Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 54
Текст из файла (страница 54)
4.33), two or more subenclosures are formed, the partitionsthemselves constituting at least part of the side walls of these subenclosures. In the conceptualprocess of forming these subenclosures, the partitions' thickness should be split down themiddle, with half of the partition being inside each of the adjacent subenclosures. With certainreservations, the heat transfer across these subenclosures can then be determined using themethods for unpartitioned enclosures described in previous sections. The reservations concern the boundary conditions applying on the outside faces of the side walls. In the section ongeometry and parameters for cavities without interior solids (which refers to Fig.
4.25), theseoutside faces were prescribed as being adiabatic surfaces. Thus, strictly speaking, for themethods of previous sections to apply, the central faces of the split partitions should be treatable as adiabatic. Under certain conditions--for example, because of symmetry--this may berealistic, although because of the possibility of corotating cells in adjacent subenclosures, eventhen the adiabatic assumption may not be strictly correct. On the other hand, the errors associated with the nonadiabatic conditions should be acceptably small. If the thermal analyst isnot satisfied with the uncertainties associated with this approximation, more refined analysesdo exist (e.g., see Asako et al.
[10]) and experimental data are available (e.g., see Cane et al.[29], Smart et al. [253]), but these will be only for certain limited methods of partitioning thatmay or may not match those of interest to the analyst.Partial Parallel Partitions. In a number of studies researchers have investigated the rectangular parallelepiped cavity of Fig.
4.25a, at 0 = 90 °, in which a partial partition is insertedparallel to and midway between the plates. In most of the experimental work, the Rayleighnumber was on the order of 101° (so the conduction boundary layers on the vertical walls arevery much smaller than the cavity dimensions) and the Prandtl number was close to 6. However, when expressed in terms of the equations below, the results should have wider applicability.
In the early studies, the partition projected some vertical distance, say Hp, either downfrom the ceiling or up from the floor, and it ran the entire width W of the cavity. For the situation in which the partition runs up from the floor, Lin and Bejan [186] found that the part ofthe cold side of the cavity that is below the top of the partition in elevation contains only avery weak circulation cell that hardly contributes to the heat transfer, whereas a strong circulatory cell runs--in a single boundary layer--up the hot plate, across the ceiling to the coldplate, down the cold plate to the elevation of the top of the partition, across the cavity to thetop of the partition, down the partition on the hot side, across the floor, and back to the hotplate.
Unless the opening is very small, this cell carries almost the entire heat transferobserved, which means that only the upper part of the cold plate receives significant heattransfer. When the partition projects down from the ceiling, an analogous pair of cells is produced. On the basis of this model Lin and Bejan developed a quantitative heat transfer modelthat fit their data and those of other workers. After some reworking, their model can beexpressed as1.5H-3/4Nu= Nunp - (Nunp - Nufp)1 - (H0 +HMIN) -3/4 +0.5(H0 +)(HMAX) -3/41.5/_/-3/41 --/_/-3/4• • MIN"~-)-1/'1K/_./-3/4v.ja• MAX(4.130)where Ho = H - lip is the height of the "window" left in the partition, HMAX= lip, and for themoment HMIN= 0.
To apply this equation, one must first evaluate the Nusselt number Nu withthe partition fully removed; this is the Nu,p (or Nu for no partition) in Eq. 4.130. Next oneevaluates the Nu with the full partition in place, i.e., with Hp = H; this is the Nule (or Nu forfull partition) in Eq. 4.130. Both Nunpand Nunpcan be evaluated using the methods of previous sections. Equation 4.130 provides an interpolation scheme to be used between these twolimiting values for Nu. The work of Nansteel and Greif [205] showed that moving the partit i o n - s o that it is no longer central but is closer to either the hot plate or the cold plate--hasonly a minor effect on the heat transfer provided the movement is H/4 or less.NATURAL CONVECTION4.63An extension can be made to Lin and Bejan's analysis to allow for there being both anupper and a lower partition, the upper partition extending down, say, distance Hu from theceiling, and the lower partition extending upward, say, distance HL from the floor, leaving anopening of height H0 = H - HL - Hr.
The result of such an analysis is again expressed by Eq.4.130, provided one now interprets HMAX as the maximum of Hu and HL and HMIN as the minimum of Hu and HL.It may be that the opening in the partition does not extend the full width W of the cavityshown in Fig. 4.25. Say it only extends over a width W0, leaving widths WL and WR on the leftand right, respectively. The flow in the parts of the cavity on the right and left of the openingshould be very similar to the flow in the corresponding complete cavity with the full partition,while the flows in the part having the opening should be very similar to the flow in the corresponding complete cavity with the opening extending the full width. Thus it is recommendedthat one determine the Nusselt number from a weighted average of the Nulp (giving it weight(WL + WR)/W) and the Nu calculated from Eq.
4.130 (giving it weight Wo/W). Karki et al. [154]have numerically investigated such enclosure in the low Rayleigh number range: 104 < Ra < 107.TRANSIENT NATURAL CONVECTIONExternal Transient ConvectionOverview. Suppose a body, such as that shown in Fig. 4.37a, is initially in equilibrium withits surroundings at T~, but at time t = 0 its surface temperature is changed impulsively to Tw.The surface heat flux rises to infinity (theoretically) and then falls off quickly with increasingt as shown in Fig. 4.37b.
Since there is a delay in initiating the fluid motion, the heat flow intothe fluid is initially by conduction, and the surface average heat flux q" follows the conductioncurve in Fig. 4.37b to the departure time to. At a later time (t > too), steady-state convection isachieved. These two regimes are called the conduction regime (0 < t < to) and the steady-stateregime (t > t~). The transition regime lies in the range to < t < too.At time ti, the heat flow by conduction matches the steady-state convective heat transferfrom the body. If conditions are met to initiate convection before ti (this will depend on Pr andRa), the heat flow falls monotonically in the transition regime, as along path A in Fig.
4.37b.Otherwise convection will not be initiated until to > ti, so the heat flow will have fallen belowthe steady-state value and must therefore recover from the undershoot in the transitionregime as shown by path B.TII~r"T"~¢-1"ooq"l i~ toSteady_T,,-Too'2V,o,e \<-.;r-!I0(a)"~"q" '~ ~ ~l~)Steadystate\iltit=(b)0tit(c)FIGURE 4.37 Transient response of heat flow from a surface (a) following a stepchange in wall temperature (b), and of the wall temperature following a suddenly appliedheat flux (c).4.64CHAPTER FOURThe correct order of magnitude of the time constant for the transient response of the fluidis given by t~.
For a flat plate (or for any body shape up to the time that the penetration distance of the heat conduction from the boundary is small compared to the radius of curvatureof the body), the conduction heat transfer is given by [244]q" =(4.131)The value of ti is found by equating this value of q" to the steady-state average heat transferq~. For example, for a vertical flat plate 0.1 m (0.32 ft) long that is subjected to an impulsive10°C (16°F) temperature change and is immersed in various fluids at 20°C (68°F), the valuesof ti are as follows: for air, ti = 0.5 s; for water, ti = 5 s; for oil (Pr = 104), ti = 8 0 s; for mercury,/i=2 s.An imposed temporal change of Tw, other than the step change, is also of interest. In fact,because the heat capacity of any body or wall is finite, a step change could never be achieved.If the time constant associated with the prescribed change in Tw is much larger than t~, the heattransfer at each instant in the transient can be accurately calculated from the steady-state natural convection equation; this is called a quasi-static transient.
From the above estimates of t~it will be appreciated that the quasi-static approximation will usually be valid for gases.The analogous problem of response to a step change in surface heat flux is depicted in Fig.4.37c. The inverse of Tw- T= (proportional to the heat transfer coefficient) again either mayfall monotonically to its steady-state value or may undershoot; the possibility of an undershoot, which corresponds to a temperature overshoot, may have serious ramifications in practical problems and has provided much of the incentive for studying the external transientproblem. As before, a step change in surface heat flux could never be achieved in practicebecause of finite surface heat capacitance.
If the time constant for changes in surface heat fluxat the surface is large compared to ti, either because of body heat capacitance or because heating is only gradually applied, the quasi-static approximation will again be accurate.Vertical Surfaces. Analyses and measurements related to transient heat transfer on verticalsurfaces have been reviewed by Ede [79].
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
