{"id":682,"date":"2017-12-20T19:05:24","date_gmt":"2017-12-20T19:05:24","guid":{"rendered":"http:\/\/box5393.temp.domains\/~quranive\/\/?p=682"},"modified":"2022-01-13T17:23:38","modified_gmt":"2022-01-13T17:23:38","slug":"quick-calculations-for-radiation-heat-transfer","status":"publish","type":"post","link":"https:\/\/mechanicsandmachines.com\/?p=682","title":{"rendered":"Quick calculations for radiation heat transfer"},"content":{"rendered":"

In certain cases, radiation heat transfer is important to include in one’s calculations.  Radiation heat transfer is nonlinear because the heat flux is proportional to the temperature to the fourth power.  I find two frequent simplifications for radiation problems quite useful.  The first approximation is to linearize and create an effective convection coefficient<\/em><\/strong>.  The second approximation is the effective emissivity<\/em><\/strong> for two bodies which are transferring heat between each other via radiation.<\/p>\n

<\/p>\n

Linearization of radiation heat transfer for effective convection coefficient<\/h6>\n

Heat transfer by radiation is nominally given by<\/p>\n

  <\/span>   <\/span>\"\[<\/p>\n

\"\dot{q}\" is the heat flux (W\/m\"^2\" in SI units), \"\sigma\" is the Stefan-Boltzmann constant (5.670 x 10\"^-8\" W\/m\"^2\"K\"^4\"), and \"\epsilon\" is the emissivity, and \"T\" and\"T_0\" are the absolute temperatures<\/em> of the surfaces.  Note: Another consideration that one must also include is the view factor which describes how well the two bodies are facing each other, but we will discuss that in a future post.<\/p>\n

When the temperatures \"T\" and \"T_0\" are not too far apart, then we can linearize the nonlinear relation above to obtain a linear convection relation of the form<\/p>\n

  <\/span>   <\/span>\"\[<\/p>\n

where \"h\" is the convection coefficient (W\/m\"^2\"K in SI units) to obtain<\/p>\n

  <\/span>   <\/span>\"\[<\/p>\n

Therefore, the effective heat transfer coefficient is<\/p>\n

  <\/span>   <\/span>\"\[<\/p>\n

where \"T_{char}\" is a characteristic temperature.  As the characteristic temperature, one may use the nominal value \"T\", \"T_0\", or some value in between, such as the mean.  Depending on the problem one may choose the value based in order to make a conservative approximation.<\/p>\n

This relation can be derived in two ways.  First, one can factor the original relation into<\/p>\n

  <\/span>   <\/span>\"\[<\/p>\n

If one defines that mean temperature as \"T_{mean}=(T+T_0)/2\" and assumes that T and T_0 are not too far apart, then \"T^2+T_0^2 \approx 2T_{mean}^2\" and one finds<\/p>\n

  <\/span>   <\/span>\"\[<\/p>\n

A second way involves using a Taylor series expansion with \"T=T_0+\delta T\".  Then the heat flux is given by<\/p>\n

  <\/span>   <\/span>\"\[<\/p>\n

Expanding and dropping higher order terms of \"\delta T^2\" and above, we obtain<\/p>\n

  <\/span>   <\/span>\"\[<\/p>\n

Above,  I wrote that a few different characteristic temperatures can be used.  Those can be obtained here depending on the expansion.  One could write \"T=T_{mean}+\delta T/2\" and \"T_0=T_{mean}-\delta_T/2\" for example.<\/p>\n

Effective emissivity between two surfaces<\/h6>\n

Suppose we have two bodies with emissivities \"\epsilon_1\" and \"\epsilon_2\" at temperatures \"T_1\" and \"T_2\"  transferring heat between each other as sketched below.  Each body emits radiation as \"\dot{q}_i=\sigma \epsilon_i T_i^4\" but the net heat transfer is not the simple difference<\/em> because the incoming and outgoing radiation between the two bodies are coupled.  <\/em>The net heat transfer is given by<\/p>\n

  <\/span>   <\/span>\"\[<\/p>\n

Therefore, the effective emissivity is<\/p>\n

  <\/span>   <\/span>\"\[<\/p>\n

which can be used with the relation<\/p>\n

  <\/span>   <\/span>\"\[<\/p>\n

One can see that if either \"\epsilon_1=1\" or \"\epsilon_2=1\" or both, then the more familiar relation for heat transfer is obtained.<\/p>\n

\"\"<\/a><\/p>\n

One can find a full derivation of this relation via this link.<\/a><\/p>\n

 <\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

In certain cases, radiation heat transfer is important to include in one’s calculations.  Radiation heat transfer is nonlinear because the heat flux is proportional to the temperature to the fourth power.  I find two frequent simplifications for radiation problems quite useful.  The first approximation is to linearize and create an effective convection coefficient.  The second … Continue reading Quick calculations for radiation heat transfer<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[8],"tags":[53,68,67,66,69,65],"series":[],"class_list":["post-682","post","type-post","status-publish","format-standard","hentry","category-fundamentals","tag-boundary-conditions","tag-convection-coefficient","tag-efffective-emissivity","tag-heat-transfer","tag-linearized-radiation","tag-radiation"],"aioseo_notices":[],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p5f9h7-b0","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mechanicsandmachines.com\/index.php?rest_route=\/wp\/v2\/posts\/682","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mechanicsandmachines.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mechanicsandmachines.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mechanicsandmachines.com\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/mechanicsandmachines.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=682"}],"version-history":[{"count":18,"href":"https:\/\/mechanicsandmachines.com\/index.php?rest_route=\/wp\/v2\/posts\/682\/revisions"}],"predecessor-version":[{"id":742,"href":"https:\/\/mechanicsandmachines.com\/index.php?rest_route=\/wp\/v2\/posts\/682\/revisions\/742"}],"wp:attachment":[{"href":"https:\/\/mechanicsandmachines.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=682"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mechanicsandmachines.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=682"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mechanicsandmachines.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=682"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/mechanicsandmachines.com\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=682"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}