Laminar thermally fully developed

Heat transfer coefficients in thermally fully developed, laminar flow... 

In the following we will show how heat transfer coefficients are calculated for thermally fully developed, laminar flow. In a corresponding manner the mass transfer coefficients with regard to fully developed concentration profile can be obtained. In order to show this fundamentally we will consider tubular flow. The explanations can easily be transferred to cover other types of channel flow. 

Table 3.2 Nusselt numbers Nu = ad JX in thermal, fully developed, laminar flow and resistance factors = Ap/(L/dh g/2w, ) in channels with different cross sections. Nut is the Nusselt number at constant wall temperature, Nuq that at constant heat flux at the wall and Re = wmdh/v is the Reynolds number.

An exclusively analytical treatment of heat and mass transfer in turbulent flow in pipes fails because to date the turbulent shear stress Tl j = —Qw w p heat flux q = —Qcpw, T and also the turbulent diffusional flux j Ai = —gwcannot be investigated in a purely theoretical manner. Rather, we have to rely on experiments. In contrast to laminar flow, turbulent flow in pipes is both hydrodynamically and thermally fully developed after only a short distance x/d > 10 to 60, due to the intensive momentum exchange. This simplifies the representation of the heat and mass transfer coefficients by equations. Simple correlations, which are sufficiently accurate for the description of fully developed turbulent flow, can be found by... 

The laminar flow inside a microchannel is considered thermally fully developed when the following equation is satisfied by the fluid temperature ... 

For a thermally fully developed laminar flow, for a fixed dynamic and thermal boundary condition and by neglecting the fluid axial conduction, (Pe oo), viscous dissipation (Br = 0), the flow work fi = 0), and electro-osmotic phenomena (Sl=f = 0), the Nusselt number depends on the cross-sectional geometiy (through the Poiseuille number e) only. 

We consider steady-state laminar and fully developed thermal and hydrodynamic single-phase flow. 

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

Because the velocity field is fully developed, the variations of U and E with R are known. The solution to Eq. (7.93) can therefore be obtained using a similar procedure to that used in Chapter 4 to solve for thermally developing laminar pipe flow, i.e., using separation of variables. Here, however, a numerical finite-difference solution procedure will be used because it is more easily adapted to the situation where the wall temperature is varying with Z. 

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)]... 

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. 

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)]... 

Hydrodynamically fully-developed laminar gaseous flow in a cylindrical microchannel with constant heat flux boundary condition was considered by Ameel et al. [2[. In this work, two simplifications were adopted reducing the applicability of the results. First, the temperature jump boundary condition was actually not directly implemented in these solutions. Second, both the thermal accommodation coefficient and the momentum accommodation coefficient were assumed to be unity. This second assumption, while reasonable for most fluid-solid combinations, produces a solution limited to a specified set of fluid-solid conditions. The fluid was assumed to be incompressible with constant thermophysical properties, the flow was steady and two-dimensional, and viscous heating was not included in the analysis. They used the results from a previous study of the same problem with uniform temperature at the boundary by Barron et al. [6[. Discontinuities in both velocity and temperature at the wall were considered. The fully developed Nusselt number relation was given by... 

Heat transfer to a laminar flow in an annulus is complicated by the fact that both the velocity and thermal profiles are simultaneously developing near the entrance and, often, over the length of the heated channel. Natural convection may also be a factor. It is usually conservative (i.e., predicted heat-transfer coefficients are lower than those experienced) to use equations for the fully developed flow. 

Isothermal Wall. Natural convection also affects the laminar thermal development in a tube with an isothermal wall. In this case the temperature differences in the fluid near the tube inlet initiate a natural convection motion, but as the fluid temperature approaches the wall temperature far downstream, the motion slows and the fully developed Nusselt number (Nujr = 3.66) is approached. 

As a result of the development of the hydrodynamic and thermal boundary layers, four types of laminar flows occur in ducts, namely, fully developed, hydrodynamically developing, thermally developing (hydrodynamically developed and thermally developing), and simultaneously developing (hydrodynamically and thermally developing). In this chapter, the term fully developed flow refers to fluid flow in which both the velocity profile and temperature profile are fully developed (i.e., hydrodynamically and thermally developed flow). In such cases, the velocity profile and dimensionless temperature profile are constant along the flow direction. The friction factor and Nusselt number are also constant. 

For a circular duct with a diameter of 2a, the characteristics of laminar flow and heat transfer for four kinds of flows, namely, fully developed, hydrodynamically developing, thermally developing, and simultaneously developing, are outlined in the following sections. 

Heat Transfer on Walls With Uniform Temperature. For this boundary condition, denoted as , temperature distribution in a circular duct for fully developed laminar flow in the absence of flow work, thermal energy sources, and fluid axial conduction has been solved by Bhatti [3] and presented by Shah and Bhatti [2], as follows ... 

Uniform Temperature at One Wall and Uniform Heat Flux at the Other. When the two walls of a parallel plate duct are subject to a thermal boundary condition such as uniform temperature at one wall and uniform heat flux at the other, the Nusselt numbers for fully developed laminar flow for qZ = 0 and q"w 0 are determined to be ... 

In this section, the friction factors and Nusselt numbers for fully developed, hydrodynami-cally developing, thermally developing, and simultaneously developing laminar flows in rectangular ducts are presented. 

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