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CHE/ME 109 Heat Transfer in Electronics

CHE/ME 109 Heat Transfer in Electronics. LECTURE 18 – FLOW IN TUBES. LAMINAR FLUID FLOW IN TUBES. FORCE BALANCE OVER A CYLINDRICAL VOLUME IN FULLY DEVELOPED LAMINAR FLOW PRESSURE FORCES = VISCOUS FORCES THE DIFFERENTIAL BALANCE IS: . LAMINAR FLUID FLOW .

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CHE/ME 109 Heat Transfer in Electronics

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  1. CHE/ME 109 Heat Transfer in Electronics LECTURE 18 –FLOW IN TUBES

  2. LAMINAR FLUID FLOW IN TUBES • FORCE BALANCE OVER A CYLINDRICAL VOLUME IN FULLY DEVELOPED LAMINAR FLOW • PRESSURE FORCES = VISCOUS FORCES • THE DIFFERENTIAL BALANCE IS:

  3. LAMINAR FLUID FLOW • INTEGRATING TWICE, WITH BOUNDARY CONDITIONS • V = 0 @ r = R (ZERO VELOCITY AT THE WALL) • (dV/dr) = 0 @ r = 0 (CENTERLINE SYMMETRY) • PARABOLIC VELOCITY PROFILE

  4. LAMINAR FLOW - MEAN VELOCITY • MEAN VELOCITY FROM THE INTEGRATED AVERAGE OVER THE RADIUS: IN TERMS OF THE MEAN VELOCITY

  5. PRESSURE DROP • PRESSURE REQUIRED TO TRANSPORT FLUID THROUGH A TUBE AT A SPECIFIED FLOW RATE IS CALLED PRESSURE DROP, ΔP • UNITS ARE TYPICALLY (PRESSURE/LENGTH PIPE) • USING RESULTS FROM THE FORCE BALANCE EQUATION, A CORRELATION FOR PRESSURE DROP AS A FUNCTION OF VELOCITY USES THE FORM: • FOR LAMINAR FLOW:

  6. GRAPHICAL VALUES

  7. PUMP WORK • REQUIRED TO TRANSPORT FLUID THROUGH A CIRCULAR TUBE IN LAMINAR FLOW:

  8. HEAT TRANSFER TO LAMINAR FLUID FLOWS IN TUBES • ENERGY BALANCE ON A CYLINDRICAL VOLUME IN LAMINAR FLOW YIELDS: • SOLUTION TO THIS EQUATION USES BOUNDARY CONDITIONS BASED ON EITHER CONSTANT HEAT FLUX OR CONSTANT SURFACE TEMPERATURE

  9. CONSTANT HEAT FLUX SOLUTIONS • BOUNDARY CONDITIONS: • AT THE WALL T = Ts @ r = R • AT THE CENTERLINE FROM SYMMETRY:

  10. CONSTANT WALL TEMPERATURE SOLUTIONS • STARTING WITH THE FLUID HEAT BALANCE IN THE FORM: • BOUNDARY CONDITIONS: • AT THE WALL: T = Ts @ r = R • AT THE CENTERLINE:

  11. CONSTANT WALL TEMPERATURE • SUBSTITUTING THE VELOCITY PROFILE INTO THIS EQUATION YIELDS AN EQUATION IN THE FORM OF AN INFINITE SERIES • RESULTING VALUES SHOW: Nu = 3.657

  12. HEAT TRANSFER IN NON-CIRCULAR TUBES • USES THE SAME APPROACH AS DESCRIBED FOR CIRCULAR TUBES • CORRELATIONS USE Re AND Nu BASED ON THE HYDRAULIC DIAMETER: • SEE TABLE 8-1 FOR LIMITING VALUES FOR f AND Nu BASED ON SYSTEM GEOMETRY AND THERMAL CONFIGURATION

  13. TURBULENT FLOW IN TUBES • FRICTION FACTORS ARE BASED ON CORRELATIONS FOR VARIOUS SURFACE FINISHES (SEE PREVIOUS FIGURE FOR f VS. Re) • FOR SMOOTH TUBES:

  14. TURBULENT FLOW • FOR VARIOUS ROUGHNESS VALUES (MEASURED BY PRESSURE DROP): • TYPICAL ROUGHNESS VALUES ARE IN TABLES 8.2 AND 8.3

  15. TURBULENT FLOW HEAT TRANSFER IN TUBES • FOR FULLY DEVELOPED FLOW DITTUS-BOELTER EQUATION: • OTHER EQUATIONS ARE INCLUDED AS (8-69) & (8-70) • SPECIAL CORRELATIONS ARE FOR LOW Pr NUMBERS (LIQUID METALS) (8-71) AND (8-72)

  16. NON-CIRCULAR DUCTS • USE THE HYDRAULIC DIAMETER: • USE THE CIRCULAR CORRELATIONS: • ANNULAR FLOWS • USE A DEFINITION FOR HYDRAULIC DIAMETER Dh = Do -Di • USE THE CIRCULAR CORRELATIONS • HAVE LIMITING VALUES FOR LAMINAR FLOW (TABLE 8-4) • HAVE LIMITING FLOWS FOR ADIABATIC WALLS (8-77 & 8-78)

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