Nusselt number equation

Very little work has been reported on vaporization under conditions of turbulent gas flow. Ingebo (61), for example, took pains to minimize approach stream turbulence. Two exceptions are the investigations of Maisel and Sherwood (83) and Fledderman and Hanson (27). Neither went so far in analysis as insertion into the Nusselt number equations of allowance for the additional relative velocity between droplet and air stream occasioned by turbulence. In the case of Maisel and Sherwood s investigation with model droplets at fixed positions, the effect would not be expected to be extreme, because at all times there was appreciable relative velocity, discounting turbulence. However, in Fledderman and Hanson s experiments the relative velocity, discounting turbulence, fell away as the droplets accelerated up to stream velocity. Thus turbulence would eventually provide the only appreciable relative velocity. The results indicate a substantial increase in vaporization rate because of the turbulence and provide some basis for gross engineering estimates. 

After the temperature distribution was obtained, following definitions were used to calculate the Nusselt number. Non-dimensionalizing the temperature by the fluid temperature at the wall instead of the wall temperature makes the boundary eondition for the eigenvalue problem easier to handle for the uniform temperature boundary condition. Then they derived the Nusselt number equations from the energy balanee at the wall so that temperature jump could be implemented. The details of this derivation ean be found in the references. 

The flow behavior of molten chocolate can also be affected by changes in processing conditions. These may lead to different values, because of the effect of non-Newtonian flow at the walls of the processing equipment. The wall shear rate is often characterized by the Nusselt number equation, which contains the so-called <5 factor. The S factor is the ratio of the wall shear rate for a non-Newtonian fluid to that of a Newtonian fluid at the same flow rate. For power law fluids, this factor can be calculated ... 

Thin-Layer Approximation. Laminar analyses often make the further approximation that the boundary layer is so thin that when the simplified equations of motion are rewritten in terms of local surface coordinates, i.e., in terms of the x and y of Fig. 4.3a, several terms normally associated with curvature effects can be dropped. The Nusselt number equation, based on solutions to such laminar thin-layer equation sets, always takes the form... 

The wall coefficient is usually correlated as a Nusselt number equation, i.e. 

The objective is to develop a model that can predict the heat transfer coefficient h or the Nusselt number. Equation (4.20) says nothing about the heat transfer coefficient h, and we need to conduct experiments to develop a model of heat flux that includes convection, heat conduction, viscosity, heat capacity, etc., so that we can predict the... 

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. 

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

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... 

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 -). 

Navier-Stokes equations, 318, 386-387 Nitrocellulose, 31 Nitroglycerine, 31-32 Normalization binary solutions, 156-157 multicomponent solutions, 157-158 Nusselt number, 118... 

An alternative equation which is in many ways more convenient has been proposed by COLBURN 16 and includes the Stanton number (St = h/Cppu) instead of the Nusselt number. This equation takes the form ... 

For mass transfer, which is considered in more detail in Chapter 10, an analogous relation (equation 10.233) applies, with the Sherwood number replacing the Nusselt number and the Schmidt number replacing the Prandtl number. 

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. 

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 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. 

Heat exchange in stirred reactors is described in [207]. By using dimensional analysis of heat flow and energy balance equations, the Nusselt number, containing hT, can be expressed as a function of the Reynolds number and the Prandtl number ... 

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 ... 

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. 

One example would be ice melting or methane hydrate dissociation when rising in seawater. Convective melting rate may be obtained by analogy to convective dissolution rate. Heat diffusivity k would play the role of mass diffusivity. The thermal Peclet number (defined as Pet = 2aw/K) would play the role of the compositional Peclet number. The Nusselt number (defined as Nu = 2u/5t, where 8t is the thermal boundary layer thickness) would play the role of Sherwood number. The thermal boundary layer (thickness 8t) would play the role of compositional boundary layer. The melting equation may be written as... 

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. 

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