BOOKCOMP, Inc. — John Wiley & Sons / Page 492 / 2nd Proofs / Heat Transfer Handbook / Bejan 492 FORCED CONVECTION: EXTERNAL FLOWS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [492], (54) Lines: 2457 to 2525 ——— -0.073pt PgVar ——— Normal Page PgEnds: T E X [492], (54) where Nu is as definedforthetwo-dimensional strip, Pe c = U o (2x s + s )/2α and A/P is the source surface area/perimeter ratio. The correlations of eqs. (6.175) and (6.176) are valid for 10 3 ≤ Pe c ≤ 10 5 , 5 ≤ (2x s + s ≤)2 s ≤ 150, and 0.2 ≤ w s / s ≤ 5, where w s is the heat source width. Additional detailed treatments of the subject have been compiled by Ortega (1996). 6.6.2 Two-Dimensional Block Array Figure 6.22 shows the pertinent dimensions, all nondimensionalized by the plate spac- ing for the two-dimensional block array. Arvizu and Moffat (1982) performed exper- iments for heat transfer from aluminum blocks in forced airflow in such channels. The parameter ranges covered are P h = 1/2, 1/4.6, and 1/7,s = 0.5, 1, 2, 3, 6, and 8, and 2200 < Re P h = UP h /ν < 12,000 P h = dimensional block height. For s ≥ 2, the heat transfer coefficients for a fixed P h almost collapse into a single curve. The effect of P h on heat transfer is large for tightly placed blocks; for s = 0.5, when P h is increased from 1/7 to 1/2, the Nusselt number almost doubles. However, Lehman and Wirtz (1985) found a near collapse of heat transfer data of various P h ranging from 1/2 to 1/6 with s = P h and P /P h = 4. The data of Lehman and Wirtz (1985) were obtained for 1000 < Re P < 12,000 and 2 3 ≤ P ≤ 2. A visualization study was also conducted by Lehman and Wirtz and it revealed modes of convection that depend on the block spacing. For close block spacing, s/P = 0.25, cavity-type flow is formed in the interblock space, indicating that the forward and back surfaces of the block do not contribute much to the heat transfer. When the spacing is widened to s/P = 1, significant cavity-channel flow interac- tions were observed. Davalath and Bayazitoglu (1987) performed numerical analysis on a three-row block array placed in a parallel-plate channel. Heat transfer correlations were derived for the cases where the following dimensions are fixed at, P = s = 0.5,P h = 0.25, 1 = 3.0, 2 = 9.5, and t = 0.1. The Reynolds number is defined as Re = U 0 H/ν, where U 0 is the average velocity in the channel (with an unobstructed cross section), H is the channel height, and ν is the fluid kinematic viscosity. The average Nusselt number is Nu = ¯ h/k, where ¯ h is the average heat transfer coefficient over the block surface, is the dimensional block length, and k f is the fluid thermal conductivity. The correlation between Nu, Re, and Pr is given in the form Nu = A · Re B · Pr C (6.177) P l l 1 l 2 s P h Flow 1 t Figure 6.22 Three-row block array studied by Davalath and Bayazitoglu (1987). BOOKCOMP, Inc. — John Wiley & Sons / Page 493 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM OBJECTS ON A SUBSTRATE 493 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [493], (55) Lines: 2525 to 2536 ——— -0.1953pt PgVar ——— Normal Page PgEnds: T E X [493], (55) TABLE 6.4 Correlating Constants A, B,andC for Use in Eq. (6.177) for the Case of Blocks on an Insulated Plate Block Number ABC 1 0.69085 0.43655 0.41081 2 0.57419 0.40087 0.37771 3 0.48004 0.39578 0.36405 TABLE 6.5 Correlating Constants A, B,andC for Use in Eq. (6.177) for the Case of Blocks on a Conducting Plate Block Number ABC 1 1.09064 0.37406 0.38605 2 0.89387 0.34568 0.33571 3 0.67149 0.36757 0.31054 TABLE 6.6 Ratio of Heat Dissipation from the Bottom Surface of a Plate to the Total Heat Dissipation by Block Percent of Total Heat k ∗ plate Dissipation in Blocks 10 46.0 5 44.3 1 32.8 The correlation constants, A, B, and C are given in Table 6.4 for the blocks on an insulated plate and in Table 6.5 for the blocks on a conducting plate having the same thermal conductivity as the block. Table 6.6 shows the ratios, expressed as a percentage, of the heat transfer rate from the bottom surface of the plate to the total heat dissipation by the block. Here k ∗ plate is the ratio of thermal conductivity of the plate to the fluid thermal conductivity. 6.6.3 Isolated Blocks Roeller et al. (1991) and Roeller and Webb (1992) performed experiments with the protruded rectangular heat sources mounted on a nonconducting substrate. The pertinent dimensions are shown in Fig. 6.23, where it is observed that H and W are the height and the width, respectively, of the channel where the heat source/substrate composite is placed, P h is the height, P the length, and P w the width of the heat BOOKCOMP, Inc. — John Wiley & Sons / Page 494 / 2nd Proofs / Heat Transfer Handbook / Bejan 494 FORCED CONVECTION: EXTERNAL FLOWS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [494], (56) Lines: 2536 to 2562 ——— 0.78311pt PgVar ——— Normal Page * PgEnds: Eject [494], (56) Figure 6.23 Rectangular block on a substrate surrounded by channel walls. source. The channel width is fixed at W = 12 mm, and the height H was allowed to vary from 7 to 30 mm. The heat source dimensions covered by the experiments were P = 12 mm; P h = 4, 8, and 12 mm, H −P h = 3, 8, and 12 mm, and P w = W = 12 mm. The heat transfer correlation is given by Nu = 0.150Re 0.632 (A ∗ ) −0.455 H P −0.727 (6.178) where Nu = ¯ hP /k and where in ¯ h = q A s ( ¯ T s − T ∞ ) ¯ h is the average heat transfer coefficient, q the heat transfer rate, A s the heat transfer area, A s = 2P h P w + P P w + 2P h P ¯ T s the average surface temperature, and T ∞ the free stream temperature. The Reyn- olds number is defined as Re = UD H /ν, where U is the average channel velocity upstream of the heat source, D H the channel hydraulic diameter at a section unob- structed by the heat source, and ν the fluid (air) kinematic viscosity. A ∗ is the fraction of the channel cross section open to flow: A ∗ = 1 − P w W P h H BOOKCOMP, Inc. — John Wiley & Sons / Page 495 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM OBJECTS ON A SUBSTRATE 495 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [495], (57) Lines: 2562 to 2587 ——— 0.25099pt PgVar ——— Normal Page PgEnds: T E X [495], (57) Airflow Support Substrate Q Q B Q A Block Figure 6.24 Heat transfer paths from a surface-mounted block to airflow. (From Nakayama and Park, 1996.) Equation (6.178) is valid for 1500 ≤ Re ≤ 10,000, 0.33 ≤ P h /P ≤ 1.0, 0.12 ≤ P w /W ≤ 1.0, and 0.583 ≤ H/P ≤ 2.5. Here, a realistic error bound is 5%. In actual situations of cooling electronic packages the heat flow generally follows two paths, one directly from the package surface to the coolant flow, and the other from the package through the lead pins or solder balls to the printed wiring board (PWB), then through the PWB, and finally, from the PWB surfaces to the coolant flow. Figure 6.24 depicts heat flows through such paths; Q is the total heat generation, Q A the direct heat transfer component, and Q B the conjugate heat transfer component through the substrate, hence Q = Q A + Q B which is due to Nakayama and Park (1996). Equation (6.178) can be used to esti- mate Q A . The ratio Q B /Q A depends on the thermal resistance between the heat source block and the substrate (the block support in Fig. 6.24 simulating the lead lines or the solder balls), the thermal conductivity and the thickness of the substrate, and the surface heat transfer coefficient. The estimation of Q B is a complex process, particularly where the lower side of the substrate is not exposed to the coolant flow, which makes Q B find its way through only the upper surface. This is the case often encountered in electronic equipment. Convective heat transfer from the upper surface is affected by flow development around the heat source block, which is three-dimensional, involv- ing a horseshoe vortex and the thermal wake shed from the block, leading to a rise in the local fluid temperature above the free stream temperature. Nakayama and Park (1996) studied such cases using a heat source block typical to electronic package, 31 mm × 31 mm × 7 mm. A good thermal bond between the block and the substrate, of the order of R = 0.01 K/W, and a high thermal conductance of the substrate, such as that of a 1-mm-thick copper plate, maximizes the contribution of conjugate heat transfer Q B to the total heat dissipation Q, raising the ratio Q B /Q to a value greater than 0.50. 6.6.4 Block Arrays Block arrays are common features of electronic equipment, particularly, large systems where a number of packages of the same size are mounted on a large printed wiring BOOKCOMP, Inc. — John Wiley & Sons / Page 496 / 2nd Proofs / Heat Transfer Handbook / Bejan 496 FORCED CONVECTION: EXTERNAL FLOWS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [496], (58) Lines: 2587 to 2605 ——— 1.927pt PgVar ——— Normal Page PgEnds: T E X [496], (58) board (PWB) and cooled by air in forced convection. Numerical analysis of three- dimensional airflow over a block array is possible only when a fully developed situation is assumed. For fully developed flow, a zone around a block is carved out, and a repeating boundary condition is assumed on the upstream and downstream faces of the zone. In general, the analysis of flow and heat transfer over an entire block array depends too much upon computational resources, and experiments are frequently the sole means of investigation. However, experiments can be costly, especially when it is desired to cover a wide range of cases where the heat dissipation varies from package to package. To reduce the demand for experimental (and computational as well) resources, a systematic methodology has been developed. Consider the block array displayed in Fig. 6.25. Block A dissipates q A and block B, q B . Assume for the moment that the other blocks are inactive; that is, they do not dissipate. The temperature of air over block B can be written as T air,B = T 0 + θ B/A q A (6.179) where T 0 is the free stream temperature and θ B/A in the second term represents the effect of heat dissipation from block A. Equation (6.179) is based on the superposi- tion of solutions that is permissible because of the linearity of the energy equation. However, the factor θ B/A results from nonlinear phenomena of dispersion of warm air from block A and is a function of the relative location of B to A and the flow velocity. To find θ B/A by experiment, block A may be energized while block B re- mains inactive (q B = 0). A measurement of the surface temperature of block B is then divided by q A . Because q B = 0, its measured surface temperature is called the adiabatic temperature. The next step is to activate block B and inactivate block A(q A = 0). The heat transfer coefficient measured in this situation is called the adiabatic heat transfer coefficient and is denoted as h ad,B . When both block A and block B are energized, the heat transfer from block B is driven by the temperature difference T s,B − T air,B , Figure 6.25 Block array on a substrate. BOOKCOMP, Inc. — John Wiley & Sons / Page 497 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM OBJECTS ON A SUBSTRATE 497 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [497], (59) Lines: 2605 to 2648 ——— 1.62717pt PgVar ——— Normal Page PgEnds: T E X [497], (59) where T s,B is the surface temperature of block B and T air,B is given by eq. (6.179). Again using the superposition principle, the heat flux at block B is q B = h ad,B T s,B − T air,B (6.180) Once θ B/A is determined, it is straightforward to estimate q B (or T s,B when q B is specified) for any value of q A from eqs. (6.179) and (6.180). Extension of this concept to a general case includes taking account of the contri- butions of all blocks upstream of block B in the equation for T air,B : T air,B = T 0 + i,j θ B/(i,j ) q (i,j) (6.181) where (i,j) is the row and column index and the summation is performed for all the packages upstream of block B. Although the foregoing concept looks convenient at first sight, it is a tedious and time-consuming task to determine θ B/(i,j ) . Except for a limited number of cases, there have been few correlations that relate θ B/(i,j ) to the geometrical and flow parameters. Moffat and Ortega (1988) summarized the work on this subject, and Anderson (1994) extended the concept to the case of conjugate heat transfer. The heat transfer data corresponding to the adiabatic heat transfer coefficient in downstream rows where the flow is fully developed were reported by Wirtz and Dykshoorn (1984). The data were correlated by the equation Nu P = 0.348Re 0.6 P (6.182) where the characteristic length for Nu and Re is the streamwise length of the block, P . 6.6.5 Plate Fin Heat Sinks While the plate stack discussed in Section 4.2.1 allows bypass flow in two-dimen- sional domain, bypass flow around an actual heat sink is three-dimensional. Numer- ical analysis of such flow is possible but very resource demanding. Ashiwake et al. (1983) developed a method that allows approximate but rapid estimation of the heat sink performance. In their formulation the bypass flow rate is estimated using the balance between the dynamic pressure in front of the fin array and the flow resistance in the interfin passages. Although the validity of the modeling was well corroborated by the experimental data, the method requires computations of the several pressure balance and heat transfer correlations, and the task of developing a more concise formulation of heat sink performance estimation remains. Presently, the performance estimation is largely in the realm of empirical art, although Ledezma et al. (1996) and Bejan and Sciubba (1992) clearly show optimum spacings which agree with the theory. The heat sink is placed in a wind tunnel and the thermal resistance is measured. The thermal resistance is defined as BOOKCOMP, Inc. — John Wiley & Sons / Page 498 / 2nd Proofs / Heat Transfer Handbook / Bejan 498 FORCED CONVECTION: EXTERNAL FLOWS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [498], (60) Lines: 2648 to 2664 ——— 0.06602pt PgVar ——— Normal Page PgEnds: T E X [498], (60) Figure 6.26 Thermal resistance of plate-fin heat sinks. (From Matsushima and Yanagida, 1993.) R = Q T b − T 0 (K/W) (6.183) where Q is the power input to the heater bonded to the bottom of the heat sink, T b the temperature at the bottom surface of the heat sink, and T 0 the airflow temperature in front of the heat sink. Figure 6.26 shows examples of thermal resistance data, where U is the free stream velocity. All the data were obtained with aluminum heat sinks having a 22 mm × 22 mm base area. Of course, there is trade-off between the heat transfer performance and the cost of heat sink. Conventional extruded heat sinks (see Fig. 6.26) are at the lowest in the cost ranking but also in the performance ranking. The heat sink having 19 thin fins (0.15 mm thick) on the 22-mm span provides low thermal resistance, particularly at high air velocities, but the manufacture of such a heat sink requires a costly process of bonding thin fins to the base. 6.6.6 Pin Fin Heat Sinks As electronic systems become compact, the path for cooling airflow is constrained. This means increased uncertainty in the direction of airflow in front of the heat sink. The performance of plate fin heat sinks degrades rapidly as the direction of air flow deviates from the orientation of the fins. The pin fin heat sink has a distinct advantage BOOKCOMP, Inc. — John Wiley & Sons / Page 499 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM OBJECTS ON A SUBSTRATE 499 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [499], (61) Lines: 2664 to 2687 ——— -0.93945pt PgVar ——— Normal Page PgEnds: T E X [499], (61) Airflow Fan L a Figure 6.27 Pin-fin fan sink assembly. over the plate fin heat sink in that its performance is relatively insensitive to the direction of the airflow. Figure 6.27 shows a scheme that exploits the advantage of pin fin heat sink to the fullest extent. A small axial fan is mounted above the fin heat sink, and air is blown from above to the heat sink. The airflow is longitudinal to those pins in the central area, and the pins in the perimeter are exposed to cross flow. Recently, the scheme has become popular for cooling CPU chips in a constrained space. The work of Wirtz et al. (1997) provides a guide for the estimation of the perfor- mance of a pin fin heat sink/fan assembly. The dimensions of the pin fin heat sinks tested by Wirtz et al. (1997) are as follows: Foot print area: 63.5 mm × 63.5 mm L in Fig. 6.26: 63.5 mm Dimensionless pin diameter: d/L = 0.05 Dimensionless pin height: a/L = 0.157 − 0.629 Fin pitch-to-diameter ratio: p/d = 2.71 − 1.46 Number of pins on a row or a column: n = 8, 10, and 14 The fan used in the experiment has overall dimensions of 52 mm × 52 mm × 10 mm, a 27-mm-diameter hub, and a 50-mm blade shroud diameter. The overall heat transfer coefficient U is defined as U = Q A T ∆T (6.184) where Q is the heat transfer rate, A T the total surface area of the heat sink, and ∆T the temperature difference between the fin base and the incoming air. The Nusselt number is defined as Nu = UL/k, where k is the fluid thermal conductivity. Two BOOKCOMP, Inc. — John Wiley & Sons / Page 500 / 2nd Proofs / Heat Transfer Handbook / Bejan 500 FORCED CONVECTION: EXTERNAL FLOWS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [500], (62) Lines: 2687 to 2735 ——— 5.62617pt PgVar ——— Normal Page * PgEnds: PageBreak [500], (62) types of correlations are proposed, one for a given pressure rise maintained by the fan, ∆p, Nu = 7.12 × 10 −4 C 0.574 ∆p a L 0.223 p d 1.72 (6.185) where in C ∆p = ρL 2 ∆p µ 2 5 × 10 6 <C ∆p < 1.5 × 10 8 µ is the dynamic viscosity of the air. The other correlation is for a given fan power P W : Nu = 3.2 × 10 −6 C 0.520 P W a L −0.205 p d 0.89 (6.186) where C P W = ρLP W µ 3 10 11 <C P W < 10 13 Wirtz et al. (1997) also reported on experimental results obtained with square and diamond-shaped pins. 6.7 TURBULENT JETS Jets are employed in a wide variety of engineering devices. In cases where the jets are located far from solid walls, they are classified as free shear flows. In most cases, how- ever, solid walls are present and affect the flow and heat transfer significantly. These flows can take many configurations. Some of the practically important cases for heat transfer include wall jets and jets impinging on solid surfaces as indicated in Fig. 6.28. Wall jets are frequently employed in turbomachinery applications and are not discussed here. Jet impingement on surfaces is of interest in materials packaging and electronics cooling. 6.7.1 Thermal Transport in Jet Impingement Due to the thin thermal and hydrodynamic boundary layers formed on the impinge- ment surface, the heat transfer coefficients associated with jet impingement are large, making these flows suitable for large heat flux cooling applications. The relationship for impingement of a jet issuing from a nozzle at a uniform velocity and at ambient temperature T e , with a surface at temperature T s , can be written as q = h(T s − T e ) (6.187) BOOKCOMP, Inc. — John Wiley & Sons / Page 501 / 2nd Proofs / Heat Transfer Handbook / Bejan 501 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [501], (63) Lines: 2735 to 2746 ——— * 528.0pt PgVar ——— Normal Page * PgEnds: PageBreak [501], (63) Main stream Jet ( ) Film cooling with tangential injectiona ( ) Transpiration coolingb ( ) Unconfined jet impingementc ( ) Confined jet impingementd ( ) Multiple confined jet impingemente Porous surface Fluid injection Nozzle Nozzle plate Nozzle plate Target surface Target surface Target surface Plenum Plenum Figure 6.28 Several configurations of jet cooling arising in applications. . = ¯ hP /k and where in ¯ h = q A s ( ¯ T s − T ∞ ) ¯ h is the average heat transfer coefficient, q the heat transfer rate, A s the heat transfer area, A s = 2P h P w + P P w + 2P h P ¯ T s the average. coolant flow. Figure 6.24 depicts heat flows through such paths; Q is the total heat generation, Q A the direct heat transfer component, and Q B the conjugate heat transfer component through the. The heat transfer coefficient measured in this situation is called the adiabatic heat transfer coefficient and is denoted as h ad,B . When both block A and block B are energized, the heat transfer