit{N}\mathit{u}}$	$\mathit{N}{\mathit{u}}_0$	$\mathit{Ra}$	B	C	D	E	p
up	$-{90}^{\circ }$	$L^{*}$	$\overline{N{u}^{*}}$	$N{u}_0^{*}$	${Ra}^{*}$	2	$1/\sqrt{8}$	1	1	$1/2$
vertical	$0^{\circ }$	$L^{\prime }$	$\overline{N{u}^{\prime }}$	$N{u}_0^{\prime }$	${Ra}^{\prime }/\mathsf{\Xi }$	$1/2$	$1/2$	$1/4$	1	$1/2$
down	$+{90}^{\circ }$	$L_R$	$\overline{N{u}_R}$	$N{u}_0^{\prime }/2$	${Ra}_R/\mathsf{\Xi }$	4	$1/2$	2	3	1

:

$$\overline{Nu}={∥N{u}_0\left[1-C\right],\sqrt[2+E]{{\left[CDN{u}_0\right]}^{3+E}{\displaystyle \frac{2}{B}}Ra}∥}_p$$

(17)

$$\mathsf{\Xi }={∥1,{\displaystyle \frac{1/2}{Pr}}∥}_{\sqrt{1/3}}N{u}_0^{*}={\displaystyle \frac{2}{\pi }}\approx 0.637N{u}_0^{\prime }={\displaystyle \frac{2^4}{\sqrt[4]{2}{\pi }^2}}\approx 1.363$$

(18)

The * superscript indicates an upward-facing surface; the ′ superscript is for vertical surfaces; the R subscript is for a downward-facing surface. Scripted $\overline{Nu}$ and $Ra$ are computed with L having the same script.

The parameters of Table 2 are as follows:

θ 

is the angle of the plate from vertical; $\theta =+{90}^{\circ }$ is face-down.

L 

is the characteristic length of a flat plate with convex perimeter:

• Face-up ($\theta =-{90}^{\circ }$) $L^{*}$ is the area-to-perimeter ratio;

  −

  For a $Y\times Z$ rectangle, $L^{*}=YZ/[2Y+2Z]$.

  −

  For a diameter d disk, $L^{*}=d/4$.

• Vertical ($\theta =0^{\circ }$) $L^{\prime }$ is the harmonic mean of the perimeter vertical spans;

  −

  Given a convex vertical plate with maximum width W whose vertical span at horizontal offset w is $H\left(w\right)$:

  $$L^{\prime }=W/{\int }_0^W{\displaystyle \frac{1}{H\left(w\right)}}\mathrm{d}w$$

  (19)

  −

  For a diameter d disk, $L^{\prime }=2d/\pi$.

  −

  For a level height H rectangle, $L^{\prime }=H$.

• Face-down ($\theta =+{90}^{\circ }$) $L_R$ is the harmonic mean of the perimeter distances to that bisector which is perpendicular to the shortest bisector;

  −

  Given a flat surface with convex perimeter defined by functions $H_{+}\left(w\right)>0$ and $H_{-}\left(w\right)<0$ within the range $0<w<W$ along the equal-area bisector which is perpendicular to the shortest equal-area bisector:

  $$L_R=4W/{\int }_0^W\left[{\displaystyle \frac{1}{|H_{+}\left(w\right)|}}+{\displaystyle \frac{1}{|H_{-}\left(w\right)|}}\right]\mathrm{d}w$$

  (20)

  −

  For a $Y\times Z$ rectangle, $L_R=min(Y,Z)/2$.

  −

  For a diameter d disk, $L_R=d/\pi$.

$\overline{\mathit{N}\mathit{u}}$

is the dimensionless convective heat transfer from Formula (17).

${\mathit{N}\mathit{u}}_0$

is the dimensionless heat conduction into motionless fluid.

$Ra$

Rayleigh number: ${Ra}^{\prime }$ is computed with vertical $L^{\prime }$; ${Ra}^{*}={Ra}^{\prime }{[L^{*}/L^{\prime }]}^3$; ${Ra}_R={Ra}^{\prime }{[L_R/L^{\prime }]}^3$. $Pr$ does not affect upward-facing heat transfer because the heated fluid flows directly upward, as does conducted heat. When heated fluid must flow along vertical and downward-facing plates, its heat transfer potential is reduced by dividing $Ra$ by $\mathsf{\Xi }$ from Formula (18).

B 

is the sum of the mean lengths of flows tangent to a wall or counter-flow on one side divided by L.

C 

is the surface area fraction responsible for flow-induced heat transfer.

D 

is the effective length of heat transfer contact with one side of the plate divided by L.

E 

is the count of ${90}^{\circ }$ changes in direction of thermally induced fluid flow.

p 

The ${\ell }^p$-norm combines the static conduction and induced convective heat flows.

4.1. Cylinder

Formula (17) is not limited to flat surfaces. Natural convection heat transfer from a cylinder can be modeled using Formula (17) with the parameters in

Table 3. Cylinder natural convection parameters.

Cylinder	$\mathsf{\vartheta }$	L	$\overline{\mathit{N}\mathit{u}}$	$\mathit{N}{\mathit{u}}_0$	$\mathit{Ra}$	B	C	D	E	p
level	$0^{\circ }$	d	$\overline{N{u}^{•}}$	$N{u}_0^{•}$	${Ra}_d/{\mathsf{\Xi }}_{•}$	$\pi /2$	$1/2$	$\pi /3$	$E^{•}$	$1/3$
vertical	${90}^{\circ }$	H	$\overline{N{u}^{\parallel }}$	$N{u}_0^{•}{\displaystyle \frac{H}{d}}$	${Ra}_H/\mathsf{\Xi }$	${\displaystyle \frac{H}{d}}$	$1/2$	${\displaystyle \frac{1}{6}}{\displaystyle \frac{d}{H}}$	1	$1/6$

.

The • superscript or subscript indicates a level cylinder; the ∥ superscript is for a vertical cylinder.

The parameters of Table 3 are as follows:

ϑ 

is the angle of the cylinder from horizontal.

L 

The height H of a vertical cylinder is its characteristic length.

The diameter d of a level circular cylinder is its characteristic length. This investigation uses the hydraulic diameter, which is 4 times the area-to-perimeter ratio of the cylinder’s cross-section, as d. Note that the diameter and hydraulic diameter are identical for a level circular cylinder.

$\overline{\mathit{N}\mathit{u}}$

is the dimensionless convective heat transfer from Formula (17).

${\mathit{N}\mathit{u}}_0$

is the dimensionless heat conduction into motionless fluid. Conduction shape factors are not well-defined with unbounded source areas, but Nusselt numbers can be. A cylinder’s $N{u}_0$ (conduction into static fluid) must distribute in two dimensions what the sphere distributes in three. Using Formula (3) $U_0$ and $L_s=d/2$:

$$N{u}_0^{•}={\left[{\displaystyle \frac{U_0{L}_s}{\pi d^{2}k}}{\displaystyle \frac{L_s}{d}}\right]}^{3/2}=2^{-3/2}\approx 0.354$$

(21)

Note that $N{u}_0^{•}/2\approx 0.177$ is smaller than the 0.36 value in prior works’ Formulas (6)–(8).

The vertical cylinder’s static conduction is the same as a level cylinder. But its characteristic length is H, not d. Consequently vertical $N{u}_0=N{u}_0^{•}H/d$ as per S. M. Yang [16] in Formula (6).

${\mathit{R}}_{\mathit{a}}$

Rayleigh number: vertical ${Ra}_H$ is computed with vertical H; level ${Ra}_d={Ra}_H{[d/H]}^3$.

${Ra}_H$ is divided by Formula (18) $\mathsf{\Xi }\left(Pr\right)$ as with a vertical plate.

The upper part of the level cylinder is unobstructed; with small $Pr$, induced fluid flow and heat transfer are reduced. The ${Ra}_d$ divisor is increased to Formula (22) ${\mathsf{\Xi }}_{•}$. Figure 2 plots $1/{\mathsf{\Xi }}_{•}$ and $1/\mathsf{\Xi }$.

$${\mathsf{\Xi }}_{•}={∥1,{\displaystyle \frac{\sqrt{1/3}}{Pr}}∥}_{\sqrt{1/3}}\mathsf{\Xi }={∥1,{\displaystyle \frac{1/2}{Pr}}∥}_{\sqrt{1/3}}$$

(22)

B 

Fluid rises the full height of the vertical cylinder; its B would be expected to be 1. Like $N{u}_0$ however, vertical B gets scaled by $H/d$.

Fluid flows tangent to the horizontal cylinder over (each) half of its perimeter in Figure 1b; $B=\pi /2$.

C 

$C=1/2$ because fluid flow with both cylinder orientations is two-dimensional.

D 

The vertical plate’s $D=1/4$ because fluid flowing by the upper half is already heated and accelerating upward. The vertical cylinder lacks vertical edges adjacent to unheated fluid. Relative to H, only a short length (1/6) of the vertical cylinder heats a significant amount of fluid:

$$D={\displaystyle \frac{1}{6}}{\displaystyle \frac{d}{H}}$$

(23)

The $d/H$ factor in Formula (23) unscales $N{u}_0=N{u}_0^{•}H/d$ in the $CDN{u}_0$ term of Formula (17).

Most of the level cylinder’s heat transfer occurs where its surface is vertical or downward-facing, not upward-facing. Two thirds of $B=\pi /2$ is $D=\pi /3$.

E 

Fluid flow induced by a vertical cylinder experiences one ${90}^{\circ }$ ($\pi /2$ rad) change in direction; its $E=1$.

Figure 1b shows convecting fluid flowing tangent to the lower half of a level cylinder. For a unit radius cylinder, fluid flowing horizontally at elevation $0<y<1$ must bend $\pi -arcsin\left(y\right)$ radians upward in order to be tangent to the cylinder. The average angle is

$${\int }_0^1\pi -arcsiny\mathrm{d}y=1+{\displaystyle \frac{\pi }{2}}\approx 2.571\mathrm{rad}$$

(24)

Fluid at mid-height is already moving upward; it requires no more bend. Fluid moving tangent to the upper part of a cylinder will require some bend, but less fluid is flowing tangent to the cylinder at the top, so $arcsinx$ is weighted by $x^2$, where x is the distance from the vertical mid-line:

$${\int }_0^1{\displaystyle \frac{\pi }{2}}-x^{2}arcsinx\mathrm{d}x={\displaystyle \frac{2+3\pi }{9}}\approx 1.269\mathrm{rad}$$

(25)

Dividing the average of Formula (24) and Formula(25) by $\pi /2$:

$$E^{•}={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{\pi }}+{\displaystyle \frac{1}{3}}+{\displaystyle \frac{2}{9\pi }}={\displaystyle \frac{5}{6}}+{\displaystyle \frac{11}{9\pi }}\approx 1.222$$

(26)

p 

Lacking vertical edges adjacent to unheated fluid, the static conduction and induced convective heat flows of a vertical cylinder cooperate much more than from a vertical plate; they combine as the ${\ell }^{1/6}$-norm.

The downward-facing part of a level cylinder preheats fluid which then flows adjacent to the vertical surfaces. The level cylinder’s heat flows cooperate more than the vertical plate’s ${\ell }^{1/2}$-norm, but less than the vertical cylinder’s ${\ell }^{1/6}$-norm. They combine as the ${\ell }^{1/3}$-norm.

4.2. Surface Conductance Formulas

Because the characteristic lengths scaling a flat plate’s $\overline{N{u}^{*}}$, $\overline{N{u}^{\prime }}$, and $\overline{N{u}_R}$ can be different, flow-mode interactions use scale-free $\overline{h^{*}}$, $\overline{h^{\prime }}$, and $\overline{h_R}$:

$$\overline{h^{*}}={\displaystyle \frac{k}{L^{*}}}{∥N{u}_0^{*}\left[1-\sqrt{{\displaystyle \frac{1}{8}}}\right],{\displaystyle \frac{{N{u}_0^{*}}^{4/3}}{4}}\sqrt[3]{{Ra}^{*}}∥}_{1/2}\approx {\displaystyle \frac{k}{L^{*}}}{∥0.411,0.137\sqrt[3]{{Ra}^{*}}∥}_{1/2}$$

(27)

$$\overline{h^{\prime }}={\displaystyle \frac{k}{L^{\prime }}}{∥{\displaystyle \frac{N{u}_0^{\prime }}{2}},{\displaystyle \frac{{N{u}_0^{\prime }}^{4/3}}{8\sqrt[3]{2}}}\sqrt[3]{{\displaystyle \frac{{Ra}^{\prime }}{\mathsf{\Xi }}}}∥}_{1/2}\approx {\displaystyle \frac{k}{L^{\prime }}}{∥0.682,0.150\sqrt[3]{{\displaystyle \frac{{Ra}^{\prime }}{\mathsf{\Xi }}}}∥}_{1/2}$$

(28)

$$\overline{h_R}={\displaystyle \frac{k}{L_R}}{∥{\displaystyle \frac{N{u}_0^{\prime }}{4}},{\displaystyle \frac{{N{u}_0^{\prime }}^{6/5}}{2^{7/5}}}\sqrt[5]{{\displaystyle \frac{{Ra}_R}{\mathsf{\Xi }}}}∥}_1\approx {\displaystyle \frac{k}{L_R}}\left[0.341+0.550\sqrt[5]{{\displaystyle \frac{{Ra}_R}{\mathsf{\Xi }}}}\right]$$

(29)

Formulas (30) and (31) model heat transfer from horizontal and vertical cylinders, respectively.

$$\overline{h^{•}}={\displaystyle \frac{k}{d}}{∥{\displaystyle \frac{N{u}_0^{•}}{2}},\sqrt[2+E^{•}]{{\left[{\displaystyle \frac{\pi N{u}_0^{•}}{6}}\right]}^{3+E^{•}}{\displaystyle \frac{{Ra}_d}{\pi {\mathsf{\Xi }}_{•}}}}∥}_{1/3}\approx k{∥{\displaystyle \frac{0.177}{d}},{\displaystyle \frac{0.118}{d}}{\left[{\displaystyle \frac{{Ra}_d}{{\mathsf{\Xi }}_{•}}}\right]}^{0.310}∥}_{1/3}$$

(30)

$$\overline{h^{\parallel }}={\displaystyle \frac{k}{H}}{∥{\displaystyle \frac{N{u}_0^{•}}{2}}{\displaystyle \frac{H}{d}},\sqrt[3]{{\left[{\displaystyle \frac{N{u}_0^{•}}{12}}\right]}^4{\displaystyle \frac{d}{H}}{\displaystyle \frac{2{Ra}_H}{\mathsf{\Xi }}}}∥}_{1/6}\approx k{∥{\displaystyle \frac{0.177}{d}},{\displaystyle \frac{0.0115}{H}}\sqrt[3]{{\displaystyle \frac{d}{H}}{\displaystyle \frac{{Ra}_H}{\mathsf{\Xi }}}}∥}_{1/6}$$

(31)

4.3. End-Caps

The flow modes of each end-cap (upper and lower, respectively) compete as the ${\ell }^{16}$-norm:

$$\overline{h_{\uparrow }}={∥\overline{h^{\prime }}(cos\vartheta {Ra}_H),{\displaystyle \frac{\overline{h^{*}}(sin\vartheta {Ra}^{*})}{1+H/d}}∥}_{16}\overline{h_{\downarrow }}={∥\overline{h^{\prime }}(cos\vartheta {Ra}_H),{\displaystyle \frac{\overline{h_R}(sin\vartheta {Ra}_R)}{1+H/d}}∥}_{16}$$

(32)

where $L^{\prime }=2d/\pi$, $L^{*}=d/4$, and $L_R=d/\pi$. The $1+H/d$ denominator models the reduction of vertical cylinder heat transfer by already heated fluid. The trigonometric functions of $\vartheta$ are explained in Section 5. Note that the end-caps and cylinder surface have different areas.

4.4. Comparison with Level Cylinder Measurements

Figure 3 compares four level cylinder theories with data-sets from Churchill and Chu [7], Goldstein et al. [11], and Heo and Chung [12]. Note that the estimated digitization accuracy of the Churchill and Chu [7] data is $\pm 10\%$.

• The “Churchill & Chu 1975: turbulent (theory)” trace is Formula (7).
• The “Churchill & Chu 1975: laminar (theory)” trace is Formula (8).
• The “Goldstein et al. 2007 (theory)” trace is Formula (16).
• The “present work” trace is $\overline{N{u}^{•}}=\overline{h^{•}}d/k$, where $\overline{h^{•}}$ is Formula (30).

Table 4. Churchill and Chu data-sets versus theories.

Study	Theory	${\mathit{Ra}}_{\mathit{d}}/{\mathsf{\Xi }}_{•}\ge$	${\mathit{Ra}}_{\mathit{d}}/{\mathsf{\Xi }}_{•}\le$	RMSRE	Bias	Scatter	#
Kutateladze [8]	turbulent	$4.3\times {10}^9$	$5.3\times {10}^{12}$	$7.4\%$	$-1.4\%$	$7.3\%$	6
Kutateladze [8]	laminar	$4.3\times {10}^9$	$5.3\times {10}^{12}$	$185.6\%$	$-172.9\%$	$67.4\%$	6
Kutateladze [8]	present	$4.3\times {10}^9$	$5.3\times {10}^{12}$	$99.7\%$	$-94.9\%$	$30.3\%$	6
all	turbulent	$7.5\times {10}^{-12}$	$5.3\times {10}^{12}$	$20.2\%$	$+8.8\%$	$18.2\%$	63
all	laminar	$7.5\times {10}^{-12}$	$5.3\times {10}^{12}$	$60.1\%$	$-23.7\%$	$55.2\%$	63
all	present	$7.5\times {10}^{-12}$	$5.3\times {10}^{12}$	$32.5\%$	$-10.4\%$	$30.8\%$	63
10 others	turbulent	$7.5\times {10}^{-12}$	$3.3\times {10}^9$	$21.1\%$	$+9.8\%$	$18.7\%$	57
10 others	laminar	$7.5\times {10}^{-12}$	$3.3\times {10}^9$	$19.0\%$	$-7.9\%$	$17.2\%$	57
10 others	present	$7.5\times {10}^{-12}$	$3.3\times {10}^9$	$10.9\%$	$-1.5\%$	$10.8\%$	57

presents statistics of the Churchill and Chu data-set measurements versus turbulent Formula (7), laminar Formula (8), and the present work’s Formula (30).

The six-point data-set which Churchill and Chu [7] attribute to Kutateladze [8] exceeds all the theory traces. The cylinder convection graph in Kutateladze [8] does not assign parameters or references to individual measurements, but asserts that $Pr\ge 1$. The large ${Ra}_d$ measurements appear to be from Hermann [18], which was “accepted as dissertation by the Technical High School at Aachen” in 1936. The two studies with large ${Ra}_d$ which Hermann cites are Wamsler [19] and Koch and Beih [20]. Hermann gives $Pr=0.74$; however, this $Pr$ difference is not sufficient to bring the measurements into line with the other Churchill and Chu data-sets.

Both the Wamsler [19] and Koch and Beih [20] data-sets were obtained from heated iron and steel pipes in air. Hermann states that the radiative heat loss ranged up to 60% of the total, and that the radiative losses of Koch and Beih had to be corrected. Having 60% of the heat loss due to thermal radiation degrades the potential accuracy of convection measurements.

Solute-transfer measurements of natural convection are capable of greater accuracy. Heo and Chung [12] estimate $0.9\%$ measurement uncertainty. In

Table 5. Level cylinder measurements versus present theory.

Source	$\mathit{Pr}$ or $\mathit{Sc}$	$\mathit{H}/\mathit{d}$	$\mathsf{\vartheta }$	RMSRE	Bias	Scatter	#
Goldstein et al. [11]	$Sc=2300$	1.48	$0^{\circ }$	$4.7\%$	$-4.4\%$	$1.6\%$	4
Heo & Chung [12]	$Sc=2094$	3.73–13.2	$0^{\circ }$	$1.4\%$	$+0.3\%$	$1.3\%$	4

the Goldstein et al. [11] and Heo and Chung [12] data sets have RMSRE values smaller than 4.8%. The $Ra/{\mathsf{\Xi }}_{•}$ range of the Goldstein et al. and Heo and Chung data-sets overlaps three of the six Kutateladze [8] points, providing further evidence justifying the exclusion of the Kutateladze data.

The remaining ten of the Churchill and Chu data-sets, spanning more than 20 orders of magnitude of $Ra/{\mathsf{\Xi }}_{•}$, have 11% RMSRE versus the present theory in Table 4, a significant improvement from the 20% RMSRE of Formula (7) and the 19% RMSRE of Formula (8).

The “Goldstein et al. 2007 (theory)” trace matches its own level cylinder data with 3.6% RMSRE but, lacking a heat conduction term, would be inaccurate when $Ra/{\mathsf{\Xi }}_{•}<10$.

4.5. Comparison with Vertical Cylinder Measurements

Figure 4 compares four vertical cylinder theories with data-sets from Al-Arabi and Khamis [9], Goldstein et al. [11], and Heo and Chung [12].

• The “vertical plate” trace is $\overline{N{u}^{\prime }}=\overline{h^{\prime }}{L}^{\prime }/k$, where $\overline{h^{\prime }}$ is Formula (28).
• The “Yang 1985” trace is Formula (6).
• The “Goldstein et al. 2007 (theory)” trace is Formula (16).
• The “present work” traces are $\overline{N{u}^{\parallel }}=\overline{h^{\parallel }}H/k$, where $\overline{h^{\parallel }}$ is Formula (31).

Table 6. Vertical cylinder measurements versus present theory.

Source	$\mathit{Pr}$ or $\mathit{Sc}$	$\mathit{H}/\mathit{d}$	$\mathsf{\vartheta }$	RMSRE	Bias	Scatter	#
Goldstein et al. [11]	$Sc=2300$	1.48	${90}^{\circ }$	$3.7\%$	$-2.6\%$	$2.6\%$	16
Heo & Chung [12]	$Sc=2094$	3.73–13.2	${90}^{\circ }$	$2.7\%$	$-0.4\%$	$2.7\%$	4
AlArabi & Khamis [9]	$Pr=0.708$	15.5–104	${90}^{\circ }$	$4.0\%$	$-0.1\%$	$4.0\%$	7

presents vertical statistics of the inclined cylinder data-sets. Comparing vertical cylinder heat transfer measurements is more difficult than with level cylinders because the surface conductance depends on $H/d$, the height-to-diameter ratio.

The “Goldstein et al. 2007 (theory)” trace matches its own level cylinder data with 3.6% RMSRE. However, the cylinder diameter d does not affect the “Goldstein et al. 2007 (theory)” Formula (16) when the cylinder is vertical. Its RMSRE for vertical Al-Arabi and Khamis data-sets exceeds 100%.

While the other formulas all raise ${Ra}_H^{1/3}$, the gentle curvature of the ${\ell }^{1/6}$-norm dominates the slopes of “present work” traces through the entire range of Figure 4. This indicates that vertical plate convection is rarely a good approximation for vertical cylinder convection.

5. Inclination

$\theta$ is the angle of a flat surface from vertical; $\theta =-{90}^{\circ }$ is face-up.

$\vartheta$ is the angle of a cylinder’s axis from horizontal.

A mass constrained to move along a line inclined at $\theta$ from vertical will experience gravitational force proportional to the projection of the gravity vector onto that line, $|cos\theta |$. Similarly, a mass constrained to move perpendicular to a plate inclined at $\theta$ from vertical will have its force scaled by $|sin\theta |$.

$Ra$ is proportional to gravitational acceleration in the direction of flow; thus plate ${Ra}^{\prime }$ is scaled by $|cos\theta |$ and ${Ra}^{*}$ and ${Ra}_R$ get scaled by $|sin\theta |$. Similarly for cylinders, ${Ra}_H$ gets scaled by $|sin\vartheta |$ and ${Ra}_d$ gets scaled by $|cos\vartheta |$.

5.1. Natural Convection from an Inclined Plate

For an inclined plate, the formula in Raithby and Hollands [21] chooses the upward-facing, downward-facing, or vertical flow mode having the maximum convective surface conductance (with each $Ra$ scaled as described above).

The upward-facing $\overline{h^{*}}$ and downward-facing $\overline{h_R}$ do not directly compete with each other, suggesting

$$\overline{h}=\left\{\begin{array}{cc}max\left(\overline{h^{\prime }}\left(\left|cos\theta \right|{Ra}^{\prime }\right),\overline{h^{*}}\left(\left|sin\theta \right|{Ra}^{*}\right)\right)\hfill & sin\theta <0\hfill \\ max\left(\overline{h^{\prime }}\left(\left|cos\theta \right|{Ra}^{\prime }\right),\overline{h_R}\left(\left|sin\theta \right|{Ra}_R\right)\right)\hfill & sin\theta \ge 0\hfill \end{array}\right.$$

(33)

However, measurements of inclined plate natural convective heat transfer revealed that, in reality, the $\theta$ transition is more gradual using the ${\ell }^{16}$-norm in Formula (34):

$$\overline{h}=\left\{\begin{array}{cc}{∥\overline{h^{\prime }}\left(\left|cos\theta \right|{Ra}^{\prime }\right),\overline{h^{*}}\left(\left|sin\theta \right|{Ra}^{*}\right)∥}_{16}\hfill & sin\theta <0\hfill \\ {∥\overline{h^{\prime }}\left(\left|cos\theta \right|{Ra}^{\prime }\right),\overline{h_R}\left(\left|sin\theta \right|{Ra}_R\right)∥}_{16}\hfill & sin\theta \ge 0\hfill \end{array}\right.$$

(34)

5.2. Natural Convection from an Inclined Cylinder

Flow along a flat surface is strongly constrained by that surface; competing flows combine with the ${\ell }^{16}$-norm ($p=16$). Flows around an inclined cylinder are less constrained but still compete, suggesting a smaller $p>1$.

However, natural convection flows around a long thin cylinder will be more competitive ($p\gg 1$) than from a cylinder where $H\approx d$. This suggests combining $\overline{h^{\parallel }}$ and $\overline{h^{•}}$ with $p=1+H/d$:

$$\overline{h}={∥\overline{h^{\parallel }}\left(\left|sin\vartheta \right|{Ra}_H\right),\overline{h^{•}}\left(\left|cos\vartheta \right|{Ra}_d\right)∥}_{1+H/d}$$

(35)

Formula (35) is suitable for $Ra/\mathsf{\Xi }\gg 1$, but doubles $\overline{h}$ when $Ra=0$. $\overline{h}$ should equal $\overline{h^{\parallel }}$ for vertical cylinders and $\overline{h^{•}}$ for level cylinders. It was established earlier that when $Ra=0$, cylinder $\overline{h}$ is insensitive to orientation; let $\overline{h_0}$ be the $Ra=0$ surface conductance.

Table 7. Cylinder asymptotic behaviors.

$\mathsf{\vartheta }$	$\overline{{\mathit{h}}^{\parallel }}$	$\overline{{\mathit{h}}^{•}}$	$\overline{{\mathit{h}}^{\parallel }}$ Coefficient	$\overline{{\mathit{h}}^{•}}$ Coefficient
${90}^{\circ }$	∞	∞	1	1
$0^{\circ }$	∞	∞	1	1
${90}^{\circ }$	∞	$\overline{h_0}$	1	0
$0^{\circ }$	$\overline{h_0}$	∞	0	1
			$1-\overline{h_0}/\overline{h^{\parallel }}$	$1-\overline{h_0}/\overline{h^{•}}$
${90}^{\circ }$	$\overline{h_0}$	$\overline{h_0}$	1	0
$0^{\circ }$	$\overline{h_0}$	$\overline{h_0}$	0	1
			${sin}^2\vartheta {\overline{h_0}}^2/\left[\overline{h^{\parallel }}\overline{h^{•}}\right]$	${cos}^2\vartheta {\overline{h_0}}^2/\left[\overline{h^{\parallel }}\overline{h^{•}}\right]$

shows the desired coefficients for asymptotic values of $\overline{h^{\parallel }}$ and $\overline{h^{•}}$:

Combining the coefficients from Table 7, Formula (36) satisfies these constraints. Plotted in Figure 5, its error relative to $\overline{h^{•}}$ Formula (30) ($\vartheta =0^{\circ }$) is negligible; its error relative to $\overline{h^{\parallel }}$ Formula (31) ($\vartheta ={90}^{\circ }$) is less than 0.4% when $H/d\ge 1/9$.

$$\begin{array}{cc}\hfill \overline{h}& ={∥\overline{h^{\parallel }}\left[1-{\displaystyle \frac{\overline{h_0}}{\overline{h^{\parallel }}}}+{\displaystyle \frac{{\overline{h_0}}^2{sin}^2\vartheta }{\overline{h^{•}}\overline{h^{\parallel }}}}\right],\overline{h^{•}}\left[1-{\displaystyle \frac{\overline{h_0}}{\overline{h^{•}}}}+{\displaystyle \frac{{\overline{h_0}}^2{cos}^2\vartheta }{\overline{h^{•}}\overline{h^{\parallel }}}}\right]∥}_{1+H/d}\hfill \\ \hfill & ={∥\overline{h^{\parallel }}-\overline{h_0}+{\displaystyle \frac{{\left[\overline{h_0}sin\vartheta \right]}^2}{\overline{h^{•}}}},\overline{h^{•}}-\overline{h_0}+{\displaystyle \frac{{\left[\overline{h_0}cos\vartheta \right]}^2}{\overline{h^{\parallel }}}}∥}_{1+H/d}\hfill \end{array}$$

(36)

5.3. Heo and Chung

Heo and Chung [12] measured copper electroplating onto a copper cylinder in a ${\mathrm{CuSO}}_4/{\mathrm{H}}_2{\mathrm{SO}}_4$ solution. They measured the mass transfer coefficient ${\overline{h}}_m$, but reported their results as ${\overline{Nu}}_d$ and ${\overline{Nu}}_H$, which scale with different characteristic lengths. The value of the mass transfer analog of k was not reported, but for the purposes of comparing theory and measurements, k is arbitrary if all conversions from $\overline{Nu}$ to $\overline{h}$ use the same k. A value of $k=1\mathrm{mW}/(\mathrm{m}·\mathrm{K})$ is used in Figure 6.

To avoid confusion, this section uses cylinder conductance $\overline{U}=\pi dH\overline{h}$ with units W/K instead of $\overline{h}$.

The cylinder and end-caps have different areas; their heat transfers must combine as conductances (W/K). The top three rows in Figure 6 were modeled as

$$\overline{h}={\displaystyle \frac{\overline{U}+\pi d^2[\overline{h_{\uparrow }}+\overline{h_{\downarrow }}]/4}{\pi [dH+d^2/2]}}$$

(37)

For the longer cylinders of the bottom two rows, $\overline{h}$ should be nearly the same as the shorter cylinders, but the measured $\overline{h}$ values for $H/d=13.24$ and $H/d=6.72$ are significantly smaller in Figure 6. Modeling only the cylinder heat transfer but including its end-cap areas yields RMSRE < 3.5%:

$$\overline{h}={\displaystyle \frac{\overline{U}}{\pi [dH+d^2/2]}}$$

(38)

Using their reported $Pr=2094$, the ${Ra}_d=5.07\times {10}^{10}$ in their table differs from the ${Ra}_d=4.96\times {10}^{10}$ and ${Gr}_d=2.37\times {10}^7$ in their figure. ${Ra}_d=5.07\times {10}^{10}$ and ${Gr}_d=2.42\times {10}^7$ are used in the $H/d=3.73$ and $H/d=6.72$ curves in Figure 6.

The parameters regarding the top row are more troubling. Their table lists ${Ra}_d=1.69\times {10}^8$ for the $d=0.010$ m cylinder. Using their reported $Pr=2094$ should result in ${Gr}_d={Ra}_d/Pr\approx 8.07\times {10}^4$. But their figures specify ${Gr}_d=6.27\times {10}^4$ and ${Ra}_d=1.31\times {10}^8$ for the $d=0.010$ m cylinder. Given these inconsistencies, the $d=0.010$ m cylinder is omitted from the present work’s summary statistics.

5.4. Al-Arabi and Khamis

Al-Arabi and Khamis [9] measured local heat transfer to air from a “nickel-electro-plated” brass cylinder heated by steam.

Text in their figures declares ${Gr}_d=2.6\times {10}^4$; however, this is significantly smaller than the ${Gr}_d=4.83\times {10}^4$ value this investigation computes from the average ambient conditions of Cairo, Egypt. The present work uses ${Gr}_d=4.83\times {10}^4$.

For the average thermal surface conductance they report $h_L$ values instead of $\overline{h}$, indicating that these are local surface conductances, not average. In an earlier paper, Al-Arabi and Salman [17] report local $h_L$ and $\overline{h}$ values, and claim that $h_L$ and $\overline{h}$ are “practically the same”. Yet this is clearly contradicted by the second figure of that paper. The only angle for which they are the same is $\vartheta ={90}^{\circ }$.

In vertical cylinder natural convection, growth of characteristic length L corresponds to growth in the direction of fluid flow. In such systems the average heat transfer can be inferred by averaging local heat transfers $h_x$ at lengths $0<x\le L$.

This fails for a horizontal cylinder because its characteristic length is the cylinder’s diameter, not its length. Figure 7 averages the $h_x$ values to produce $\overline{h}$, and confirms that this averaging works only for the vertical cylinder $\vartheta ={90}^{\circ }$. Only the vertical cylinder is included in the present work’s summary statistics.

5.5. Goldstein et al.

Goldstein et al. [11] measured copper electroplating onto three cylinders at four angles in a ${\mathrm{CuSO}}_4/{\mathrm{H}}_2{\mathrm{SO}}_4$ solution. The cylinders were 78.8 mm in diameter and had lengths 49.9 mm, 116.4 mm, and 184.4 mm. They presented measurements without identifying the cylinder used in each trial. The present analysis treats all as having length 116.4 mm, which succeeds because the present work’s $0.63\le H/d\le 2.34$ traces in Figure 4 are converging above $Ra/{\mathsf{\Xi }}_{•}>{10}^8$.

The value of the mass transfer analog of k was not reported, but for the purposes of comparing theory and measurements, k is arbitrary if all conversions from $\overline{Nu}$ to $\overline{h}$ use the same k. A value of $k=1\mathrm{mW}/(\mathrm{m}·\mathrm{K})$ is used in Figure 8.

6. Discussion

Using the thermodynamics-based analysis pioneered by Jaffer [1], this investigation derived novel Formulas (30), (31) and (36) predicting the natural convective heat transfer from level, vertical, and inclined cylinders, respectively, given length H, diameter d, inclination angle $\vartheta$, ${Ra}_d$, and the fluid’s $Pr$ and k, where $H/d\ge 1/9$.

These formulas enable the direct calculation of cylinder convective heat-transfer estimates at any inclination, avoiding the need for measurements of experimental prototypes or finite-element computations.

End-cap heat-transfer Formulas (32) were also proposed and tested on two of the Heo and Chung [12] data-sets, yielding combined RMSRE less than 2.2%.

Table 8. Measurements versus present theory.

Source	$\mathit{Pr}$ or $\mathit{Sc}$	$\mathit{H}/\mathit{d}$	$\mathsf{\vartheta }$	RMSRE	Bias	Scatter	#
Goldstein et al. [11]	$Sc=2300$	1.48	$0^{\circ }$	$4.7\%$	$-4.4\%$	$1.6\%$	4
Goldstein et al. [11]	$Sc=2300$	1.48	${30}^{\circ }$	$3.7\%$	$-1.3\%$	$3.5\%$	11
Goldstein et al. [11]	$Sc=2300$	1.48	${60}^{\circ }$	$3.5\%$	$+2.0\%$	$2.9\%$	11
Goldstein et al. [11]	$Sc=2300$	1.48	${90}^{\circ }$	$3.7\%$	$-2.6\%$	$2.6\%$	16
Heo & Chung [12]	$Sc=2094$	$7.4$	$0^{\circ }$–${90}^{\circ }$	$1.9\%$	$+1.4\%$	$1.3\%$	13
Heo & Chung [12]	$Sc=2094$	$3.7$	$0^{\circ }$–${90}^{\circ }$	$2.1\%$	$+0.7\%$	$2.0\%$	13
Heo & Chung [12]	$Sc=2094$	13	$0^{\circ }$–${90}^{\circ }$	$2.1\%$	$+1.9\%$	$0.9\%$	9
Heo & Chung [12]	$Sc=2094$	$6.7$	$0^{\circ }$–${90}^{\circ }$	$3.4\%$	$-2.6\%$	$2.2\%$	9
AlArabi & Khamis [9]	$Pr=0.708$	15.5–104	${90}^{\circ }$	$4.0\%$	$-0.1\%$	$4.0\%$	7

summarizes the present theory’s conformance with 93 inclined cylinder measurements having $1.48<H/d<104$ at angles $0^{\circ }\le \vartheta \le {90}^{\circ }$ in nine data-sets from three peer-reviewed studies, yielding (data-set) RMSRE values between 1.9% and 4.7%.

On 57 level cylinder measurements from Churchill and Chu [7] in

Table 9. Churchill and Chu versus theories.

Source	Theory	${\mathit{Ra}}_{\mathit{d}}/{\mathsf{\Xi }}_{•}\ge$	${\mathit{Ra}}_{\mathit{d}}/{\mathsf{\Xi }}_{•}\le$	RMSRE	Bias	Scatter	#
Churchill and Chu [7]	turbulent	$7.5\times {10}^{-12}$	$3.3\times {10}^9$	$21.1\%$	$+9.8\%$	$18.7\%$	57
Churchill and Chu [7]	laminar	$7.5\times {10}^{-12}$	$3.3\times {10}^9$	$19.0\%$	$-7.9\%$	$17.2\%$	57
Churchill and Chu [7]	present	$7.5\times {10}^{-12}$	$3.3\times {10}^9$	$10.9\%$	$-1.5\%$	$10.8\%$	57

spanning more than 20 orders of magnitude of $Ra$, the present Formula (30) has 11% RMSRE, a significant improvement from the 21% and 19% RMSRE of prior works’ Formulas (7) and (8).

6.1. Laminar and Turbulent Flows

Heo and Chung [12] claimed that natural convection from four of their five cylinders was turbulent. Goldstein et al. [11] claimed that most of their data was for laminar natural convection, but that their single correlation was “reasonably accurate” on all their data. If they are correct about the flow modes induced by their cylinders, then the present Formula (36) $1.9\%\le \mathrm{RMSRE}\le 4.8\%$ performance in Table 8 includes both laminar and turbulent natural convection.

${Ra}_d$ in level cylinder Formula (30) has neither the 1/4 exponent commonly attributed to laminar flow nor the 1/3 exponent attributed to turbulent flow, but an intermediate exponent of $1/[2+E^{•}]\approx 0.310$.

6.2. Local Convective Heat Transfer

The technique of calculating average heat transfer $\overline{h}$ by averaging local heat transfers $h_x$ at lengths $0<x\le L$ works only when $h_x$ is constant across the width of a plate or the circumference of the cylinder. For natural convection, this is only when the plate or cylinder is vertical.

Otherwise, each $h_{x,y}$ must be averaged, requiring a two-dimensional local convection model. Lacking such a model, only the vertical cylinder data-set of Al-Arabi and Khamis [9] is valid.

Thermodynamic constraints can be powerful tools, but apply only to complete systems. While a thermodynamic constraint would be intrinsic to a perfect molecular simulation, such a simulation is impractical. Furthermore, thermodynamic constraints might well not survive the truncation errors of digital computation.

6.3. Short Cylinders

Around a short ($H\ll d$) level cylinder the fluid flow is not restricted to the vertical plane, resulting in larger heat transfers than predicted by Formula (36). Formula (36) should work for small $H/d$ ratios as long as H is the width of a heated band embedded in a longer, insulated cylinder as shown in Figure 9.

6.4. Non-Circular Cylinders

For a convex cylinder with hydraulic diameter d, vertical Formula (31) is expected to predict heat transfer correctly.

For a level convex cylinder, Table 3’s natural convection parameters B, D, and E need to be reevaluated. B should be the cross-section’s perimeter length squared divided by its area. D will be less than B; for a circular cross-section $D=2B/3$. Parameter E (average bend divided by $\pi /2$ rad) can be calculated by the method of Formulas (24), (25) and (26).

If the cross-section lacks bilateral symmetry, then the convection should be calculated separately for each side of the cross-section, split along the line connecting its highest and lowest point of the cross-section.

6.5. Rough Cylinders

Jaffer and Jaffer [22] made natural convective heat transfer measurements of a 0.305 m square plate with 3 mm root-mean-square height of roughness at angles between $-{90}^{\circ }$ and $+{90}^{\circ }$. Those measurements matched Formula (34) with 3% RMSRE, providing evidence that flat surface natural convection is insensitive to roughness which is much smaller than its characteristic length. A similar test conducted with a rough cylinder would ascertain whether the natural convective flows from cylinders are also insensitive.