Detailed Description

This calculator estimates how much heat a hot or cold vertical cylindrical tank exchanges with the surrounding air and the equivalent annual energy that must be supplied to the heating or cooling systems to compensate for those losses. Two cases are supported:

Insulated tank — the tank wall is wrapped in a layer of insulation; heat conducts through the tank wall and the insulation annulus before being dissipated to the air at the jacket outer surface.
Bare (uninsulated) tank — no insulation is present; heat is dissipated directly from the tank wall outer surface.

The calculation uses U.S. customary units throughout. Temperatures are in degrees Rankine, lengths in feet, thermal conductivities in BTU/(hr·ft·°F), and gravitational acceleration g = 32.174 ft/s².

The calculation follows these steps:

Evaluate air properties (thermal conductivity, density, kinematic viscosity, and specific heat) at the ambient temperature using fourth-order polynomial fits to tabulated data.
Compute the thermal diffusivity of air from the polynomial-fitted properties.
Compute the Rayleigh number (Ra) from the outer surface temperature (jacket or bare tank wall), the ambient temperature, and the air properties.
Derive the natural convection heat transfer coefficient at the outer surface using the Ra^(1/3) empirical correlation.
Compute the overall heat transfer coefficient using a cylindrical multi-layer resistance model that accounts for the inner convective resistance, the tank wall conductive resistance, and (if insulated) the insulation conductive resistance and the outer air-film resistance.
Compute the total heat loss as the sum of the convective–conductive and radiative components.
Scale the heat loss by operating hours and heating system efficiency to obtain annual heat loss.

Relevant formulas and symbol definitions are documented below.

Air Property Polynomial Fits

Fourth-order polynomial fits for air properties as a function of temperature.

Air properties are evaluated at the ambient temperature using curve fits to tabulated air data in U.S. customary units. Each property is expressed as a fourth-order polynomial in temperature \(T\) (°R):

\begin{equation}\label{eq:insulated-tank-poly} p(T) = c_4 T^4 + c_3 T^3 + c_2 T^2 + c_1 T + c_0\end{equation}

The following properties are computed (with scaling applied after polynomial evaluation):

Property	Unit	Scaling
Specific heat, \(c_p\)	BTU/(lb·°F)	× 1
Density, \(\rho\)	lb/ft³	× 10⁻²
Kinematic viscosity, \(\nu\)	ft²/s	× 10⁻⁴ / 3600
Thermal conductivity, \(k\)	BTU/(hr·ft·°F)	× 1

Thermal diffusivity is derived from these fitted properties:

\begin{equation}\label{eq:insulated-tank-thermal-diffusivity} \alpha = \frac{k}{\rho \, c_p}\end{equation}

Symbols

\(\alpha\)	Thermal diffusivity of air \([\unit{ \foot\squared\per\hour}]\)
\(k\)	Thermal conductivity of air \([\unit{ \btu\per\hour\foot\degreeFahrenheit}]\)
\(\rho\)	Density of air \([\unit{ \pound\per\foot\cubed}]\)
\(c_p\)	Specific heat of air \([\unit{ \btu\per\pound\degreeFahrenheit}]\)

Rayleigh Number (U.S. Customary)

Rayleigh number for natural convection along a vertical cylinder in U.S. customary units.

The Rayleigh number characterises the strength of buoyancy-driven natural convection at the tank outer surface. It combines the buoyancy force (from the temperature difference between the surface and the ambient air) with the fluid's resistance to motion and heat diffusion.

\begin{equation}\label{eq:insulated-tank-rayleigh} \mathrm{Ra} = \frac{g \,\beta \,\lvert T_s - T_\infty \rvert\, d^3}{\nu\,\alpha} \end{equation}

Note: This formula uses gravitational acceleration g = 32.174 ft/s² (U.S. customary). The thermal expansion coefficient \(\beta\) is approximated as \(1 / T_\infty\) for an ideal gas.

Symbols

\(\mathrm{Ra}\)	Rayleigh number \([\unit{ \unitless}]\)
\(g\)	Gravitational acceleration (32.174) \([\unit{ \foot\per\second\squared}]\)
\(\beta\)	Volumetric thermal expansion coefficient of air (1/ \(T_\infty\)) \([\unit{ \per\degreeRankine}]\)
\(T_s\)	Outer surface temperature (jacket or bare tank wall) \([\unit{ \degreeRankine}]\)
\(T_\infty\)	Ambient air temperature \([\unit{ \degreeRankine}]\)
\(d\)	Tank outer diameter \([\unit{ \foot}]\)
\(\nu\)	Kinematic viscosity of air at ambient temperature \([\unit{ \foot\squared\per\second}]\)
\(\alpha\)	Thermal diffusivity of air at ambient temperature \([\unit{ \foot\squared\per\second}]\)

Natural Convection Coefficient

Natural convection heat transfer coefficient for a vertical cylinder.

This empirical correlation applies to turbulent natural convection along a vertical surface. It uses the cube-root dependence on Rayleigh number that is characteristic of the turbulent regime.

\begin{equation}\label{eq:insulated-tank-natural-convection} h_{nat} = 0.125 \,\mathrm{Ra}^{1/3} \frac{k}{d} \end{equation}

Symbols

\(h_{nat}\)	Natural convection heat transfer coefficient \([\unit{ \btu\per\hour\foot\squared\degreeRankine}]\)
\(\mathrm{Ra}\)	Rayleigh number \([\unit{ \unitless}]\)
\(k\)	Thermal conductivity of air at ambient temperature \([\unit{ \btu\per\hour\foot\degreeFahrenheit}]\)
\(d\)	Tank outer diameter \([\unit{ \foot}]\)

Insulated Tank — Overall Heat Transfer Coefficient

Overall heat transfer coefficient for an insulated cylindrical tank.

Heat flows in series through four resistances: the inner convective resistance (tank contents to tank wall), the conductive resistance of the tank wall, the conductive resistance of the insulation annulus, and the outer air-film resistance (natural convection at the jacket outer surface). Each conductive resistance is expressed per unit tank-diameter basis, accounting for the cylindrical geometry.

\begin{equation}\label{eq:insulated-tank-insulated-coeff} U_{ins} = \frac{1} {\dfrac{r_i}{r_{o,ins}}\cdot\dfrac{1}{h_{nat}} + \dfrac{r_i}{k_{ins}}\ln\!\left(\dfrac{r_{o,ins}}{r_{o,tank}}\right) + \dfrac{r_i}{k_{tank}}\ln\!\left(\dfrac{r_{o,tank}}{r_i}\right) + \dfrac{1}{h_{inner}}} \end{equation}

Note: The inner convection coefficient \(h_{inner}\) = 10 BTU/(hr·ft²·°F) is a standard assumption for natural convection of process fluid inside the tank.

Symbols

\(U_{ins}\)	Overall heat transfer coefficient (insulated) \([\unit{ \btu\per\hour\foot\squared\degreeFahrenheit}]\)
\(r_i\)	Tank inner radius ( \(d/2\)) \([\unit{ \foot}]\)
\(r_{o,tank}\)	Tank outer radius ( \(r_i + t_{tank}\)) \([\unit{ \foot}]\)
\(r_{o,ins}\)	Insulation outer radius ( \(r_{o,tank} + t_{ins}\)) \([\unit{ \foot}]\)
\(h_{nat}\)	Natural convection coefficient at jacket outer surface \([\unit{ \btu\per\hour\foot\squared\degreeFahrenheit}]\)
\(k_{ins}\)	Thermal conductivity of insulation \([\unit{ \btu\per\hour\foot\degreeFahrenheit}]\)
\(k_{tank}\)	Thermal conductivity of tank wall \([\unit{ \btu\per\hour\foot\degreeFahrenheit}]\)
\(h_{inner}\)	Inner surface convection coefficient (10 BTU/(hr·ft²·°F)) \([\unit{ \btu\per\hour\foot\squared\degreeFahrenheit}]\)

Bare Tank — Overall Heat Transfer Coefficient

Overall heat transfer coefficient for a bare (uninsulated) cylindrical tank.

Without insulation, heat flows through two resistances: the inner convective resistance and the tank wall conductive resistance. The outer air-film resistance acts directly on the tank wall outer surface.

\begin{equation}\label{eq:insulated-tank-bare-coeff} U_{bare} = \frac{1} {\dfrac{r_i}{r_{o,tank}}\cdot\dfrac{1}{h_{nat}} + \dfrac{r_i}{k_{tank}}\ln\!\left(\dfrac{r_{o,tank}}{r_i}\right) + \dfrac{1}{h_{inner}}} \end{equation}

Symbols

\(U_{bare}\)	Overall heat transfer coefficient (bare) \([\unit{ \btu\per\hour\foot\squared\degreeFahrenheit}]\)
\(r_i\)	Tank inner radius ( \(d/2\)) \([\unit{ \foot}]\)
\(r_{o,tank}\)	Tank outer radius ( \(r_i + t_{tank}\)) \([\unit{ \foot}]\)
\(h_{nat}\)	Natural convection coefficient at tank outer surface \([\unit{ \btu\per\hour\foot\squared\degreeFahrenheit}]\)
\(k_{tank}\)	Thermal conductivity of tank wall \([\unit{ \btu\per\hour\foot\degreeFahrenheit}]\)
\(h_{inner}\)	Inner surface convection coefficient (10 BTU/(hr·ft²·°F)) \([\unit{ \btu\per\hour\foot\squared\degreeFahrenheit}]\)

Convective–Conductive Heat Loss

Total convective and conductive heat loss through the tank system.

The overall coefficient, lateral tank surface area, and temperature difference are combined to produce the convective–conductive heat loss. The result is scaled by 10⁻⁵ to produce the output unit.

\begin{equation}\label{eq:insulated-tank-conv-cond} q_{cc} = U \cdot A \cdot \Delta T \end{equation}

Symbols

\(q_{cc}\)	Convective–conductive heat loss \([\unit{ \btu\per\hour}]\)
\(U\)	Overall heat transfer coefficient ( \(U_{ins}\) or \(U_{bare}\)) \([\unit{ \btu\per\hour\foot\squared\degreeFahrenheit}]\)
\(A\)	Lateral tank surface area ( \(\pi,d,H\)) \([\unit{ \foot\squared}]\)
\(\Delta T\)	Temperature difference (surface – ambient) \([\unit{ \degreeRankine}]\)

Radiative Heat Loss

Radiation heat loss from the tank outer surface using the Stefan–Boltzmann law.

Computes the radiated heat flux using the U.S. customary Stefan–Boltzmann constant and the emissivity of the outermost surface (jacket or bare tank wall).

\begin{equation}\label{eq:insulated-tank-radiation} q_{rad} = \sigma\,\varepsilon\,A\,(T_s^4 - T_\infty^4) \end{equation}

Symbols

\(q_{rad}\)	Radiative heat loss \([\unit{ \btu\per\hour}]\)
\(A\)	Lateral tank surface area ( \(\pi,d,H\)) \([\unit{ \foot\squared}]\)
\(\sigma\)	Stefan–Boltzmann constant (1.713441 × 10⁻⁹) \([\unit{ \btu\per\hour\foot\squared\degreeRankine\tothe{4}}]\)
\(\varepsilon\)	Surface emissivity (jacket or bare tank) \([\unit{ \unitless}]\)
\(T_s\)	Outer surface temperature (jacket or bare tank wall) \([\unit{ \degreeRankine}]\)
\(T_\infty\)	Ambient air temperature \([\unit{ \degreeRankine}]\)

Annual Heat Loss

Annual heat loss for the tank, adjusted for system efficiency.

The total instantaneous heat loss is multiplied by annual operating hours and divided by a factor of 10 and the heating system efficiency to convert to the annual energy equivalent the heating system must supply.

\begin{equation}\label{eq:insulated-tank-annual} Q_{annual} = \frac{q_{total} \cdot t_{op}}{\eta} \end{equation}

Symbols

\(Q_{annual}\)	Annual heat loss (efficiency-adjusted) \([\unit{ \btu}]\)
\(q_{total}\)	Total instantaneous heat loss ( \(q_{cc} + q_{rad}\)) \([\unit{ \btu\per\hour}]\)
\(t_{op}\)	Annual operating hours \([\unit{ \hour\per\year}]\)
\(\eta\)	Heating system efficiency (0–1) \([\unit{ \unitless}]\)

Modules
	Air Property Polynomial Fits
	Fourth-order polynomial fits for air properties as a function of temperature.

	Rayleigh Number Formula
	Rayleigh number for natural convection along a vertical cylinder in U.S. customary units.

	Natural Convection Coefficient Formula
	Natural convection heat transfer coefficient for a vertical cylinder.

	Insulated Tank — Overall Heat Transfer Coefficient
	Overall heat transfer coefficient for an insulated cylindrical tank.

	Bare Tank — Overall Heat Transfer Coefficient
	Overall heat transfer coefficient for a bare (uninsulated) cylindrical tank.

	Convective–Conductive Heat Loss
	Total convective and conductive heat loss through the tank system.

	Radiative Heat Loss
	Radiation heat loss from the tank outer surface using the Stefan–Boltzmann law.

	Annual Heat Loss Formula
	Annual heat loss for the tank, adjusted for system efficiency.

Files
file	insulated_tank_reduction.h
	Declares structs and functions for the Insulated Tank Reduction Calculator.Calculates heat loss and annual energy cost for a hot cylindrical tank, with or without thermal insulation.

Namespaces
namespace	insulated_tank_reduction
	Insulated tank heat loss calculations for treasure hunt measures.

Classes
struct	insulated_tank_reduction::InsulatedTankInput
	Input parameters for the insulated tank heat loss calculation. More...

struct	insulated_tank_reduction::InsulatedTankOutput
	Output results for the insulated tank heat loss calculation. More...

Functions
InsulatedTankOutput	insulated_tank_reduction::calculate (const InsulatedTankInput &input)
	Calculates heat loss and annual heat loss for a tank.

InsulatedTankOutput	insulated_tank_reduction::insulatedTankHeatLoss (const InsulatedTankInput &input)
	Calculates heat loss for a tank wrapped in insulation.

InsulatedTankOutput	insulated_tank_reduction::bareTankHeatLoss (const InsulatedTankInput &input)
	Calculates heat loss for an uninsulated (bare) tank.

double	insulated_tank_reduction::rayleighNumber (double thermal_expansion, double surface_temperature, double ambient_temperature, double diameter, double kin_viscosity, double thermal_diffusivity)
	Computes the Rayleigh number for natural convection along a vertical cylinder in U.S. customary units.

double	insulated_tank_reduction::naturalConvectionCoefficient (double rayleigh, double conductivity, double diameter)
	Computes the natural convection heat transfer coefficient for a vertical cylinder using the Ra^(1/3) empirical correlation.

Function Documentation

◆ bareTankHeatLoss()

InsulatedTankOutput insulated_tank_reduction::bareTankHeatLoss ( const InsulatedTankInput & input )

Air properties are evaluated at the ambient temperature. The tank fluid temperature is used as the surface temperature driving the Rayleigh number and radiation term. The resistance model includes only the tank wall and the outer air film.

See also: Bare Tank — Overall Heat Transfer Coefficient

Parameters

[in] input InsulatedTankInput describing the bare tank system (insulation_thickness and insulation_conductivity are ignored).

Returns: InsulatedTankOutput with heat loss and annual heat loss.

◆ calculate()

InsulatedTankOutput insulated_tank_reduction::calculate ( const InsulatedTankInput & input )

Dispatches to insulatedTankHeatLoss or bareTankHeatLoss based on whether insulation_thickness is positive.

See also: Insulated Tank Reduction Calculator

Parameters

[in] input InsulatedTankInput with tank geometry, material properties, operating conditions, and insulation parameters.

Returns: InsulatedTankOutput with heat loss and annual heat loss.

◆ insulatedTankHeatLoss()

InsulatedTankOutput insulated_tank_reduction::insulatedTankHeatLoss ( const InsulatedTankInput & input )

Air properties are evaluated at the ambient temperature. The outer surface temperature (insulation jacket) drives the Rayleigh number and radiation term. A cylindrical multi-layer resistance model (tank wall + insulation annulus + outer air film) yields the overall heat transfer coefficient, which is combined with a radiation term to produce total heat loss.

See also: Insulated Tank — Overall Heat Transfer Coefficient

Parameters

[in] input InsulatedTankInput describing the insulated tank system.

Returns: InsulatedTankOutput with heat loss and annual heat loss.

◆ naturalConvectionCoefficient()

double insulated_tank_reduction::naturalConvectionCoefficient	(	double	rayleigh,
		double	conductivity,
		double	diameter
	)

Applies h = 0.125 × Ra^(1/3) × k / d, valid for turbulent natural convection along a vertical surface.

See also: Natural Convection Coefficient Formula

Parameters

[in]	rayleigh	Rayleigh number \([\unit{\unitless}]\)
[in]	conductivity	Thermal conductivity of air \([\unit{\btu\per\hour\foot\degreeFahrenheit}]\)
[in]	diameter	Tank outer diameter \([\unit{\foot}]\)

Returns: Natural convection heat transfer coefficient \([\unit{\btu\per\hour\foot\squared\degreeFahrenheit}]\)

◆ rayleighNumber()

double insulated_tank_reduction::rayleighNumber	(	double	thermal_expansion,
		double	surface_temperature,
		double	ambient_temperature,
		double	diameter,
		double	kin_viscosity,
		double	thermal_diffusivity
	)

Uses gravitational acceleration g = 32.174 ft/s².

See also: Rayleigh Number Formula

Parameters

[in]	thermal_expansion	Volumetric thermal expansion coefficient of air (1/T_ambient) \([\unit{\per\degreeRankine}]\)
[in]	surface_temperature	Outer surface temperature \([\unit{\degreeRankine}]\)
[in]	ambient_temperature	Ambient air temperature \([\unit{\degreeRankine}]\)
[in]	diameter	Tank inner diameter \([\unit{\foot}]\)
[in]	kin_viscosity	Kinematic viscosity of air \([\unit{\foot\squared\per\second}]\)
[in]	thermal_diffusivity	Thermal diffusivity of air \([\unit{\foot\squared\per\second}]\)

Returns: Dimensionless Rayleigh number \([\unit{\unitless}]\)

Detailed Description

Air Property Polynomial Fits

Rayleigh Number (U.S. Customary)

Natural Convection Coefficient

Insulated Tank — Overall Heat Transfer Coefficient

Bare Tank — Overall Heat Transfer Coefficient

Convective–Conductive Heat Loss

Radiative Heat Loss

Annual Heat Loss

Modules

Files

Namespaces

Classes

Functions

Function Documentation

◆ bareTankHeatLoss()

◆ calculate()

◆ insulatedTankHeatLoss()

◆ naturalConvectionCoefficient()

◆ rayleighNumber()