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unit volume. SCHEMATIC: Fin dimensions Convection Cross section Length Number of coefficient Design w x w (mm) L (mm) fins (W/m2\u22c5K) A 1 x 1 30 6 x 9 125 B 3 x 3 7 14 x 17 375 ASSUMPTIONS: (1) Steady-state conditions, (2) One-dimensional conduction in fins, (3) Convection coefficient is uniform over fin and prime surfaces, (4) Fin tips experience convection, and (5) Constant properties. ANALYSIS: Following the treatment of Section 3.6.5, the overall efficiency of the array, Eq. (3.98), is t t o max t b q q q hA \u3b7 \u3b8 = = (1) where At is the total surface area, the sum of the exposed portion of the base (prime area) plus the fin surfaces, Eq. 3.99, t f bA N A A= \u22c5 + (2) where the surface area of a single fin and the prime area are ( ) 2fA 4 L W w= × + (3) b cA b1 b2 N A= × \u2212 \u22c5 (4) Combining Eqs. (1) and (2), the total heat rate for the array is t f f b b bq N hA hA\u3b7 \u3b8 \u3b8= + (5) where \u3b7f is the efficiency of a single fin. From Table 4.3, Case A, for the tip condition with convection, the single fin efficiency based upon Eq. 3.86, f f f b q hA \u3b7 \u3b8 = (6) Continued... PROBLEM 3.137 (Cont.) where ( ) ( )f sinh(mL) h mk cosh(mL) q M cosh(mL) h mk sinh(mL) + = + (7) ( ) ( )1/ 2 1/ 2 2c b c cM hPkA m hP kA P 4w A w\u3b8= = = = (8,9,10) The single fin effectiveness, from Eq. 3.81, f f c b q hA \u3b5 \u3b8 = (11) Additionally, we want to compare the performance of the designs with respect to the array volume, vol ( )f f fq q q b1 b2 L\u2032\u2032\u2032 = \u2200 = \u22c5 \u22c5 (12) The above analysis was organized for easy treatment with equation-solving software. Solving Eqs. (1) through (11) simultaneously with appropriate numerical values, the results are tabulated below. Design qt qf \u3b7o \u3b7f \u3b5f qf (W) (W) (W/m3) A 113 1.80 0.804 0.779 31.9 1.25×106 B 165 0.475 0.909 0.873 25.3 7.81×106 COMMENTS: (1) Both designs have good efficiencies and effectiveness. Clearly, Design B is superior because the heat rate is nearly 50% larger than Design A for the same board footprint. Further, the space requirement for Design B is four times less (\u2200 = 2.12×10-5 vs. 9.06×10-5 m3) and the heat rate per unit volume is 6 times greater. (2) Design A features 54 fins compared to 238 fins for Design B. Also very significant to the performance comparison is the magnitude of the convection coefficient which is 3 times larger for Design B. Estimating convection coefficients for fin arrays (and tube banks) is discussed in Chapter 7.6. Of concern is how the fins alter the flow past the fins and whether the convection coefficient is uniform over the array. (3) The IHT Extended Surfaces Model, for a Rectangular Pin Fin Array could have been used to solve this problem. PROBLEM 3.138 KNOWN: Geometrical characteristics of a plate with pin fin array on both surfaces. Inner and outer convection conditions. FIND: (a) Heat transfer rate with and without pin fin arrays, (b) Effect of using silver solder to join the pins and the plate. SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) Constant k, (3) Negligible radiation. PROPERTIES: Table A-1: Copper, T \u2248 315 K, k = 400 W/m\u22c5K. ANALYSIS: (a) The heat rate may be expressed as ,i ,o t,o(c),i w t,o(c),o T T q R R R \u221e \u221e \u2212 = + + where ( ) 1t,o(c) o(c) tR hA\u3b7 \u2212= , f f o(c) t 1 NA 1 1 A C \u3b7\u3b7 = \u2212 \u2212\uf8eb \uf8f6\uf8ec \uf8f7\uf8ed \uf8f8 , t f bA NA A= + , ( )f p c pA D L D L D 4\u3c0 \u3c0= \u2248 + , ( )2 2 2b c,b pA W NA W N D 4\u3c0= \u2212 = \u2212 , ( )1/ 2cf p c tanh mL , m 4h kD mL \u3b7 = = , Continued... PROBLEM 3.138 (Cont.) ( )1 f f t,c c,bC 1 hA R A\u3b7 \u2032\u2032= + , and w w 2 L R W k = . Calculations may be expedited by using the IHT Performance Calculation, Extended Surface Model for the Pin Fin Array. For t,cR\u2032\u2032 = 0, C1 = 1, and with W = 0.160 m, Rw = 0.005 m/(0.160 m)2 400 W/m\u22c5K = 4.88 × 10-4 K/W. For the prescribed array geometry, we also obtain c,bA = 1.26 × 10 -5 m 2 , fA = 2.64 × 10-4 m2, bA = 2.06 × 10 -2 m 2 , and At = 0.126 m2. On the outer surface, where ho = 100 W/m 2 \u22c5K, m = 15.8 m-1, f\u3b7 = 0.965, o\u3b7 = 0.970 and t,oR = 0.0817 K/W. On the inner surface, where ih = 5 W/m 2 \u22c5K, m = 3.54 m-1, f\u3b7 = 0.998, o\u3b7 = 0.999 and t,oR = 1.588 K/W. Hence, the heat rate is ( ) ( )4 65 20 C q 26.94W 1.588 4.88 10 0.0817 K W\u2212 \u2212 = = + × + $ < Without the fins, ( ) ( ) ( ) ( ) ,i ,o 4i w w o w T T 65 20 C q 5.49W 1 h A R 1 h A 7.81 4.88 10 0.39 \u221e \u221e \u2212 \u2212 \u2212 = = = + + + × + $ < Hence, the fin arrays provide nearly a five-fold increase in heat rate. (b) With use of the silver solder, o(c),o\u3b7 = 0.962 and t,o(c),oR = 0.0824 K/W. Also, o(c),i\u3b7 = 0.998 and t,o(c),iR = 1.589 K/W. Hence ( ) ( )4 65 20 C q 26.92W 1.589 4.88 10 0.0824 K W\u2212 \u2212 = = + × + $ < Hence, the effect of the contact resistance is negligible. COMMENTS: The dominant contribution to the total thermal resistance is associated with internal conditions. If the heat rate must be increased, it should be done by increasing hi. PROBLEM 3.139 KNOWN: Long rod with internal volumetric generation covered by an electrically insulating sleeve and supported with a ribbed spider. FIND: Combination of convection coefficient, spider design, and sleeve thermal conductivity which enhances volumetric heating subject to a maximum centerline temperature of 100°C. SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) One-dimensional radial heat transfer in rod, sleeve and hub, (3) Negligible interfacial contact resistances, (4) Constant properties, (5) Adiabatic outer surface. ANALYSIS: The system heat rate per unit length may be expressed as ( )2 1o sleeve hub t,o T Tq q r R R R \u3c0 \u221e \u2212 \u2032 = = \u2032 \u2032 \u2032+ + \ufffd where ( )1 o sleeve s ln r r R 2 k\u3c0 \u2032 = , ( )2 1 4 hub r ln r r R 3.168 10 m K W 2 k\u3c0 \u2212 \u2032 = = × \u22c5 , t,o o t 1R hA\u3b7 \u2032 = \u2032 , ( )fo f t NA1 1 A \u3b7 \u3b7\u2032= \u2212 \u2212 \u2032 , ( )f 3 2A 2 r r\u2032 = \u2212 , ( )t f 3A NA 2 r Nt\u3c0\u2032 \u2032= + \u2212 , ( ) ( ) 3 2 f 3 2 tanh m r r m r r \u3b7 \u2212= \u2212 , ( )1/ 2rm 2h k t= . The rod centerline temperature is related to T1 through ( ) 2 o o 1 qrT T 0 T 4k = = + \ufffd Calculations may be expedited by using the IHT Performance Calculation, Extended Surface Model for the Straight Fin Array. For base case conditions of ks = 0.5 W/m\u22c5K, h = 20 W/m2\u22c5K, t = 4 mm and N = 12, sleeveR\u2032 = 0.0580 m\u22c5K/W, t,oR\u2032 = 0.0826 m\u22c5K/W, \u3b7f = 0.990, q\u2032 = 387 W/m, and q\ufffd = 1.23 × 10 6 W/m3. As shown below, \ufffdq may be increased by increasing h, where h = 250 W/m2\u22c5K represents a reasonable upper limit for airflow. However, a more than 10-fold increase in h yields only a 63% increase in q\ufffd . Continued... PROBLEM 3.139 (Cont.) 0 50 100 150 200 250 Convection coefficient, h(W/m^2.K) 1E6 1.2E6 1.4E6 1.6E6 1.8E6 2E6 H ea t g en er at io n, q do t(W /m ^3 ) t = 4 mm, N = 12, ks = 0.5 W/m.K The difficulty is that, by significantly increasing h, the thermal resistance of the fin array is reduced to 0.00727 m\u22c5K/W, rendering the sleeve the dominant contributor to the total resistance. Similar results are obtained when N and t are varied. For values of t = 2, 3 and 4 mm, variations of N in the respective ranges 12 \u2264 N \u2264 26, 12 \u2264 N \u2264 21 and 12 \u2264 N \u2264 17 were considered. The upper limit on N was fixed by requiring that (S - t) \u2265 2 mm to avoid an excessive resistance to airflow between the ribs. As shown below, the effect of increasing N is small, and there is little difference between results for the three values of t. 12 14 16 18 20 22 24 26 Number of ribs, N 2 2.02 2.04 2.06 2.08 2.1 H ea t g en er at io n, q do tx 1E -6 (W /m t = 2 mm, N: 12 - 26, h = 250 W/m^2.K t