{"id":3230,"date":"2016-03-22T21:08:30","date_gmt":"2016-03-22T21:08:30","guid":{"rendered":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/?p=3230"},"modified":"2016-03-22T21:08:30","modified_gmt":"2016-03-22T21:08:30","slug":"forced-convectionlaminar-flow-in-tubes","status":"publish","type":"post","link":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/forced-convectionlaminar-flow-in-tubes\/","title":{"rendered":"FORCED CONVECTION:LAMINAR FLOW IN TUBES"},"content":{"rendered":"<div class=\"jbvoh6a0dae04a8b33\" ><script type=\"text\/javascript\">\n\tatOptions = {\n\t\t'key' : '61e5902552e2353963d8d2f1bd1f4a8f',\n\t\t'format' : 'iframe',\n\t\t'height' : 250,\n\t\t'width' : 300,\n\t\t'params' : {}\n\t};\n<\/script>\n<script type=\"text\/javascript\" src=\"\/\/www.highperformanceformat.com\/61e5902552e2353963d8d2f1bd1f4a8f\/invoke.js\"><\/script><\/div><style type=\"text\/css\">\r\n@media screen and (min-width: 1201px) {\r\n.jbvoh6a0dae04a8b33 {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (min-width: 993px) and (max-width: 1200px) {\r\n.jbvoh6a0dae04a8b33 {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (min-width: 769px) and (max-width: 992px) {\r\n.jbvoh6a0dae04a8b33 {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (min-width: 768px) and (max-width: 768px) {\r\n.jbvoh6a0dae04a8b33 {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (max-width: 767px) {\r\n.jbvoh6a0dae04a8b33 {\r\ndisplay: block;\r\n}\r\n}\r\n<\/style>\r\n<p align=\"justify\"><font size=\"5\"><b>LAMINA<\/b><b>R FLOW IN TUBES<\/b><\/font>  <\/p>\n<h5 align=\"justify\"><font size=\"5\">Reconsider steady laminar flow of a fluid in a circular tube of radius <i>R. <\/i>The fluid properties r, <i>k<\/i><i>, <\/i>and <i>C<\/i><i>p <\/i>are constant, and the work done by viscous stresses is negligible. The fluid flows along the <i>x<\/i>-axis with velocity &#8216;V. The <\/font><font size=\"5\">flow is fully developed so that &#8216;V is independent of <i>x <\/i>and thus &#8216;V = &#8216;V(<i>r<\/i>). Noting that energy is transferred by mass in the <i>x<\/i>-direction, and by conduction in the <i>r<\/i>-direction (heat conduction in the <i>x<\/i>-direction is assumed to be negligible), the steady-flow energy balance for a cylindrical shell element of thickness <i>dr <\/i>and length <i>d<\/i><i>x <\/i>can be expressed as (Fig. 19\u201330)<\/font><\/h5>\n<p align=\"justify\"><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/FORCED-CONVECTION-04392.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-right-width: 0px; margin: 0px auto; padding-left: 0px; padding-right: 0px; display: block; float: none; border-top-width: 0px; border-bottom-width: 0px; border-left-width: 0px; padding-top: 0px\" title=\"FORCED CONVECTION-0439\" border=\"0\" alt=\"FORCED CONVECTION-0439\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/FORCED-CONVECTION-0439_thumb2.jpg\" width=\"553\" height=\"289\"><\/a><\/p>\n<p align=\"justify\"><font size=\"5\">which states that the rate of net energy transfer to the control volume by mass flow is equal to the net rate of heat conduction in the radial direction.<\/font><\/p>\n<p align=\"justify\"><font size=\"5\"><b>Constan<\/b><b>t Surface Heat Flux<\/b><\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\">For fully developed flow in a circular pipe subjected to constant surface heat <\/font><font size=\"5\">flux, we have, from Eq. 19\u201350,<\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\"><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/FORCED-CONVECTION-04402.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-right-width: 0px; margin: 0px auto; padding-left: 0px; padding-right: 0px; display: block; float: none; border-top-width: 0px; border-bottom-width: 0px; border-left-width: 0px; padding-top: 0px\" title=\"FORCED CONVECTION-0440\" border=\"0\" alt=\"FORCED CONVECTION-0440\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/FORCED-CONVECTION-0440_thumb2.jpg\" width=\"304\" height=\"38\"><\/a><\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\">If heat conduction in the <i>x<\/i>-direction were considered in the derivation of Eq. 19\u201364, it would give an additional term aa2<i>T<\/i>\/a<i>x<\/i>2, which would be equal to zero since a<i>T<\/i>\/a<i>x <\/i>= constant and thus <i>T <\/i>= <i>T<\/i>(<i>r<\/i>). Therefore, the assumption that there is no axial heat conduction is satisfied exactly in this case.<\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\">Substituting Eq. 19\u201365 and the relation for velocity profile (Chap. 14) into Eq. 19\u201364 gives<\/font><\/p>\n<p align=\"justify\"><font size=\"5\"><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/FORCED-CONVECTION-04412.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-right-width: 0px; margin: 0px auto; padding-left: 0px; padding-right: 0px; display: block; float: none; border-top-width: 0px; border-bottom-width: 0px; border-left-width: 0px; padding-top: 0px\" title=\"FORCED CONVECTION-0441\" border=\"0\" alt=\"FORCED CONVECTION-0441\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/FORCED-CONVECTION-0441_thumb2.jpg\" width=\"347\" height=\"111\"><\/a><\/font><\/p>\n<p align=\"justify\"><font size=\"5\">The desired solution to the problem is obtained by applying the boundary conditions a<i>T<\/i>\/a<i>x <\/i>= 0 at <i>r <\/i>= 0 (because of symmetry) and <i>T <\/i>= <i>T<\/i><i>s <\/i>at <i>r <\/i>= <i>R.<\/i><i> <\/i>We get<\/font><\/p>\n<p align=\"justify\"><font size=\"5\"><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/FORCED-CONVECTION-04422.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-right-width: 0px; margin: 0px auto; padding-left: 0px; padding-right: 0px; display: block; float: none; border-top-width: 0px; border-bottom-width: 0px; border-left-width: 0px; padding-top: 0px\" title=\"FORCED CONVECTION-0442\" border=\"0\" alt=\"FORCED CONVECTION-0442\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/FORCED-CONVECTION-0442_thumb2.jpg\" width=\"401\" height=\"231\"><\/a><\/font><\/p>\n<h5 align=\"justify\"><font size=\"5\">Therefore, for fully developed laminar flow in a circular tube subjected to constant surface heat flux, the Nusselt number is a constant. There is no dependence on the Reynolds or the Prandtl numbers.<\/font><\/h5>\n<p align=\"justify\"><font size=\"5\"><b>Constan<\/b><b>t Surface Temperature<\/b><\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\">A similar analysis can be performed for fully developed laminar flow in a cir<\/font><font size=\"5\">cular tube for the case of constant surface temperature <i>T<\/i><i>s<\/i><i>. <\/i>The solution procedure in this case is more complex as it requires iterations, but the Nusselt number relation obtained is equally simple (Fig. 19\u201331):<\/font>  <\/p><div class=\"ioogc6a0dae04a8de3\" ><script type=\"text\/javascript\">\n\tatOptions = {\n\t\t'key' : '0c1eb4c533eaedb7b996f49a5a4983a9',\n\t\t'format' : 'iframe',\n\t\t'height' : 300,\n\t\t'width' : 160,\n\t\t'params' : {}\n\t};\n<\/script>\n<script type=\"text\/javascript\" src=\"\/\/www.highperformanceformat.com\/0c1eb4c533eaedb7b996f49a5a4983a9\/invoke.js\"><\/script><\/div><style type=\"text\/css\">\r\n@media screen and (min-width: 1201px) {\r\n.ioogc6a0dae04a8de3 {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (min-width: 993px) and (max-width: 1200px) {\r\n.ioogc6a0dae04a8de3 {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (min-width: 769px) and (max-width: 992px) {\r\n.ioogc6a0dae04a8de3 {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (min-width: 768px) and (max-width: 768px) {\r\n.ioogc6a0dae04a8de3 {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (max-width: 767px) {\r\n.ioogc6a0dae04a8de3 {\r\ndisplay: block;\r\n}\r\n}\r\n<\/style>\r\n<div class=\"flujv6a0dae04a8c9d\" ><script async src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-0778475562755157\"\n     crossorigin=\"anonymous\"><\/script>\n<!-- 300x600 hydraulics-and-pneumatics -->\n<ins class=\"adsbygoogle\"\n     style=\"display:inline-block;width:300px;height:600px\"\n     data-ad-client=\"ca-pub-0778475562755157\"\n     data-ad-slot=\"3735577695\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script><\/div><style type=\"text\/css\">\r\n@media screen and (min-width: 1201px) {\r\n.flujv6a0dae04a8c9d {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (min-width: 993px) and (max-width: 1200px) {\r\n.flujv6a0dae04a8c9d {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (min-width: 769px) and (max-width: 992px) {\r\n.flujv6a0dae04a8c9d {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (min-width: 768px) and (max-width: 768px) {\r\n.flujv6a0dae04a8c9d {\r\ndisplay: block;\r\n}\r\n}\r\n@media screen and (max-width: 767px) {\r\n.flujv6a0dae04a8c9d {\r\ndisplay: block;\r\n}\r\n}\r\n<\/style>\r\n\n<p align=\"justify\"><font size=\"5\"><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/FORCED-CONVECTION-04432.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-right-width: 0px; margin: 0px auto; padding-left: 0px; padding-right: 0px; display: block; float: none; border-top-width: 0px; border-bottom-width: 0px; border-left-width: 0px; padding-top: 0px\" title=\"FORCED CONVECTION-0443\" border=\"0\" alt=\"FORCED CONVECTION-0443\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/FORCED-CONVECTION-0443_thumb2.jpg\" width=\"349\" height=\"37\"><\/a><\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\">The thermal conductivity <i>k <\/i>for use in the Nu relations above should be evalu<\/font><font size=\"5\">ated at the bulk mean fluid temperature, which is the arithmetic average of the mean fluid temperatures at the inlet and the exit of the tube. For laminar flow, the effect of <i>surface roughness <\/i>on the friction factor and the heat transfer co<\/font><font size=\"5\">efficient is negligible.<\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\"><b>Lamina<\/b><b>r Flow in Noncircular Tubes<\/b><\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\">The Nusselt number relations are given in Table 19\u20133 for <i>fully developed lam<\/i><\/font><font size=\"5\"><i>inar flow <\/i>in tubes of various cross sections. The Reynolds and Nusselt numbers for flow in these tubes are based on the hydraulic diameter <i>D<\/i><i>h <\/i>= 4<i>A<\/i><i>c <\/i>\/<i>p, <\/i>where <i>A<\/i><i>c <\/i>is the cross-sectional area of the tube and <i>p <\/i>is its perimeter. Once the Nusselt number is available, the convection heat transfer coefficient is determined from <i>h <\/i>= <i>k<\/i>Nu\/<i>D<\/i><i>h<\/i><i>.<\/i><\/font>  <\/p>\n<p align=\"justify\"><em><font size=\"5\"><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/FORCED-CONVECTION-04442.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-right-width: 0px; margin: 0px auto; padding-left: 0px; padding-right: 0px; display: block; float: none; border-top-width: 0px; border-bottom-width: 0px; border-left-width: 0px; padding-top: 0px\" title=\"FORCED CONVECTION-0444\" border=\"0\" alt=\"FORCED CONVECTION-0444\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/FORCED-CONVECTION-0444_thumb2.jpg\" width=\"176\" height=\"189\"><\/a><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/FORCED-CONVECTION-04452.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-right-width: 0px; margin: 0px auto; padding-left: 0px; padding-right: 0px; display: block; float: none; border-top-width: 0px; border-bottom-width: 0px; border-left-width: 0px; padding-top: 0px\" title=\"FORCED CONVECTION-0445\" border=\"0\" alt=\"FORCED CONVECTION-0445\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/FORCED-CONVECTION-0445_thumb2.jpg\" width=\"375\" height=\"433\"><\/a><\/font><\/em>  <\/p>\n<p align=\"justify\"><font size=\"5\"><strong>Developing Laminar Flow in the Entrance Region<\/strong><\/font>  <\/p>\n<h5 align=\"justify\"><font size=\"5\">For a circular tube of length <i>L <\/i>subjected to constant surface temperature, the <\/font><font size=\"5\">average Nusselt number for the <i>thermal entrance region <\/i>can be determined from (Edwards et al., 1979)<\/font><\/h5>\n<p align=\"justify\"><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/FORCED-CONVECTION-04462.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-right-width: 0px; margin: 0px auto; padding-left: 0px; padding-right: 0px; display: block; float: none; border-top-width: 0px; border-bottom-width: 0px; border-left-width: 0px; padding-top: 0px\" title=\"FORCED CONVECTION-0446\" border=\"0\" alt=\"FORCED CONVECTION-0446\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/FORCED-CONVECTION-0446_thumb2.jpg\" width=\"362\" height=\"38\"><\/a><\/p>\n<p align=\"justify\"><font size=\"5\">Note that the average Nusselt number is larger at the entrance region, as expected, and it approaches asymptotically to the fully developed value of 3.66 as <i>L <\/i>\u00ae x. This relation assumes that the flow is hydrodynamically developed when the fluid enters the heating section, but it can also be used approximately for flow developing hydrodynamically.<\/font><\/p>\n<p align=\"justify\"><font size=\"5\">When the difference between the surface and the fluid temperatures is large, it may be necessary to account for the variation of viscosity with temperature.<\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\">The average Nusselt number for developing laminar flow in a circular tube in that case can be determined from [Sieder and Tate (1936)]<\/font>  <\/p>\n<p align=\"justify\"><font size=\"5\"><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/FORCED-CONVECTION-04472.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-right-width: 0px; margin: 0px auto; padding-left: 0px; padding-right: 0px; display: block; float: none; border-top-width: 0px; border-bottom-width: 0px; border-left-width: 0px; padding-top: 0px\" title=\"FORCED CONVECTION-0447\" border=\"0\" alt=\"FORCED CONVECTION-0447\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/FORCED-CONVECTION-0447_thumb2.jpg\" width=\"416\" height=\"484\"><\/a><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/FORCED-CONVECTION-04481.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-right-width: 0px; margin: 0px auto; padding-left: 0px; padding-right: 0px; display: block; float: none; border-top-width: 0px; border-bottom-width: 0px; border-left-width: 0px; padding-top: 0px\" title=\"FORCED CONVECTION-0448\" border=\"0\" alt=\"FORCED CONVECTION-0448\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/FORCED-CONVECTION-0448_thumb1.jpg\" width=\"290\" height=\"484\"><\/a><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/FORCED-CONVECTION-0449.jpg\"><img decoding=\"async\" loading=\"lazy\" style=\"background-image: none; border-right-width: 0px; margin: 0px auto; padding-left: 0px; padding-right: 0px; display: block; float: none; border-top-width: 0px; border-bottom-width: 0px; border-left-width: 0px; padding-top: 0px\" title=\"FORCED CONVECTION-0449\" border=\"0\" alt=\"FORCED CONVECTION-0449\" src=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-content\/uploads\/2016\/03\/FORCED-CONVECTION-0449_thumb.jpg\" width=\"371\" height=\"219\"><\/a><\/font>  <\/p>\n<p align=\"justify\">\n<p align=\"justify\"><font size=\"5\"><\/font><\/p>\n","protected":false},"excerpt":{"rendered":"<p>LAMINAR FLOW IN TUBES Reconsider steady laminar flow of a fluid in a circular tube of radius R. The fluid properties r, k, and Cp are constant, and the work done by viscous stresses is negligible. The fluid flows along the x-axis with velocity &#8216;V. The flow is fully developed so that &#8216;V is independent [&hellip;]<br \/><a href=\"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/forced-convectionlaminar-flow-in-tubes\/\" class=\"more-link\" >Continue reading&#8230;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[],"_links":{"self":[{"href":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-json\/wp\/v2\/posts\/3230"}],"collection":[{"href":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-json\/wp\/v2\/comments?post=3230"}],"version-history":[{"count":1,"href":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-json\/wp\/v2\/posts\/3230\/revisions"}],"predecessor-version":[{"id":3231,"href":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-json\/wp\/v2\/posts\/3230\/revisions\/3231"}],"wp:attachment":[{"href":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-json\/wp\/v2\/media?parent=3230"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-json\/wp\/v2\/categories?post=3230"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/machineryequipmentonline.com\/hydraulics-and-pneumatics\/wp-json\/wp\/v2\/tags?post=3230"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}