Download Numerical Model Validation, cont.

Analysis of Fluid Flow in Axial Re-entrant
Grooves with Application to Heat Pipes
Vikrant Damle
B.S., Pune University, 1999
Advisor: Dr. Scott K. Thomas
Outline
•
Motivation
•
Introduction
•
Mathematical Model
•
Numerical Model
•
Numerical Model Validation
•
Parametric Analysis
•
Effect of Groove Fill Amount
•
Capillary Limit Analysis for a Re-entrant Groove Heat Pipe
•
Conclusions
Motivation
•
Previous researchers assumed that the pressure drop
within the liquid in a re-entrant groove could be modeled
as flow within a smooth tube
(Poiseuille number, Po = f Re =16)
•
Based on previous studies of flow in grooves with shear
stress at the liquid-vapor interface, it was postulated that
this assumption could lead to significant errors in pressure
drop calculations
•
To the authors’ knowledge, the flow in re-entrant grooves
has never been modeled in the open literature
Introduction
• Heat pipes provide high heat transfer rates with selfregulating cooling characteristics
• For optimal performance, the capillary pumping pressure
should be high with low axial pressure drop
– Small groove openings for small meniscus radii
– Large hydraulic diameter
– Minimize liquid-vapor interaction
• Re-entrant grooves give good results due to their geometry
Introduction, cont.
Re-entrant grooves located around
the pipe circumference
Monogroove heat pipe using a
single re-entrant groove
Mathematical Model
• Purpose
– Analyze the fully-developed flow in a re-entrant groove by determining
velocity profiles as function of groove geometry, applied liquid-vapor shear
stress and groove fill amount
• Assumptions
– Steady state, fully developed laminar flow
– Constant properties
– Shear stress at the liquid-vapor interface is uniform across the meniscus
Mathematical Model, cont.
Mathematical Model, cont.
Mathematical Model, cont.
Mathematical Model, cont.
Mathematical Model, cont.
Mathematical Model, cont.
Mathematical Model, cont.
Mathematical Model, cont.
Mathematical Model, cont.
Mathematical Model, cont.
Mathematical Model, cont.
Mathematical Model, cont.
Mathematical Model, cont.
Numerical Model
• A finite element code was used to solve the elliptic Poisson
equation
• The fluid flow problem was solved as a heat conduction
problem
– Flat plate of uniform thickness, steady state, constant properties, uniform
internal volumetric heat generation
• Results were grid independent to <1% when the number of
elements were doubled
• The numerical model was validated using existing solutions
in the archival literature
Numerical Model, cont.
Numerical Model Validation
Comparison of present solution with Shah and London
Circular sector duct
The present solution is in agreement with
Shah and London with a maximum
difference of 1.4%
Po vs 2alpha
Numerical Model Validation, cont.
Comparison of present solution with DiCola
The maximum difference is 1.2% for tau_lv =
- 0.1, 0.0 and 1.0, and 0.1 < beta < 1.0
Rectangular groove
Po vs beta
Numerical Model Validation, cont.
Comparison of present solution with Romero and Yost
Triangular groove
For gamma = 5o and 60o and 0.1o < phi <
80o, the maximum difference was 2.6%
Po vs phi
Numerical Model Validation, cont.
Comparison of present solution with Thomas et al.
Sinusoidal groove
(beta = 0.5, Wl*/2 = 0.25)
phi = 72.34o
(Flat meniscus)
Po vs tau_lv
tau_lv = 2.0
Po vs phi
Numerical Model Validation, cont.
Comparison of present solution with Thomas et al.
Trapezoidal groove
(beta = 1.0, theta = 30o)
phi = 60o
(Flat meniscus)
tau_lv = 5.0
Po vs tau_lv
Po vs phi
Numerical Model Validation, cont.
• Agreement between the present solution and by Thomas et
al. for sinusoidal and trapezoidal grooves is excellent when
the liquid surface is flat
• As phi decreases, the agreement is poor
• This is due to the approximation used by Thomas et al.
(countercurrent shear stress normal to z* for liquid
meniscus)
• Using the finite element method, it is possible to apply
countercurrent shear stress normal to the liquid meniscus for
any value of meniscus radius
• Thus solution obtained by finite element method is more
accurate
Parametric Analysis
• Independent variables
–
–
–
–
Liquid-vapor shear stress
Slot width
Groove height
Fillet radius
• Dependent variables
– Mean velocity
– Poiseuille number
– Volumetric flow rate
Parametric Analysis, cont.
tau_lv = 0.0
tau_lv = -2.5
(No shear stress)
(Countercurrent shear
stress)
•Maximum velocity
inside circular region
•Maximum velocity
less than tau_lv=0.0
•Liquid at the interface
forced in the opposite
direction
(To scale: H* = 1.75,
Hl* = 2.75, Rf* = 0.1,
W*/2 = 0.5, phi = 90o)
Parametric Analysis, cont.
1.0 < H* < 4.0
Po vs tau_lv
Mean velocity vs tau_lv
•
•
•
•
•
Volumetric flow rate vs tau_lv
Mean velocity is linear with tau_lv
Mean velocity decreases with tau_lv due to
increase in flow resistance
Po increases monotonically with tau_lv (Po
~1/v_mean)
Po increases dramatically for H* < 1.5 (l-v
interface is closer to circular region)
Flow rate decreases with tau_lv due to
decrease in v_mean
(Hl* = H* + 1, Rf* = 0.1, W*/2 = 0.5, phi = 90o)
Parametric Analysis, cont.
0.05 < W*/2 < 0.90
Mean velocity vs H*
Po vs H*
•
•
•
Volumetric flow rate vs H*
Mean velocity is weak function of H* for
range of half slot width
The Po approaches 16 as H* tends to 1 and
W*/2 tends to 0 (Smooth circular tube
solution)
Flow rate increases with H*
(Hl* = H* + 1, Rf* = 0.1, phi = 90o, tau_lv = 0.0)
Parametric Analysis, cont.
1.0 < H* < 4.0
Mean velocity vs W*/2
Po vs W*/2
•
•
•
Volumetric flow rate vs W*/2
Mean velocity affected by slot width more
significantly as the groove height
increases
Po increases substantially with slot width
and becomes nearly constant
Volumetric flow rate is a monotonic
function of slot width
(Hl* = H* + 1, Rf* = 0.1, phi = 90o, tau_lv = 0.0)
Parametric Analysis, cont.
0.1 < W*/2 < 0.5
Mean velocity vs Rf*
Po vs Rf*
Mean velocity, Po and volumetric flow
rate are weak functions of fillet radius
Volumetric flow rate vs Rf*
(H* = 2.0, Hl* = 3.0, phi = 90o, tau_lv = 0.0)
Parametric Analysis, cont.
0.0 < Rf* < 1.0
W*/2 = 0.1
W*/2 = 0.2
W*/2 = 0.3
W*/2 = 0.4
W*/2 = 0.5
Effect of Groove Fill Amount
For Evaporation
• Groove is initially full (phi = 90o)
phi_0 = 10o
phi_0 = 40o
• Contact angle decreases until phi =
phi_0 (minimum contact angle)
• Meniscus detaches from top of groove
• In fillet region, liquid cross-sectional
area decreases and meniscus radius
increases dramatically
• In the lower circular region, meniscus
may become convex instead of
concave, depending on phi
(To scale: H* = 1.75, W*/2 = 0.5, Rf* = 0.1)
Effect of Groove Fill Amount, cont.
Liquid cross-sectional area vs Hl*
•
Area decreases dramatically in the fillet
region for small change in height of the
meniscus attachment point.
•
For smaller values of phi_0, decrease in
the liquid area is more significant in fillet
and circular region
Liquid cross-sectional area vs Hl*
0 < phi_0 < 40o
Meniscus radius vs Hl*
•
Rm* is constant in the fillet region
•
Rm* increases dramatically in the
circular region
Meniscus radius vs Hl*
Effect of Groove Fill Amount, cont.
Mean velocity vs Hl*
•
As liquid recedes into the groove, mean
velocity increases to maximum and then
decreases to zero
•
Po is relatively constant in slot region,
decreases in the fillet region, increases in
circular region
•
Flow rate decreases steadily in slot region
and then decreases rapidly in fillet region
0 < phi_0 < 40o
Volumetric flow rate vs Hl*
Po vs Hl*
Effect of Groove Fill Amount, cont.
•
As liquid recedes into the groove, mean
velocity increases to maximum and then
approaches zero
•
Po is nearly constant
•
Flow rate for all the meniscus contact
angles studied here nearly collapse to a
single curve
Mean velocity vs Al*/Ag*
0 < phi_0 < 40o
Volumetric flow rate vs Al*/Ag*
Po vs Al*/Ag*
Capillary Limit Analysis for a
Re-entrant Groove Heat Pipe
• Objective
– Develop an analytical capillary limit prediction model using the results of the
numerical analysis
• Assumptions
–
–
–
–
Fluid properties vary with temperature
Meniscus radius and liquid height constant along heat pipe length
Zero gravity condition
Negligible liquid-vapor shear stress
Capillary Limit Analysis, cont.
Re-entrant Groove Heat Pipe Specifications
Evaporator length
Le = 15.2 E-02 m
Adiabatic length
La = 8.2 E-02 m
Condenser length
Lc = 15.2 E-02 m
Radius of the heat pipe vapor space
Rv = 8.59 E-03 m
Radius of circular portion of the groove
R = 0.8 E-03 m
Groove height
H = 1.4 E-03 m
Slot half-width
W/2 = 0.4 E-03 m
Number of grooves
Ng = 15
Operating temperature
Tsat = 60oC
Capillary Limit Analysis, cont.
Ethanol
•
Capillary limit attains maximum value in
the slot region
•
Decreases dramatically in the circular
region
•
Shows the critical nature of fluid fill
amount in heat pipes with re-entrant
groove
Heat transport vs groove fill ratio
Water
Heat transport vs groove fill ratio
Conclusions
• The finite element solution was faster and more accurate than
previous method
• Easy to apply the shear stress boundary conditions
• Poiseuille number was relatively unaffected by fillet radius in
comparison with groove height and width
• Volumetric flow rate was fairly constant with slot half width
for groove height ranging from 1.0 < H* < 4.0
• The capillary limit attained maximum value in slot region and
decreased dramatically as meniscus receded into circular
region