Similar flows forced convection

Sedov LI (1993) Similarity and dimensional methods in mechanics, 10th edn. CRC, Boca Raton Shah RK, London AL (1978) Laminar flow forced convection in duct. Academic, New York Shapiro AK (1953) The dynamics and thermodynamics of compressible fluid flow. Wiley, New York... 

Consider a heated vertical plate in a quiescent fluid. The plate heats the fluid in its neighborhood, which then becomes lighter and moves upward. The force resulting from the product of gravity and density difference and causing this upward motion is called buoyancy. The fluid moving under the effect of buoyancy develops a vertical boundary layer about the plate. Within the boundary layer the temperature decreases from the plate temperature to the fluid temperature, while the velocity vanishes on the plate walls and beyond the boundary layer and has a maximum in between (Fig. 5.13). Actually, in a manner similar to forced convection, the momentum boundary layer of natural convection is expected to be thicker for larger Prandtl numbers than the thermal boundary layer. However, the characteristic velocity for the enthalpy flow across should be scaled relative to Ss rather than 5,... 

In the present note we use the same methods to find similar expressions for convective flows freely-ascending under the action of gravity and density differences. We direct the x-axis upward in the direction of the flow. Instead of conservation of momentum (which will not exist because of the action of buoyancy forces), we write the conservation of heat flux... 

The prime variables should be selected so that each dimensionless product characterizes some distinct feature of the flow. For example, the forced velocity, U, should be used as one prime variable since it determines whether or not the problem involves forced convection. Similarly, the buoyancy variable (3g(Tw - T/) will determine the importance of free convective effects and should also be used as a prime variable. Also, since h is the variable whose value is required, it should be used as a prime variable. The fourth prime variable will be taken as cp. This choice is not as obvious as the others but stems from the fact that cp will determine the thermal capacity of the fluid and will, therefore, influence the relation between the velocity and temperature fields. 

The flows over a series of bodies of the same geometrical shape will be similar, i.e., will differ from each other only in scale, if the Reynolds and Prandtl numbers are the same in all the flows. From this it follows that the Nusselt number in forced convection will depend only on the Reynolds and Prandtl numbers. 

As was done in dealing with forced convective flow over a uniform temperature plate, it is assumed that the velocity and temperature profiles are similar at all values of x, i.e., that ... 

Available analyses of turbulent natural convection mostly rely in some way on the assumption that the turbulence structure is similar to that which exists in turbulent forced convection, see [96] to [105]. In fact, the buoyancy forces influence the turbulence and the direct use of empirical information obtained from studies of forced convection to the analysis of natural convection is not always appropriate. This will be discussed further in Chapter 9. Here, however, a discussion of one of the earliest analyses of turbulent natural convective boundary layer flow on a flat plate will be presented. This analysis involves assumptions that are typical of those used in the majority of available analyses of turbulent natural convection. 

To proceed further, relationships for the wall shear stress, tw> and the wall heat transfer rate, qw, must be assumed. It is consistent with the assumption that the flow near the wall in a turbulent natural convective boundary layer is similar to that in a turbulent forced convective boundary layer to assume that the expressions for tw and qw that have been found to apply in forced convection should apply in natural convection. It will therefore be assumed here that the following apply in a natural convective boundary layer ... 

Before turning to a discussion of other methods of solving the laminar boundary layer equations for combined convection, a series-type solution aimed at determining the effects of small forced velocities on a free convective flow will be considered. In the analysis given above to determine the effect of weak buoyancy forces on a forced flow, the similarity variables for forced convection were applied to the equations for combined convection. Here, the similarity variables that were previously used in obtaining a solution for free convection will be applied to these equations for combined convection. Therefore, the following similarity variable is introduced ... 

In a similar way. because in forced convective flow the turbulent heat transfer rate is given by ... 

Even though the fluid motion is the result of density variations, these variations are quite small, and a satisfactory solution to the problem may be obtained by assuming incompressible flow, that is, p = constant. To effect a solution of the equation of motion, we use the integral method of analysis similar to that used in the forced-convection problem of Chap. 5. Detailed boundary-layer analyses have been presented in Refs. 13, 27, and 32. 

The Grashof number may be interpreted physically as a dimensionless group representing the ratio of the buoyancy forces to the viscous forces in the free-convection flow system. It has a role similar to that played by the Reynolds number in forced-convection systems and is the primary variable used as a criterion for transition from laminar to turbulent boundary-layer flow. For air in free convection on a vertical flat plate, the critical Grashof number has been observed by Eckert and Soehngen [1] to be approximately 4 x 10". Values ranging between 10" and 109 may be observed for different fluids and environment turbulence levels. ... 

Convection, sometimes identified as a separate mode of heat transfer, relates to the transfer of heat from a bounding surface to a fluid in motion, or to the heat transfer across a flow plane within the interior of the flowing fluid. If the fluid motion is induced by a pump, a blower, a fan, or some similar device, the process is called forced convection. If the fluid motion occurs as a result of the density difference produced by the temperature difference, the process is called free or natural convection. 

At high enough qualities and mass fluxes, however, it would be expected that the nucleate boiling would be suppressed and the heat transfer would be by forced convection, analogous to that for the evaporation for pure fluids. Shock [282] considered heat and mass transfer in annular flow evaporation of ethanol water mixtures in a vertical tube. He obtained numerical solutions of the turbulent transport equations and carried out calculations with mass transfer resistance calculated in both phases and with mass transfer resistance omitted in one or both phases. The results for interfacial concentration as a function of distance are illustrated in Fig. 15.112. These results show that the liquid phase mass transfer resistance is likely to be small and that the main resistance is in the vapor phase. A similar conclusion was reached in recent work by Zhang et al. [283] these latter authors show that mass transfer effects would not have a large effect on forced convective evaporation, particularly if account is taken of the enhancement of the gas mass transfer coefficient as a result of interfacial waves. 

Heat transfer to a digitized flow in a microchannel is similar in many ways to single-phase forced convection in microchannels. The thermal boundary conditions that exist are the same however, due to the unique rolling-type flow, DHT behaves in a significantly different fashion. In Fig. 3, the temperature field shows that heat is convected by the vortices and circulates within the droplet. As a droplet rolls down a heated microchannel, cool fluid from the center of the droplet is continually transported to the outer edges of the droplet while hot fluid at the wall is convected inward. Heat gradually diffuses... 

Ambient air flow or forced convection can also be a source of energy for distributed environmental sensors or similar autonomous devices exposed to a flow. The airflow can induce the rotation of a microturbine (similar to large-scale wind mills) or the oscillation of a structure. This mechanical energy can then be converted to electricity by electromagnetic, piezoelectric, or electrostatic principles. 

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