Fully Developed Heat Transfer

Thermally fully developed heat transfer do to electro-osmotic fluid transport in micro parallel plate channel and micro mbe has been recently investigated by [21]. The dimensionless temperature profile and corresponding Nusselt number have been determined for imposed constant wall heat flux and constant temperature. The complement paper [22] study the effect of viscous dissipation. These two papers gives important physical details and references. The analyses of both papers is based on the classical simplifying assumptions that are avoided in the book by Mikhailov and Ozisik [20]. 

Marcos and Bergles [189] correlated their data for fully developed heat transfer for water and ethylene glycol in glass and metal tubes by the equation... 

FIGURE 5.41 Fully developed heat transfer characteristics for longitudinal turbulent flow between a triangular array and a square array [261]. 

Fully Developed Heat Transfer—Purely Viscous Fluids... 

The solution of Eq. 10.55 describing the conservation of energy requires the solution of the momentum equation for a specified constitutive relationship. The previous section provides this information for a power-law fluid. This section will treat the fully developed heat transfer... 

Fully-developed heat transfer to power-law fluids in laminar flow... 

Near the entrance of a pipe where the fluid is being heated, the temperature profile is not fully developed and the local coefficient h is greater than the fully developed heat-transfer coefficient for turbulent flow. At the entrance itself where no temperature gradient has been established, the value of h is infinite. The value of h drops rapidly and is approximately the same as at L/D = 60, where L is the entrance length. These relations for turbulent flow inside a pipe are as follows where the entrance is an abrupt contraction. 

The fully developed velocity distribution in micro parallel plate channel and micro tube are well known [19]. Using a fully developed velocity one could investigate thermally developing heat transfer and its limiting case - thermally developed heat transfer. The corresponding solutions for electro-osmotic flow in micro parallel plate channel and micro tube are special cases from the general results given in the book by [20]. 

S. Piva, An Analytical Approach to Fully Developed Heating of Laminar Flows in Circular Pipes, Int. Comm. Heat Mass Transfer, (22/6) 815-824,1995. 

The measured dimensionless heat transfer factors jH (that is, St Pr ) are shown in Fig. 10.28 as a function of the Reynolds number Re for concentrations ranging from 10 to 1000 wppm polyacrylamide [37, 93]. These measurements were made at xld equal to 430, which corresponds approximately to thermally fully developed conditions as shown in the figure. The asymptotic values of the fully established heat transfer coefficients are reached at a concentration of 500 wppm of polyacrylamide, whereas less than 50 wppm was required to reach... 

The minimum velocity requited to maintain fully developed turbulent flow, assumed to occur at Reynolds number (R ) of 8000, is inside a 16-mm inner diameter tube. The physical property contribution to the heat-transfer coefficient inside and outside the tubes are based on the following correlations (39) ... 

Limiting Nusselt numbers for laminar flow in annuli have been calculated by Dwyer [Nucl. Set. Eng., 17, 336 (1963)]. In addition, theoretical analyses of laminar-flow heat transfer in concentric and eccentric annuh have been published by Reynolds, Lundberg, and McCuen [Jnt. J. Heat Ma.s.s Tran.sfer, 6, 483, 495 (1963)]. Lee fnt. J. Heat Ma.s.s Tran.sfer, 11,509 (1968)] presented an analysis of turbulent heat transfer in entrance regions of concentric annuh. Fully developed local Nusselt numbers were generally attained within a region of 30 equivalent diameters for 0.1 < Np < 30, lO < < 2 X 10, 1.01 <... 

In some convection equations, such as for turbulent pipe flow, a special correction factor is used. This factor relates to the heat transfer conditions at the flow inlet, where the flow has not reached its final velocity distribution and the boundary layer is not fully developed. In this region the heat transfer rate is better than at the region of fully developed flow. 

For the common problem of heat transfer between a fluid and a tube wall, the boundary layers are limited in thickness to the radius of the pipe and, furthermore, the effective area for heat flow decreases with distance from the surface. The problem can conveniently be divided into two parts. Firstly, heat transfer in the entry length in which the boundary layers are developing, and, secondly, heat transfer under conditions of fully developed flow. Boundary layer flow is discussed in Chapter 11. 

This expression is applicable only to the region of fully developed flow. The heat transfer coefficient for the inlet length can be calculated approximately, using the expressions given in Chapter 11 for the development of the boundary layers for the flow over a plane surface. It should be borne in mind that it has been assumed throughout that the physical properties of the fluid are not appreciably dependent on temperature and therefore the expressions will not be expected to hold accurately if the temperature differences are large and if the properties vary widely with temperature. 

In addition to momentum, both heat and mass can be transferred either by molecular diffusion alone or by molecular diffusion combined with eddy diffusion. Because the effects of eddy diffusion are generally far greater than those of the molecular diffusion, the main resistance to transfer will lie in the regions where only molecular diffusion is occurring. Thus the main resistance to the flow of heat or mass to a surface lies within the laminar sub-layer. It is shown in Chapter 11 that the thickness of the laminar sub-layer is almost inversely proportional to the Reynolds number for fully developed turbulent flow in a pipe. Thus the heat and mass transfer coefficients are much higher at high Reynolds numbers. 

The application of the analogies to the problems of heat and mass transfer to plane surfaces and to pipe walls for fully developed flow is discussed later. 

In fully developed flow, equations 12.102 and 12.117 can be used, but it is preferable to work in terms of the mean velocity of flow and the ordinary pipe Reynolds number Re. Furthermore, the heat transfer coefficient is generally expressed in terms of a driving force equal to the difference between the bulk fluid temperature and the wall temperature. If the fluid is highly turbulent, however, the bulk temperature will be quite close to the temperature 6S at the axis. 

Zhao YH, Masuoka T, Tsuruta T (2002) Unified theoretical prediction of fully developed nucleate boiling and critical heat flux based on a dynamic microlayer model. Int J Heat Mass Transfer 45 3189-3197... 

The micro-channels utilized in engineering systems are frequently connected with inlet and outlet manifolds. In this case the thermal boundary condition at the inlet and outlet of the tube is not adiabatic. Heat transfer in a micro-tube under these conditions was studied by Hetsroni et al. (2004). They measured heat transfer to water flowing in a pipe of inner diameter 1.07 mm, outer diameter 1.5 mm, and 0.600 m in length, as shown in Fig. 4.2b. The pipe was divided into two sections. The development section of Lj = 0.245 m was used to obtain fully developed flow and thermal fields. The test section proper, of heating length Lh = 0.335 m, was used for collecting the experimental data. 

Chakraborty S (2006) Analytical solutions of Nusselt number for thermally fully developed flow in microtubes under a combined action of electroosmotic forces and imposed gradients. Int J Heat Mass Transfer 49 810-813... 

The values of the Nusselt and the Poiseuille numbers for heat transfer and friction for fully developed laminar flows through specifled channels are presented in Table 7.1 (Shah and London 1978). 

For high values of the Reynolds number, the mean value of the Nusselt number does not differ significantly from the theoretical value for fully developed flow. On the contrary, at low Re the effects of conjugate heat transfer on the mean value of... 

In the flow region between laminar and fully developed turbulent flow heat-transfer coefficients cannot be predicted with certainty, as the flow in this region is unstable, and the transition region should be avoided in exchanger design. If this is not practicable the coefficient should be evaluated using both equations 12.11 and 12.13 and the least value taken. 

Liquid core temperature and velocity distribution analysis. BankofT (1961) analyzed the convective heat transfer capability of a subcooled liquid core in local boiling by using the turbulent liquid flow equations. He found that boiling crisis occurs when the core is unable to remove the heat as fast as it can be transmitted by the wall. The temperature and velocity distributions were analyzed in the singlephase turbulent core of a boiling annular flow in a circular pipe of radius r. For fully developed steady flow, the momentum equation is given as... 

The interaction of parametric effects of solid mass flux and axial location is illustrated by the data of Dou et al. (1991), shown in Fig. 19. These authors measured the heat transfer coefficient on the surface of a vertical tube suspended within the fast fluidized bed at different elevations. The data of Fig. 19 show that for a given size particle, at a given superficial gas velocity, the heat transfer coefficient consistently decreases with elevation along the bed for any given solid mass flux Gs. At a given elevation position, the heat transfer coefficient consistently increases with increasing solid mass flux at the highest elevation of 6.5 m, where hydrodynamic conditions are most likely to be fully developed, it is seen that the heat transfer coefficient increases by approximately 50% as Gv increased from 30 to 50 kg/rrfs. 

Use of medium-scale heat flow calorimeter for separate measurement of reaction heat removed via reaction vessel walls and via reflux condenser system, under fully realistic processing conditions, with data processing of the results is reported [2], More details are given elsewhere [3], A new computer controlled reaction calorimeter is described which has been developed for the laboratory study of all process aspects on 0.5-2 1 scale. It provides precise data on reaction kinetics, thermochemistry, and heat transfer. Its features are exemplified by a study of the (exothermic) nitration of benzaldehyde [4], A more recent review of reaction safety calorimetry gives some comment on possibly deceptive results. [5],... 

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