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Heat Pipe with Accurate Liquid and Gas Properties
Introduction
Heat pipes are designed to transfer heat efficiently through vaporization, mass transfer, and condensation of a working fluid. They are found in a wide variety of applications where thermal control is of importance, with cooling of electronics being a prominent example.
Inside a heat pipe, the temperature difference between the hot and cold sides together with the temperature dependence of the vapor pressure, induce a pressure difference across the vapor chamber. The pressure difference, in turn, drives the vapor from the hot to the cold side. The vaporization acts as a heat sink at the vapor–wick interface at the hot side, and conversely, the condensation as a heat source, at the cold side. This model demonstrates how the laminar flow in the vapor chamber of the heat pipe can be coupled to the liquid phase transport through the porous wick, and how thermodynamic properties of water can be obtained from COMSOL’s built-in thermodynamics database. The importance of vapor transport is compared to the conductive heat transfer in the pipe wall. The former dominates the latter by several orders of magnitude.
Model Definition
Heat pipes exits in variety of different shapes. Tubular ones are probably the most common type however. Here we will look at an axisymmetric model of a copper tube with a porous copper wick and a vapor chamber. The heat pipe has a contact surface at the bottom, which is to be connected to the source of the heat to be removed. At the top of the pipe, a similar contact surface for a heat sink is used. The latter often corresponds to a finned metal structure that can easily be cooled by a fan. The geometry used including the different parts is visualized in Figure 1 below.
Figure 1: Overview of the heat pipe geometry.
Before setting up the model, we will investigate under what conditions our assumption of a saturated wick holds.
For heat pipes operating near ambient conditions, the capillary pressure, Δpc, is usually the limiting factor (Ref. 1):
(1)
Here, σ is the surface tension and rc is the capillary radius. At the capillary limit, this pressure equals those pressure needed to drive the vapor, the static pressure due to gravity, and the pressure needed to drive liquid through the wick in the manner of
(2)
For most applications, we can neglect all but the liquid term, which can be obtained from Darcy’s law:
(3)
where μl is the dynamic viscosity of the liquid, Leff is the effective length of the heat pipe, K is the permeability of the wick, Aw is the cross-sectional area of the wick, and is the volumetric flow rate. The latter is governed by the rate of evaporation:
(4)
where, is the heat transfer rate, ρ is the density of the liquid, ΔHvap is the latent heat of vaporization (with dimensionality of energy per mass). Inserting Equation 2-4 into Equation 1, and neglecting Δpv and Δpg yields
(5)
Evaluating Equation 5 with K =1·10-9 m2, Aw =1·10-4 m2, ΔHvap = 2.5·106 J/kg, ρ = 1·103 kg/m3, σ = 7·10-2 N/m, μ = 1·10-3 Pa·s, L = 0.15 m, and rc = 3.1·10-5 m, gives a value of 7.5 kW. In the model, a modest heat transfer rate of 30 W will be used, thus far from the capillary limit. For comparison, a CPU of a typical consumer PC produces on the order of 10–100 W.
Physics Setup
A Laminar Flow interface is used to solve for the laminar flow in the vapor cavity. It is subjected to single boundary condition, apart from the axial symmetry line. The pressure is prescribed to equal the saturated vapor pressure at the cavity–wick interface.
(6)
This implies that the water and vapor phase are assumed to be in equilibrium at this position. The vapor pressure increases with temperature, which is what drives the vapor from the high temperature region to the low temperature region. For the liquid flow in the porous wick, a Brinkman Equations interface is used. The velocity in the wick at cavity–wick interface is computed from the vapor flow rate on the cavity side
(7)
The pressure level is fixed using a pressure point constraint on the solid wall in the middle of the geometry.
For heat transfer in all parts of the geometry, the tube wall, the wick, and the vapor cavity, a Heat Transfer in Porous Media interface is used. It includes domain features for each domain type.
Material Properties
Material properties are created using the Thermodynamics node. A vapor system using the ideal gas law is set up for the vapor phase, while a liquid system using IAPWS models (Ref. 2) is created for the liquid phase in the wick. To describe the saturation pressure a vapor pressure function is created for the liquid system. In order to easily apply properties in the model, two materials are created using the Generate Material option available for thermodynamic systems. Copper from the material library is used for the properties in the tube wall.
Results and Discussion
As a first step, analyze the temperature when conduction is the only means of energy transport in the pipe. This corresponds to using a dry wick, with no liquid water, and negligible natural convection in the vapor. The resulting temperature is seen in Figure 2 below.
Figure 2: Temperature of heat pipe with dry wick.
Note that the temperature at the heat source is almost 100°C higher than at the heat sink. In an application where temperature sensitive parts are present (electronics, plastics, and so on) such a high temperature would be detrimental.
In the second simulation, the wick is assumed to be saturated with liquid water, corresponding to a heat pipe running at its design point. The resulting temperature profile, seen in Figure 3, looks dramatically different.
Figure 3: Temperature of heat pipe with saturated wick.
The predicted temperature difference between heat sink and heat source is now less than 2°C. And the surface of the heat pipe outside of the contact areas is essentially isothermal. In Figure 4, the computed velocity fields, in both fluids, and the temperature throughout the geometry, are plotted next to each other.
Figure 4: Fluid velocities — lg(|u| / m·s-1) and temperature in the heat pipe running at the design point.
The heat transfer process can be decomposed into different contributions, by computing line integrals across the cavity, wick, and casing at the middle of the pipe. In Table 1, the relative importance of the different heat transfer mechanisms in the saturated heat pipe.
Table 1: Results from the Energy Balance Evaluation Group.
Process
Heat transfer rate / W
Conductive heat transfer in casing
4·10-5
Conductive heat transfer in wick
3·10-5
Latent heat transfer in cavity
30
It is seen that, at normal operating conditions, vapor mass transfer (and its associated phase changes) is the completely dominating mechanism by which the heat pipe transfers heat.
References
1. I. Shishido, I. Oishi, and S. Ohtani, “Capillary Limit in Heat Pipes,” J. Chem. Eng. Japan, vol. 17, no. 2, pp. 179–186, 1986.
2. W. Wagner and H.J. Kretzschmar IAPWS industrial formulation 1997 for the thermodynamic properties of water and steam. International steam tables: properties of water and steam based on the industrial formulation IAPWS-IF97, pp. 7–150, 2008.
Application Library path: Chemical_Reaction_Engineering_Module/Thermodynamics/heat_pipe
Note: This model is included in the booklet Introduction to the Liquid & Gas Properties Module.
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D Axisymmetric.
2
In the Select Physics tree, select Fluid Flow > Single-Phase Flow > Laminar Flow (spf).
3
Right-click and choose Add Physics.
4
In the Select Physics tree, select Fluid Flow > Porous Media and Subsurface Flow > Brinkman Equations (br).
5
Right-click and choose Add Physics.
6
In the Select Physics tree, select Heat Transfer > Porous Media > Heat Transfer in Porous Media (ht).
7
Right-click and choose Add Physics.
8
Click  Study.
9
In the Select Study tree, select General Studies > Stationary.
10
Click  Done.
Global Definitions
Parameters 1
Start by reading in a set of parameters defining the dimensions and specific properties.
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file heat_pipe_parameters.txt.
Geometry 1
Define a cylinder with rounded ends, use a sector of a circle, three rectangles and a mirror plane.
Circle 1 (c1)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, locate the Size and Shape section.
3
In the Radius text field, type r_outer.
4
In the Sector angle text field, type 90.
5
Locate the Position section. In the z text field, type length/2.
6
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
w_casing
Layer 2
w_wick
Rectangle 1 (r1)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type r_outer + w_contact.
4
In the Height text field, type l_heatsource.
5
Locate the Position section. In the z text field, type length/2-l_heatsource.
6
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
w_casing + w_contact
Layer 2
w_wick
7
Select the Layers to the right checkbox.
8
Clear the Layers on bottom checkbox.
Rectangle 2 (r2)
1
Right-click Rectangle 1 (r1) and choose Duplicate.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type r_outer.
4
Locate the Position section. In the z text field, type length/2-l_heatsource*2.
5
Locate the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
w_casing
Rectangle 3 (r3)
1
Right-click Rectangle 2 (r2) and choose Duplicate.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Height text field, type length/2-l_heatsource*2.
4
Locate the Position section. In the z text field, type 0.
At this point, the geometry primitives for the upper half of the heat pipe has been added, go ahead and add a mirror plane to obtain a complete heat pipe.
Mirror 1 (mir1)
1
In the Geometry toolbar, click  Transforms and choose Mirror.
2
Click in the Graphics window and then press Ctrl+A to select all objects.
3
In the Settings window for Mirror, locate the Input section.
4
Select the Keep input objects checkbox.
5
Locate the Normal Vector to Line of Reflection section. In the r text field, type 0.
6
In the z text field, type 1.
Use a quadrilateral mesh for the middle part of the heat pipe and triangular meshes for the ends. To facilitate this mesh transition, define mesh control edges; these can be referenced during meshing, while not contributing to splitting of domains in other contexts.
Mesh Control Edges 1 (mce1)
1
In the Geometry toolbar, click  Virtual Operations and choose Mesh Control Edges.
2
On the object fin, select Boundaries 9, 13, 25, 29, 38, and 42 only.
Analogously, ignore some other edges completely, not only during meshing.
Ignore Edges 1 (ige1)
1
In the Geometry toolbar, click  Virtual Operations and choose Ignore Edges.
2
On the object mce1, select Boundaries 5, 7, 11, 13, 17, 19, 23, and 24 only.
Ignore Edges 2 (ige2)
1
In the Geometry toolbar, click  Virtual Operations and choose Ignore Edges.
2
On the object ige1, select Boundaries 12, 14, 18, and 19 only.
3
In the Settings window for Ignore Edges, locate the Input section.
4
Clear the Ignore adjacent vertices checkbox.
Ignore Vertices 1 (igv1)
1
In the Geometry toolbar, click  Virtual Operations and choose Ignore Vertices.
2
On the object ige2, select Points 9, 10, 12, and 13 only.
3
In the Geometry toolbar, click  Build All.
Vapor Cavity
1
In the Geometry toolbar, click  Selections and choose Explicit Selection.
It is a good practice to introduce explicit selections with descriptive names, these will help when making selections in the physics interfaces later on. Therefore, define selections for the three domains, and a handful of boundaries of interest.
2
In the Settings window for Explicit Selection, type Vapor Cavity in the Label text field.
3
On the object igv1, select Domains 3 and 4 only.
Porous Copper Wick
1
In the Geometry toolbar, click  Selections and choose Explicit Selection.
2
In the Settings window for Explicit Selection, type Porous Copper Wick in the Label text field.
3
On the object igv1, select Domains 2 and 5 only.
Solid Copper Tube Casing
1
In the Geometry toolbar, click  Selections and choose Explicit Selection.
2
In the Settings window for Explicit Selection, type Solid Copper Tube Casing in the Label text field.
3
On the object igv1, select Domains 1 and 6 only.
Heat Sink
1
In the Geometry toolbar, click  Selections and choose Explicit Selection.
2
In the Settings window for Explicit Selection, type Heat Sink in the Label text field.
3
Locate the Entities to Select section. From the Geometric entity level list, choose Boundary.
4
On the object igv1, select Boundary 21 only.
Heat Source
1
In the Geometry toolbar, click  Selections and choose Explicit Selection.
2
In the Settings window for Explicit Selection, locate the Entities to Select section.
3
From the Geometric entity level list, choose Boundary.
4
On the object igv1, select Boundary 20 only.
5
In the Label text field, type Heat Source.
Cross Section of Cavity
1
In the Geometry toolbar, click  Selections and choose Explicit Selection.
2
In the Settings window for Explicit Selection, type Cross Section of Cavity in the Label text field.
3
Locate the Entities to Select section. From the Geometric entity level list, choose Boundary.
4
On the object igv1, select Boundary 5 only.
Cross Section of Wick
1
In the Geometry toolbar, click  Selections and choose Explicit Selection.
2
In the Settings window for Explicit Selection, type Cross Section of Wick in the Label text field.
3
Locate the Entities to Select section. From the Geometric entity level list, choose Boundary.
4
On the object igv1, select Boundary 10 only.
Cross Section of Casing
1
In the Geometry toolbar, click  Selections and choose Explicit Selection.
2
In the Settings window for Explicit Selection, type Cross Section of Casing in the Label text field.
3
Locate the Entities to Select section. From the Geometric entity level list, choose Boundary.
4
On the object igv1, select Boundary 13 only.
Inner Wick Boundary
1
In the Geometry toolbar, click  Selections and choose Explicit Selection.
2
In the Settings window for Explicit Selection, type Inner Wick Boundary in the Label text field.
3
Locate the Entities to Select section. From the Geometric entity level list, choose Boundary.
4
On the object igv1, select Boundaries 8 and 9 only.
All Wick Boundaries
1
In the Geometry toolbar, click  Selections and choose Explicit Selection.
2
In the Settings window for Explicit Selection, type All Wick Boundaries in the Label text field.
3
Locate the Entities to Select section. From the Geometric entity level list, choose Boundary.
4
On the object igv1, select Boundaries 8, 9, 11, and 12 only.
Next, a union selection of the cross sections is created. Those cross sections have no physical importance, and will only be used for analysis in the postprocessing stage. Therefore, hide them during the setup of the model.
All Cross Sections
1
In the Geometry toolbar, click  Selections and choose Union Selection.
2
In the Settings window for Union Selection, type All Cross Sections in the Label text field.
3
Locate the Geometric Entity Level section. From the Level list, choose Boundary.
4
Locate the Input Entities section. Click  Add.
5
In the Add dialog, in the Selections to add list, choose Cross Section of Cavity, Cross Section of Wick, and Cross Section of Casing.
6
Click OK.
Definitions
Hide for Geometry 1
1
In the Model Builder window, expand the Component 1 (comp1) > Definitions node.
2
Right-click View 1 and choose Hide for Geometry.
3
In the Settings window for Hide for Geometry, locate the Selection section.
4
From the Geometric entity level list, choose Boundary.
5
From the Selection list, choose All Cross Sections.
Mesh 1
The following steps will set up a mesh with its element size controlled by a global parameter, and use a different mesh for the middle segment of the heat pipe.
Mapped 1
1
In the Mesh toolbar, click  Mapped.
2
In the Settings window for Mapped, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domains 4, 7–9, 11, and 12 only.
Distribution 1
1
Right-click Mapped 1 and choose Distribution.
2
In the Settings window for Distribution, locate the Distribution section.
3
From the Distribution type list, choose Predefined.
4
In the Number of elements text field, type length/r_outer/mesh_factor.
5
In the Element ratio text field, type 10.
6
Select the Symmetric distribution checkbox.
7
Click  Build Selected.
Free Triangular 1
In the Mesh toolbar, click  Free Triangular.
Size
1
In the Model Builder window, click Size.
2
In the Settings window for Size, locate the Element Size section.
3
Click the Custom button.
4
Locate the Element Size Parameters section. In the Maximum element size text field, type 0.9*min(w_casing, w_wick)*mesh_factor.
5
In the Minimum element size text field, type 0.3*min(w_casing, w_wick)*mesh_factor.
Boundary Layers 1
1
In the Mesh toolbar, click  Boundary Layers.
2
In the Settings window for Boundary Layers, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domains 2–5 and 8–11 only.
Boundary Layer Properties
1
In the Model Builder window, click Boundary Layer Properties.
2
In the Settings window for Boundary Layer Properties, locate the Boundary Selection section.
3
From the Selection list, choose All Wick Boundaries.
4
Click  Build All.
Materials
You need to add materials for the metal casing and the wick, as well as for the working fluid (both in gaseous and liquid form). Copper will be added from the built-in materials. For water vapor and liquid water, two thermodynamic systems are created, from which materials are then generated. Note that you need to add copper to global materials for it to be accessible in the porous material node, which will be added to Component 1. To use this global Copper material in Component 1 for the Solid Copper Tube Casing domain selection, a material link will be added.
Add Material
1
In the Materials toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in > Copper.
4
Right-click and choose Add to Global Materials.
5
In the Materials toolbar, click  Add Material to close the Add Material window.
Global Definitions
For water vapor use an ideal gas model.
1
In the Physics toolbar, click  Thermodynamics and choose Thermodynamic System.
Select System
1
Go to the Select System window.
2
Click the Next button in the window toolbar.
Select Species
1
Go to the Select Species window.
2
In the Species list, select water (7732-18-5, H2O).
3
Click  Add Selected.
4
Click the Next button in the window toolbar.
Select Thermodynamic Model
1
Go to the Select Thermodynamic Model window.
2
From the Gas phase model list, choose Ideal gas.
3
Click the Finish button in the window toolbar.
Global Definitions
Gas System 1 (pp1)
Right-click Global Definitions > Thermodynamics > Gas System 1 (pp1) and choose Generate Material.
Select Phase
1
Go to the Select Phase window.
2
Click the Next button in the window toolbar.
Select Species
1
Go to the Select Species window.
2
Click  Add All.
3
Click the Next button in the window toolbar.
Select Properties
1
Go to the Select Properties window.
2
Click the Next button in the window toolbar.
Define Material
1
Go to the Define Material window.
2
From the Component list, choose Global.
3
Click the Finish button in the window toolbar.
Global Definitions
1
In the Physics toolbar, click  Thermodynamics and choose Thermodynamic System.
For liquid water, use the IAPWS model.
Select System
1
Go to the Select System window.
2
From the Phase list, choose Vapor-liquid.
3
Click the Next button in the window toolbar.
Select Species
1
Go to the Select Species window.
2
In the Species list, select water (7732-18-5, H2O).
3
Click  Add Selected.
4
Click the Next button in the window toolbar.
Select Thermodynamic Model
1
Go to the Select Thermodynamic Model window.
2
Click the Finish button in the window toolbar.
Global Definitions
Vapor-Liquid System 1 (pp2)
Right-click Global Definitions > Thermodynamics > Vapor-Liquid System 1 (pp2) and choose Species Property.
Select Properties
1
Go to the Select Properties window.
2
From the Amount base unit list, choose kg.
3
In the list, select Heat of vaporization (J/kg).
4
Click  Add Selected.
5
In the list, select Ln vapor pressure, Pa.
6
Click  Add Selected.
7
Click the Next button in the window toolbar.
Select Phase
1
Go to the Select Phase window.
2
Click the Next button in the window toolbar.
Select Species
1
Go to the Select Species window.
2
Click  Add All.
3
Click the Next button in the window toolbar.
Species Property Overview
1
Go to the Species Property Overview window.
2
Click the Finish button in the window toolbar.
Global Definitions
Vapor pressure of water
1
In the Home toolbar, click  Functions and choose Global > Analytic.
2
In the Settings window for Analytic, type Vapor pressure of water in the Label text field.
3
In the Function name text field, type pH2O.
4
Locate the Definition section. In the Expression text field, type exp(LnVaporPressure_water22(T)).
5
In the Arguments text field, type T.
6
Locate the Units section. In the table, enter the following settings:
Argument
Unit
T
K
7
In the Function text field, type Pa.
8
Locate the Plot Parameters section. In the table, enter the following settings:
Plot
Argument
Lower limit
Upper limit
Fixed value
Unit
√
T
273.15
373.15
0
K
Vapor-Liquid System 1 (pp2)
Right-click Vapor-Liquid System 1 (pp2) and choose Generate Material.
Select Phase
1
Go to the Select Phase window.
2
From the list, choose Liquid.
3
Click the Next button in the window toolbar.
Select Species
1
Go to the Select Species window.
2
Click  Add All.
3
Click the Next button in the window toolbar.
Select Properties
1
Go to the Select Properties window.
2
Click the Next button in the window toolbar.
Define Material
1
Go to the Define Material window.
2
Click the Finish button in the window toolbar.
Global Definitions
Two cases will be investigated: a dry wick, and a saturated wick. By introducing a material switch, later interfaces can refer to this switch, which can either refer to vapor or liquid. Its state will be controlled from the Study nodes.
Fluid in Wick
1
In the Model Builder window, under Global Definitions right-click Materials and choose Material Switch.
2
In the Settings window for Material Switch, type Fluid in Wick in the Label text field.
Liquid: water 1 (pp2mat1)
In the Model Builder window, right-click Liquid: water 1 (pp2mat1) and choose Copy.
Liquid: water 1 (sw1.pp2mat1)
In the Model Builder window, right-click Fluid in Wick (sw1) and choose Paste Material.
Gas: water 1 (pp1mat1)
In the Model Builder window, under Global Definitions > Materials right-click Gas: water 1 (pp1mat1) and choose Copy.
Gas: water 1 (sw1.pp1mat1)
In the Model Builder window, right-click Fluid in Wick (sw1) and choose Paste Material.
Materials
Add a Porous Material for use in the wick, two subfeatures are added to the Porous Material node, one for the fluid, and one for the solid.
Porous Material 1 (pmat1)
1
In the Model Builder window, under Component 1 (comp1) right-click Materials and choose More Materials > Porous Material.
2
In the Settings window for Porous Material, locate the Geometric Entity Selection section.
3
From the Selection list, choose Porous Copper Wick.
Fluid 1 (pmat1.fluid1)
1
Right-click Porous Material 1 (pmat1) and choose Fluid.
2
In the Settings window for Fluid, locate the Fluid Properties section.
3
From the Material list, choose Fluid in Wick (sw1).
Solid 1 (pmat1.solid1)
1
In the Model Builder window, right-click Porous Material 1 (pmat1) and choose Solid.
2
In the Settings window for Solid, locate the Solid Properties section.
3
In the θs text field, type 1-wick_porosity.
The material for the wick is now defined, continue to add links to water vapor and copper.
Water Vapor
1
In the Model Builder window, under Component 1 (comp1) right-click Materials and choose More Materials > Material Link.
2
In the Settings window for Material Link, type Water Vapor in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose Vapor Cavity.
4
Locate the Link Settings section. From the Material list, choose Gas: water 1 (pp1mat1).
Copper Metal
1
Right-click Materials and choose More Materials > Material Link.
2
In the Settings window for Material Link, type Copper Metal in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose Solid Copper Tube Casing.
Now that the geometry, selections and materials are in place, go on to set up the physics.
Multiphysics
Nonisothermal Flow 1 (nitf1)
In the Physics toolbar, click  Multiphysics Couplings and choose Domain > Nonisothermal Flow.
Nonisothermal Flow 2 (nitf2)
1
In the Physics toolbar, click  Multiphysics Couplings and choose Domain > Nonisothermal Flow.
2
In the Settings window for Nonisothermal Flow, locate the Coupled Interfaces section.
3
From the Fluid flow list, choose Brinkman Equations (br).
Laminar Flow (spf)
1
In the Model Builder window, under Component 1 (comp1) click Laminar Flow (spf).
2
In the Settings window for Laminar Flow, locate the Domain Selection section.
3
From the Selection list, choose Vapor Cavity.
4
Locate the Physical Model section. From the Compressibility list, choose Compressible flow (Ma<0.3).
5
In the pref text field, type p_ref.
Apply a pressure condition on the cavity boundary and set it equal to the vapor pressure of water. Not suppressing backflow allows vapor to enter at the hot side and exit at the cold side.
Inlet 1
1
In the Physics toolbar, click  Boundaries and choose Inlet.
2
In the Settings window for Inlet, locate the Boundary Selection section.
3
From the Selection list, choose Inner Wick Boundary.
4
Locate the Boundary Condition section. From the list, choose Pressure.
5
Locate the Pressure Conditions section. In the p0 text field, type pH2O(T)-p_ref.
6
Clear the Suppress backflow checkbox.
Initial Values 1
1
In the Model Builder window, click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values section.
3
In the p text field, type p_ref.
Brinkman Equations (br)
1
In the Model Builder window, under Component 1 (comp1) click Brinkman Equations (br).
2
In the Settings window for Brinkman Equations, locate the Domain Selection section.
3
From the Selection list, choose Porous Copper Wick.
4
Locate the Physical Model section. From the Compressibility list, choose Compressible flow (Ma<0.3).
5
In the pref text field, type p_ref.
Porous Matrix 1
1
In the Model Builder window, under Component 1 (comp1) > Brinkman Equations (br) > Porous Medium 1 click Porous Matrix 1.
2
In the Settings window for Porous Matrix, locate the Matrix Properties section.
3
From the κ list, choose User defined. In the associated text field, type wick_permeability.
The mass flux of water through the cavity–wick interface must be equal in both the Laminar Flow interface, as well as the Brinkman Equations interface. Using the ratios of densities between vapor and liquid, you can apply a velocity boundary condition on the wick.
Inlet 1
1
In the Physics toolbar, click  Boundaries and choose Inlet.
2
In the Settings window for Inlet, locate the Boundary Selection section.
3
From the Selection list, choose Inner Wick Boundary.
4
Locate the Velocity section. Click the Velocity field button.
5
Specify the u0 vector as
u*spf.rho/br.rho
r
w*spf.rho/br.rho
z
Apply a pressure constraint of 0[Pa] to the point on the r axis at the interface between the wick and the casing:
Pressure Point Constraint 1
1
In the Physics toolbar, click  Points and choose Pressure Point Constraint.
2
Select Point 9 only.
3
In the Settings window for Pressure Point Constraint, locate the Pressure Constraint section.
4
In the p0 text field, type pH2O(T)-p_ref.
Heat Transfer in Porous Media (ht)
In the Model Builder window, under Component 1 (comp1) click Heat Transfer in Porous Media (ht).
Solid 1
1
In the Physics toolbar, click  Domains and choose Solid.
2
In the Settings window for Solid, locate the Domain Selection section.
3
From the Selection list, choose Solid Copper Tube Casing.
Fluid 1
1
In the Physics toolbar, click  Domains and choose Fluid.
2
In the Settings window for Fluid, locate the Domain Selection section.
3
From the Selection list, choose Vapor Cavity.
Porous Matrix 1
Use copper as the solid phase in the porous material. Because the material contains properties for the dense bulk material, you need to specify that the material properties are bulk properties and therefore need to be scaled by the porosity.
1
In the Model Builder window, under Component 1 (comp1) > Heat Transfer in Porous Media (ht) > Porous Medium 1 click Porous Matrix 1.
2
In the Settings window for Porous Matrix, locate the Matrix Properties section.
3
From the Define list, choose Solid phase properties.
If you had a predefined material which had data for its porous state, say “copper sponge”, you would have kept the “Define” choice under Matrix Properties as “Dry bulk properties”.
Next, specify the boundary condition corresponding to the heat source.
Heat Flux 1
1
In the Physics toolbar, click  Boundaries and choose Heat Flux.
2
In the Settings window for Heat Flux, locate the Boundary Selection section.
3
From the Selection list, choose Heat Source.
4
Locate the Heat Flux section. In the q0 text field, type phi_in.
The boundary condition for the heat sink is set up analogously, but with a convective heat-flux condition. That is, the heat flux out of the pipe through the heat sink is proportional to the temperature difference between the heat sink and the external environment. The magnitude of the proportionality constant (the heat-transfer coefficient) depends on external flow conditions such as the presence of an external fan (and its speed), the external surface area, and the geometry of fins (if present).
Heat Flux 2
1
In the Physics toolbar, click  Boundaries and choose Heat Flux.
2
In the Settings window for Heat Flux, locate the Boundary Selection section.
3
From the Selection list, choose Heat Sink.
4
Locate the Heat Flux section. From the Flux type list, choose Convective heat flux.
5
In the h text field, type h_conv.
The heat associated with the phase change of water, removes heat at the hot side (evaporation) and contributes heat at the cold side (condensation), the amount of energy involved is the heat of vaporization. Add a boundary heat source at the cavity/wick interface for this process.
Boundary Heat Source 1
1
In the Physics toolbar, click  Boundaries and choose Boundary Heat Source.
2
In the Settings window for Boundary Heat Source, locate the Boundary Selection section.
3
From the Selection list, choose Inner Wick Boundary.
4
Locate the Boundary Heat Source section. In the Qb text field, type (u*spf.nr+w*spf.nz)*HeatOfVaporization_water21(T)*spf.rho.
Study 1 - Dry Wick
1
In the Model Builder window, click Study 1.
2
In the Settings window for Study, type Study 1 - Dry Wick in the Label text field.
In order to select the state of our material switch "Fluid in Wick" to correspond to the dry case, a material sweep is added to the study, but is only given a single case, the index (1) of our first material in the switch, "Gas: water 1".
Material Sweep
1
In the Study toolbar, click  Material Sweep.
2
In the Settings window for Material Sweep, locate the Study Settings section.
3
Click  Add.
4
In the table, enter the following settings:
Switch
Cases
Case numbers
Fluid in Wick (sw1)
User defined
2
Step 1: Stationary
1
In the Model Builder window, click Step 1: Stationary.
2
In the Settings window for Stationary, locate the Physics and Variables Selection section.
3
In the Solve for column of the table, under Component 1 (comp1), clear the checkboxes for Laminar Flow (spf) and Brinkman Equations (br).
4
In the Study toolbar, click  Compute.
Results
Temperature - Dry Wick
Once the solver is done, look at the temperature profile along the heat pipe for this "dry" case.
1
In the Settings window for 2D Plot Group, type Temperature - Dry Wick in the Label text field.
2
Click to expand the Title section. From the Title type list, choose Manual.
3
In the Title text area, type Temperature.
4
Locate the Color Legend section. Select the Show maximum and minimum values checkbox.
5
Select the Show units checkbox.
Surface 1
1
In the Model Builder window, expand the Temperature - Dry Wick node, then click Surface 1.
2
In the Settings window for Surface, locate the Expression section.
3
From the Unit list, choose °C.
4
In the Temperature - Dry Wick toolbar, click  Plot.
This is Figure 2 from the earlier Results section. Note that for the parameters at hand, the pipe would get quite hot.
Set the preferred unit of temperature to degrees Celsius for future plots.
Preferred Units 1
1
In the Results toolbar, click  Configurations and choose Preferred Units.
2
In the Settings window for Preferred Units, locate the Units section.
3
Click  Add Physical Quantity.
4
In the Physical Quantity dialog, select General > Temperature (K) in the tree.
5
Click OK.
6
In the Settings window for Preferred Units, locate the Units section.
7
In the table, enter the following settings:
Quantity
Unit
Preferred unit
Temperature
K
°C
Remove one of the plots before moving on.
Root
Now solve the model for the normal case, with actual liquid in the wick.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies > Stationary.
4
Right-click and choose Add Study.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 2 - Saturated Wick
In the Settings window for Study, type Study 2 - Saturated Wick in the Label text field.
Material Sweep
1
In the Study toolbar, click  Material Sweep.
2
In the Settings window for Material Sweep, locate the Study Settings section.
3
Click  Add.
4
In the table, enter the following settings:
Switch
Cases
Case numbers
Fluid in Wick (sw1)
User defined
1
Step 1: Stationary
Solve Study 2 in two steps: first without the Brinkman Equations and then with all interfaces active. Solving in this order will give the Brinkman Equations an initial guess close to the final solution, making the process more robust and efficient.
Step 2: Stationary 1
In the Model Builder window, under Study 2 - Saturated Wick right-click Step 1: Stationary and choose Duplicate.
Step 1: Stationary
1
In the Settings window for Stationary, locate the Physics and Variables Selection section.
2
In the Solve for column of the table, under Component 1 (comp1), clear the checkbox for Brinkman Equations (br).
3
In the Study toolbar, click  Get Initial Value.
Results
Velocity (spf) and Temperature (ht)
By requesting initial values, default plot groups are created which you can now modify.
1
In the Settings window for 2D Plot Group, type Velocity (spf) and Temperature (ht) in the Label text field.
2
Locate the Title section. From the Title type list, choose Manual.
3
In the Title text area, type Fluid Velocity and Temperature.
4
In the Parameter indicator text field, type Fluid in Wick = Liquid : water.
5
Locate the Color Legend section. Select the Show units checkbox.
Surface 1 - Fluid Velocity, lg(|u|)
1
In the Model Builder window, expand the Velocity (spf) and Temperature (ht) node, then click Surface.
2
In the Settings window for Surface, type Surface 1 - Fluid Velocity, lg(|u|) in the Label text field.
3
Locate the Expression section. In the Expression text field, type log10(ht.uz^2 + ht.ur^2)/2.
4
Locate the Coloring and Style section. From the Color table list, choose Prionace.
5
From the Color table transformation list, choose Reverse.
Arrow Surface 1
In the Model Builder window, right-click Velocity (spf) and Temperature (ht) and choose Arrow Surface.
Arrow Surface 1, Surface 1 - Fluid Velocity, lg(|u|)
Right-click and choose Duplicate.
Surface 2 - Temperature
1
In the Settings window for Surface, type Surface 2 - Temperature in the Label text field.
2
Locate the Expression section. In the Expression text field, type T.
3
Locate the Coloring and Style section. From the Color table list, choose HeatCamera.
4
From the Color table transformation list, choose None.
Use a deformation to allow showing two plots side by side.
Deformation 1
1
Right-click Surface 2 - Temperature and choose Deformation.
2
In the Settings window for Deformation, locate the Expression section.
3
In the r-component text field, type r_outer.
4
Locate the Scale section.
5
Select the Scale factor checkbox. In the associated text field, type 3.
Arrow Surface 1 - Vapor Flow
1
In the Model Builder window, under Results > Velocity (spf) and Temperature (ht) click Arrow Surface 1.
2
In the Settings window for Arrow Surface, type Arrow Surface 1 - Vapor Flow in the Label text field.
3
Locate the Arrow Positioning section. Find the r grid points subsection. In the Points text field, type 9.
4
Locate the Coloring and Style section. From the Arrow length list, choose Logarithmic.
5
Select the Scale factor checkbox. In the associated text field, type 0.0035.
6
From the Color list, choose Black.
Arrow Surface 2 - Liquid Flow
1
In the Model Builder window, click Arrow Surface 2.
2
In the Settings window for Arrow Surface, type Arrow Surface 2 - Liquid Flow in the Label text field.
3
Locate the Expression section. In the r-component text field, type u2.
4
In the z-component text field, type w2.
5
Locate the Arrow Positioning section. Find the r grid points subsection. In the Points text field, type 5.
6
Locate the Coloring and Style section.
7
Select the Scale factor checkbox. In the associated text field, type 50.
8
From the Color list, choose Black.
Line 1 - Material Boundaries
1
In the Model Builder window, right-click Velocity (spf) and Temperature (ht) and choose Line.
2
In the Settings window for Line, type Line 1 - Material Boundaries in the Label text field.
3
Locate the Expression section. In the Expression text field, type 1.
4
Locate the Coloring and Style section. From the Coloring list, choose Uniform.
5
From the Color list, choose Black.
Deformation 1
1
Right-click Line 1 - Material Boundaries and choose Deformation.
2
In the Settings window for Deformation, locate the Expression section.
3
In the r-component text field, type r_outer.
4
Locate the Scale section.
5
Select the Scale factor checkbox. In the associated text field, type 3.
Study 2 - Saturated Wick
Solver Configurations
In the Model Builder window, expand the Study 2 - Saturated Wick > Solver Configurations node.
Solution 4 (sol4)
1
In the Model Builder window, expand the Study 2 - Saturated Wick > Solver Configurations > Solution 4 (sol4) > Stationary Solver 1 node.
2
Right-click Stationary Solver 1 and choose Fully Coupled.
3
In the Settings window for Fully Coupled, click to expand the Results While Solving section.
4
Select the Plot checkbox.
5
From the Plot group list, choose Velocity (spf) and Temperature (ht).
6
In the Model Builder window, right-click Stationary Solver 2 and choose Fully Coupled.
7
In the Settings window for Fully Coupled, locate the Results While Solving section.
8
Select the Plot checkbox.
9
From the Plot group list, choose Velocity (spf) and Temperature (ht).
Results
Surface
1
In the Model Builder window, expand the Velocity, 3D (spf) node, then click Surface.
2
In the Settings window for Surface, locate the Coloring and Style section.
3
From the Color table list, choose Prism.
Surface
1
In the Model Builder window, expand the Results > Velocity, 3D (br) node, then click Surface.
2
In the Settings window for Surface, locate the Coloring and Style section.
3
From the Color table list, choose Prism.
To aid the analysis of the results, create a series of line integrals of the heat flux across boundaries, in an Evaluation Group. Let the line integrals run across the middle section of the pipe (on the r-axis), and the contact surfaces for heat source and heat sink.
Energy Balance
1
In the Results toolbar, click  Evaluation Group.
2
In the Settings window for Evaluation Group, type Energy Balance in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 2 - Saturated Wick/Parametric Solutions 2 (sol6).
4
Locate the Transformation section. Select the Transpose checkbox.
Heat Sink
1
Right-click Energy Balance and choose Integration > Line Integration.
2
In the Settings window for Line Integration, type Heat Sink in the Label text field.
3
Locate the Selection section. From the Selection list, choose Heat Sink.
4
Click Replace Expression in the upper-right corner of the Expressions section. From the menu, choose Component 1 (comp1) > Heat Transfer in Porous Media > Boundary fluxes > ht.ndflux - Normal conductive heat flux - W/m².
5
Locate the Expressions section. In the table, enter the following settings:
Expression
Unit
Description
ht.ndflux
W
sink: ndflux
Heat Source
1
Right-click Heat Sink and choose Duplicate.
2
In the Settings window for Line Integration, type Heat Source in the Label text field.
3
Locate the Selection section. From the Selection list, choose Heat Source.
4
Locate the Expressions section. In the table, enter the following settings:
Expression
Unit
Description
ht.ndflux
W
source: ndflux
Casing
1
Right-click Heat Source and choose Duplicate.
2
In the Settings window for Line Integration, type Casing in the Label text field.
3
Locate the Selection section. From the Selection list, choose Cross Section of Casing.
4
Locate the Expressions section. In the table, enter the following settings:
Expression
Unit
Description
ht.ndflux
W
casing: ndflux
Wick
1
Right-click Casing and choose Duplicate.
2
In the Settings window for Line Integration, type Wick in the Label text field.
3
Locate the Selection section. From the Selection list, choose Cross Section of Wick.
4
Locate the Expressions section. In the table, enter the following settings:
Expression
Unit
Description
ht.ndflux
W
wick: ndflux
Cavity
1
Right-click Wick and choose Duplicate.
2
In the Settings window for Line Integration, type Cavity in the Label text field.
3
Locate the Selection section. From the Selection list, choose Cross Section of Cavity.
4
Locate the Expressions section. In the table, enter the following settings:
Expression
Unit
Description
ht.ndflux
W
cavity: ndflux
w*spf.rho*HeatOfVaporization_water21(T)
W
cavity: latent heat
To investigate how the phase change along the wick transfers thermal energy, create a line graph plotting this heat flux along the cavity–wick boundary.
Latent heat flux from phase change
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Latent heat flux from phase change in the Label text field.
Line Graph 1
1
Right-click Latent heat flux from phase change and choose Line Graph.
2
In the Settings window for Line Graph, locate the Data section.
3
From the Dataset list, choose Study 2 - Saturated Wick/Parametric Solutions 2 (sol6).
4
Locate the Selection section. From the Selection list, choose Inner Wick Boundary.
5
Locate the y-Axis Data section. In the Expression text field, type (u*spf.nr + w*spf.nz)*spf.rho*HeatOfVaporization_water21(T).
6
Select the Description checkbox. In the associated text field, type (<B>u</B>\cdot<B>n</B>)\rho\DELTA H<sub>vap</sub>.
7
Click to expand the Title section. From the Title type list, choose Manual.
8
In the Title text area, type Latent heat flux from phase change.
9
Locate the x-Axis Data section. From the Parameter list, choose Expression.
10
In the Expression text field, type z.
Study 2 - Saturated Wick
In the Study toolbar, click  Compute.
Results
Energy Balance
1
In the Model Builder window, under Results click Energy Balance.
2
In the Energy Balance toolbar, click  Evaluate.
Temperature (ht)
1
In the Model Builder window, click Temperature (ht).
2
In the Settings window for 2D Plot Group, locate the Title section.
3
From the Title type list, choose Manual.
4
In the Title text area, type Temperature.
5
Locate the Color Legend section. Select the Show maximum and minimum values checkbox.
6
Select the Show units checkbox.
Temperature (ht)
1
In the Model Builder window, expand the Temperature (ht) node, then click Results > Temperature (ht).
2
In the Temperature (ht) toolbar, click  Plot.
This is Figure 3 from the results section earlier. It is striking how even the temperature is throughout the heat pipe. Also the maximum temperature is dramatically lower compared to the dry wick case investigated previously.