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112 CHAPTER 5 1949 Guide1 Table 5. Solutions fob Some Steady-State Thebmal Conduction Problems*- t> Expressions for the resistance R entering into No. the equation: q - At/R (Btu per hour) i. Flat wall or curved wall if curvature is small (wall thickness less than 0.1 of inside dia meter). R - *x Surface area.A Radial flow through a right circular cylinder. at 2wkN .(See footnote c). The buried cylinder. '^ Long cylinder trl Of length. N Radial flow In a hollow sphere. oh.,(v7^) R" trkN m txkN . For > 3, a satisfactory approximation U: 2xkN 2xiN The straight fin or rod heated at one end. Conduction k cross'Section d/______area. A Finned surface of area BB. ftrftanhmft. (see footnotes d and e). For ml > 2.3.,tanh m L * 1 m \/h*p/kA A -- conduction cross-section area. P perimeter of cross-section A. h% -- unit conductance to the surroundings from the fin surface, ft thermal conductivity fin material. At -- wall temperature--ambient temperature + <) R - ^^tanh!'+* jffB ktf /2~fta. VkA " V ki At defined as in Case 5 above. * The dimensions to be employed in these solutions are: length of dimension p,L,r *s feet* units of ft = (hour) (square foot) (Fahrenheit degree for one foot thickness); units of ft, Btu per (hour) (souare foot) (Fahrenheit degree); units of area, A ** square feet. b Pui* c*?111*1 conductivity, ft, in these solutions should be taken at the average material temperature (see table 2). 0 Log* * = 2.303 logu *. d This expression can also be employed as an approximation for tapered fins or of annuls fin* by em ploying average magnitudes of A and p. * tanh is the hyperbolic tangent. Fundamentals of Heat Transfer 113 The resultant resistance of Ec and Rr acting in parallel (see Fig. 4) can now be evaluated as 1__ L , i = 1 - 4---- = 4.54 Btu per (hour) (Fahrenheit degree). ft ~ ft+ ft . 0.312 0.75 ^ ft = 0.22 hr Fahrenheit degree per Btu. The over-all resistance, Rt, surroundings to cold water, is the sum of A 4- Rt -f- R3 + R* = 4-1 hr F deg per Btu for 1 ft length of pipe. Note" that the controlling resistances are Rs and Rt and that neglect of both Ri and Ri would not significantly influence the total resistance, Rt. On the basis of this resistance calculation the heat transfer from the surroundings to the cold water may be evaluated as: -- = ---- = ^ = 21 Btu per (hour) (foot) N Rt 4.1 or about 0.175 tons of refrigeration per 100 ft of pipe. ' since the calculation is based on a 1 ft pipe length : $r. = 21 Btu per hour. The temperature drops through the various resistances are now readily evaluated by Equation 12 as: t. -- la a>r to insulation surface = ft q,, = 0.22 X 21 = 4.6 F. la -- <! through the insulation = ft q,, = 3.9 X 21 = 82 F. -- 1.1 through the pipe wall = ft qn ** 8.5 X 10-4 X 21 = 0.02 F. Ui -- It pipe wall to cold water * ft qr0 = 2.8 X 10"* X 21 = 0.06 F. The solution was obtained on the assumption that the air temperature and the outside temperature differed by 20 deg. In order to obtain a slightly better estimate of the rate of heat transfer the numerical solution should be repeated using the temperatures calculated from the previous listed temperature differences. The foregoing problem serves to illustrate a general method of solving steady-state heat transfer problems. There are many problems which cannot be approximated by steady-state solutions. For instance, the problem of pipe line insulation in transient service; the behavior of auto matically controlled thermoflow circuits; or. the periodic absorption of solar energy by roof and wall structures during the day and nocturnal radiation to the cold sky at night. The transient heat transfer problem differs from the steady-state in that energy storage fates need to be con sidered. Thus thermal capacity in addition to resistance effects is signifi cant. The. vector sum of the thermal capacitance and resistance is the thermal impedance. It is not within the scope of this chapter to deal with these problems. There are, however, solutions available in graphical form for certain special cases. Also a general approximate method may be employed which is analogous, to the treatment of capacity-resistance lumped parameter electrical circuits. REFERENCES * Absorption and Extraction, by T. K. Sherwood (McGraw-Hill Co., 1937). * The Transmission of Heat by Radiation and Convection, by Griffith and Davis (Special Report No. 9, 1922, Department of Scientific and Industrial Research, His Majesty's Stationery Office, London, England). 1 Heat Insulation in Air Conditioning, by R. H. Heilman (Industrial and Engineertng Chemistry, Vol. 28, July,.1936, p, 782).