Nusselt numbers fully developed

Nume is the Nusselt number for developing flow, Num is that for hydrodynamically fully developed laminar flow. 

Limiting Nusselt numbers for laminar flow in annuli have been calculated by Dwyer [Nucl. Set. Eng., 17, 336 (1963)]. In addition, theoretical analyses of laminar-flow heat transfer in concentric and eccentric annuh have been published by Reynolds, Lundberg, and McCuen [Jnt. J. Heat Ma.s.s Tran.sfer, 6, 483, 495 (1963)]. Lee fnt. J. Heat Ma.s.s Tran.sfer, 11,509 (1968)] presented an analysis of turbulent heat transfer in entrance regions of concentric annuh. Fully developed local Nusselt numbers were generally attained within a region of 30 equivalent diameters for 0.1 < Np < 30, lO < < 2 X 10, 1.01 <... 

Chakraborty S (2006) Analytical solutions of Nusselt number for thermally fully developed flow in microtubes under a combined action of electroosmotic forces and imposed gradients. Int J Heat Mass Transfer 49 810-813... 

The values of the Nusselt and the Poiseuille numbers for heat transfer and friction for fully developed laminar flows through specifled channels are presented in Table 7.1 (Shah and London 1978). 

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... 

It is given as the duct length required for the Nusselt number to fall within a 5% interval of the fully developed value. Again, a dimensionless quanhty... 

Assuming an initially fully developed parabolic velocity profile. Re = 1000, and Pr = 5, calculate and plot the Nusselt number as a function of the inverse Graetz number,... 

Systematically find the expression for the temperature distribution and the Nusselt number for laminar flow between two large parallel plates in the region of fully developed velocity and temperature profiles for a uniformly applied wall heat flux. 

Consider a fully developed steady-state laminar flow of a constant-property fluid through a circular duct with a constant heat flux condition imposed at the duct wall. Neglect axial conduction and assume that the velocity profile may be approximated by a uniform velocity across the entire flow area (i.e., slug flow). Obtain an expression for the Nusselt number. 

Consider the fully-developed flow of a viscous fluid in a circular duct of radius a<,. Without neglecting viscous dissipation, derive an expression for the Nusselt number if the boundary condition at r = ao, is T = Tw < Tm, where T,n is the mean temperature of the fluid. 

Now, in fully developed flow it is usually convenient to utilize the mean fluid temperature, Tmt rather than the center line temperature in defining the Nusselt number. This mean or bulk temperature is given, its explained in Chapter 1, by ... 

This chapter has been concerned with the analysis of laminar flows in ducts with various cross-sectional shapes. If the flow is far from the inlet to the duct or from anything else causing a disturbance in the flow, a fully developed state is reached in many situations, the basic characteristics of the flow in this state not changing with distance along the duct. If the diffusion of heat down the duct can be neglected, which is true in most practical situations, it was shown that in such fully developed flows, the Nusselt number based on the difference between the local wall and bulk mean temperatures is constant. Values of the Nusselt number for fully developed flow in ducts of various cross-sectional shape were discussed. 

Consider fully developed laminar flow fluid through a circular pipe with a uniform wall heat flux. If heat is generated uniformly in the fluid, perhaps as the result of a chemical reaction, at a rate of q per unit volume, determine the value of the Nusselt number based on the difference between the wall temperature and the mean fluid temperature in the pipe. 

Derive an expression for the Nusselt number in fully developed laminar slug flow through an annulus when the inner and outer surfaces of the annulus have diameters of Dj and D0 respectively and when there is a uniform heat flux applied at the inner surface and when the outer surface is adiabatic. 

In some situations it is possible to find the heat transfer rate with adequate accuracy by assuming that the velocity is constant across the duct. Le to assume that so-called slug flow exists. Find the temperature distribution and the Nusselt number in sllug flow in a plane duct when the thermal field is fully developed and when there is a uniform wall heat flux. 

Consider fully developed flow in a plane duct in which uniform heat fluxes qw and qwi are applied at the two walls. Derive expressions for the temperature distribution in the duct and the Nusselt number. 

Use the Reynolds analogy to derive an expression for the Nusselt number for fully developed turbulent flow in an annulus in which the inner wall is heated to a uniform temperature and the outer wall is adiabatic. Assume that the friction factor can be derived by introducing the hydraulic diameter concept. 

The so-called Taylor-Prandtl analogy was applied to boundary layer flow in Chapter 6. Use this analogy solution to derive an expression for the Nusselt number in fully developed turbulent pipe flow. 

Derive the value for the Nusselt number for fully developed flow through a porous medium-filled pipe with a uniform heat flux at the wall. 

The heat-transfer coefficient calculated from this relation is the average value over the entire length of tube. Note that the Nusselt number approaches a constant value of 3.66 when the tube is sufficiently long. This situation is similar to that encountered in the constant-heat-flux problem analyzed in Chap. 5 [Eq. (5-107)], except that in this case we have a constant wall temperature instead of a linear variation with length. The temperature profile is fully developed when the Nusselt number approaches a constant value. 

Kays [36] and Sellars, Tribus, and Klein (Ref. 3, Chap. 5) have calculated the local and average Nusselt numbers for laminar entrance regions of circular tubes for the case of a fully developed velocity profile. Results of these analyses are shown in Fig. 6-5 in terms of the inverse Graetz number, where... 

Fig. 5-5 Local and average Nusselt numbers tor circular tube thermal entrance regions in fully developed laminar flow.

For laminar flow (ReD < 2100) that is fully developed, both hydro-dynamically and thermally, the Nusselt number has a constant value. For a uniform wall temperature, NuD = 3.66. For a uniform heat flux through the tube wall, NuD = 4.36. In both cases, the thermal conductivity of the fluid in NuD is evaluated at Tb. The distance x required for a fully developed laminar velocity profile is given by [(x/D)/ReD] 0.05. The distance x required for fully developed velocity and thermal profiles is obtained from [(x/D)/(ReD Pr)] = 0.05. 

For a constant wall temperature, a fully developed laminar velocity profile, and a developing thermal profile, the average Nusselt number is estimated by [Hausen, Mlg. Waermetech., 9, 75 (1959)]... 

For fully developed turbulent flow of liquid metals, the Nusselt number depends on the wall boundary condition. For a constant wall temperature [Notter and Sleicher, Chem. Eng. Science, 27,2073 (1972)],... 

Flow in Noncircular Ducts The length scale in the Nusselt and Reynolds numbers for noncircular ducts is the hydraulic diameter, D), = 4AJp, where A, is the cross-sectional area for flow and p is the wetted perimeter. Nusselt numbers for fully developed laminar flow in a variety of noncircular ducts are given by Mills (Heat Transfer, 2d ed., Prentice-Hall, 1999, p. 307). For turbulent flows, correlations for round tubes can be used with D replaced by l. ... 

B Obtain analytic relations for the velocity profile, pressure drop, friction factor, and Nusselt number in fully developed laminar How, and B Determine the friction factor and Nusselt number in fully developed turbulent flow using empirical relations, and calculate the pressure drop and heat transfer rdte. 

The Nusselt number reaches a constant value at a distance of less than 10 diameters, and thus the flow can be assumed to be fully developed for V > iOD. 

The Nusselt numbers for the uniform. surface temperature and uniform surface heat flux conditions are identical in the fully developed regions, and nearly identical in the entrance regions. Therefore, Nusselt number... 

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. 

A similar analysis can be performed for fully developed laminar flow in a circular tube for the case of constant surface temperature Tj. The solution procedure in this case is more complex as it requires iterations, but the Nusselt number relation obtained is equally simple (Fig. 8-22) ... 

Tlie friction factor/and the Nusselt number relations are given in Table 8-1 for fully developed laminar flow in tubes of various cross sections. The Reynolds and Nu.sselt numbers for flow in these tubes are based on the hydraulic diameter D/, - 4AJp, where is the cross sectional area of the tube and p is its perimeter. Once the Nusselt number is available, the convection heat transfer coefficient is determined from h = / Nu/D. ... 

Nusselt number and friction factor for fully developed laminar flow in tubes of various cross sections (D/, = Re = V, JO v, and Nu = hU lk)... 

Note that this Nusselt number is considerably higher than the fully developed value of 3.66. Then,... 

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