Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 75
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However, Lehmanand Wirtz (1985) found a near collapse of heat transfer data of various Ph rangingfrom 1/2 to 1/6 with s = Ph and P /Ph = 4. The data of Lehman and Wirtz (1985)were obtained for 1000 < ReP < 12,000 and 23 ≤ P ≤ 2.A visualization study was also conducted by Lehman and Wirtz and it revealedmodes 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 theforward 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 interactions were observed.Davalath and Bayazitoglu (1987) performed numerical analysis on a three-rowblock array placed in a parallel-plate channel. Heat transfer correlations were derivedfor the cases where the following dimensions are fixed at, P = s = 0.5, Ph =0.25, 1 = 3.0, 2 = 9.5, and t = 0.1.
The Reynolds number is defined as Re =U0 H /ν, where U0 is the average velocity in the channel (with an unobstructed crosssection), H is the channel height, and ν is the fluid kinematic viscosity. The averageNusselt number is Nu = h̄/k, where h̄ is the average heat transfer coefficientover the block surface, is the dimensional block length, and kf is the fluid thermalconductivity.
The correlation between Nu, Re, and Pr is given in the formNu = A · ReB · PrCFlow(6.177)1Phl1tPlsl2Figure 6.22 Three-row block array studied by Davalath and Bayazitoglu (1987).BOOKCOMP, Inc. — John Wiley & Sons / Page 492 / 2nd Proofs / Heat Transfer Handbook / Bejan[492], (54)Lines: 2457 to 2525———-0.073pt PgVar———Normal PagePgEnds: TEX[492], (54)HEAT TRANSFER FROM OBJECTS ON A SUBSTRATE123456789101112131415161718192021222324252627282930313233343536373839404142434445493TABLE 6.4 Correlating Constants A, B, and C for Use in Eq. (6.177)for the Case of Blocks on an Insulated PlateBlockNumberABC1230.690850.574190.480040.436550.400870.395780.410810.377710.36405TABLE 6.5 Correlating Constants A, B, and C for Use in Eq.
(6.177)for the Case of Blocks on a Conducting PlateBlockNumberABC1231.090640.893870.671490.374060.345680.367570.386050.335710.31054[493], (55)Lines: 2525 to 2536———-0.1953pt PgVarTABLE 6.6 Ratio of Heat Dissipationfrom the Bottom Surface of a Plate tothe Total Heat Dissipation by Block∗kplatePercent of Total HeatDissipation in Blocks105146.044.332.8The correlation constants, A, B, and C are given in Table 6.4 for the blocks onan insulated plate and in Table 6.5 for the blocks on a conducting plate having thesame thermal conductivity as the block. Table 6.6 shows the ratios, expressed as apercentage, of the heat transfer rate from the bottom surface of the plate to the total∗heat dissipation by the block.
Here kplateis the ratio of thermal conductivity of theplate to the fluid thermal conductivity.6.6.3Isolated BlocksRoeller et al. (1991) and Roeller and Webb (1992) performed experiments withthe protruded rectangular heat sources mounted on a nonconducting substrate. Thepertinent dimensions are shown in Fig. 6.23, where it is observed that H and W arethe height and the width, respectively, of the channel where the heat source/substratecomposite is placed, Ph is the height, P the length, and Pw the width of the heatBOOKCOMP, Inc.
— John Wiley & Sons / Page 493 / 2nd Proofs / Heat Transfer Handbook / Bejan———Normal PagePgEnds: TEX[493], (55)494123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: EXTERNAL FLOWS[494], (56)Figure 6.23 Rectangular block on a substrate surrounded by channel walls.Lines: 2536 to 2562———0.78311pt PgVar———source. The channel width is fixed at W = 12 mm, and the height H was allowed toNormal Pagevary from 7 to 30 mm. The heat source dimensions covered by the experiments wereP = 12 mm; Ph = 4, 8, and 12 mm, H − Ph = 3, 8, and 12 mm, and Pw = W = 12 * PgEnds: Ejectmm.
The heat transfer correlation is given byNu = 0.150Re0.632∗ −0.455(A )HP−0.727[494], (56)(6.178)where Nu = h̄P /k and where inh̄ =qAs (T̄s − T∞ )h̄ is the average heat transfer coefficient, q the heat transfer rate, As the heat transferarea,As = 2Ph Pw + P Pw + 2Ph PT̄s the average surface temperature, and T∞ the free stream temperature. The Reynolds number is defined as Re = UDH /ν, where U is the average channel velocityupstream of the heat source, DH the channel hydraulic diameter at a section unobstructed by the heat source, and ν the fluid (air) kinematic viscosity. A∗ is the fractionof the channel cross section open to flow:A∗ = 1 −BOOKCOMP, Inc.
— John Wiley & Sons / Page 494 / 2nd Proofs / Heat Transfer Handbook / BejanPw PhW HHEAT TRANSFER FROM OBJECTS ON A SUBSTRATE123456789101112131415161718192021222324252627282930313233343536373839404142434445Airflow495QABlockQBSupportQSubstrateFigure 6.24 Heat transfer paths from a surface-mounted block to airflow. (From Nakayamaand Park, 1996.)Equation (6.178) is valid for 1500 ≤ Re ≤ 10,000, 0.33 ≤ Ph /P ≤ 1.0, 0.12 ≤Pw /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 followstwo paths, one directly from the package surface to the coolant flow, and the otherfrom 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 coolantflow.
Figure 6.24 depicts heat flows through such paths; Q is the total heat generation,QA the direct heat transfer component, and QB the conjugate heat transfer componentthrough the substrate, henceQ = QA + QBwhich is due to Nakayama and Park (1996). Equation (6.178) can be used to estimate QA .The ratio QB /QA depends on the thermal resistance between the heat source blockand the substrate (the block support in Fig. 6.24 simulating the lead lines or the solderballs), the thermal conductivity and the thickness of the substrate, and the surface heattransfer coefficient. The estimation of QB is a complex process, particularly wherethe lower side of the substrate is not exposed to the coolant flow, which makes QBfind its way through only the upper surface.
This is the case often encountered inelectronic equipment. Convective heat transfer from the upper surface is affected byflow development around the heat source block, which is three-dimensional, involving a horseshoe vortex and the thermal wake shed from the block, leading to a risein 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, 31mm × 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, suchas that of a 1-mm-thick copper plate, maximizes the contribution of conjugate heattransfer QB to the total heat dissipation Q, raising the ratio QB /Q to a value greaterthan 0.50.6.6.4Block ArraysBlock arrays are common features of electronic equipment, particularly, large systemswhere a number of packages of the same size are mounted on a large printed wiringBOOKCOMP, Inc.
— John Wiley & Sons / Page 495 / 2nd Proofs / Heat Transfer Handbook / Bejan[495], (57)Lines: 2562 to 2587———0.25099pt PgVar———Normal PagePgEnds: TEX[495], (57)496123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: EXTERNAL FLOWSboard (PWB) and cooled by air in forced convection. Numerical analysis of threedimensional airflow over a block array is possible only when a fully developedsituation 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 facesof the zone. In general, the analysis of flow and heat transfer over an entire block arraydepends too much upon computational resources, and experiments are frequently thesole means of investigation. However, experiments can be costly, especially whenit is desired to cover a wide range of cases where the heat dissipation varies frompackage to package.
To reduce the demand for experimental (and computational aswell) resources, a systematic methodology has been developed.Consider the block array displayed in Fig. 6.25. Block A dissipates qA and blockB, qB . Assume for the moment that the other blocks are inactive; that is, they do notdissipate. The temperature of air over block B can be written asTair,B = T0 + θB/A qAwhere T0 is the free stream temperature and θB/A in the second term represents theeffect of heat dissipation from block A. Equation (6.179) is based on the superposition of solutions that is permissible because of the linearity of the energy equation.However, the factor θB/A results from nonlinear phenomena of dispersion of warmair from block A and is a function of the relative location of B to A and the flowvelocity.
To find θB/A by experiment, block A may be energized while block B remains inactive (qB = 0). A measurement of the surface temperature of block B isthen divided by qA . Because qB = 0, its measured surface temperature is called theadiabatic temperature.The next step is to activate block B and inactivate block A(qA = 0). The heattransfer coefficient measured in this situation is called the adiabatic heat transfercoefficient and is denoted as had,B . When both block A and block B are energized,the heat transfer from block B is driven by the temperature difference Ts,B − Tair,B ,Figure 6.25 Block array on a substrate.BOOKCOMP, Inc.
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