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)