Heat transfer parameter, dimensionless

The three unitary heat transfer parameters descriptive of equipment efficiency can now be redefined in terms of the above dimensionless integrals ... 

By using the three-dimensionless integrals 01 Zl5 and -C>, the unitary heat-transfer parameters can now be quantified three... 

Solutions for the heat-transfer problem, which arises in the use of packed beds as direct-contact recuperative heat exchangers, were presented by Furnas in 1930. The parameter Njj is the number of heat-transfer units. For heat transfer, the dimensionless time t is the heat capacity of the gas times the amount of gas that has passed through the bed divided by the total bed capacity. For Nu = co, the breakthrough curve of T /Tq vs. t would be a vertical line at t = 1.0, just as for mass transfer. The defining equations are... 

P clet number Fe Re Fr = p CpVL k Dimensionless independent heat transfer parameter (ratio of heat transfer by convection to conduction) Forced convection... 

Heat Transfer. Fundamental solutions for boundary conditions of the first, second, and third kinds for fully developed flow in concentric annular ducts are given in Table 5.14. The nomenclature used in describing the corresponding solutions can best be explained with reference to the specific heat transfer parameters G) and 0 which are the dimensionless duct wall and fluid bulk mean temperature, respectively. The superscript k denotes the type of the fundamental solution according to the four types of boundary conditions described in the section entitled Four Fundamental Thermal Boundary Conditions. Thus, k = 1,2, 3, or 4. The subscript l in Gj 1 refers to the particular wall at which the temperature is evaluated / = i or o when the temperature is evaluated at the inner or the outer wall. The subscript j in G) 1 refers... 

As seen in many of the correlations for fluid flow and heat transfer, many dimensionless groups, such as the Reynolds number and Prandtl number, occur in these correlations. Dimensional analysis is often used to group the variables in a given physical situation into dimensionless parameters or numbers which can be useful in experimentation and correlating data. 

Relation between mass- and heat-transfer parameters. In order to use the unsteady-state heat-conduction charts in Chapter 5 for solving unsteady-state diffusion problems, the dimensionless variables or parameters for heat transfer must be related to those for mass transfer. In Table 7.1-1 the relations between these variables are tabulated. For K 1.0, whenever appears, it is given as Kk, and whenever c, appears, it is given as cJK. 

In an adsorbent bed of relatively small diameter, heat exchange through the column wall becomes appreciable. In this case, the overall heat transfer model introduced in 8.2(ii) can replace Eq. (8-54). Then similar calculations ae possible by taking the overall heat transfer coefficient ho and/or the radius of the column / as a parameter to examine this effect. Calculation using the same example considering this effect is shown in Fig. 8.9. Obviously the heat transfer parameter, (hoi R )(CftPfUol L), which is derived from the dimensionless form of the heat balance equation, determines the effect of heat transfer through the wall. 

The dimensionless heat transfer parameters Pe, Pe and Bi are jointly estimated by fitting Eqn. (11) to measured radial temperature profiles at several bed depths by minimisation of the sum of squares function... 

This chapter reviews the various types of impellers, die flow patterns generated by diese agitators, correlation of die dimensionless parameters (i.e., Reynolds number, Froude number, and Power number), scale-up of mixers, heat transfer coefficients of jacketed agitated vessels, and die time required for heating or cooling diese vessels. 

The effect of axial conduction on heat transfer in the fluid in the micro-channel can be characterized by a dimensionless parameter... 

The problem of axial conduction in the wall was considered by Petukhov (1967). The parameter used to characterize the effect of axial conduction is P = (l - dyd k2/k ). The numerical calculations performed for q = const, and neglecting the wall thermal resistance in radial direction, showed that axial thermal conduction in the wall does not affect the Nusselt number Nuco. Davis and Gill (1970) considered the problem of axial conduction in the wall with reference to laminar flow between parallel plates with finite conductivity. It was found that the Peclet number, the ratio of thickness of the plates to their length are important dimensionless groups that determine the process of heat transfer. 

A general case of heat transfer under the conditions of combined action of electro-osmotic forces and imposed pressure gradient was considered by Chakra-borty (2006). The analysis showed that in this case the Nusselt number depends not only on parameters z and S, but also on an additional dimensionless group, which is a measure of the relative significance of the pressure gradient and osmotic forces. 

Two-phase flows in micro-channels with an evaporating meniscus, which separates the liquid and vapor regions, have been considered by Khrustalev and Faghri (1996) and Peles et al. (1998, 2000). In the latter a quasi-one-dimensional model was used to analyze the thermohydrodynamic characteristics of the flow in a heated capillary, with a distinct interface. This model takes into account the multi-stage character of the process, as well as the effect of capillary, friction and gravity forces on the flow development. The theoretical and experimental studies of the steady forced flow in a micro-channel with evaporating meniscus were carried out by Peles et al. (2001). These studies revealed the effect of a number of dimensionless parameters such as the Peclet and Jacob numbers, dimensionless heat transfer flux, etc., on the velocity, temperature and pressure distributions in the liquid and vapor regions. The structure of flow in heated micro-channels is determined by a number of factors the physical properties of fluid, its velocity, heat flux on... 

Bunimovich et al. (1995) lumped the melt and solid phases of the catalyst but still distinguished between this lumped solid phase and the gas. Accumulation of mass and heat in the gas were neglected as were dispersion and conduction in the catalyst bed. This results in the model given in Table V with the radial heat transfer, conduction, and gas phase heat accumulation terms removed. The boundary conditions are different and become identical to those given in Table IX, expanded to provide for inversion of the melt concentrations when the flow direction switches. A dimensionless form of the model is given in Table XI. Parameters used in the model will be found in Bunimovich s paper. 

The interfacial heat transfer coefficient can be evaluated by using the correlations described by Sideman (S5), and then the dimensionless parameter Ni can be calculated. If the Peclet numbers are assumed to be infinite, Eqs. (30) can be applied to the design of adiabatic cocurrent systems. For nonadiabatic systems, the wall heat flux must also be evaluated. The wall heat flux is described by Eqs. (32) and the wall heat-transfer coefficient can be estimated by Eq. (33) with... 

The scale models must be carefully designed. Failure to match the important dimensionless parameters will lead to erroneous simulation results. Modeling can be extended to particle convective heat transfer. Wear or erosion of in-bed surfaces can be qualitatively studied, although quantitative assessment requires the identification and simulation of additional wear-related parameters. 

The parameter C in Eq. (25) is a dimensionless parameter inversely proportional to the average residence time of single particles on the heat transfer surface. It is suggested that this parameter be treated as an empirical constant to be determined by comparison with actual data in fast fluidized beds. The lower two dash lines in Fig. 17 represent predictions by Martin s model, with C taken as 2.0 and 2.6. It is seen that an appropriate adjustment of this constant would achieve reasonable agreement between prediction and data. 

Dimensionless numbers in mixing, 16 685 used in convection heat-transfer analysis, 73 246-247 Dimensionless parameter, external mass transfer resistance and, 25 290-292 Dimensionless reactor design formulation, 21 350... 

In the gas/vapour phase the dimensionless distance tj ranges from 0 to 1, where tj — 1 corresponds to the position of the interface. In the liquid phase this parameter ranges from 0 to 1 for the mass transfer film and from 0 to Le for the heat transfer film. Hence, rj = 0 corresponds to the position of the interface and rj = I and t] = Le correspond, respectively, to the boundaries of the mass and heat transfer film. The mass and energy fluxes can now be calculated by solving the differential equations (4) and (8)-(12) subject to the boundary conditions (15). Due to the non-linearities a numerical solution procedure has been used which will be discussed subsequently. 

The second method uses dimensionless numbers to predict scale-up parameters. The use of dimensionless numbers simplifies design calculations by reducing the number of variables to consider. The dimensionless number approach has been used with good success in heat transfer calculations and to some extent in gas dispersion (mass transfer) for mixer scale-up. Usually, the primary independent variable in a dimensionless number correlation is Reynolds number ... 

The type of dimensionless representation of the material function affects the (extended) pi set within which the process relationship is formulated (for more information see Ref. 5). When the standard representation is used, the relevance list must include the reference density po instead of p and incorporate two additional parameters po. Tq. This leads to two additional dimensionless numbers in the process characteristics. With regard to the heat transfer characteristics of a mixing vessel or a smooth straight pipe, Eq. (27), it now follows that... 

For practical purposes, heat-transfer engineers often use empirical or semi-empirical correlations to predict h values. These formulations are usually based on the dimensionless numbers described before. In this case, the appropriate formulation should be used to prevent significant errors. If dimensionless correlations are applicable under conditions of gas extraction, then heat-transfer coefficients can be determined from these correlations and the influence of parameter variations may be derived also from them. 

---