Document 2RVnXB50Jv31x0z3GGaO9Bye6

68 CHAPTER 4 1965 Guide And-Data Book tube-:or longitudinal fins inside the tubes. .Since;the..'heat transfer- coefficient -for the- refrigerant ;side- isl.generally smaller than the coefficient for-the water, side, the addition of fins on:the refrigerant,side leads to compact.designs and to appreciable improvement in performance.- i - . When fins are inside the tubes, they tend to spread the liquid-refrigerant over.-a larger area,-thus giving a film' of small thickness'which results in higher, heat transfer-coeffi cients. High fin efficiency is important for boiling- once .the coefficient-is reduced by lower-temperature difference;,-espe cially, .-if this shifts boilingrfrom the nucleate-regime to .the free convection regime (Fig. 7). -'V--i-'v!' '...When fins are inside tubes, the refrigerant -pressure ^drop can be,high. Thfo.will result in reduced available temperature difference and lower-heat ^transfer. It can-be:avoided 1-by carefully determining- the 1 number, of parallel refrigerant passes-which'give optimum loading for best' overalLibeat transfer. -i. i . , ..- OVERALL; HEAT TRANSFER' . >. .- ''In most of1 the steady state heat' transfer problems1 en countered in practice,' more than one'of'the heat''transfer modes maybe:irivolved simultaneously. Therefore,- it is convenient to combine the various heat transfer coefficients' into an overall coefficient in oriler that the total heat transfer may be calculated from the terminal temperatures. 'The 'solution to this problem is made much simpler if one employs the con cept of the thermal circuit and thermal resistance.:.. Local Overall-Coefficient of Heat . Transfer-- Resistance Method .. >. j: Consider an application where heat is transferred'from one ffuid'to.another by-.a three-step steady-state process: from a warmer 'fluid to a solid wall, through the solid wall, and thence to a colder fluid. It is customary, .to.employ en overall coefficient of.heat transfer, U, based on toe'overall difference between^ the\bulk temperatures of the'two fluids, ;ii--I* defined'as follows: .. ' ' '\ q - UAiix -h) \ (22) tthere A is the surface area. Since Equation-22 is essentially a definition of U,' the surface area 'A'upon'which U is baaed is arbitrary, and should always be specified in-referring to.'(7. The temperature drops across each part of the heat flow path are as follows: \ --_ \ <i? -- fa - qRi . fa,-. fa " qRt I .<*--<* " ?R*............. ;v-' where fa and fa are the surface temperatures of the wall on the warm side and cold side respectively,'and Ri, R,, and 72* are the thermal resistances. Since the same Quantity of heat;flows tothrough each thermal resistance, these^equations canbe.com bined give: ' -1 * --^-- " UA " T. (23) As ` demonstrated above, the. equations 1 involved are analogous to those for electrical circuits; i.e., when there is a thermal current flowing through several resistances in series, the resistances are.additive. : . .. R. fi,+R, + R.+ + R. (24) Similarly, conductance is the redprocal of reSstanoe, and for heat-flow through. Beveral resistances in parallelthe con ductances are additive: _ For conduction through a solid material, the thermal resistance is a function..of the mean length of the heat flow path, the thermal conductivity of the material, and the mean cross-sectional area normal to'the flow, Equation 2.'Equa tions for various configurations are given in Table 3. , - In the case of; convection,..(h thermal - resistance is in versely proportional to the .convection .coefficient,Ac, and the surface area to which it applies. . -The. thermal resistance for radiation is written similarly to that for convection. ; ' ; j; . ' KA (27> The term radiation-coefficient K has no physical significance as such, being introduced primarily for convenience in compu tations. In general, it will be.a.very complex function' of the temperatures, radiation, properties, and geometrical arrange mentof the enclosure and the body.in question. For-the sim ple case of a small body-in a large enclosure, thc;radiation coefficient- can be determined from Equation 12 as.follows: - * = r'/-T ~ 0'-<7',' + 7V)<T` + r*> <2S> toftere' ' <i -- emissivity'of the enclosed body. Ti = absolute temperature of enclosed body. absolute temperature of enclosure. Equation 28, with -- 1, is plotted in Fig. 20. The use of these relations for resistance and conductance makes possible toe solution of many practical heat transfer problems. As dfocused in Chapter 24, the practical analyses of heat'transfer in building walls and pipe coverings are usually computed by this method. ' ' A complete analysis by toe resistance method is well illus trated by considering the heat transfer from the air outside to the cold water inside an insulated pipe. The temperature gradients and the nature of the resistance analysis are indi cated by the two sketches of Fig. 21. ' Since air is sensibly transparent to radiation, there will be some heat transfer by both radiation and convection to the outer'insulation surface. The`mechanisms act in parallel on the-air side. The total transfer by radiation and convection thftn pftgaftw through toe Simulating layer and the pipe wall'by thermal conduction, and-thence by convection and radiation into the cold water stream.! (Radiation is' not significant on the water side as liquids-are7sensibly-opaque to-radiation, although-water transmits energy in toe visible region.) The contact resistance between the insulation and-the pipe wail is assumed negligible:' ' ; Referring to'Fig. 21,'the heat transferred for a given length L of pipe,' ?n(Btu/hr) may be thought of as flowing through 'toe parallel-resistances fi, and Re, associated-with the insula tion surface radiation and convection coefficients. -The flow then proceeds - through the resistance offered' to' thermal conduction'by the'insulation R*, through the pipe wall re sistance R*, and into the water stream through the convection resistance Ri.-Note the analogy to the direct current electrical f ! Heat Transfers circuit problem. A temperature (potential) drop is required to overcome these resistances to toe flow of thermal current. The total or.overall resistance to heat transfer R is the sum mation of tiie individual resistances: - +-vm R. Rt R.+ R, + Ri - .(29) where the resultant parallel resistance' R fo-bbtained 'from Provided the individual resistances may. be. evaluated,- the total resistance, can be obtaiped from thin relation. The heat transfer fonthe length of pipe, L, (ft) can be established by the relation:,. 5 >i(Btu/hr) - (fc - O'/R. " ` (31) For a unit.length,of the pipe the heat transfer rate is, WL(Btu/hr ft).- (t, -Q/R.L ; '-(32) The temperature drop, At, through an individual resistance may then be calculated from the relation: ' s i . -At -Rg, toftere R is the resistance in question. (33) 11 69, Mean Temperature Difference When heat is exchanged between two fluids flowing'con tinuously through a heat exchanger the local temperature difference. At, varies along toe flow path. Two methods may be used to account for this variation. In the first, the heat transfer is calculated iming tl familiar rate equation: q = UAAU (34) tohere U is toe overall coefficient of heat transfer from fluid to fluid, A fo an area associated with the coefficient U, and At,, fo toe necessary mean temperature difference. For paraltpl flow or counter-flow heat exchangers and for any exchanger in which one of the fluid's temperature' is sub stantially constant, the mean temperature difference is:' Afa All Ati log,-- .2.3 logi* --Alt Ait (35) where AA and Alt are the temperature differences between the fluids at inlet and exit of the heat exchanger. At, fo often called toe logarithmic mean temperature difference. Equation 35 for At, fo strictly true only if the overall coeffi cient and the specific heats of the fluids are constant through the heat exchanger, and there are no heat losses. In most practical cases, these assumptions are reasonably well approximated. A procedure for dealing with cases with a variable overall coefficient U fo outlined by McAdams.* Calculations using Equation 34 and At, are found con venient when the terminal temperatures are known. In many cases, however, the temperatures of the fluids leaving the exchanger are not known. To avoid trial-and-error calcula tions, an alternate method, originally proposed by Nusselt, has been recommended by London and Kays.** It involves the use of 3 non-dimensional parameters, defined as follows: 1. Exchanger Heat Transfer Effectiveness (q). where Ck(l* - <u) _ C(t, - <,) ' " (Wiu - " CWI* - w (36) Ck -- (tocp)i -- the hot fluid capacity rate, Btu per (hour) - (Fahrenheit degree). . C -- (ice,), a the oold fluid capacity rate, Btu per (hour) . - (Fahrenheit degree). ,, C~i. the smaller of the two capacity rates. ti " terminal temperature of hot fluid, Fahrenheit degrees. Subscript i indicates entering condition, and subscript o indicates leaving condition. I, es terminal temperature of cold fluid, Fahrenheit degrees. 2. Number of Exchanger Heat Transfer Units (NTU). . <NTU) ~r(trJ.u<dAy ' (-7) tofiere A fo the same heat transfer area used in defining the overall coefficient U. 3. Capacity Rate Ratio (Z). : (38) In general it fo possible to express the heat transfer effec? tiveness for a given exchanger as a function of toe number of transfer units arid .the capacity rate'ratio: e * f(NTU, Z, flow arrangement) (39)