Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 55
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For negligible solid heat capacitance, results of analyses are discussed for step changes in wall temperature and in surface heat flux, and for periodic specified temperature or flux, as well as for other prescribed variations. Most recentanalyses have confirmed the estimate of Siegel [252] that the local time for departure from theconduction regime at distance position x from the leading of a vertical flat plate after a stepchange in wall temperature isto = 1.8(1.5 + Pr) 1/2, [g~(TwxT~)] 11/2(4.132a)The steady-state regime is attained in time t.o, wheret = 5.24(0.952 + pr)l/2 [x]1/2[g~(TwT=)](4.132b)Gebhart and Adams [105, 106, 108] have predicted the average wall temperature responseof a vertical flat plate following a sudden application of internal heating to the wall (appliedby switching on a current).
The plate was assumed to be isothermal in any horizontal plane,and the average (over the length of the plate) wall temperature was reported. The predictionsand measurements for air and water are reproduced in Fig. 4.38. In Fig. 4.38, ~, x*, and Q aredimensionless average surface temperature, time, and thermal capacitance defined in Fig.4.38 by Gebhart and Adams.
In these relations b and M are Prandtl-number-dependentparameters given in Table 4.12, Tw is the average wall temperature (the unknown), P" in thedefinition of Ra* is the rate of internal generation of energy per unit heat transfer surfacearea A, and C~' in the definition of Q is the total heat capacitance of the wall per unit of heatNATURAL CONVECTION4.651.0 m°-,o.7"F!/17/~///0.~-~"~:t/- - Theoretical,., .curvesJ0.029~~Data,~TirO,Water.Q:3o.a -Q:O.OZ9= 0171~'/_,,"c~,IvoiLo : o.oo~4IIII_J_254675. Q = 0.0014r */Q__ -T.(t)- T~Q= ~C (, ~b/ L R a *T~ (pCp).)~/5~' (T~-Too),_.=Tw'atr* : ~'~(bRa*)2/5Ra* = g/3uakP'' L-4FIGURE 4.38 Temperature response of a vertical plate to a stepheat input including the effects of thermal capacitance (from Gebhartand Adams [108]).transfer surface area. All the data collapse into a single curve for Q > 1 because the quasistatic approximation applies in this range of Q.H o r i z o n t a l Wires.
Attention is restricted to horizontal cylinders (see Fig. 4.39 for nomenclature) t h a t a r e nearly isothermal on any plane perpendicular to the axis of the cylinder (thiswill occur if h D / k w < 0.1, where h is the average heat transfer coefficient on the outside surface and kw is the thermal conductivity of the wire). If the wire is suddenly heated by switching on a current, its temperature response in the conduction regime for a given heat capacityratio (pCp)w/pCp can be found [30, 218]. The trends are schematically illustrated in Fig.
4.39 fordifferent values of Ra*.The end of the conduction regime at time xo (see Fig. 4.39 for definition of x) is associatedwith the top-heavy instability (see the section on horizontal rectangular parallelepiped andcircular cylinder cavities) that occurs along the separation line on the top of the cylinder [272].The dimensionless time xo for air, water, and alcohol was adequately correlated by"co = 80.2(Ra*) -2/3(4.133)where P", which appears in the definition of Ra* in Fig. 4.39, is the internal generation rateper unit surface area.In the transition regime ('co < "r, < "r,=), if the conduction heat transfer at 'r,v exceeds thesteady-state heat transfer for high Ra*, Nu falls monotonically as shown in Fig. 4.39.
For smallT A B L E 4.12Constants for the Gebhart Model (Fig. 4.38)PrMb X 10 40.010.721.01010010001.881.411.7940.21.7971.21.7787.41.771371.761184.66CHAPTERFOURPw, Cp wrDIncreasing Ra*TOoNu =RamP" D(Tw-Too) kg,8 P" D4yakr = 4atD2FIGURE 4.39 Temporalresponse of the Nusselt number to suddenly applied internalheating in a horizontal wire.Ra* the heat transfer at 'co falls below the steady-state value, causing the wall temperature toovershoot its steady-state value. The value of Ra* at which the steady-state heat transfer isthe same as the heat transfer at the departure time "co is defined as Ra], and the corresponding 'co is denoted in the figure as 'co; in this case the transition regime is virtually nonexistent.Because the existence of an overshoot in the wall temperature is of crucial concern, it isdesirable to know both Ra] and 'co.
The value of Ra] can be calculated from the followingequation, which fits the prediction of Parsons and Mulligan [218] for intermediate values ofPr (air, water, alcohol):Ra] = [20, 20C3/2]min2pCpC" - -(pc~)w(4.134)where the minimum of the two quantities in brackets is to be used. Subscript w refers to thewire, and pCp is the volumetric heat capacitance of the fluid, x0 is the value of "co evaluatedfrom Eq. 4.133 at Ra = Ra~.Parsons and Mulligan [218] have correlated the time to reach steady state, 'coo,by the equation(Xoo- %) = 2.5 Ixo- %1(4.135)Internal Transient ConvectionOverview.
This section deals with the transient heating or cooling of a fluid in an enclosure,such as a tank. Applications include storage of cryogenic fluids and transients in startup orshutdown of operating systems. Literature reviews by Clark [59] and Hess and Miller [135]are available.Response to Step Change in Wall Temperature.Following a sudden change in wall temperature there is an initial transient in which conduction dominates, as described in the previoussection. This heat transfer depends on the shape of the enclosure but is given roughly by Eq.4.131 at small enough time.Following the conduction regime there is a transition period, during which convectionbecomes established, to a quasi-steady regime in which the bulk temperature of the fluid inthe enclosure gradually approaches the wall temperature.
It is heat transfer in this quasisteady regime that is the subject of this section.NATURAL CONVECTION4.67Spheres. Schmidt [242] measured heat transfer to the fluid inside a sphere following whatwas essentially a step change in wall temperature. Fluids tested were water and three alcohols:methyl, ethyl, and butyl. For 2 × 108 < Ra < 5 x 1011 the Nusselt number was correlated byNu =q"D-(Tw- 73k- 0.098 Ra °345Ra =gfS(Tw wT)D 3(4.136)mwhere Tw is the wall temperature and T is the average temperat__ure of the fluid within thesphere. Properties are to be evaluated at T.
This equation (with T replaced by Too)agrees towithin 18 percent with Eq. 4.49, with constants in Table 4.3a, for steady turbulent heat transfer from the external surface of a sphere. No corresponding data seem to be available for thelaminar flow regime.Cylinders. Evans and Stefany [92] measured the heat transfer to fluids in short cylindrical enclosures and, with length-to-diameter ratios 0.75 < L/D < 2.0, following an abruptincrease or decrease in wall temperature.
Measurements were obtained for both horizontallyand vertically oriented cans, 7 < Pr < 7000 and 6 x 105 < Ra < 6 × 109, and all the data in thequasi-steady regime were correlated to within about 10 percent byNu =q"L--(Tw- r)k= 0.55 RaTMRa = g~(Tw- Ti)L 3(4.137)vamIn this equation T is the mean fluid temperature at any time, Tw is the wall temperature, Ti isthe mean temperature of the quiescent fluid before the transient is initiated, and L is thelength of the cylinder.
All physical properties are evaluated at Tw. The length scales L and Dwere not sufficiently different to affect the correlation much, and L was arbitrarily chosen.The heat transfer coefficient q"/(Tw- T) remained constant throughout the quasi-steadyperiod. Hiddink et al. [138] found that Eq. 4.137 also correlated data for heating only thebottom and side walls of vertical cylinders provided the cylinder diameter replaced L, and acoefficient of 0.52 rather than 0.55 was used. Their experiments were performed for lengthto-diameter ratios of 0.25 to 2.0 and for 5 < Pr < 83,000.Hauf and Grigull [133-135] precisely measured the natural convection heat transfer insidea tube following a step change in the temperature of a fluid in forced convection over the outside of the tube. In this case the heat transfer coefficient on the outer surface is constantthroughout the transient, and the heat capacity of the wall plays an important role.
Cheng etal. [50] have studied conditions leading to the formation of ice inside horizontal tubes (without throughflow), also with uniform heat transfer coefficient between the outside boundaryand a cold environment.mUniform Flux or Linear Rise in Boundary Temperature. If there is a uniform heat flux tothe fluid in the enclosure, all temperatures increase linearly with time in the quasi-steadyregime, but velocities and spatial gradients become virtually independent of time. A similarquasi-steady regime also exists when the boundary temperature is isothermal but increaseslinearly with time.The equations governing the fluid motion and heat transfer in these quasi-steady regimesare (if the property values and boundary conditions are the same) identical to those forsteady-state convection in the same geometry with a uniform internal generation of energy[73].
The heat transfer equations from one situation can therefore be readily transferred tothe other by replacing the constant pcp 3T/3t by the internal generation rate q" (in W/m 3 orBtu/h.ft3).Heat transfer to water in a long cylindrical tube subjected to a linearly increasing wall temperature was measured by Dearer and Eckert [73]. Their data in the laminar quasi-steadyregime can be closely represented by/(Nu = 8 + 3 In ( ( N u t _ 3-i~ ~)max)(4.138)4.68CHAPTER FOURwhere Nu v is given by Eq.
4.45a and where diameter and the difference between the wall temperature and the mean fluid temperature (Tw- T) are the length scale and temperature difference used in Nu and Ra, and • = 10 -3°. The first term in Eq. 4.138 represents the conductionsolution, while the second term is the laminar Nusselt number for a cylinder corrected for curvature effects.
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