Document 857p8y08wBxNykoDY7xbyv7Ly

50 CHAPTER 5 1959 Guide- Table 2____Approximate Unit Conductances for Thermal Convection for Several Flow Systems {Concluded) Heat framfn. Equatimi* and Hi Umdi of AppCcotton FrM Convection* 11 Ref. 6 Free convection past a heated horizontal cylinder. Equation for Air h. -- .271 (0`(0" where 10* < Not < 10* 12 Ref. 6 ij f 1 Free convection past a 1 1 single vertical surface. kJ Equation for Air (y "(")" where 10* < Not < 19* where 10* < Nar < 10** 13 Ref. 5 Jk U-- i --J Free convection past a heated horizontal surface (face up). 0.478 Equation for Air Ref. 5 r-- *--i Free convection past a heated Equation for Air --(0"(r horizontal surface (face down). where 10* < Not < 107 inhere 10` < Al,,. < W Noaendofvre and Dteoftsreo* for Tobfo 2 c, = heat capacity at constant pressure, Btu per (pound) (Fahrenheit degree). D = cylinder diameter, feet. / * subscript denoting film. g = body force per unit mass, feet per hour per hour. (For static system on earth, g = 32.2 X 3600* feet i = a dimension of the system, feet. m -- a subscript denoting mean. P = pressure, atmospheres. Pm x pressure (atmospheric), atmospheres. t -- temperature, Fahrenheit. T -- temperature, Fahrenheit, absolute. per hour per hour.) G *= 3600 uw> = mass flow per unit cross-sectional area normal to flow, pounds per (hour) (square foot of flow cross-section). NCt = Grashof modulus, dimensionless (Nor " D*p*0aig/fi*). h, = average unit thermal convective conductance from the leading edge of surface to the position x, Btu per (hour) (square foot) (Fahrenheit degree). *" local unit thermal convective conductance, at the position x from the leading edge of surface, Btu per (hour) (square foot) (Fahrenheit degree). k = thermal conductivity, Btu per (hour) (square foot) (Fahrenheit degree per foot thickness). u -- fluid velocity, feet per second. V -- volume, cubic feet, x = a dimension of the system, feet. KS>"0 -- coefficient of cubical expansion (0 TM for perfect gases 0 = \/T. &l h difference between wall and fluid temperatures, Fahrenheit degrees. m = fluid viscosity, pounds per (hour) (foot). p -- density, pounds per cubic foot, co = infinity, referring the quantity to a point not directly affected by the phenomenon in question. Fluid properties should be vluted at the arithmetic bum fluid temperature, t, - (W/sc + (/) diridad by 3. __ . .. __ b These exwexiooa are suitable appirudmationa to longitudinal flow is other than right oreular eyiindert, provided the hydreulm diameter a employee! me toe cos' duit dimension poremeter. For ooo^rculmr eroes sections, the hydraulic diameter is equal to four times the erros-sectional area dnndedby the wetted pen*"' __ * For low rmtre of teat transfer by free convection the exponent decreases towards xero, and for higher rates, increases towards 0.33. The above equations employing an exponent equal to 0-25 are applicable in the intermediate range indicated. = the geometrical factor which is dimensionless and 1. This factor accounts for the shape and relative position of the two surfaces. The value ol FA -- 1 may be used in the eases of large parallel planes, long conceQtric cylinders or smaller bodies in large enclosures. -- the enussivity factor which is also dimensionless and S 1. This factor accounts for the absorption and emission characteristics of the surfaces for the radiation which exists. Emissivities or absorptivities (e) for many com mon surfaces, are given in Table 3. The value of Pg for large parallel planes, long concentric cylinders, or large enclosed bodies is 1 + (1/e, + l/x -- 1)- The radiation under black-body conditions, or for an emissivity of 1.0, is given in Table 47 for cold surfaces as low as --39 F to warmer surfaces as high as 139 F. Some net radiation exchange solutions for several common radia tion systems are given in Table 5. There are several methods by which the geometrical HeatTransfer 51 factor Fa can be determined. One method involves the use of a mechanical geometrical integrator (Reference 8). Photo graphic and other methods are given in References 9 and 10. Equivalent Conductance for Radiation Although Equation 5 is a suitable equation for describing radiant exchange, it is not convenient for computations where other modes of energy transfer are operative. For such cases, it is convenient to define an equivalent conductance for radiation -by the equation: q, - KA(t, - tt) (6) The conductance A, thus defined is a function of the shapeemissivity factor, as well as the temperatures of the radia tor and receiver. Fig. 7 shows a plot of the equivalent con ductance for two black bodice (i.e., with emissivities equal to unity) which exchange energy only with one another. Combined Convection and Radiation It should be noted that the previous equations and-tables give the heat transfer by convection and by radiation com puted separately. In many practical cases it is desirable to treat convection and radiation as a single combined process, using a first-power equation: 0rt " -- <,) (7) where q,e is the total heat flow due to radiation and convec tion, in Btu per hour. Values of hre, the surface or film con ductance for combined radiation and convection, are given in Table 3 and Fig. 4 of Chapter 9. Complete tables for the combined heat transfer of steam and hot water radiators, pipes, coverings, etc., will be found in the appropriate chap ters. HEAT-ROW RESISTANCE In most of the steady-state heat transfer problems en countered in air conditioning applications, more than one of the heat transfer mechanisms are effective, and the ther mal-current flows through several resistances in series or in parallel. In using the resistance concept, the calculations involved are analogous to the application of Ohm's Law in electricity, viz., the heat flow or thermal current is directly proportional to the thermal potential or temperature differ ence, and inversely proportional to the thermal resistance: Following the electrical analogy, when there is a thermal current flowing through several resistances in series, the re sistances are additive: Rr = R, + Rt + R* + - + ft. (9) Similarly, conductance is the reciprocal of resistance, and for heat flow through several resistances in parallel, the conductances are additive: tical analyses of heat transfer in building walls, in fin-tube coils, and in pipe coverings, are usually computed by this method. The same resistance analysis may be applied to complicated steady-state conduction problems. Table 6 gives the. resistances in six common cases of steady-state conduc tion. A complete analysis by the resistance method is well il lustrated by considering the heat transfer from the air out side to the cold water inside of an insulated pipe. The tem perature gradients and the nature of the resistance analysis are indicated by the two sketches of Fig. 8. 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 convection then passes through the insulating layer and the pipe wall by thermal conduction, and thence by convection and radia tion into main cold water streams. (Radiation is not signifi cant on the water side as liquids are sensibly opaque to radi ation, although water transmits energy in the visible re gion.) The contact resistance between the insulation and the pipe wall is presumed to be equal to zero. Referring to Fig. 8, the heat transferred for a given length N of pipe, ?rc, Btu per hour, may be thought of as flowing through the parallel resistances ft, and R,, associated with the insulation surface radiation' and convection transfer. Then the flow is through the resistance offered to thermal conduction by the insulation, ft,, through the pipe wall resistance, ft,, and into the water stream through the con vection resistance, fti. Note the analogy to the direct cur rent electrical circuit problem. A temperature (potential) drop is required to overcome these resistances to the flow of thermal current. The total resistance to heat transfer, Rt, (hour) (Fahrenheit degrees) per Btu, is the summation of the individual resistances: Rt TM fti -+- ft* ri* ft, + ft, (11) where the result-ant parallel resistance ft is obtained from R, (12) Provided the individual resistances may be evaluated, the total resistance can be obtained from this relation. Then the heat transfer for the length of pipe (N, ft) con be es tablished by the relation q,, (Btu per hour) = ^ ftr (13) For a unit length of the pipe the heat transfer rate is ~ Btu per (hour) (foot) - ^ (14) Cr-;-^; + s, + a + ' (10) Practical Heat Transfer Problems The use of these relations for resistance and conductance makes possible the solution of many practical heat transfer problems. As discussed in Chapters 9, 23, and 32, the prac Fig. 3 .... Radiation Between Surfaces