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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
• INTEGRATING TWICE, WITH BOUNDARY CONDITIONS• V = 0 @ r = R (ZERO VELOCITY AT THE WALL)• (dV/dr) = 0 @ r = 0 (CENTERLINE SYMMETRY)• PARABOLIC VELOCITY PROFILE
LAMINAR FLOW - MEAN VELOCITY
• MEAN VELOCITY FROM THE INTEGRATED AVERAGE OVER THE RADIUS:
IN TERMS OF THE MEAN VELOCITY
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:
GRAPHICAL VALUES
PUMP WORK
• REQUIRED TO TRANSPORT FLUID THROUGH A CIRCULAR TUBE IN LAMINAR FLOW:
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
CONSTANT HEAT FLUX SOLUTIONS
• BOUNDARY CONDITIONS: • AT THE WALL T = Ts @ r = R• AT THE CENTERLINE FROM SYMMETRY:
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:
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
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
TURBULENT FLOW IN TUBES
• FRICTION FACTORS ARE BASED ON CORRELATIONS FOR VARIOUS SURFACE FINISHES (SEE PREVIOUS FIGURE FOR f VS. Re)
• FOR SMOOTH TUBES:
TURBULENT FLOW
• FOR VARIOUS ROUGHNESS VALUES (MEASURED BY PRESSURE DROP):
• TYPICAL ROUGHNESS VALUES ARE IN TABLES 8.2 AND 8.3
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)
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)