Heat transfer coefficient velocity distribution

Laminar Flow Although heat-transfer coefficients for laminar flow are considerably smaller than for turbulent flow, it is sometimes necessary to accept lower heat transfer in order to reduce pumping costs. The heat-flow mechanism in purely laminar flow is conduction. The rate of heat flow between the walls of the conduit and the fluid flowing in it can be obtained analytically. But to obtain a solution it is necessary to know or assume the velocity distribution in the conduit. In fully developed laminar flow without heat transfer, the velocity distribution at any cross section has the shape of a parabola. The velocity profile in laminar flow usually becomes fully established much more rapidly than the temperature profile. Heat-transfer equations based on the assumption of a parabolic velocity distribution will therefore not introduce serious errors for viscous fluids flowing in long ducts, if they are modified to account for effects caused by the variation of the viscosity due to the temperature gradient. The equation below can be used to predict heat transfer in laminar flow. 

Figure 5.37a-d illustrates a typical temperature distribution in the range of the angle 0 < 0 < 180° (where 0 = 0° is at the top of the tube). The heat flux was q = 8,000 W/m, the superficial gas velocity was Uqs = 36 m/s. The superficial liquid velocities were 0.016, 0.027, 0.045 and 0.099 m/s, respectively. The flow moves from the right to the left. The color shades are indicative of the wall temperature. Comparison to simultaneous visual observations shows that the distribution of heat transfer coefficient at Uls = 0.0016 m/s corresponds to dryout on the upper part of the pipe. 

No and Kazimi (1982) derived the wall heat transfer coefficient for the forced-convective two-phase flow of sodium by using the momentum-heat transfer analogy and a logarithmic velocity distribution in the liquid film. The final form of their correlation is expressed in terms of the Nusselt number based on the bulk liquid temperature, Nuft ... 

The data of Fig. 20 also point out an interesting phenomenon—while the heat transfer coefficients at bed wall and bed centerline both correlate with suspension density, their correlations are quantitatively different. This strongly suggests that the cross-sectional solid concentration is an important, but not primary parameter. Dou et al. speculated that the difference may be attributed to variations in the local solid concentration across the diameter of the fast fluidized bed. They show that when the cross-sectional averaged density is modified by an empirical radial distribution to obtain local suspension densities, the heat transfer coefficient indeed than correlates as a single function with local suspension density. This is shown in Fig. 21 where the two sets of data for different radial positions now correlate as a single function with local mixture density. The conclusion is That the convective heat transfer coefficient for surfaces in a fast fluidized bed is determined primarily by the local two-phase mixture density (solid concentration) at the location of that surface, for any given type of particle. The early observed parametric effects of elevation, gas velocity, solid mass flux, and radial position are all secondary to this primary functional dependence. 

As opposed to the relatively uniform bed structure in dense-phase fluidization, the radial and axial distributions of voidage, particle velocity, and gas velocity in the circulating fluidized bed are very nonuniform (see Chapter 10) as a result the profile for the heat transfer coefficient in the circulating fluidized bed is nonuniform. 

Figure 12.17. Effect of gas velocity on the radial distribution of the heat transfer coefficient in a circulating fluidized bed (from Bi et at., 1989).

A transparent gas flows into and out of a black circular tube of length L and diameter D. The gas has a mean velocity um, specific heat at constant pressure cp and density p. The wall of the tube is thin, and the outer surface is insulated. The tube wall is heated electrically and a uniform input of heat is provided per unit area, per unit time. Determine the local wall temperature distribution along the tube length. Assume that the convective heat transfer coefficient h between the gas and the inside of the tube is constant. 

Air at a temperature of 10°C enters a plane duct with a distance of 6 cm between the two surfaces of the duct. The mean velocity of the air in the duct is 40 m/s. The walls of the duct are maintained at a uniform temperature of 40°C. Assuming that the velocity and temperature distributions are uniform at the entrance to the duct, determine how the mean heat transfer coefficient varies with distance along the duct. 

Using the linear-velocity profile in Prob. 5-2 and a cubic-parabola temperature distribution [Eq. (5-30)], obtain an expression for heat-transfer coefficient as a function of the Reynolds number for a laminar boundary layer on a flat plate. 

The effect of gas velocity on radial distribution of heat transfer coefficient is shown in Figs 9 and 10. With increasing gas velocity the heat transfer coefficients decrease. For the lower bed sections (see Fig. 10) the radial distributions are mainly affected by solids concentrations, and for the higher bed sections this trend changes significantly. 

Figure 20 indicates that the ratio of hjh, between probes B and A, is in the range of 1.1 to 1.8 and keeps almost constant along the axial direction of the probe. The ratio of hjhx is large at low gas velocities, and when the probes are located at the center of the bed (see Fig. 20). This implies that the distribution of heat transfer coefficients, even with rings, along the radial direction of the bed becomes flatter at a higher bed level where solids concentration is low. Stated alternatively, with increasing gas velocity the influence of the rings on heat transfer coefficient diminishes.

Two-fluid simulations have also been performed to predict void profiles (Kuipers et al, 1992b) and local wall-to-bed heat transfer coefficients in gas fluidized beds (Kuipers et al., 1992c). In Fig. 18 a comparison is shown between experimental (a) and theoretical (b) time-averaged porosity distributions obtained for a 2D air fluidized bed with a central jet (air injection velocity through the orifice 10.0 m/s which corresponds to 40u ). The experimental porosity distributions were obtained with the aid of a nonintrusive light transmission technique where the principles of liquid-solid fluidization and vibrofluidization were employed to perform the necessary calibration. The principal differences between theory and experiment can be attributed to the simplified solids rheology assumed in the hydrodynamic model and to asymmetries present in the experiment. 

Shell-and-tube exchanger with reactants and catalyst inside the tubes, 250 to 400 m /m. Tube diameter <50 mm. Gas with fixed bed of catalyst use high mass gas velocity to improve heat transfer kg/s m > 1.35. To ensure good gas distribution and negligible backmixing, Pe > 2 height/catalyst particle diameter H/D > 100 and D/D < 0.10. Gas velocity 3 to 10 m/s residence time 0.6 to 2 s. Heat transfer coefficient U = 0.05 kW/m K. For fast reactions, catalyst pore diffusion mass transfer may control if catalyst diameter is >1.5 mm. Liquids with fixed bed of catalyst to minimize backmixing, Pe> use UD > 200 and D/D <0.10. Liquid velocity 1 to 2 m/s residence time 2 to 6 s. Heat transfer coefficient U = 0.5 kW/m -K. 

For a calorimeter experiment, an insulated cylindrical shell of diameter Do = 102 cm is placed concentrically around a 2 = 2 m long furnace of diameter D, = 100 cm (Fig. 6P-4). The distribution of flame heat flux q" x) [W/m2] acting axisymetrically on the inner surface of the furnace wall is linear with a maximum of q l = 210,000 W/m2 in the middle of the furnace. Heat is removed by water flowing coaxially between the furnace and the shell. The water inlet temperature and velocity are 7 = 285 K and V = 1 m/s, respectively. Neglecting the fumace-wall thickness, determine (a) the heat transfer coefficient on the outer surface of the furnace, (b) the longitudinal temperature of the furnace wall. 

The heat transfer coefficient for the gas convective component can be regarded as comparable to that at incipient fluidization conditions. By assuming /igc = hmt, Xavier and Davidson [54] simulated the fluidization system, considering a pseudofluid with the apparent thermal conductivity Ka of the gas-solid medium flowing at the same superficial velocity and the same inlet and outlet temperatures as the gas. They found the temperature distribution in the bed as... 

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