Heat Transfer Models

The HeatTransferModels namespace provides various model functions relevant to heat transfer.

Cylindrical Thermal Conductance

The function cylindricalThermalConductance computes the thermal conductance across a cylindrical medium using a steady thermal resistance analysis (Incropera and DeWitt, 2002):

(1)var element = document.getElementById("moose-equation-26024697-c67a-4d6c-93f7-6649291d34db");katex.render("\\mathcal{H} = \\frac{k}{\\bar{r} \\ln(r_o / r_i)} \\,,", element, {displayMode:true,throwOnError:false});

where

$var element = document.getElementById("moose-equation-ee062eba-787b-4b77-b7c6-382009f3edcd");katex.render("r_i", element, {displayMode:false,throwOnError:false});$ is the inner surface radius,
$var element = document.getElementById("moose-equation-95cdfcc2-d342-476a-84f8-f39dedb2a3cc");katex.render("r_o", element, {displayMode:false,throwOnError:false});$ is the outer surface radius,
$var element = document.getElementById("moose-equation-821d2c14-fcf1-4bac-8068-499aea8ed53d");katex.render("\\bar{r}", element, {displayMode:false,throwOnError:false});$ is the arithmetic mean radius, and
$var element = document.getElementById("moose-equation-0ede6afc-80cf-42db-9305-72e6df592f47");katex.render("k", element, {displayMode:false,throwOnError:false});$ is the medium thermal conductivity.

Cylindrical Gap Conduction Heat Flux

The function cylindricalGapConductionHeatFlux computes the heat flux $var element = document.getElementById("moose-equation-8e9110fb-1237-4d48-b10a-299ae6df46ff");katex.render("q", element, {displayMode:false,throwOnError:false});$ at a point across a cylindrical gap due to conduction, using the thermal conductance given in Cylindrical Thermal Conductance:

var element = document.getElementById("moose-equation-59d73138-b641-4c8e-a96d-6d840c207acb");katex.render("q = \\mathcal{H} (T_i - T_o) \\,,", element, {displayMode:true,throwOnError:false});

where

$var element = document.getElementById("moose-equation-96097b73-0bb8-4e03-96b0-247adb53a8c9");katex.render("T_i", element, {displayMode:false,throwOnError:false});$ is the inner surface temperature,
$var element = document.getElementById("moose-equation-a0944f1f-b88e-47b9-9510-904d237843a2");katex.render("T_o", element, {displayMode:false,throwOnError:false});$ is the outer surface temperature, and
$var element = document.getElementById("moose-equation-572bf00b-c716-4cc3-9b1c-162a5d7eac76");katex.render("\\mathcal{H}", element, {displayMode:false,throwOnError:false});$ is computed from Eq. (1).

Note that the convention here is that a positive heat flux corresponds to heat moving from the inner surface to the outer surface.

Cylindrical Gap Radiation Heat Flux

The function cylindricalGapRadiationHeatFlux computes heat flux $var element = document.getElementById("moose-equation-2cd3e6ad-d796-492b-9787-751c59b03ffa");katex.render("q", element, {displayMode:false,throwOnError:false});$ at a point across a cylindrical gap due to radiation, assuming opaque, gray, diffuse surfaces with infinitely long, concentric cylinders (Incropera and DeWitt, 2002):

var element = document.getElementById("moose-equation-fe082b24-9ee5-4679-a8aa-5b01e9cd5c6d");katex.render("q = \\frac{\\sigma (T_i^4 - T_o^4)}{\\mathcal{R}} \\,,", element, {displayMode:true,throwOnError:false});

var element = document.getElementById("moose-equation-679056ea-ad2f-4ac9-b1d4-d79a02f3c49e");katex.render("\\mathcal{R} = \\frac{1}{\\epsilon_i} + \\frac{r_i}{r_o} \\left( \\frac{1 - \\epsilon_o}{\\epsilon_o} \\right) \\,,", element, {displayMode:true,throwOnError:false});

where $var element = document.getElementById("moose-equation-0ab6a07a-2f2f-4b0a-879a-0b257080c6bc");katex.render("\\sigma", element, {displayMode:false,throwOnError:false});$ is the Stefan-Boltzmann constant.

Note that the convention here is that a positive heat flux corresponds to heat moving from the inner surface to the outer surface.

References

F. P. Incropera and D. P. DeWitt. Fundamentals of Heat and Mass Transfer. John Wiley & Sons, 5th edition, 2002.

@book{Incropera2002,
    author = "Incropera, F. P. and DeWitt, D. P.",
    city = "New York",
    edition = "5th",
    publisher = "John Wiley \& Sons",
    title = "Fundamentals of Heat and Mass Transfer",
    year = "2002"
}

Cylindrical Thermal Conductance
Cylindrical Gap Conduction Heat Flux
Cylindrical Gap Radiation Heat Flux
References