Heat Nusselt number

A comparable curve for the case where the HI condition prevails on one or more walls with the other wall adiabatic is shown as Fig. 10.12. From Figs. 10.11 and 10.12 it is interesting to note for a constant finite value of the aspect ratio that the highest Nusselt number for both the T and HI boundary conditions occurs for the case where the two long walls are heated (2L). The one-long-wall-heated Nusselt number (1L) falls below the Nusselt value for four heated walls. 

Nusse/t Number. Empidcal correlations can be obtained for a particular size of tube diameter and particular flow conditions. To generalize such results and to apply the correlations to different sizes of equipment and different flow conditions, the heat-transfer coefficient, Z, is traditionally nondimensionalized by the use of the Nusselt number, Nu named after Wilhelm Nusselt,... 

Correlations for Convective Heat Transfer. In the design or sizing of a heat exchanger, the heat-transfer coefficients on the inner and outer walls of the tube and the friction coefficient in the tube must be calculated. Summaries of the various correlations for convective heat-transfer coefficients for internal and external flows are given in Tables 3 and 4, respectively, in terms of the Nusselt number. In addition, the friction coefficient is given for the deterrnination of the pumping requirement. 

The convective heat-transfer coefficient and friction factor for laminar flow in noncircular ducts can be calculated from empirically or analytically determined Nusselt numbers, as given in Table 5. For turbulent flow, the circular duct data with the use of the hydrauhc diameter, defined in equation 10, may be used. 

NUh2 is the Nusselt number for uniform heat flux boundary condition along the flow direction and periphery. 

Heat transfer in static mixers is intensified by turbulence causing inserts. For the Kenics mixer, the heat-transfer coefficient b is two to three times greater, whereas for Sulzer mixers it is five times greater, and for polymer appHcations it is 15 times greater than the coefficient for low viscosity flow in an open pipe. The heat-transfer coefficient is expressed in the form of Nusselt number Nu = hD /k as a function of system properties and flow conditions. 

Limiting Nusselt numbers for slug-flow annub may be predicted (for constant heat flux) from Trefethen (General Discu.s.sion.s on Heat Tran.sfer, London, ASME, New York, 1951, p. 436) ... 

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 <... 

A droplet Nusselt number = 2, corresponding to pure conduction (Reynolds number = 0) to infinity, is employed for evaluating the coefficient of heat transfer. 

For low values of the Reynolds number, such as 10, where sn eamline flow should certainly apply, the Nusselt number has a value of about 2, and a typical value of the average heat transfer coefficient is 10 ". For a Reynolds number of 104, where the gas is certainly in turbulent flow, the value of the Nusselt number is typically 20. Hence there is only a difference of a factor of ten in the heat transfer coefficient between tlrese two extreme cases. 

The Nusselt number for the heat transfer between a gas and a solid particle of radius d, is given by the Ranz-Marshall equation... 

The inside film heat transfer eoeffieient (h ) ean be ealeulated from the following Nusselt number eorrelation ... 

Flemeon is the first standard reference book that presents the equations for calculating thermal updrafts. These equations are repeated and expanded in other standard reference books, including Heinsohn, Goodfellow, and the ACGIFl Industrial Ventilation Manual.These equations are derived from the more accurate formulas for heat transfer (Nusselt number) at natural convection (where density differences, due to temperature differences, provide the body force required to move the fluid) and both the detailed and the simplified formulas can be found in handbooks on thermodynamics (e.g., Perry--, and ASHRAE -). 

Re) nolds number, dimensionless Nusselt number, (heat transfer)... 

X = distance film has fallen g = gravitational constant Pi = liquid density = latent heat of vaporization JL = liquid viscosity k = liquid thermal conductivity AT = temperature difference = (Tb bbi,p i -NrUj = local Nusselt number, h x/k, h = local heat transfer coefficient... 

The heat and mass transfer coefficients are given in the literature usually in terms of the Sherwood and Nusselt numbers... 

The correlation studies of heat and mass transfer in pellet beds have been investigated by many, usually in terms of the. /-factors (113-115). According to Chilton and Colburn the two. /-factors are equal in value to one half of the Fannings friction factor / used in the calculation of pressure drop. The. /-factors depend on the Reynolds number raised to a factor varying from —0.36 to —0.68, so that the Nusselt number depends on the Reynolds number raised to a factor varying from 0.64 to 0.32. In the range of the Reynolds number from 10 to 170 in the pellet bed, jd should vary from 0.5 to 0.1, which yields a Nusselt number from 4.4 to 16.1. The heat and mass transfer to wire meshes has received much less attention (110,116). The correlation available shows that the /-factor varies as (Re)-0-41, so that the Nusselt number varies as (Re)0-69. In the range of the Reynolds number from 20 to 420, the j-factor varies from 0.2 to 0.05, so that the Nusselt number varies from 3.6 to 18.6. The Sherwood number for CO is equal to 1.05 Nu, but the Sherwood number for benzene is 1.31 Nu. 

When the gas velocities are increased, both the Reynolds number and the Nusselt number would increase, while the ratio Nu/Re decreases with (Re) to the —0.4 to —0.6 power. An increase in gas velocities would improve on the heat and mass transfer coefficients from gas to wall, but would also increase the fraction of heat that is not given up to the wall and the fraction of benzene that never goes near the wall due to the reduction in residence time. 

The Biot number is essentially the ratio of the resistance to heat transfer within the particle to that within the external fluid. At first sight, it appears to be similar in form to the Nusselt Number Nu where ... 

The minimum value of the Nusselt Number for which equation 9.216 applies is 3.5. Reynolds Numbers in the range 2000-10,000 should be avoided in designing heat exchangers as the flow is then unstable and coefficients cannot be predicted with any degree of accuracy. If this cannot be avoided, the lesser of the values predicted by Equations 9.214 and 9.216 should be used. 

By comparing equations 11.61 and 11.66, it is seen that the local Nusselt number and the heat transfer coefficient are both some 36 per cent higher for a constant surface heat flux as compared with a constant surface temperature. 

The heat transfer correlations are considered separately in the laminar and turbulent regimes in Figs. 2.21 and 2.22, respectively. The dependence of the Nusselt number on the Reynolds number is stronger in all the micro-channel predictions compared to conventional results, as indicated by the steeper slopes of the former Choi et al. (1991) predict the strongest variation of Nusselt number with Re. The predictions for all cases by Peng et al. (1996) also fall below those for a conventional channel. 

In general, the axial heat conduction in the channel wall, for conventional size channels, can be neglected because the wall is usually very thin compared to the diameter. Shah and London (1978) found that the Nusselt number for developed laminar flow in a circular tube fell between 4.36 and 3.66, corresponding to values for constant heat flux and constant temperature boundary conditions, respectively. 

However, for flow in micro-channels, the wall thickness can be of the same order of channel diameter and will affect the heat transfer significantly. For example, Choi et al. (1991) reported that the average Nusselt numbers in micro-channels were much lower than for standard channels and increased with the Reynolds number. 

DC current was supplied through the development and test sections for direct heating. The outer temperature on the heated wall was measured by means of an infrared radiometer. Experiments were carried out in the range of Re = 10-450. The average Nusselt number was calculated using the average temperature of the inner tube wall and mean temperature of the fluid at the inlet and outlet of the tube. 

The dependence of the local Nusselt number on non-dimensional axial distance is shown in Fig. 4.3a. The dependence of the average Nusselt number on the Reynolds number is presented in Fig. 4.3b. The Nusselt number increased drastically with increasing Re at very low Reynolds numbers, 10 < Re < 100, but this increase became smaller for 100 < Re < 450. Such a behavior was attributed to the effect of axial heat conduction along the tube wall. Figure 4.3c shows the dependence of the relation N /N on the Peclet number Pe, where N- is the power conducted axially in the tube wall, and N is total electrical power supplied to the tube. Comparison between the results presented in Fig. 4.3b and those presented in Fig. 4.3c allows one to conclude that the effect of thermal conduction in the solid wall leads to a decrease in the Nusselt number. This effect decreases with an increase in the... 

Reynolds number. It should be stressed that the heat transfer coefficient depends on the character of the wall temperature and the bulk fluid temperature variation along the heated tube wall. It is well known that under certain conditions the use of mean wall and fluid temperatures to calculate the heat transfer coefficient may lead to peculiar behavior of the Nusselt number (see Eckert and Weise 1941 Petukhov 1967 Kays and Crawford 1993). The experimental results of Hetsroni et al. (2004) showed that the use of the heat transfer model based on the assumption of constant heat flux, and linear variation of the bulk temperature of the fluid at low Reynolds number, yield an apparent growth of the Nusselt number with an increase in the Reynolds number, as well as underestimation of this number. 

Adams et al. (1998) investigated turbulent, single-phase forced convection of water in circular micro-channels with diameters of 0.76 and 1.09 mm. The Nusselt numbers determined experimentally were higher than those predicted by traditional Nusselt number correlations such as the Gnielinski correlation (1976). The data suggest that the extent of enhancement (deviation) increases as the channel diameter decreases. Owhaib and Palm (2004) investigated the heat transfer characteristics... 

Warrier et al. (2002) conducted experiments of forced convection in small rectangular channels using FC-84 as the test fluid. The test section consisted of five parallel channels with hydraulic diameter = 0.75 mm and length-to-diameter ratio Lh/r/h = 433.5 (Fig. 4.5d and Table 4.4). The experiments were performed with uniform heat fluxes applied to the top and bottom surfaces. The wall heat flux was calculated using the total surface area of the flow channels. Variation of single-phase Nusselt number with dimensionless axial distance is shown in Fig. 4.6b. The numerical results presented by Kays and Crawford (1993) are also shown in Fig. 4.6b. The measured values agree quite well with the numerical results. 

Qu et al. (2000) carried out experiments on heat transfer for water flow at 100 < Re < 1,450 in trapezoidal silicon micro-channels, with the hydraulic diameter ranging from 62.3 to 168.9pm. The dimensions are presented in Table 4.5. A numerical analysis was also carried out by solving a conjugate heat transfer problem involving simultaneous determination of the temperature field in both the solid and fluid regions. It was found that the experimentally determined Nusselt number in micro-channels is lower than that predicted by numerical analysis. A roughness-viscosity model was applied to interpret the experimental results. 

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