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

• 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

Outline

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 self-regulating 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

Monogroove heat pipe using a single re-entrant groove

Re-entrant grooves located around the pipe circumference

Introduction, cont.

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

Circular sector duct

Comparison of present solution with Shah and London

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

Rectangular groove

Po vs beta

The maximum difference is 1.2% for tau_lv = - 0.1, 0.0 and 1.0, and 0.1 < beta < 1.0

Numerical Model Validation, cont.

Triangular groove

Comparison of present solution with Romero and Yost

Po vs phi

For gamma = 5o and 60o and 0.1o < phi < 80o, the maximum difference was 2.6%

Numerical Model Validation, cont.

Po vs tau_lv Po vs phi

Comparison of present solution with Thomas et al.

Sinusoidal groove

(beta = 0.5, Wl*/2 = 0.25)

phi = 72.34o

(Flat meniscus) tau_lv = 2.0

Numerical Model Validation, cont.

Po vs tau_lv Po vs phi

Comparison of present solution with Thomas et al.

Trapezoidal groove(beta = 1.0, theta = 30o)

phi = 60o

(Flat meniscus)tau_lv = 5.0

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(No shear stress)

tau_lv = -2.5(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.

Mean velocity vs tau_lv

Volumetric flow rate vs tau_lv

Po 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

1.0 < H* < 4.0

(Hl* = H* + 1, Rf* = 0.1, W*/2 = 0.5, phi = 90o)

Parametric Analysis, cont.

Mean velocity vs H*

Volumetric flow rate vs H*

Po 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*

0.05 < W*/2 < 0.90

(Hl* = H* + 1, Rf* = 0.1, phi = 90o, tau_lv = 0.0)

Parametric Analysis, cont.

Po vs W*/2

Volumetric flow rate vs W*/2

Mean velocity 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

1.0 < H* < 4.0

(Hl* = H* + 1, Rf* = 0.1, phi = 90o, tau_lv = 0.0)

Parametric Analysis, cont.

Mean velocity vs Rf* Po vs Rf*

Volumetric flow rate vs Rf*

Mean velocity, Po and volumetric flow rate are weak functions of fillet radius

0.1 < W*/2 < 0.5

(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

phi_0 = 10o phi_0 = 40o

For Evaporation

(To scale: H* = 1.75, W*/2 = 0.5, Rf* = 0.1)

• Groove is initially full (phi = 90o)

• 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

Effect of Groove Fill Amount, cont.

Liquid cross-sectional area vs Hl*

Meniscus radius vs Hl*

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

Meniscus radius vs Hl*• Rm* is constant in the fillet region

• Rm* increases dramatically in the circular region

0 < phi_0 < 40o

Effect of Groove Fill Amount, cont.

Mean velocity vs Hl*

Volumetric flow rate vs Hl* Po 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

Effect of Groove Fill Amount, cont.

Po vs Al*/Ag*Volumetric flow rate vs Al*/Ag*

Mean velocity vs Al*/Ag*

• 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

0 < phi_0 < 40o

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.

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

Re-entrant Groove Heat Pipe Specifications

Capillary Limit Analysis, cont.

Heat transport vs groove fill ratio

Heat transport vs groove fill ratio

• 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

Ethanol

Water

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