Nusselt number internal

Correlations for Convective Heat Transfer. In the design or sizing of a heat exchanger, the heat-transfer coefficients on the inner and outer walls of the tube and the friction coefficient in the tube must be calculated. Summaries of the various correlations for convective heat-transfer coefficients for internal and external flows are given in Tables 3 and 4, respectively, in terms of the Nusselt number. In addition, the friction coefficient is given for the deterrnination of the pumping requirement. 

Figure 3 illustrates some additional capability of the flow code. Here no pressure gradient is Imposed (this is then drag or "Couette flow only), but we also compute the temperatures resulting from Internal viscous dissipation. The shear rate in this case is just 7 — 3u/3y — U/H. The associated stress is.r — 177 = i/CU/H), and the thermal dissipation is then Q - r7 - i/CU/H). Figure 3 also shows the temperature profile which is obtained if the upper boundary exhibits a convective rather than fixed condition. The convective heat transfer coefficient h was set to unity this corresponds to a "Nusselt Number" Nu - (hH/k) - 1. 

This chapter has been concerned with flows in wb ch the buoyancy forces that arise due to the temperature difference have an influence on the flow and heat transfer values despite the presence of a forced velocity. In extemai flows it was shown that the deviation of the heat transfer rate from that which would exist in purely forced convection was dependent on the ratio of the Grashof number to the square of the Reynolds number. It was also shown that in such flows the Nusselt number can often be expressed in terms of the Nusselt numbers that would exist under the same conditions in purely forced and purely free convective flows. It was also shown that in turbulent flows, the buoyancy forces can affect the turbulence structure as well as the momentum balance and that in turbulent flows the heat transfer rate can be decreased by the buoyancy forces in assisting flows whereas in laminar flows the buoyancy forces essentially always increase the heat transfer rate in assisting flow. Some consideration was also given to the effect of buoyancy forces on internal flows. 

Figure 4 Normalised mass G(a), energy H(a), average enthalpy increase K(a) and transition mass flux g(a) of a droplet for the cases of (a) enthalpy improved 3EM and (b) for standard 3EM. Ambient temperature dFiame = 900°C, initial temperature 910 = 270°C, transition temperature STrans = 420 °C, initial droplet diameter di,0= 400mm, Nusselt number 18 and amplification factor of the heat conduction coefficient due to internal forced convection IWkmoiec = 2.72 (Beer [5]), residence time t=30 ms.

One-dimensional conduction, i-e between the external and internal wall only is the implicit assumption to calculate the Nusselt number from experimental data. In the case of minichannels whose wall thickness can be of the same order of magnitude as the hydraulic diameter this hypothesis may be questionable. 

Nnu Nusselt number, hX/k or hd/k for vessel wall or internal pipe, respectively... 

FIGURE 4.39 Temporal response of the Nusselt number to suddenly applied internal heating in a horizontal wire. 

The Brinkmann number Br = ( yuli)l[k TKm - Te)] is introduced to account for the influence of viscous dissipation, such as heating or cooling of the fluid due to internal friction in high-velocity flow, highly viscous fluid, or in cases in which viscous dissipation cannot be ignored. When viscous dissipation is considered, the asymptotic Nusselt number in a very long pipe, found by Ou and Cheng [5], is 9.6 and independent of the Brinkmann number. 

Internally finned tubes are ducts with internal longitudinal fins. These tubes are widely used in compact heat exchangers. The friction factor-Reynolds number product and the Nusselt number for such internally finned tubes, designated as (/ Re), and Nu/,c> respectively, are computed from the following definitions ... 

An elliptical duct with four internal longitudinal fins mounted on the major and minor axes, as shown in Fig. 5.48, has been analyzed by Dong and Ebadian [275] for fully developed laminar flow and heat transfer. In this analysis, the fins are considered to have zero thickness. The thermal boundary condition is applied to the duct wall, and / is defined as a ratio of Ha a = Hbib. The friction factors and Nusselt numbers for fully developed laminar flow are given in Table 5.52. 

TABLE 5.52 Friction Factors and Nusselt Numbers for Fully Developed Flow in an Elliptical Duct With Internal Fins [275]... 

It is interesting to note that some compound attempts are unsuccessful. Masliyah and Nan-dakumar [361], for example, found analytically that average Nusselt numbers for internally finned coiled tubes were lower than they were for plain coiled tubes. 

T. W. Jackson, K. R. Purdy, and C. C. Oliver, The Effects of Resonant Acoustic Vibrations on the Nusselt Number for a Constant Temperature Horizontal Tube, International Developments in Heat Transfer, pp. 483-489, ASME, New York, 1961. 

FIGURE 18.13 A comparison between the predicted and correlated local Nusselt numbers for the UWT and UHF cases [71], published with permission of ASME International. 

S. Faggiani and W. Grassi, Round Liquid Jet Impingement Heat Transfer Local Nusselt Numbers in the Region with Non-Zero Pressure Gradient, in G. Hestroni (ed.) Proceedings of the 9th International Heat Transfer Conference, 4, pp. 197-202, Hemisphere, New York, 1990. 

To adopt a common procedure here, the external film coefficient is expressed in terms of the Nusselt number. The internal coefficients, however, are given indirectly by the transfer efficiency, E , representing the fractional approach to the maximum possible heat transfer. Thus, by definition. 

If potential flow and constant surface temperature are assumed, an equation analogous to Eq. (18) is obtained for the internal Nusselt number. Note, however, that the reference velocity in the internal Peclet number is the drop velocity. Similar results will be obtained from the penetration theory, according to which the film is assumed infinite with respect to the depth of heat penetration during the short contact time of a fluid element sliding over the interface. Licht and Pansing (L13) report West s equation, based on the transient film concept, for the case of mass transfer through the combined film resistance. In terms of the overall heat-transfer resistance, l/U (= l//jj + I/he) and if the contact time is that required for the drop to traverse a distance equal to its diameter. West s equations yield... 

The simplest model for droplet evaporation is based on an equilibrium uniform-state model for an isolated droplet [28-30]. Miller et al. [31] investigated different models for evaporation accounting for nonequilibrium effects. Advanced models considering internal circulation, temperature variations inside the droplet, and effects of neighboring droplets [30] may alter the heating rate (Nusselt number) and the vaporization rates (Sherwood number). For the uniform-state model, the Lagrangian equations governing droplet temperature and mass become [28-30]... 

To correlate internal mixing with heat transfer, the Nusselt number (Nu=h dj l) of the reactor is related to the Reynolds (Re=pNds/fi) and Prandtl Pr=juCp/l) number according to... 

For example, for an incompressible liquid = 0) with negligible viscous dissipation (Br = 0) and no-slip condition at the walls (ij = 0) without significant fixed external forces (f = 0) and internal heat sources (S = 0) under an H3 thermal boundary condition, the average Nusselt number can be considered as a function of the following dimensionless quantities ... 

The solution of Eqs. 22 and 23a with the appropriate dynamic and thermal boundary conditions allows one to obtain the velocity and temperature distribution inside the microchannel for laminar fully developed flows. The analytical solution of Eqs. 22 and 23a has been obtained only for a few cross-sectional geometries. The numerical approach enables the calculation of the local and the average Nusselt number by means of which the internal convective heat transfer coefficients in microchannels can be computed. 

Consider the hydrogenation of benzene, which is exothermic with a heat of reaction 50 kcal/mol. For a catalyst pellet containing 58% Ni on Kieselguhr (Harshaw A1-0104T), the effective thermal conductivity and diffusivity are 3.6 lO cal/(cm s K) and 0.052 cm /s, respectively. The fluid bulk concentration of benzene is 5.655 x 10 mol/cm, and the fluid bulk temperature is 412 K. The characteristic length of the pellet is 0.296 cm. The observed rate for the reaction is 2.258 10 mol/(gcats) and the density of the catalyst is 1.88 g/cm. The modified Sherwood and Nusselt numbers are 215 and 10.8, respectively. Estimate the internal and external temperature differences. (Note The experimental values of the internal and external temperatures are 35 and 40 K, respectively. The difference between the surface and bulk fluid temperature is 6-7 K, Froment and Bischoff, 1979)... 

In the case of airlift reactors, the flow pattern may be similar to that in bubble columns or closer to that two-phase flow in pipes (when the internal circulation is good), in which case the use of suitable correlations developed for pipes may be justified [55]. Blakebrough et al. studied the heat transfer characteristics of systems with microorganisms in an external loop airlift reactor and reported an increase in the rate of heat transfer [56], In an analytical study, Kawase and Kumagai [57] invoked the similarity between gas sparged pneumatic bioreactors and turbulent natural convection to develop a semi-theoretical framework for the prediction of Nusselt number in bubble columns and airlift reactors the predictions were in fair agreement with the limited experimental results [7,58] for polymer solutions and particulate slurries. 

For fully developed forced convective cooling in smooth internal passages, at turbulent Reynolds numbers, the Colburn equation [44] can be used to estimate the Nusselt number ... 

---