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

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

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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

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

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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:

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GRAPHICAL VALUES

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

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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

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

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CONSTANT WALL TEMPERATURE S OLUTIONS STARTING WITH THE FLUID HEAT BALANCE IN THE FORM: BOUNDARY CONDITIONS: AT THE WALL: T = T s @ r = R AT THE CENTERLINE:

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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

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HEAT TRANSFER IN NON-CIRCULAR T UBES 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 B ASED ON SYSTEM GEOMETRY AND THERMAL CONFIGURATION

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

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

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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)

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NON-CIRCULAR DUCTS USE THE HYDRAULIC DIAMETER: USE THE CIRCULAR CORRELATIONS: ANNULAR FLOWS USE A DEFINITION FOR HYDRAULIC DIAMETER D h = D o -D i 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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