Representation of Heat-Transfer Coefficients

Representation of Heat-Transfer Coefficients Heat-transfer coefficients are usually expressed in two ways (1) dimensionless relations and (2) dimensional equations. Both approaches are used below. The dimensionless form of the heat-transfer coefficient is the Nusselt 

5-5 Nomenclature for (a) counterflow and (b) parallel flow heat exchangers for use with Eq. (5-32). 

Natural convection occurs when a fluid is in contact with a solid surface of different temperature. Temperature differences create the density gradients that drive natural or free convection. In addition to the Nusselt number mentioned above, the key dimensionless parameters for natural convection include the Rayleigh number Ra = p AT gx3/ va and the Prandtl number Pr = v/a. The properties appearing in Ra and Pr include the volumetric coefficient of expansion p (K-1) the difference AT between the surface (Ts) and free stream (Te) temperatures (K or °C) the acceleration of gravity g(m/s2) a characteristic dimension x of the surface (m) the kinematic viscosity v(m2/s) and the thermal diffusivity a(m2/s). The volumetric coefficient of expansion for an ideal gas is p = 1/T, where T is absolute temperature. For a given geometry, 

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Finally, it should also be pointed out that in heat conduction problems the dimensionless representation and the combination of the influencing quantities into dimensionless numbers are not as significant as in the representation and determination of heat transfer coefficients in 1.1.4. In the following sections we will frequently refrain from making the heat conduction problem dimensionless and will only present the solution of a problem in a dimensionless form by a suitable combination of variables and influencing quantities. 

Figure 9 Qualitative representation of the effect on radial temperature profiles of varying heat transfer coefficients. (After Ref. 9.)...

The temperature held is dependent on this number when heat transfer takes place into a fluid. The Biot number has the same form as the Nusselt number defined by (1.36). There is however one very significant difference, A in the Biot number is the thermal conductivity of the solid whilst in the Nusselt number A is the thermal conductivity of the fluid. The Nusselt number serves as a dimensionless representation of the heat transfer coefficient a useful for its evaluation, whereas the Biot number describes the boundary condition for thermal conduction in a solid body. It is the ratio of L0 to the subtangent to the temperature curve within the solid body, cf. Fig. 2.4, whilst the Nusselt number is the ratio of a (possibly different choice of) characteristic length L0 to the subtangent to the temperature profile in the boundary layer of the fluid. 

The Colburn y-factor is another representation of the heat-transfer coefficient and arises from a boundary layer theory model. It is defined as... 

The proper representation of macroscopic transport properties, particularly the heat transfer coefficient, is a major problem in the predictive modeling of spinning and other free-surface processing flows. Heat transfer coefficients are typically obtained from experiments on nondeforming wires, and the extension to a deforming surface with a variable cross section is not obvious. Data obtained on real spinlines require either infrared or intrusive contact temperatme... 

The representation of the calorimeter by mathematical models described by a set of heat balance equations has long traditions. In 1942 King and Grover [22] and then Jessup [23] and Chumey et al. [24] used this method to explain the fact that the calculated heat capacity of a calorimetric bomb as the sum of the heat capacities of particular parts of the calorimeter was not equal to the experimentally determined heat capacity of the system. Since that time, many papers have been published on this field. For example, Zielenkiewicz et al. applied systems of heat balance equations for two and three distinguished domains [25 8] to analyze various phenomena occurring in calorimeters with a constant-temperature external shield Socorro and de Rivera [49] studied microeffects on the continuous-injection TAM microcalorimeter, while Kumpinsky [50] developed a method or evaluating heat-transfer coefficients in a heat flow reaction calorimeter. 

Pumparound sections usually contain from 4 ft to 9 ft of packed depth. The traditional method for calculating bed depth is by use of Equation 6-20. This equation is a simplified representation of a complex group of heat and mass transfer processes. A considerable amount of industrial experience has led to the development of satisfactory empirical equations for the calculation of overall heat transfer coefficients. 

The variation of efficiencies is due to interaction phenomena caused by the simultaneous diffusional transport of several components. From a fundamental point of view one should therefore take these interaction phenomena explicitly into account in the description of the elementary processes (i.e. mass and heat transfer with chemical reaction). In literature this approach has been used within the non-equilibrium stage model (Sivasubramanian and Boston, 1990). Sawistowski (1983) and Sawistowski and Pilavakis (1979) have developed a model describing reactive distillation in a packed column. Their model incorporates a simple representation of the prevailing mass and heat transfer processes supplemented with a rate equation for chemical reaction, allowing chemical enhancement of mass transfer. They assumed elementary reaction kinetics, equal binary diffusion coefficients and equal molar latent heat of evaporation for each component. 

An exclusively analytical treatment of heat and mass transfer in turbulent flow in pipes fails because to date the turbulent shear stress Tl j = —Qw w p heat flux q = —Qcpw, T and also the turbulent diffusional flux j Ai = —gwcannot be investigated in a purely theoretical manner. Rather, we have to rely on experiments. In contrast to laminar flow, turbulent flow in pipes is both hydrodynamically and thermally fully developed after only a short distance x/d > 10 to 60, due to the intensive momentum exchange. This simplifies the representation of the heat and mass transfer coefficients by equations. Simple correlations, which are sufficiently accurate for the description of fully developed turbulent flow, can be found by... 

The Boltzmann integro-differential kinetic equation written in terms of statistical physics became the foundation for construction of the structure of physical kinetics that included derivation of equations for transfer of matter, energy and charges, and determination of kinetic coefficients that entered into them, i.e. the coefficients of viscosity, heat conductivity, diffusion, electric conductivity, etc. Though the interpretations of physical kinetics as description of non-equilibrium processes of relaxation towards the state of equilibrium are widespread, the Boltzmann interpretations of the probability and entropy notions as functions of state allow us to consider physical kinetics as a theory of equilibrium trajectories. These trajectories as well as the trajectories of Euler-Lagrange have the properties of extremality (any infinitesimal part of a trajectory has this property) and representability in the form of a continuous sequence of states of rest. These trajectories can be used to describe the behavior of (a) isolated systems that spontaneously proceed to final equilibrium (b) the systems for which the differences of potentials with the environment are fixed (c) and non-homogeneous systems in which different parts have different values of the same intensive parameters. 

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