Nusselt heat transfer equations

By eliminating the film thickness 5 using a = AL/ , the heat transfer equation for Nusselt s film condensation theory can also be written as... 

Equation (21.46) is analogous to the Colburn form of the heat-transfer equations (12.52) and (12.53). A second dimensionless equation, analogous to the Nusselt form of the heat-transfer equation is obtained by multiplying Eq, (21.46) by DG/ti)(jifpD ). This gives... 

Blending of Laminar and Turbulent Nusselt Numbers. The previous two sections provided heat transfer equations for the cases where there is laminar heat transfer from the entire body, (Nue), and turbulent heat transfer from the entire body, (Nu,). To obtain a fit to heat transfer data over the entire range of Ra, the blending equation of Churchill and Usagi [54] is used ... 

Substituting calculated values of Reynolds number and Prandtl number in the film heat transfer equation. The Nusselt number, Nu, ... 

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. 

Natural convection occurs when a solid surface is in contact with a fluid of different temperature from the surface. Density differences provide the body force required to move the flmd. Theoretical analyses of natural convection require the simultaneous solution of the coupled equations of motion and energy. Details of theoretical studies are available in several general references (Brown and Marco, Introduction to Heat Transfer, 3d ed., McGraw-HiU, New York, 1958 and Jakob, Heat Transfer, Wiley, New York, vol. 1, 1949 vol. 2, 1957) but have generally been applied successfully to the simple case of a vertical plate. Solution of the motion and energy equations gives temperature and velocity fields from which heat-transfer coefficients may be derived. The general type of equation obtained is the so-called Nusselt equation hL I L p gp At cjl... 

Vertical Tubes For the following cases Reynolds number < 2100 and is calculated by using F = Wp/ KD. The Nusselt equation for the heat-transfer coefficient for condensate films may be written in the following ways (using liquid physical properties and where L is the cooled lengm and At is — t,) ... 

For liorizontal in-tube condensation at low flow rates Kern s modification (Process Heat Transfer, McGraw-Hill, New York, 1950) of the Nusselt equation is vahd ... 

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

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 -). 

H = heat transfer coefficient ratio, h /hN h i = effective heat transfer film coefficient, Btu/hr-ff-°F hNu condensing film coefficient by Nusselt equation Btu/hr-ff-°F... 

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. 

Experimental and numerical analyses were performed on the heat transfer characteristics of water flowing through triangular silicon micro-channels with hydraulic diameter of 160 pm in the range of Reynolds number Re = 3.2—84 (Tiselj et al. 2004). It was shown that dissipation effects can be neglected and the heat transfer may be described by conventional Navier-Stokes and energy equations as a common basis. Experiments carried out by Hetsroni et al. (2004) in a pipe of inner diameter of 1.07 mm also did not show effect of the Brinkman number on the Nusselt number in the range Re = 10—100. 

Equation (4.12) indicates the effect of viscous dissipation on heat transfer in micro-channels. In the case when the inlet fluid temperature, To, exceeds the wall temperature, viscous dissipation leads to an increase in the Nusselt number. In contrast, when To < Tv, viscous dissipation leads to a decrease in the temperature gradient on the wall. Equation (4.12) corresponds to a relatively small amount of heat released due to viscous dissipation. Taking this into account, we estimate the lower boundary of the Brinkman number at which the effect of viscous dissipation may be observed experimentally. Assuming that (Nu-Nuo)/Nuo > 10 the follow-... 

A number of authors have considered channel cross-sections other than rectangular [102-104]. Figure 2.17 shows some examples of cross-sections for which friction factors and Nusselt numbers were computed. In general, an analytical solution of the Navier-Stokes and the enthalpy equations in such channel geometries would be involved owing to the implementation of the wall boundary condition. For this reason, usually numerical methods are employed to study laminar flow and heat transfer in channels with arbitrary cross-sectional geometry. 

If the Nusselt number given by equation 12.13 is less than 3.5, it should be taken as 3.5. In laminar flow the length of the tube can have a marked effect on the heat-transfer rate for length to diameter ratios less than 500. 

The basic equations for filmwise condensation were derived by Nusselt (1916), and his equations form the basis for practical condenser design. The basic Nusselt equations are derived in Volume 1, Chapter 9. In the Nusselt model of condensation laminar flow is assumed in the film, and heat transfer is assumed to take place entirely by conduction through the film. In practical condensers the Nusselt model will strictly only apply at low liquid and vapour rates, and where the flowing condensate film is undisturbed. Turbulence can be induced in the liquid film at high liquid rates, and by shear at high vapour rates. This will generally increase the rate of heat transfer over that predicted using the Nusselt model. The effect of vapour shear and film turbulence are discussed in Volume 1, Chapter 9, see also Butterworth (1978) and Taborek (1974). 

The equation given by Bromley (1950) can be used to estimate the heat-transfer coefficient for film boiling on tubes. Heat transfer in the film-boiling region will be controlled by conduction through the film of vapour, and Bromley s equation is similar to the Nusselt equation for condensation, where conduction is occurring through the film of condensate. 

The heat transfer coefficient to the vessel wall can be estimated using the correlations for forced convection in conduits, such as equation 12.11. The fluid velocity and the path length can be calculated from the geometry of the jacket arrangement. The hydraulic mean diameter (equivalent diameter, de) of the channel or half-pipe should be used as the characteristic dimension in the Reynolds and Nusselt numbers see Section 12.8.1. 

The convective mass transfer coefficient hm can be obtained from correlations similar to those of heat transfer, i.e. Equation (1.12). The Nusselt number has the counterpart Sherwood number, Sh = hml/Di, and the counterpart of the Prandtl number is the Schmidt number, Sc = p/pD. Since Pr k Sc k 0.7 for combustion gases, the Lewis number, Le = Pr/Sc = k/pDcp is approximately 1, and it can be shown that hm = hc/cp. This is a convenient way to compute the mass transfer coefficient from heat transfer results. It comes from the Reynolds analogy, which shows the equivalence of heat transfer with its corresponding mass transfer configuration for Le = 1. Fire involves both simultaneous heat and mass transfer, and therefore these relationships are important to have a complete understanding of the subject. 

The maximum values of the heat transfer coefficients, and of the corresponding Nusselt numbers, may be predicted satisfactorily from equation 6.58 by differentiating with respect to voidage and putting the derivative equal to zero. 

The Nusselt number with respect to the tube Nu(= hdt/k) is expressed as a function of four dimensionless groups the ratio of tube diameter to length, the ratio of tube to particle diameter, the ratio of the heat capacity per unit volume of the solid to that of the fluid, and the tube Reynolds number, Rec = (ucdtp/p,). However, equation 6.59 and other equations quoted in the literature should be used with extreme caution, as the value of the heat transfer coefficient will be highly dependent on the flow patterns of gas and solid and the precise geometry of the system. 

There is apparently an inherent anomaly in the heat and mass transfer results in that, at low Reynolds numbers, the Nusselt and Sherwood numbers (Figures. 6.30 and 6.27) are very low, and substantially below the theoretical minimum value of 2 for transfer by thermal conduction or molecular diffusion to a spherical particle when the driving force is spread over an infinite distance (Volume 1, Chapter 9). The most probable explanation is that at low Reynolds numbers there is appreciable back-mixing of gas associated with the circulation of the solids. If this is represented as a diffusional type of process with a longitudinal diffusivity of DL, the basic equation for the heat transfer process is ... 

Since the dimensionless equations and boundary conditions governing heat transfer and dilute-solution mass transfer are identical, the solutions to these equations in dimensionless form are also identical. Profiles of dimensionless concentration and temperature are therefore the same, while the dimensionless transfer rates, the Sherwood number (Sh = kL/ ) for mass transfer, and the Nusselt number (Nu = hL/K ) for heat transfer, are identical functions of Re, Sc or Pr, and dimensionless time. Most results in this book are given in terms of Sh and Sc the equivalent results for heat transfer may be found by simply replacing Sh by Nu and Sc by Pr. 

Once the system of equations has been solved, the nondimensional temperature gradient can be easily evaluated at the surface, providing the Nusselt number. It should be expected that the heat transfer depends on the boundary-layer thickness, which in turn depends on the flow field, which is principally governed by the Reynolds number. Figure 6.9 shows a correlation between the Nusselt number and the Reynolds number that was obtained by solving the nondimensional system for several Reynolds numbers. 

The Sherwood number is a nondimensional mass-transfer coefficient that is analogous to the Nusselt number for heat transfer. For the situation of A being dilute in B, the mass transfer at the stagnation surface is derived from the solution to the species equation by... 

Values of overall coefficients of heat transfer are collected in Tables 17.10-17.12. Two sets of formulas for tank-side film coefficients are in Tables 17.13 and 17.14. They relate the Nusselt number to the Reynolds and Prandtl numbers and several other factors. In the equation for jacketed tanks, for example,... 

The parameters (2 + 0.60 Re1/2 Sc1/3) and (2 -f 0.60 ReI/2Pr1/3) represent the Nusselt numbers for mass and heat transfer, respectively (87). Equation 3 derives from Froessling directly, while Spalding substituted thermal diffusivity for molecular diffusivity to establish the basis for Equation 4. 

Labuntsov (L2), 1957 Heat transfer to condensate films on vertical and horizontal surfaces. In laminar region, Nusselt equations are corrected for (a) inertia effects, (b) variation of physical properties with temperature, (c) effects of waves. In turbulent region various universal velocity profiles are used. Results compared with experimental data. 

Since the particles are so small in most cases, the Nusselt number based on the diameter reaches its lower limiting value of 2, i.e., Nu = (h2rs)/kg = 2 when k is the thermal conductivity of the gas and h is the heat transfer coefficient. Then the preceding equation becomes ... 

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