Thermal Entry Length

The thermal entry length should be considered by comparison between experimental and numerical results. 

The Graetz problem considers the thermal entry of an incompressible fluid in a circular pipe with a fixed velocity profile. The situation is illustrated in Fig. 4.16. The Graetz problem is a classic problem in fluid mechanics, and one that permits an analytic solution. After some hydrodynamic entry length, the velocity profile approaches a steady profile that is,... 

The temperature and species profiles also have entry-region behavior. The fully coupled entry-region problem is easily formulated and can be solved using the method of lines. The details of the entry-region profiles depend on species and thermal boundary conditions as well as fluid properties. The entry length and the corresponding profile development also depend on the channel geometry. 

The Circular Tube Thermal-Entry-Length, with Hydrodynamically Fully Developed Laminar Flow... 

Figure 8.4 Sketch for the thermal-entry length problem.

Table 8.1 Infinite-series solution functions for the circular tube constant surface temperature thermal-entry length.

Azer, N.Z., Thermal Entry Length for Turbulent Row of Liquid Metals in Pipes with Constant Wall Heat Rux , Trans. ASME Serv. C, J. Heat Transfer, Vol. 90. pp. 483-485, 1968. 

We start this chapter with a general physical description of internal flow, and the average velocity and average temperature. We continue with the discussion of the hydrodynamic, and thermal entry lengths, developing flow, and fully developed flow. We then obtain the velocity and temperature profiles for fully developed laminar flow, and develop relations for the friction factor and Nusselt nmnber. Hinally we present empirical relations for developing and full developed flows, and demonstrate their use. 

B Have a visual understanding of different flow regions in internal flow, such as Ihe entry and the fully developed flow regions, and calculate hydrodynamic and thermal entry lengths,... 

Th regioii of flow over which the thermal boundary layer develops and re.iches (he tube center i.s called the thermal entrance region, and the length of this region is called the thermal entry length L,. Flow in the thermal... 

During laminar flow in a tube, the magnitude of the dimensionless Prandtl number Pr is a measure of the relative growth of the velocity and thermal boundary layers. For fluids with Pr = I, such as gases, the two boundary layers essentially coincide with each other. For fluids with Pr > I, such as oils, the velocity boundary layer outgrows the thermal boundary layer. As a result, the hydrodynamic entry length is smaller than the thermal entry length. The opposite is tnie for fluids with Pr < 1 such as liquid metals. 

The hydrodynamic entry length is usually taken to be the distance from the lube entrance where the wall shear. stress (and thus the fficliou factor) reaches within about 2 percent of the fully developed value. In laminar flow, the hydrodynamic and thermal entry lengths are given approximately as (see Kays and Crawford (1993) and Shah and Bhatli (1987)]... 

In turbulent flow, the intense iqjxing during random fluctuations usually overshadows the effects of molecular diffusion, and therefore the hydrodynamic and thermal entry lengths ate of about the same size and independent of the PflLndil number. The hydrodynamic entry length for turbulent flow can be detennined from [see Bbatti and Shah (1987) and 7.hi-qing (1982)]... 

The entry length is much shorter in turbulent flow, as expected, and its dependence on the Reynolds number is weaker. In many lube flotvs of practical interest, the entrance effects become insignificant beyond a tube length of 10 diameters, and the hydrodynamic and thermal entry lengths are approximately taken to be... 

C How is the thermal entry length defined for flow in a tube In what region is the flow in a tube fully developed ... 

S-18C Consider the flow of mercury (a liquid meial) in a tube. How will the hydrodynamic and thermal entry lengths compare if the flow is laminar How would they compare if (he flow uere turbulent ... 

The unusual behavior of Nu decreasing with increasing Re in the laminar regime in microchannels may alter the status of thermal development and hence the conventional thermal entry length, since the variation of the heat transfer coefficient along the flow is a variation of the boundary condition. 

The effect of any variation of the boundary condition on thermal entry length has not been explained. 

The entrance length z/d for a fully developed velocity profile, for a concentration profile, or for a corresponding thermal entry length are given as 0.05 Re, 0.05 Re Sc, 0.05 Re Pr, respectively. The symbols are summarized in Table 11. 

Laminar Flow. Thermal entry length solutions with developed velocity profiles are summarized in Refs. 19 and 44 for a large number of practically important flow passage geometries with extensive comparisons. 

Turbulent Flow. The thermal entry length solutions for smooth ducts for several cross-sectional geometries have been summarized [46]. As for laminar flow, the Nusselt numbers in the thermal region are higher than those in the fully developed region. However, unlike laminar flow, Nu,x and NuxHi are very nearly the same for turbulent flow. The local and mean Nusselt numbers for a circular tube with and boundary conditions are [46] ... 

The hydrodynamic and thermal entry length problem can be solved either by employing the boundary layer-type idealizations or by considering the complete set of equations [1]. 

However, also larger factors of 0.08 and 0.09 [9] have been found for microchannels. The thermal entry length Lj may be calculated by... 

The thermal entry length 1th for laminar flow can be calculated by [40]... 

With the Hausen correlation [13], Nusselt numbers in the thermal entry length can be calculated ... 

The hybrid solution procedure described in the previous section is computationally more demanding than one that does not rely on the CFD package to predict the heat transfer from the exhaust gas. In fact, this simpler approach was adopted in die early stages of the project, the heat transfer process was modelled using a mean heat transfer coefficient estimated from correlations for convective heat transfer in annuli. However, it was soon realized that this method has a high degree of uncertainty when the heat transfer process takes place under unsteady-state conditions and when the thermal entry length spreads over an appreciable extent of the domain. These conditions are always met in the application under study. 

For laminar flow, the thermal entry length is approximately given by (Cengel, 2002) ... 

Thermal entry length and thermally fully developed region. 

For internal flows, thermal boundary layers develop from both top and bottom surfaces and develop into two regions thermal entry length and thermal fully developed regions similar to hydrodynamic internal flow as shown in Figure 6.8. 

Thermal entry length The length required for the dimensionless temperature profile to become fully developed. 

Criterion for thermal entry length for laminar flow ... 

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