{"id":63,"date":"2022-12-30T18:15:16","date_gmt":"2022-12-30T18:15:16","guid":{"rendered":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/chapter\/nusselt-number\/"},"modified":"2023-01-08T22:59:38","modified_gmt":"2023-01-08T22:59:38","slug":"nusselt-number","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/chapter\/nusselt-number\/","title":{"raw":"5.5 Nusselt Number","rendered":"5.5 Nusselt Number"},"content":{"raw":"
\r\n

The Nusselt number (Nu<\/em>) is equivalent to the Sherwood number but applies to heat transport. It is defined in Equation\u00a031 (Elder, 1967a).<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle Nu=\\frac{Q_{h}H}{k_{T}\\Delta T}[\/latex]<\/td>\r\n(31)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

where:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Q<\/em>h<\/em><\/sub><\/td>\r\n=<\/td>\r\nheat flow across the source boundary (M\/T3<\/sup>), e.g., W\u00a0m\u22122<\/sup><\/td>\r\n<\/tr>\r\n
k<\/em>T<\/em><\/sub><\/td>\r\n=<\/td>\r\nthermal conductivity (M\/(L\u0398)), e.g., W\u00a0m\u22121<\/sup>K\u22121<\/sup><\/td>\r\n<\/tr>\r\n
\u2206T<\/em><\/td>\r\n=<\/td>\r\ntemperature difference over the height H<\/em> (\u0398), e.g., K<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Laboratory and computational experiments have shown that there is a relationship between the Rayleigh number and the Nusselt number (Cheng, 1979). This is illustrated in Figure\u00a017. If the Rayleigh number is below the critical value, there is no convective heat transfer and so Nu<\/em>\u00a0=\u00a01 (conductive heat transfer only). As the Rayleigh number increases, so does the Nusselt number, which means that more heat is being transported convectively. This is because at high Rayleigh numbers, the convective cells become slimmer and thereby more efficient in transferring heat. Modeling by Holzbecher (1996) showed that the exact relationship varies depending on the aspect ratio of the domain as illustrated in Figure\u00a017, that is, the larger the height\/width ratio, the higher the heat transfer at high Rayleigh numbers.<\/a><\/p>\r\n

\"Graph<\/p>\r\n

Figure\u00a0<\/strong>17<\/strong>\u00a0-\u00a0Relationship between the Rayleigh (Ra<\/em>) and Nusselt (Nu<\/em>) numbers originally published by Cheng (1978). The lines represent model-based relationships by Holzbecher (1996) for domains of different height\/width ratios H\/W, as shown schematically to the right of the graph. The critical Rayleigh number for each H\/W value is indicated by an arrow below the horizontal axis. Original figure reprinted from Advances in Heat Transfer, 14, P. Cheng, Heat Transfer in Geothermal Systems, Copyright (1979), with permission from Elsevier. Model data courtesy of Ekkehard Holzbecher.<\/p>\r\n\r\n<\/div>","rendered":"

\n

The Nusselt number (Nu<\/em>) is equivalent to the Sherwood number but applies to heat transport. It is defined in Equation\u00a031 (Elder, 1967a).<\/p>\n\n\n\n
<\/td>\n\"\displaystyle Nu=\frac{Q_{h}H}{k_{T}\Delta T}\"<\/td>\n(31)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n\n\n
Q<\/em>h<\/em><\/sub><\/td>\n=<\/td>\nheat flow across the source boundary (M\/T3<\/sup>), e.g., W\u00a0m\u22122<\/sup><\/td>\n<\/tr>\n
k<\/em>T<\/em><\/sub><\/td>\n=<\/td>\nthermal conductivity (M\/(L\u0398)), e.g., W\u00a0m\u22121<\/sup>K\u22121<\/sup><\/td>\n<\/tr>\n
\u2206T<\/em><\/td>\n=<\/td>\ntemperature difference over the height H<\/em> (\u0398), e.g., K<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Laboratory and computational experiments have shown that there is a relationship between the Rayleigh number and the Nusselt number (Cheng, 1979). This is illustrated in Figure\u00a017. If the Rayleigh number is below the critical value, there is no convective heat transfer and so Nu<\/em>\u00a0=\u00a01 (conductive heat transfer only). As the Rayleigh number increases, so does the Nusselt number, which means that more heat is being transported convectively. This is because at high Rayleigh numbers, the convective cells become slimmer and thereby more efficient in transferring heat. Modeling by Holzbecher (1996) showed that the exact relationship varies depending on the aspect ratio of the domain as illustrated in Figure\u00a017, that is, the larger the height\/width ratio, the higher the heat transfer at high Rayleigh numbers.<\/a><\/p>\n

\"Graph<\/p>\n

Figure\u00a0<\/strong>17<\/strong>\u00a0–\u00a0Relationship between the Rayleigh (Ra<\/em>) and Nusselt (Nu<\/em>) numbers originally published by Cheng (1978). The lines represent model-based relationships by Holzbecher (1996) for domains of different height\/width ratios H\/W, as shown schematically to the right of the graph. The critical Rayleigh number for each H\/W value is indicated by an arrow below the horizontal axis. Original figure reprinted from Advances in Heat Transfer, 14, P. Cheng, Heat Transfer in Geothermal Systems, Copyright (1979), with permission from Elsevier. Model data courtesy of Ekkehard Holzbecher.<\/p>\n<\/div>\n","protected":false},"author":1,"menu_order":10,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-63","chapter","type-chapter","status-publish","hentry"],"part":131,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/63","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":3,"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/63\/revisions"}],"predecessor-version":[{"id":355,"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/63\/revisions\/355"}],"part":[{"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/parts\/131"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/63\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/wp\/v2\/media?parent=63"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/chapter-type?post=63"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/wp\/v2\/contributor?post=63"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/wp\/v2\/license?post=63"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}