Convection in a circular tube

C Consider laminar forced convection in a circular tube. Will the heal flux be higher near the inlet of the tube or near the... 

Ameel, T.A., Wang, X., Barron, R.F. and Warrington, R.O., Laminar Forced Convection in a Circular Tube with Constant Heat Flux and Slip Flow, Mieroseale Ther-mophys. Eng, 1(4), 1997, 303-320. 

T.A. Ameel, R.F. Barron, X.M. Wang, and R.O. Warrington, Lantinar forced convection in a circular tube with constant heat flux and shp flow. Microscale Thermophysical Engineering 1, 303-320 (1997). 

T. A. Ameel, et at. Laminar forced convection in a circular tube with constant heat... 

The dispersion of a non-reactive solute in a circular tube of constant cross-section in which the flow is laminar is described by the convective-diffusion equation... 

SOLUTION Water is healed by steam in a circular tube. The tube length required to heat the water to a specified temperature is to be determined. Assumptions 1 Steady operating conditions exist. 2 Fluid properties are constant. 3 The convection heat transfer coefficient is constant. 4 The conduction resistance of copper tube is negligible so that (he inner surface temperature of the tube is equal to the condensation temperature of steam. 

Heat convection for gaseous flow in a circular tube in the slip flow regime with uniform temperature boundary condition was solved in [23]. The effects of the rarefaction and surface accommodation coefficients were considered. They defined a fictitious extrapolated boundary where the fluid velocity does not slip by scaling the velocity profile with a new variable, the shp radius, pj = l/(l + 4p.,Kn), where is a function of the momentum accommodation coefficient, and defined as p, =(2-F,j,)/F,j,. Therefore, the velocity profile is converted to the one used for the... 

Let us briefly consider convective mass transfer accompanied by a surface reaction in a circular tube. Laminar steady-state fluid flow in a circular tube of radius a with Poiseuille velocity profile is outlined in Subsection 1.5-3. For... 

J. A. Sabbagh, A. Aziz, A. S. El-Ariny, and G. Hamad, Combined Free and Forced Convection in Inclined Circular Tubes, J. Heat Transfer (98) 322-324,1976. 

Ideally the output of a loop Injector used In liquid chromatography or flow Injection analysis would be a sharp concentration pulse. However, this Is unlikely to be the case because of various dispersive forces which act on the concentration plug. Convective flow under laminar conditions In a circular tube, such as the outlet tube of a loop Injector, tends to be much slower near the walls of the tube than In the center and this will distort the Initial shape of the Injected materials (8,9). In addition, radial and axial diffusion of material In the tube can alter its initial shape. The degree of dispersion can be evaluated (9) by the Peclet nusd>er (P,) and the reduced time (r). These values are defined as... 

Currently, analytical approaches are still the most preferred tools for model reduction in microfluidic research community. While it is impossible to enumerate all of them in this chapter, we will discuss one particular technique - the Method of Moments, which has been systematically investigated for species dispersion modeling [9, 10]. The Method of Moments was originally proposed to study Taylor dispersion in a circular tube under hydrodynamic flow. Later it was successfully applied to investigate the analyte band dispersion in microfluidic chips (in particular electrophoresis chip). Essentially, the Method of Moments is employed to reduce the transient convection-diffusion equation that contains non-uniform transverse species velocity into a system of simple PDEs governing the spatial moments of the species concentration. Such moments are capable of describing typical characteristics of the species band (such as transverse mass distribution, skew, and variance). 

Now consider a fluid at a uniform temperature entering a circular tube whose surface is maintained at a different temperature. This time, the fluid particles in the layer in contact with the surface of the tube assume the surface temperature. Tins initiates convection heat transfer in the tube and Ihe development of a thermal hoimdaiy layer along the tube. The thickness of this boundary layer also increases in tfle flow direction until Ihe boundary layer reaches the tube center and thus fills the entire tube, as sliown in Fig. 8-7. 

Let us discuss qualitative specific features of convective heat and mass transfer in a turbulent flow through a circular tube and plane channel in the region of stabilized flow. Experimental evidence indicates that several characteristic regions with different temperature profiles can be distinguished. At moderate Prandtl numbers (0.5 < Pr < 2.0), the structure and sizes of these regions are similar to those of the wall layer and the core of the turbulent stream considered in Section 1.6. 

H. H. Al-Ali, and M. S. Selim, Analysis of Laminar Flow Forced Convection Heat Transfer with Uniform Heating in the Entrance Region of a Circular Tube, Can. J. Chem. Eng., (70) 1101-1107, 1992. 

Axial dispersion in packed beds, and Taylor dispersion of a tracer in a capillary tube, are described by the same form of the mass transfer equation. The Taylor dispersion problem, which was formulated in the early 1950s, corresponds to unsteady-state one-dimensional convection and two-dimensional diffusion of a tracer in a straight tube with circular cross section in the laminar flow regime. The microscopic form of the generalized mass transfer equation without chemical reaction is... 

Kuznetsov, AV, Nield, DA., 2009. Thermally developing forced convection in a porous medium occupied by a rarefied gas parallel plate channel or circular tube with walls at constant heat flux. Transp. Porous Media 76, 345-362. 

Nield, D.A., Kuznetsov, A.V., Xiong, M., 2004a. Thennally developing forced convection in a porous medium parallel plate channel or circular tube with isothermal walls. J. Porous Media 7, 19 27. 

A variety of studies can be found in the literature for the solution of the convection heat transfer problem in micro-channels. Some of the analytical methods are very powerful, computationally very fast, and provide highly accurate results. Usually, their application is shown only for those channels and thermal boundary conditions for which solutions already exist, such as circular tube and parallel plates for constant heat flux or constant temperature thermal boundary conditions. The majority of experimental investigations are carried out under other thermal boundary conditions (e.g., experiments in rectangular and trapezoidal channels were conducted with heating only the bottom and/or the top of the channel). These experiments should be compared to solutions obtained for a given channel geometry at the same thermal boundary conditions. Results obtained in devices that are built up from a number of parallel micro-channels should account for heat flux and temperature distribution not only due to heat conduction in the streamwise direction but also conduction across the experimental set-up, and new computational models should be elaborated to compare the measurements with theory. 

The convective and nucleate boiling heat transfer coefficient was the subject of experiments by Grohmann (2005). The measurements were performed in microtubes of 250 and 500 pm in diameter. The nucleate boiling metastable flow regimes were observed. Heat transfer characteristics at the nucleate and convective boiling in micro-channels with different cross-sections were studied by Yen et al. (2006). Two types of micro-channels were tested a circular micro-tube with a 210 pm diameter, and a square micro-channel with a 214 pm hydraulic diameter. The heat transfer coefficient was higher for the square micro-channel because the corners acted as effective nucleation sites. 

The key analysis of hydrodynamic dispersion of a solute flowing through a tube was performed by Taylor [149] and Aris [150]. They assumed a Poiseuille flow profile in a tube of circular cross-section and were able to show that for long enough times the dispersion of a solute is governed by a one-dimensional convection-diffusion equation ... 

Circular Isothermal Fins on a Horizontal Tube. Tsubouchi and Masuda [269] measured the heat transfer by natural convection in air from circular fins attached to circular tubes, as in the configuration shown in Fig. 4.23/ Correlations for the heat transfer from the tips of the fins (see the figure for definition), and from the cylinder plus vertical fin surfaces, were reported separately. 

H. Miyazaki, Combined Free and Forced Convective Heat Transfer and Fluid Flow in a Rotating Curved Circular Tube, Int. J. Heat Mass Transfer (14) 1295-1309,1971. 

Figure 4-13 represents a history of reactor development. The most well-known type of reactor is shown in Figure 4-13a, which is the horizontal reactor. Horizontal reactors are well studied and understood. These reactors produce good materials and devices. Such reactors have been scaled to hold several 50 mm wafers. These reactors require sufficient gas flows to counter buoyancy driven convection (hot gases rise) and to counter reactor depletion along the flow path. In-position rotation of the wafer minimizes depletion effects. Dramatic increases in wafer numbers have come about by spreading the linear horizontal tube into a circular symmetric device as described below. 

G.I. Taylor (1953, 1954) first analyzed the dispersion of one fluid injected into a circular capillary tube in which a second fluid was flowing. He showed that the dispersion could be characterized by an unsteady diffusion process with an effective diffusion coefficient, termed a dispersion coefficient, which is not a physical constant but depends on the flow and its properties. The value of the dispersion coefficient is proportional to the ratio of the axial convection to the radial molecular diffusion that is, it is a measure of the rate at which material will spread out axially in the system. Because of Taylor s contribution to the understanding of the process of miscible dispersion, we shall, as is often done, refer to it as Taylor dispersion. 

The picture described is that of convective-diffusion of finite size spherical Brownian particles through a circular capillary. In consequence, this may be looked upon as a generalizaton of the Taylor problem for point size particles (Brenner Edwards 1993). A detailed analysis of this problem based on Brenner s moment analysis method has been carried out by Brenner Gaydos (1977), taking into account the tube wall effects on the motions of the particles. Neglecting wall interactions, the essential element of the chromatographic technique can be illustrated by a simple calculation for the average velocity of a particle. 

In regard to nonsteady tube flows. Mason et al. have observed both inward and outward radial migration of rigid, neutrally buoyant spheres in oscillatory (S9b, G9b) and pulsatile (Tl) flows in circular tubes at frequencies up to 3 cps, at which frequencies inertial effects are likely to be important. We refer here to inertial effects arising from the local acceleration terms in the Navier-Stokes equations, rather than from the convective acceleration terms. In the oscillatory case the spheres (a/R 0.10) attained equilibrium positions at about P = 0.85. Important Reynolds numbers here are those based upon mean tube velocity for one-half cycle and upon frequency. Nonneutrally buoyant spheres in oscillatory flow migrate permanently to the tube axis, irrespective of whether they are denser or lighter than the fluid (K4a). 

The heating or cooling of process streams is frequently required. Chapter 6 discusses the fundamentals of convective heat transfer to non-Newtonian fluids in circular and non-circular tubes imder a range of boundary and flow conditions. Limited information on heat transfer from variously shaped objects - plates, cylinders and spheres - immersed in non-Newtonian fluids is also included here. 

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