Nusselt number correlation equations

In these equations, Tj is the temperature at coordinates (r, t) in the macrograin, pcp is the average value of the heat capacity per unit volume of the macroparticle, is the effective thermal diffiisivity in the macrograin and (—Affr) the enthalpy of polymerization. In Equation 2.141, the parameter hp is the average convective heat transfer coefficient, usually calculated from a Nusselt number correlation. Early works tended to use the well-known Ranz-Marshall correlation for evaporation from a droplet however, it has been... 

First the dimensionless characteristics such as Re and Pr in forced convection, or Gr and Pr in free convection, have to be determined. Depending on the range of validity of the equations, an appropriate correlation is chosen and the Nu value calculated. The equation defining the Nusselt number is... 

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 experimental results obtained for a wide range of systems(96-99) are correlated by equation 6.58, in terms of the Nusselt number (Nu = hd/k) for the particle expressed as a function of the Reynolds number (Re c = ucdp/fx) for the particle, the Prandtl number Pr for the liquid, and the voidage of the bed. This takes the form ... 

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. 

Most of the data available to substantiate Equations 1 to 4 pertain to single droplets in still air. Therefore, in most cases, the Nusselt number is 2, and Froessling s correlation for the Nusselt number in terms of the Reynolds and Schmidt numbers is neither verified nor cast in doubt. Data of Ranz and Marshall (102) do afford substantiation in particular, they verify Equation 4. 

Using the values of the local Nusselt number given in Fig. 6-11, obtain values for the average Nusselt number as a function of the Reynolds number. Plot the results as log Nu versus log Re, and obtain an equation which represents all the data. Compare this correlation with that given by Eq. (6-17) and Table 6-2. 

In turbulent flow, wall roughness increases the heat transfer coefficient h by a factor of 2 or more [Dipprey and Saber.sky (1963)]. The convection heat transfer coefficient for rough tubes can be calculated approximately from the Nusselt number relations such as Eq. 8-71 by using the friction factor determined from the Moody chart or the Colebrook equation. However, this approach is not very accurate since there is no further increase in h with/for /> 4/sn,ooih [Norris (1970)1 and correlations developed specifically for rough tubes should be used when more accuracy is desired. 

Coefficients of heat transfer by natural convection from bodies of various shapes, chiefly plates and cylinders, are correlated in terms of Grashof, Prandtl, and Nusselt numbers. Table 8.9 covers the most usual situations, of which heat losses to ambient air are the most common process. Simplified equations are shown for air. Transfer of heat by radiation is appreciable even at modest temperatures such data are presented in combination with convective coefficients in item 16 of this table. 

This is the appropriate correlation to use when there is heat or mass (i.e., substitute Nu by Sh) transfer from a sphere immersed in a stagnant film is studied, Nu = 2. The second term in (5.294) accounts for convective mechanisms, and the relation is derived from the solution of the boundary layer equations. For higher Re3molds numbers the Nusselt number is set equal to the relation resulting from the boundary layer analysis of a flat plate ... 

The process of formulating mesoscale models from the microscale equations is widely used in transport phenomena (Ferziger Kaper, 1972). For example, heat transfer between the disperse phase and the fluid depends on the Nusselt number, and mass transfer depends on the Sherwood number. Correlations for how the Nusselt and Sherwood numbers depend on the mesoscale variables and the moments of the NDF (e.g. mean particle temperature and mean particle concentration) are available in the literature. As microscale simulations become more and more sophisticated, modified correlations that are based on the microscale results will become more and more common (Beetstra et al, 2007 Holloway et al, 2010 Tenneti et al, 2010). Note that, because the kinetic equation requires mesoscale models that are valid locally in phase space (i.e. for a particular set of mesoscale variables) as opposed to averaged correlations found from macroscale variables, direct numerical simulation of the microscale model is perhaps the only way to obtain the data necessary in order for such models to be thoroughly validated. For example, a macroscale model will depend on the average drag, which is denoted by... 

It was stated that the most important result of the previous section is the correlation, at the leading order of approximation, between the Nusselt and Peclet numbers, namely, equation (9-230) ... 

We have considered the thermal boundary-layer problem in this chapter for an arbitrary 2D body with no-slip boundary conditions for Re 1 and Pr (or Sc) either arbitrarily large or small. If we assume that we have a body of the exact same shape, but the surface of which is a slip surface (e.g., it is an interface, so that the surface tangential velocity is not zero), the form of the correlation between Nusselt number and Pr will change for Pr 1. Solve this problem, i.e., derive the governing boundary-layer equation, and show that it has a similarity solution. What is the resulting form of the heat transfer correlation among Nu, Re, and Pr ... 

These equations apply to ordinary fluids (not liquid metals) and ignore radiative transfer. Equation 5.35 is rarely used and applies to very low Re or very long mbes. No correlation is available for the transition region, but Equation 5.36 should provide a lower limit on hdt/K in the transition region. The dimensionless number, hdt /k, is the Nusselt number, Nu. 

For single horizontal cylinders, the heat-transfer coefficient can be correlated by an equation containing three dimensionless groups, the Nusselt number, the Prandtl number, and the Grashof number, or specifically. 

FLOW INSIDE PIPES. Correlations for mass transfer to the inside wall of a pipe are of the same form as those for heat transfer, since the basic equations for diffusion and conduction are similar. For laminar flow, the Sherwood number shows the same trends as the Nusselt number, with a limiting value of 3.66 for very long tubes and a one-third-power dependence on flow rate for short tubes. The solution shown in Figure 12.2 can be used for Agh if the Graetz number is based on the dilfusivity or on the Schmidt number as follows ... 

Raithby and Hollands, in the chapter on natural convection, have developed correlation equations for external convection from isothermal bodies. In the correlation equations, the conduction Nusselt number is based on the shape factors developed in these sections. 

Heat Transfer in Smooth Circular Ducts. For gases and liquids (Pr > 0.5), very little difference exists between the Nusselt number for uniform wall temperature and the Nusselt number for uniform wall heat flux in smooth circular ducts. However, for Pr < 0.1, there is a difference between NuT and NuH- Table 5.11 presents the fully developed turbulent flow Nusselt number in a smooth circular duct for Pr > 0.5. The correlation proposed by Gnielinski [69] is recommended for Pr > 0.5, as are those suggested by Bhatti and Shah [45]. In this table, the / in the equation is calculated using the Prandtl [52]-von Karman [53]-Nikuradse [43] Cole-brook [54] Filonenko [55] or Techo et al. [56] correlations shown in Table 5.8. 

Heat Transfer in Rough Circular Ducts The Nusselt number for a complete, rough flow regime in a circular duct is given in Table 5.12. The term/in this table denotes the friction factor for fully rough flow. It is given by the Nikuradse [60] correlation shown in Table 5.9. The recommended equations for practical calculations are those correlations by Bhatti and Shah [45] shown in Table 5.12. 

Bhatti and Shah [45] and Sparrow and Lin [133] have performed a comparison of Nusselt numbers predicted using Eq. 5.222 or other equations for parallel plate ducts and the Nusselt number calculated using the equation for circular ducts replacing 2a with the hydraulic diameter of the parallel plate duct. It was concluded that the Nusselt number for parallel plate ducts can be determined using the circular duct correlations. 

For most engineering calculations of friction factors and Nusselt numbers for fully developed flow in rectangular ducts, it is sufficiently accurate to use the circular duct correlations by replacing the circular duct diameter 2a with the hydraulic diameter Dh = 4abf(a + b) or with D/, defined by the following equations to consider the shape effect [168] ... 

Equation 5.286 is valid for 0.7 < Pr < 5 and 20 < De < 705. This correlation agrees quite well with the experimental data for air for the boundary condition [224]. It also represents the Nusselt number for the boundary condition [229] quite well. 

It should be noted again that the effect of the wall thermal boundary condition on the Nus-selt number for coils is not significant for the fluids with Pr > 0.7. Equation 5.286 can also be used for the , , and thermal boundary conditions. Furthermore, the appropriate correlation for circular cross section coiled tubes can be adopted with the substitution of the appropriate hydraulic diameter for 2a to calculate the Nusselt number when the parameters are out of the application range as is the case in Eq. 5.286. 

Convective heat transfer from a flat surface to a row of impinging, submerged air jets formed by square-edged orifices having a length/diameter ratio of unity has been measured [89], Local Nusselt numbers were averaged over the spanwise direction, and averaged values were correlated by the equation... 

Jet-induced crossflow has been found to have an important effect on impingement heat transfer [82, 92, 93]. In order to delineate its influence on average convective coefficients more clearly, Obot and Trabold have identified three crossflow schemes, referred to minimum, intermediate, and maximum, and correlated their experimental data. The best heat transfer performance was obtained with the minimum crossflow scheme. Intermediate and complete crossflow was associated with varying degrees of degradation. The average Nusselt numbers for air were represented by the equation... 

Using a long parallel-plate nozzle to produce fully developed turbulent jets, Wolf et al. [103] correlated their stagnation zone Nusselt number data to an accuracy of 10 percent by the equation... 

The correlation is based on Re . from 17,000 to 79,000 and Pr between 2.8 and 5.0. Vader et al. [104] used a converging nozzle to produce uniform velocity profile water jets. These nozzles were intended to suppress but not to eliminate turbulence. The stagnation zone Nusselt numbers were correlated by an equation... 

In this study, we were interested in the hydrodynamics on the one hand and its influence on the heat or mass transfer on the other hand. Recent numerical results based on the resolution of the Navier-Stokes equations and the heat equation were used to confront various correlations of the literature for the drag coefficient and the Nusselt number (or Sherwood number). 

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