Dimensionless number Nusselt

American engineers are probably more familiar with the magnitude of physical entities in U.S. customary units than in SI units. Consequently, errors made in the conversion from one set of units to the other may go undetected. The following six examples will show how to convert the elements in six dimensionless groups. Proper conversions will result in the same numerical value for the dimensionless number. The dimensionless numbers used as examples are the Reynolds, Prandtl, Nusselt, Grashof, Schmidt, and Archimedes numbers. 

Nac = Reynolds number, dimensionless Nu = Nusselt number, (heat transfer)... 

Dimensionless numbers (Reynolds number = udip/jj., Nusselt number = hd/K, Schmidt number = c, oA, etc.) are the measures of similarity. Many correlations between them (known also as scale-up correlations) have been established. The correlations are used for calculations of effective (mass- and heat-) transport coefficients, interfacial areas, power consumption, etc. 

L/pj-A)(S/psA), liquid-solids velocity ratio, dimensionless Number of heat-transfer stages, dimensionless = hdp/kg, Nusselt number, dimensionless Pressure drop, gm-wt/cm2 = Cpu kg, Prandtl number, dimensionless = dpiipj U, Reynolds number, dimensionless S Mass velocity of solids, gm/cirf sec... 

Both methods yield dimensionless groups, which correspond to dimensionless numbers (1), e.g.. Re, Reynolds number Fr, Froude number Nu, Nusselt number Sh, Sherwood number Sc, Schmidt number etc. (2). The classical principle of similarity can then be expressed by an equation of the form ... 

Dijfusional dimensionless numbers The Peclet, Prandtl, Schmidt, Sherwood, and Nusselt number are the most common ones. 

Heat transfer and its counterpart diffusion mass transfer are in principle not correlated with a scale or a dimension. On a molecular level, long-range dimensional effects are not effective and will not affect the molecular carriers of heat. One could say that physical processes are dimensionless. This is essentially the background of the so-called Buckingham theorem, also known as the n-theorem. This theorem states that a product of dimensionless numbers can be used to describe a process. The dimensionless numbers can be derived from the dimensional numbers which describe the process (for example, viscosity, density, diameter, rotational speed). The amount of dimensionless numbers is equal to the number of dimensional numbers minus their basic dimensions (mass, length, time and temperature). This procedure is the background for the development of Nusselt correlations in heat transfer problems. It is important to note that in fluid dynamics especially laminar flow and turbulent flow cannot be described by the same set of dimensionless correlations because in laminar flow the density can be neglected whereas in turbulent flow the viscosity has a minor influence [144], This is the most severe problem for the scale-up of laminar micro results to turbulent macro results. 

Sebastian (2,6) following Frank-Kamenetskii (43) arrived at the results depicted on Fig. 11.7, where the Nusselt (Nu) dimensionless number Nu = hRH/k, k is the thermal conductivity, tH is the characteristic time for removing the reaction-generated heat by conduction... 

Hence, in experimental heat transfer, the number of variables to be studied is significantly reduced. The Nusselt number or heat transfer coefficient is correlated to only two dimensionless numbers. 

Dimensionless numbers help in convection heat transfer engineering Used to compare relative values in the practice of engineering In convection, there is the Eckert number and the Prandtl number, There is also the Reynolds number, Peclet number and Nusselt number. 

The surface tension can also sometimes influence the heat transfer rate and in this case another new dimensionless number will be involved in defining the Nusselt number. 

If, in the case of gas flows, the heat transfer coefficient is assumed to depend on the speed of sound, a, in the gas in addition to the variables considered in this chapter, And the additional dimensionless number on which the Nusselt number will depend. 

Now that we have introduced the heat convection coefficient, we will define our first dimensionless number, the Nusselt number, which is used in heat transfer studies. We represent the size of a particular plant part by a characteristic dimension d, which for a flat plate is the quantity / in Equation 7.10 and for a cylinder or sphere is the diameter. This leads to... 

Dimensionless numbers have proved useful for analyzing relationships between heat transfer and boundary layer thickness for leaves. In particular, the Nusselt number increases as the Reynolds number increases for example, Nu experimentally equals 0.97 Re0-5 for flat leaves (Fig. 7-9). By Equations 7.18 and 7.19, d/8bl is then equal to 0.97 (vd/v)V2y so for air temperatures in the boundary layer of 20 to 25°C, we have... 

This section will present one of the possible physical interpretations of these important dimensionless numbers. First, to show the meaning of Nusselt number, we consider the heat transfer flux in the x direction in the case of a pure molecular mechanism compared with the heat transfer characterizing the process when convection is important. The corresponding fluxes are then written as ... 

Considering these Biot numbers, we can observe that they are similar to the Nusselt and Sherwood numbers. The only difference between these dimensionless numbers is the transfer coefficient property characterizing the Biot numbers transfer kinetics for the external phase (a x heat transfer coefficient for the external phase, k ex- mass transfer coefficient for the external phase). We can conclude that the Biot number is an index of the transfer resistances of the contacting phases. 

In convection studies,-it is common practice to nondimensionalize the governing equations and combine the variables, which group together into dimensionless numbers in order to reduce the number of total variables. It is also common practice to nondimensionalize the heat transfer coefficient h with the Nusselt number, defined as... 

Various correlations exist for mass and heat transfer coefficients in terms of dimensionless numbers. Table 8.7 surveys the most appropriate ones for laboratory fixed-bed reactors [8,17-19]. The Sherwood number, Sh, and the Nusselt number, Nu, express the ratio of total mass transfer to diffusive mass transfer, and the ratio of total heat transfer to conductive heat transfer, respectively. Values of kf and h for gases in laboratory systems range from 0.1 to 10 mf " s ... 

The calculation of a leads back to the determination of the Nusselt number. According to (1.38) Nu is dependent on the dimensionless temperature field, and therefore it must be clarified which dimensionless numbers determine the dimensionless temperature t +. To this end, instead of using the fundamental differential equations — this will be done in chapter 3 — we establish a list of the physical quantities and then use this to derive the dimensionless numbers. 

The geometric and flow conditions are not the only parameters which have a considerable influence on the relationship between the Nusselt number and the other dimensionless numbers. The thermal boundary conditions also affect heat transfer. An example of this is, with the same values of Re and Pr in parallel flow over a plate, we have different Nusselt numbers for a plate kept at constant wall temperature dw, and for a plate with a constant heat flux qw at the wall, where the surface temperature adjusts itself accordingly. 

The determination of heat transfer coefficients with the assistance of dimensionless numbers has already been explained in section 1.1.4. This method can also be used for mass transfer, and as an example we will take the mean Nusselt number Num = amL/ in forced flow, which can be represented by an expression of the form... 

Consider the natural convection from a horizontal cylinder rotating with an angular frequency to (Fig. 5P-9). The peripheral surface temperature of the cylinder is Tm and the ambient temperature is To,. The diameter of the cylinder is D. Assuming that the natural convection resulting from rotation and that from gravity can be superimposed, express the Nusselt number in terras of the appropriate dimensionless numbers. 

In Chapter 5, following some dimensional arguments, we learned that the independent dimensionless numbers characterizing buoyancy driven flows are the Rayleigh number and the Prandtl number (Ra, Pr), and the heat transfer in (Nusselt number Nu for) natural convection is governed by... 

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. 

The Nusselt number Nu is the relevant dimensionless number which is directly proportional to the film heat transfer coefficient, in this case of the reaction mixture. 

At normal gravity, the Nusselt munber, Nu (hD/kj), can be obtained from these two dimensionless numbers, from which hj (kWm °C )> the heat transfer coefficient, can be calculated. Then the heat that is to be removed from the system can be calculated as... 

Heat transfer reduction, also expressed as a percentage, is generally obtained by comparing the Nusselt number (a dimensionless number defined as the ratio of convective to conductive heat transfer across a boundary) of a DR surfactant solution (Nu) with that of the solvent (Nu. ) at the same solvent Reynolds number ... 

The Nusselt number, Nu, is a dimensionless number wherein C, x, and y are constants determined by experiment or experience for specific fluids, configurations, and... 

From a local heat-transfer perspective, the outcome of this characterization is mainly in the form of empirical correlations for the Nusselt dimensionless number Nu = hDjk), summarized in Table 21.3. Most correlations were derived from experiments within the context of IC engines, except for the dynamic correlation reported in Panao and Moreira [21], which considers the transient characteristics of droplets of an intermittent spray along an injection cycle, and a newly introduced dimensionless parameter X corresponding to the average number of droplets impinging in the vicinity of each other [24]. 

There are two types of correlations for estimating the heat transfer coefficient for condensation inside vertical tube. In the first type of correlations, the local heat transfer coefficient is expressed in the form of a degradation factor defined as the ratio of the experimental heat transfer coefficient (when noncondensable gas is present) and pure steam heat transfer coefficient (Kuhn et al. [1997]). The correlations in general are the functions of local noncondensable gas mass fraction and mixture Reynolds number (or condensate Reynolds number). In the other type of correlations, the local heat transfer coefficient is expressed in the form of dimensionless numbers. In these correlations, local Nusselt number is expressed as a function of mixed Re5molds number, Jakob number, noncondensable gas mass fraction, condensate Reynolds number, and so on. 

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