Thermal, fully developed, laminar flow

In tubular flow, M = gwmR2ir, dAq = 2rn dr and therefore the adiabatic mixing temperature, under the assumption g = const, is 

For the temperature profile for a thermal fully developed flow, from 

This says that the temperature increase at the wall in a thermal fully developed flow changes with the length x in the same way as the difference between the wall and the adiabatic mixing temperature. The temperature profile that satisfies this condition is of the general form 

Subtraction of both equations yields with the abbreviation r+ =r/R  

This relationship serves in many publications as the definition of a thermally fully developed flow. It is, as we have already seen, a result of the fact that the heat transfer coefficient reaches its asymptotic, constant end value downstream. 

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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.

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. 

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

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

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. 

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 ... 

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. 

A cardioid duct is shown in Fig. 5.63. Fully developed laminar flow and heat transfer under the boundary condition have been analyzed by Tyagi [294]. The/Re and NuHj values derived from this analysis are 5.675 and 4.208, respectively. The Nusselt number for the thermal boundary condition was found to be 4.097 [1]. 

For the T3 boundary condition, the average Nusselt number for fully developed laminar flow with negligible external volume forces (f xt, 2 = 0), axial heat conduction (Pe oo), viscous dissipation (Br = 0), flow work (jS = 0), and thermal energy sources (S = 0) within the fluid... 

Worsoe-Schmidt PM. Heat transfer in the thermal entrance region of circular tubes and annular passages with fully developed laminar flow. International Journal of Heat and Mass Transfer 1967 10 541-551. 

The equations given in this chapter for Nu are only valid for a fully developed laminar flow, that is, without a hydrodynamic entrance region. Similar considerations can be made for the heat transfer in a tube with both a thermal and a hydrodynamic entrance region, whereby the differences in the values of Nu and Nu are mostly small (VDI, 2002). 

Under the h)potheses that Pe = Br = / = 5 =/ = 0, from the equality in Eq. (40) it is evident that this factor is a constant for the fully developed laminar flow of a specified fluid. Since the dimensionless quantities j and / are independent of the scale of the geometry (Dh), the area goodness factor j/f for different microchannels represents the influence of the geometric factors on the pressure losses and the heat transfer and allows the comparison of the thermal hydraulic performances of microchannels having different cross-sections. In the viscous dissipation section a comparison of the thermal performances of rectangular and trapezoidal microchannels by means of the area goodness factor is presented. 

RELAP5-3D has two laminar flow correlations from which the analyst may choose. The first is an exact solution for fully developed laminar flow in a circular tube (Sellars et al, 1956) with a uniform wall heat flux and constant thermal properties. This is a constant Nusselt number solution of the form ... 

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. 

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... 

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... 

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. 

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

Micro reactors permit high-throughput screening of process chemistries imder controlled conditions, unlike most conventional macroscopic systems [2], In addition, extraction of kinetic parameters from sensor data is possible, as heat and mass transfer can be fully characterized due to the laminar-flow condihons applied. More uniform thermal condihons can also be utilized. Further, reactor designs can be developed in this way that have specific research and development funchons. 

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