Heat mean Nusselt number

Once 6 has been determined, the heat flux at the wall and the mean temperature can be found and the mean Nusselt number can then be found. Exact solutions for values of n up to 4 can be relatively easily obtained and approximate solutions for higher values of n can be obtained. The variation of the mean Nusselt number with Z given by these solutions is shown in Fig. 4.IS. 

Nuh and Nuc being the mean Nusselt numbers, based on VV for the hot and the cold walls, respectively. With adiabatic end walls, because a steady state situation is being considered, these two values should have the same numerical value, i.e., the rate at which heat is transferred from the hot wall to the fluid should be equal to the rate at which heat is transferred from the fluid to thi cold wall. Small differences usually exist between the values of Nuh and Nuc given by the numerical solution due to the small numerical round-off errors and due to the finite convergence criterion used in the numerical solution. 

The mean heat transfer coefficient am from (1.33) is also independent of x+ and z+. The mean Nusselt number Num, which contains am, is only a function of... 

The determination of heat transfer coefficients with the assistance of dimensionless numbers has already been explained in section 1.1.4. This method can also be used for mass transfer, and as an example we will take the mean Nusselt number Num = amL/ in forced flow, which can be represented by an expression of the form... 

The Nusselt number is equal to the dimensionless temperature gradient at the wall. It is a universal function of x+, Re and Pr for every fluid with a body of a given shape. The mean Nusselt number is independent of x+, as it is the integral mean value over the heat transfer surface... 

The mean Nusselt number comes from the mean heat transfer coefficient... 

In the same way as shown in the previous section, the heat transfer coefficient and from that the mean Nusselt number Nume = amed/A (the index e stands for entry flow) can be obtained from the temperature profile. The Nusselt number can be calculated from an empirical equation of the form... 

In simultaneous heat and mass transfer in binary mixtures, mean mass transfer coefficients can likewise be found using the equations from the previous sections. Once again this requires that the mean Nusselt number Num is replaced by the mean Sherwood number Shm, and instead of the Grashof number a modified Grashof number is introduced, in which the density p(p,T, ) is developed into a Taylor series,... 

The mean Nusselt number, the mean heat transfer coefficient and the mean heat flow in the laminar region according to section 3.7.4, No. 1, are... 

Rohsenow et al. [4.19] rearranged (4.51) by introducing dimensionless quantities and then determined the heat transfer coefficient a = AL/<5 and also the mean heat transfer coefficient am from the calculated film thickness. Fig. 4.14 shows as a result of this the mean Nusselt number... 

For smooth particles of an arbitrary shape in ideal fluid (this model is used, say, to describe heat exchange between particles and liquid metals at Pr -C 1 and Re 3> 1) and in the absence of regions with closed streamlines, the mean Nusselt number can be calculated by the formula... 

Convection is another possible means of heat loss. The importance of convection can be found in terms of the mean Nusselt number Nu, which, at low flow rates, is given in terms of the Prandtl number Pr (the ratio between the momentum and thermal dififiisivities) and the Grashof number Gr (the ratio between buoyant and viscous forces) by [51]... 

Heat Transfer on Walls With Uniform Heat Flux. The temperature profile and the local and mean Nusselt numbers for thermally developing flow in a circular duct with uniform wall heat flux are provided by Siegel et al. [25] as follows ... 

The mean Nusselt number Num for thermally developing flow with uniform wall temperature or uniform wall heat flux conditions can be calculated using Al-Arabi s [95] correlation ... 

The local Nusselt number is displayed in Fig. 5.23 for Pr = 0.0,0.01,0.7,10, and °° when one wall of the parallel plate duct is insulated and the other wall is subjected to uniform heat flux heating [140]. Included in Fig. 5.23 are the results for Pr = obtained from the concentric annular duct corresponding to r = 1. The local and mean Nusselt numbers for Pr = 0 were obtained by Bhatti [34]. 

FIGURE 5.23 Local and mean Nusselt numbers for simultaneously developing flow in a flat duct with uniform heat flux at one wall and the other wall insulated [34,140]. 

Thermally Developing Flow. Wibulswas [160] and Aparecido and Cotta [161] have solved the thermal entrance problem for rectangular ducts with the thermal boundary condition of uniform temperature and uniform heat flux at four walls. However, the effects of viscous dissipation, fluid axial conduction, and thermal energy sources in the fluid are neglected in their analyses. The local and mean Nusselt numbers Nu j, Num T, and Nu hi and Num Hi obtained by Wibulswas [160] are presented in Tables 5.32 and 5.33. 

A generalized correlation of mean Nusselt number for turbulent heat transfer in an isothermal circular duct with inserted tape was developed by Manglik and Bergles [273] based on the experimental data. It is expressed as ... 

In the Hausen correlation, the mean Nusselt number is calculated as a sum of two terms the first term (Nu) is the fully developed value of the Nusselt number its value can be calculated by using the data quoted in the above sections. The second term takes into account the effects of the thermal entrance region. Rosehnow and Choi [7] give the value of the coefficients K, K, and b for circular channels in the case of prescribed wall heat flux (H boundary condition) and developed profile of velocity, Nu is equal to 48/11 (see Eq. 43) with Br = 5 = 0, = 0.023,... 

On the contrary, under the H boundary condition (see Convective Heat Transfer in Microchannels ) with a fully developed velocity profile hydrodynamically fully developed and thermally developing flow Fig. 3), the mean Nusselt number for circular microtubes as a functitMi of the dimensionless axial coordinate z for laminar flows can be calculated as follows ... 

For laminar flows, the value of the mean Nusselt number depends on the thermal boundary condition considered a description of the most common thermal boundary conditions for microchannels can be found in convective heat transfer in microchannels (i.e., T, HI, H2 boundary conditions). 

Rectangular Cross Section In Tables 4 and 5, the mean Nusselt numbers for hydrodynamically fully developed and thermally developing flow in rectangular microchannels are reported as a function of the dimensionless axial coordinate z and of the aspect ratio p for T and HI boundary conditions (with four heated sides), respectively. 

By averaging the local Nusselt number along the heated perimeter, one can obtain the thermally fully developed mean Nusselt number ... 

For laminar flow and completely developed radial velocity and temperature profiles the local heat transfer is constant and the mean Nusselt number reaches an asymptotic value, given by Nu. The same value is obtained for the asymptotic Sherwood number, Sh characterizing the mean mass transfer between the fluid and the channel wall (see Chapter 6). 

In turbulent flow, the boundary conditions constant wall temperature and constant heat flux lead to approximately the same mean Nusselt numbers. Correlations in the far turbulent regime (Re> 10 ) are noted here. The hydrodynamic entry length is approximately independent of Re, so that an approximation for fully turbulent flow after length x can be made for... 

From the temperature profile given by Equation 7.58, the heat flux at the wall, the total rate of heat transfer, and the bulk temperature of the fluid at the exit can be evaluated. Also, using the arithmetic mean of terminal temperature differences together with the definition of heat transfer coefficient, the arithmetic mean Nusselt number can be expressed as a function of the Graetz number. Similarly, the logarithmic Nusselt number can also be obtained as a function of the Graetz number [12]. 

In a recent book by Molerus and Wirth (1997), the recommended heat transfer correlations for packed beds can be summarized as follows. For fully developed laminar flow, an approximation formula for the mean Nusselt number, derived from the pipe flow analogy, was proposed as... 

DC current was supplied through the development and test sections for direct heating. The outer temperature on the heated wall was measured by means of an infrared radiometer. Experiments were carried out in the range of Re = 10-450. The average Nusselt number was calculated using the average temperature of the inner tube wall and mean temperature of the fluid at the inlet and outlet of the tube. 

Reynolds number. It should be stressed that the heat transfer coefficient depends on the character of the wall temperature and the bulk fluid temperature variation along the heated tube wall. It is well known that under certain conditions the use of mean wall and fluid temperatures to calculate the heat transfer coefficient may lead to peculiar behavior of the Nusselt number (see Eckert and Weise 1941 Petukhov 1967 Kays and Crawford 1993). The experimental results of Hetsroni et al. (2004) showed that the use of the heat transfer model based on the assumption of constant heat flux, and linear variation of the bulk temperature of the fluid at low Reynolds number, yield an apparent growth of the Nusselt number with an increase in the Reynolds number, as well as underestimation of this number. 

For high values of the Reynolds number, the mean value of the Nusselt number does not differ significantly from the theoretical value for fully developed flow. On the contrary, at low Re the effects of conjugate heat transfer on the mean value of... 

The heat transfer coefficient to the vessel wall can be estimated using the correlations for forced convection in conduits, such as equation 12.11. The fluid velocity and the path length can be calculated from the geometry of the jacket arrangement. The hydraulic mean diameter (equivalent diameter, de) of the channel or half-pipe should be used as the characteristic dimension in the Reynolds and Nusselt numbers see Section 12.8.1. 

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