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CHE/ME 109 Heat Transfer in Electronics

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CHE/ME 109 Heat Transfer in Electronics

LECTURE 18 –FLOW IN TUBES

- FORCE BALANCE OVER A CYLINDRICAL VOLUME IN
FULLY DEVELOPED LAMINAR FLOW

- PRESSURE FORCES = VISCOUS FORCES
- THE DIFFERENTIAL BALANCE IS:

- INTEGRATING TWICE, WITH BOUNDARY CONDITIONS
- V = 0 @ r = R (ZERO VELOCITY AT THE WALL)
- (dV/dr) = 0 @ r = 0 (CENTERLINE SYMMETRY)
- PARABOLIC VELOCITY PROFILE

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

- 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:

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

- 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

- BOUNDARY CONDITIONS:
- AT THE WALL T = Ts @ r = R
- AT THE CENTERLINE FROM SYMMETRY:

- STARTING WITH THE FLUID HEAT BALANCE IN THE FORM:
- BOUNDARY CONDITIONS:
- AT THE WALL: T = Ts @ r = R
- AT THE CENTERLINE:

- SUBSTITUTING THE VELOCITY PROFILE INTO THIS EQUATION YIELDS AN EQUATION IN THE FORM OF AN INFINITE SERIES
- RESULTING VALUES SHOW: Nu = 3.657

- 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

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

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

- 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)

- 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)

- USE A DEFINITION FOR HYDRAULIC DIAMETER
- HAVE LIMITING FLOWS FOR ADIABATIC WALLS (8-77 & 8-78)