Dimensional analysis heat transfer

In forced convection, the heat transferred per unit area per unit time q, is determined by a linear dimension which characterizes the surface /, the temperature difference between the surface and the fluid, AT, the viscosity rj, the density p, and the velocity u, of the fluid, its conductivity k, and its specific heat Cp. Dimensional analysis yields Eq. (32)... 

Entrance andExit SpanXireas. The thermal design methods presented assume that the temperature of the sheUside fluid at the entrance end of aU tubes is uniform and the same as the inlet temperature, except for cross-flow heat exchangers. This phenomenon results from the one-dimensional analysis method used in the development of the design equations. In reaUty, the temperature of the sheUside fluid away from the bundle entrance is different from the inlet temperature because heat transfer takes place between the sheUside and tubeside fluids, as the sheUside fluid flows over the tubes to reach the region away from the bundle entrance in the entrance span of the tube bundle. A similar effect takes place in the exit span of the tube bundle (12). 

This implies that the LMTD or M I D as computed in equations 20 through 26 may not be a representative temperature difference between the two heat-transferring fluids for aU tubes. The effective LMTD or M ID would be smaller than the value calculated, and consequentiy would require additional heat-transfer area. The tme value of the effective M I D may be determined by two- or three-dimensional thermal—hydrauUc analysis of the tube bundle. Baffle—Tube Support PlateXirea. The portion of a heat-transfer tube that passes through the flow baffle—tube support plates is usuaUy considered inactive from a heat-transfer standpoint. However, this inactive area must be included in the determination of the total length of the heat-transfer tube. 

Advanced two- and three-dimensional computer analysis methods are used today in the analyses of all critical components to verify aerodynamic, heat transfer, and mechanical performance. Additionally, the reduction of leakage paths in the compressor, as well as in the gas turbine expander, results in further plant efficiency improvements. At the compressor inlet, an advanced inlet flow design improves efficiency by reducing pressure loss. Rotor air cooler heat utilization and adt anccd blade and vane cooling arc also used. 

Clearly, the maximum degree of simplification of the problem is achieved by using the greatest possible number of fundamentals since each yields a simultaneous equation of its own. In certain problems, force may be used as a fundamental in addition to mass, length, and time, provided that at no stage in the problem is force defined in terms of mass and acceleration. In heat transfer problems, temperature is usually an additional fundamental, and heat can also be used as a fundamental provided it is not defined in terms of mass and temperature and provided that the equivalence of mechanical and thermal energy is not utilised. Considerable experience is needed in the proper use of dimensional analysis, and its application in a number of areas of fluid flow and heat transfer is seen in the relevant chapters of this Volume. 

Obtain, by dimensional analysis, a functional relationship for the heat transfer coeflicien for forced convection at the inner wall of an annulus through which a cooling liquid is flowing. 

Obtain by dimensional analysis a functional relationship for the wall heat transfer coefficient for a fluid flowing through a straight pipe of circular cross-section. Assume that the effects of natural convection can be neglected in comparison with those of forced convection. 

By dimensional analysis, derive a relationship for the heat transfer coefficient h for natural convection between a surface and a fluid on the assumption that the coefficient is a function of the following variables ... 

Kroeker CJ, Soliman HM, Ormiston SJ (2004) Three-dimensional thermal analysis of heat sinks with circular cooling micro-channels. Int J Heat Mass Transfer 47 4733 744 Lee PS, Garimella SV, Liu D (2005) Investigation of heat transfer in rectangular micro-channels. Int J Heat Mass Transfer 48 1688-1704... 

Lelea D, Nishio S, Takano K (2004) The experimental research on micro-tube heat transfer and fluid flow of distilled water. Int J Heat Mass Transfer 47 2817-2830 Li J, Peterson GP, Cheng P (2004) Three-dimensional analysis of heat transfer in a micro-heat sink with single phase flow. Int J Heat Mass Transfer 47 4215-4231 Lin TY, Yang CY (2007) An experimental investigation by method of fluid crystal thermography. Int. J. Heat Mass Transfer 50(23-24) 4736-4742... 

Qu W, Mudawar I (2002) Analysis of three-dimensional heat transfer in micro-channel heat sinks. Int J Heat Mass Transfer 45 3973-3985... 

Wallis GB (1969) One-dimensional two-phase flow. McGraw-HUl, New York Wayner PC, Kao YK, LaCroix LV (1976) The interline heat transfer coefiicient of an evaporating wetting film. Int 1 Heat Mass Transfer 19 487-492 Weisberg A, Bau HH, Zemel IN (1992) Analysis of micro-channels for integrated cooling. Int 1 Heat Mass Transfer 35 2465-2472... 

The quasi-one-dimensional model used in the previous sections for analysis of various characteristics of fiow in a heated capillary assumes a uniform distribution of the hydrodynamical and thermal parameters in the cross-section of micro-channel. In the frame of this model, the general characteristics of the fiow with a distinct interface, such as position of the meniscus, rate evaporation and mean velocities of the liquid and its vapor, etc., can be determined for given drag and intensity of heat transfer between working fluid and wall, as well as vapor and wall. In accordance with that, the governing system of equations has to include not only the mass, momentum and energy equations but also some additional correlations that determine... 

However, the most complex analysis is that in which heat transfer through the reactor walls is taken into account. This type of operation must be employed when it is necessary to supply or remove energy from the system so as to moderate the temperature excursions that would otherwise follow. It is frequently employed in industrial reactors and, to model such systems, one must often resort to two-dimensional models of the reactor that allow the concentration and temperature to vary in both the radial and axial directions. In the analysis of such systems, we make incremental calculations across the diameter of a given longitudinal segment of the packed bed reactor, and then proceed to repeat the process for successive longitudinal increments. 

The stagnant layer analysis offers a pedagogical framework for presenting the essence of diffusive burning. For the most part the one-dimensional stagnant layer approximated a two-dimensional boundary layer in which 6 = <5(x), with x the flow direction. For a convective boundary layer, the heat transfer coefficient, hc, is defined as... 

According to Summerfield (Ref 12, p 442) The ideal rocket motor analysis rests on the following simplifications (a) the proplnt gas obeys the perfect gas law (b) its specific heat is constant, independent of temp (c) the flow is parallel to the axis of the motor and uniform in every plane normal to the axis, thus constituting a one-dimensional problem (d) there is no frictional dissipation in the chamber or nozzle (e) there is no heat transfer to the motor walls (f) the flow velocity in the chamber before the nozzle entrance is zero (g)combustion or heat addition is completed in the chamber at constant pressure and does not occur in the nozzle and (h) the process is steady in time. ... 

Conventional dimensional analysis employs single length and time scales. Correlations are thus obtained for the mass or heat transfer coefficients in terms of the minimum number of independent dimensionless groups these can generally be represented by power functions such as... 

The second approach consists in deliberately abandoning certain similarity criteria and checking the effect on the entire process. This technique was used by Gerhard Damkbhler (1908-1944) in his trials to treat a chemical reaction in a catalytic fixed-bed reactor by means of dimensional analysis. Here the problem of a simultaneous mass and heat transfer arises—they are two processes that obey completely different fundamental principles ... 

Heat-Transfer Analysis Thermal-Capillary Models. Numerous analyses of various aspects of heat transfer in the CZ system have been reported many of these are cited by either Kobayashi (143) or Derby and Brown (144). The analyses vary in complexity and purpose, from the simple one-dimensional or fin approximations designed to give order-of-magni-tude estimates for the axial temperature gradient in the crystal (98) to complex system-oriented calculations designed to optimize heater design and power requirements (145,146). The system-oriented, large-scale calculations include radiation between components of the heater and the crucible assemblies, as well as conduction and convection. 

Billig (98) realized this point and used a one-dimensional heat-transfer analysis for the crystal with assumptions about the temperature field in the melt to derive the the following relationship that has been used heavily in qualitative discussions of crystal growth dynamics ... 

SiH4 — epitaxial Si 225 Two-dimensional, axisymmetric flow and heat transfer analysis of detailed chemistry model with 17 species and 27 elementary reactions combined with similarity solution for flow problem. 

Ga(CH3)3 + AsH3 —> GaAs 195, 198 Two-dimensional, axisymmetric flow, mass- and heat-transfer analysis, reactor geometry effects, flow transitions, and mass-transport-limited growth. 

Two approaches have been used to describe the effect of concentration polarization. One has its origins in the dimensional analysis used to solve heat transfer problems. In this approach the resistance to permeation across the membrane and the resistance in the fluid layers adjacent to the membrane are treated as resistances in series. Nothing is assumed about the thickness of the various layers or the transport mechanisms taking place. 

Single screw extruder operating curves. The conveying characteristics of a single screw extruder can also be analyzed by use of dimensional analysis. Pawlowski [6,7] used dimensional analysis and extensive experimental work to fully characterize the conveying and heat transfer characteristics of single screw extruders, schematically depicted in Fig. 4.7. 

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