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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

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## Analysis of Fluid Flow in Axial Re-entrant Grooves with Application to Heat Pipes

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**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 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**Introduction, cont.**Re-entrant grooves located around the pipe circumference Monogroove heat pipe using a single re-entrant groove**MathematicalModel**• 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**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 Validation**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% Circular sector duct 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) tau_lv = 2.0 Po vs tau_lv 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 = -2.5 (Countercurrent shear stress) tau_lv = 0.0 (No 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 • 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 Volumetric flow rate vs tau_lv (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* • 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* Volumetric flow rate vs 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 • 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 Volumetric flow rate vs W*/2 (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 = 0.3 W*/2 = 0.4 W*/2 = 0.5 W*/2 = 0.2**Effect of Groove Fill Amount**For Evaporation • 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 phi_0 = 10o phi_0 = 40o (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.**• 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 Mean velocity vs Hl* 0 < phi_0 < 40o Po vs Hl* Volumetric flow rate 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 Po vs Al*/Ag* Volumetric flow rate 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**Capillary Limit Analysis, cont.**• 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 Heat transportvs groove fill ratio Water Heat transportvs 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

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