{"id":20435,"date":"2018-12-02T13:25:49","date_gmt":"2018-12-02T13:25:49","guid":{"rendered":"http:\/\/sitepourvtc.com\/?page_id=20435"},"modified":"2023-02-15T08:08:39","modified_gmt":"2023-02-15T08:08:39","slug":"what-is-nusselt-number","status":"publish","type":"page","link":"https:\/\/sitepourvtc.com\/nuclear-engineering\/heat-transfer\/introduction-to-heat-transfer\/characteristic-numbers\/what-is-nusselt-number\/","title":{"rendered":"What is Nusselt Number"},"content":{"rendered":"
The Nusselt number<\/strong> is a dimensionless number closely related to the\u00a0P\u00e9clet number<\/a>. Both\u00a0numbers are used to describe the ratio of the thermal energy convected<\/strong> to the fluid to the thermal energy conducted<\/strong> within the fluid. Nusselt number<\/strong> is equal to the dimensionless temperature gradient<\/strong> at the surface, and it provides a measure of the convection heat transfer occurring at the surface.<\/div><\/div>\n

The Nusselt number<\/strong> is named after a German engineer Wilhelm Nusselt. The conductive component is measured under the same conditions as the heat convection but with stagnant fluid. The Nusselt number<\/strong> is to the thermal boundary layer what the friction coefficient is to the velocity boundary layer. Thus, the Nusselt number is defined as:<\/p>\n

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

where:<\/p>\n

k<\/em><\/strong>f<\/sub><\/em><\/strong> is a thermal conductivity<\/strong><\/a> of the fluid [W\/m.K]\n

L <\/em><\/strong>is the characteristic length<\/strong><\/p>\n

h <\/em><\/strong>is the convective heat transfer coefficient<\/strong> [W\/m2<\/sup>.K]\n

For illustration, consider a fluid layer of thickness L<\/strong> and temperature difference \u0394T<\/strong>. Heat transfer through the fluid layer will be by convection when the fluid involves some motion and conduction when the fluid layer is motionless.<\/p>\n

In the case of conduction<\/a>, the heat flux <\/a>can be calculated using Fourier\u2019s law of conduction<\/a>. In the case of convection, the heat flux can be calculated using Newton\u2019s law of cooling. Taking their ratio gives:<\/p>\n

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

The preceding equation defines the Nusselt number<\/strong>. Therefore, the Nusselt number<\/strong> represents the enhancement of heat transfer through a fluid layer due to convection relative to conduction <\/strong>across the same fluid layer. A Nusselt number<\/strong> of Nu=1<\/strong> for a fluid layer represents heat transfer across the layer by pure conduction<\/strong>. The larger the Nusselt number<\/strong>, the more effective the convection. A larger Nusselt number corresponds to more effective convection, with turbulent flow typically in the 100\u20131000 range. The Nusselt number<\/strong> is usually a function of the Reynolds number<\/a> and the Prandtl number<\/a> for turbulent flow.<\/p>\n

\n

Correlations – Single-phase Fluid Flow<\/h2>\n

External Laminar Flow<\/h3>\n

The average Nusselt number<\/strong> <\/a>over the entire plate is determined by:<\/p>\n

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

This relation gives the average heat transfer coefficient <\/a><\/strong>for the entire plate when the flow is laminar<\/a> over the entire plate.<\/p>\n

Internal Laminar Flow<\/h3>\n

Constant Surface Temperature<\/strong><\/p>\n

In laminar flow<\/strong><\/a> in a tube with constant surface temperature, both the friction factor and the heat transfer coefficient<\/strong><\/a> remain constant in the fully developed region.<\/p>\n

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

Constant Surface Heat Flux<\/strong><\/p>\n

Therefore, for fully developed laminar flow<\/strong> in a circular tube subjected to constant surface heat flux<\/a>, the Nusselt number is a constant. There is no dependence on the Reynolds<\/a> or the Prandtl numbers<\/a>.<\/p>\n

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

External Turbulent Flow<\/h3>\n

The average Nusselt number<\/strong> <\/a>over the entire plate is determined by:<\/p>\n

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

This relation gives the average heat transfer coefficient<\/strong><\/a> for the entire plate only when the flow is turbulent<\/strong> <\/a>over the entire plate or when the laminar flow<\/a> region of the plate is too small relative to the turbulent flow region.<\/p>\n

Internal Turbulent Flow – Dittus-Boelter<\/h3>\n

See also: Dittus-Boelter Equation<\/a><\/p>\n

For fully developed (hydrodynamically and thermally)\u00a0turbulent flow<\/a>\u00a0in a smooth circular tube, the local\u00a0Nusselt number<\/a>\u00a0may be obtained from the well-known\u00a0Dittus-Boelter equation<\/strong>. The Dittus-Boelter equation<\/strong>\u00a0is easy to solve but is less accurate when there is a large temperature difference across the\u00a0fluid<\/a> and is less accurate for rough tubes (many commercial applications) since it is tailored to smooth tubes.<\/p>\n

\"Dittus-Boelter<\/a><\/p>\n

The\u00a0Dittus-Boelter correlation<\/strong><\/a>\u00a0may be used for small to moderate temperature differences, Twall<\/sub>\u00a0\u2013 Tavg<\/sub>, with all properties evaluated at an, averaged temperature Tavg<\/sub>.<\/p>\n

For flows characterized by large property variations, the corrections (e.g., a viscosity correction factor\u00a0\u03bc\/\u03bc<\/strong>wall<\/sub><\/strong>) must be taken into account, for example, as Sieder and Tate<\/a> recommend.<\/p>\n

Nusselt Number for Liquid Metal Reactors<\/h2>\n

See also: Nusselt Number for Liquid Metal Reactors<\/a><\/p>\n

For liquid metals<\/strong><\/a>, the Prandtl number<\/strong> <\/a>is very small, generally in the range from 0.01 to 0.001. <\/strong>This means that the thermal diffusivity<\/strong><\/a>, which is related to the rate of heat transfer by conduction<\/strong><\/a>, unambiguously dominates<\/strong>. This very high thermal diffusivity results from metals\u2019 very high thermal conductivity, which is about 100 times higher than that of water. The Prandtl number<\/strong> for sodium at a typical operating temperature in the Sodium-cooled fast reactors is about 0.004. For this case, the thermal boundary layer development is much more rapid than that of the velocity boundary layer (\u03b4t<\/sub> >> \u03b4), and it is reasonable to assume uniform velocity throughout the thermal boundary layer.<\/p>\n

Heat transfer coefficients for sodium flow through the fuel channel are based on the Prandtl number <\/strong>and P\u00e9clet number<\/a>. <\/strong>Pitch-to-diameter (P\/D) also enters many calculations of heat transfer in liquid metal reactors. Convective heat transfer correlations<\/strong> are usually presented in terms of Nusselt number versus P\u00e9clet number<\/strong>. The typical P\u00e9clet number for normal operation is from 150 to 300 in the fuel bundles. As for other flow regimes, \u00a0the Nusselt number and a given correlation can be used to determine the convective heat transfer coefficient.<\/p>\n

<\/span>Graber-Rieger Correlation<\/div>
\"Nusselt<\/a><\/div><\/div>
<\/span>FFTF Correlation<\/div>
\"Nusselt<\/a><\/div><\/div><\/div>\n

Example – Nusselt Number – Cladding Surface Temperature<\/h2>\n

\"Convection<\/a>Cladding<\/strong> is the outer layer of the fuel rods, standing between the reactor coolant<\/strong> and the nuclear fuel<\/strong> <\/a>(i.e., fuel pellets<\/strong>). It is corrosion-resistant material with a low absorption cross-section for thermal neutrons<\/a>, usually zirconium alloy<\/strong>. Cladding<\/strong> prevents radioactive fission products from escaping the fuel matrix into the reactor coolant and contaminating it. Cladding constitutes one of the barriers in the \u2018defense-in-depth <\/b>\u2018approach; therefore, its coolability<\/strong> is one of the key safety aspects.<\/p>\n

Consider the fuel cladding of inner radius r<\/strong>Zr,2<\/sub><\/strong> = 0.408 cm<\/strong> and outer radius r<\/strong>Zr,1<\/sub><\/strong> = 0.465 cm<\/strong>. Compared to fuel pellet, there is almost no heat generation in the fuel cladding (cladding is slightly heated by radiation<\/a>). All heat generated in the fuel must be transferred via conduction<\/strong><\/a> through the cladding, and therefore the inner surface is hotter than the outer surface.<\/p>\n

Assume that:<\/p>\n