pipeflowmodellibrarymanual.pdf

VERSION 4.3b

Pipe Flow ModuleModel Library Manual

C o n t a c t I n f o r m a t i o n

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Part number: CM022804

P i p e F l o w M o d u l e M o d e l L i b r a r y M a n u a l 19982013 COMSOL

Protected by U.S. Patents 7,519,518; 7,596,474; and 7,623,991. Patents pending.

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Version: May 2013 COMSOL 4.3b

Solved with COMSOL Multiphysics 4.3b

2 0 1 3 C O M S O L

D i s c h a r g i n g T ank

Introduction

This tutorial model illustrates how to calculate the pressure drop and initial flow rate in a pipe system connected to water tank. The Pipe Flow interface contains ready-to-use friction models accounting for the surface roughness of pipes as well as pr

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essure losses in bends and valves.

odel Definition

ater from a tank flows through a total of 105 m of pipe to be discharged through an en ball valve. The pipes are 15 cm in diameter and made out of galvanized iron. The

ater level is 10 m above the point of discharge.

gure 1: Water flows through a pipe system with two 90 bends and discharges through an en ball valve.

he model example is taken from Ref. 1.

H E M O M E N T U M E Q U A T I O N

side a stretch of pipe section, the momentum balance solved is:

(1)

d the continuity equation

(2)

25 m

60 m 30 m

15 m

p fD 2dh----------u u F+=

Au 0=


2 | D I S C H A R G I N

The term on the left-hand side of Equation 1 is the pressure gradient along the tangential direction (flow direction) of a pipe stretch. The first term on the right-hand side represents the pressure drop due to viscous shear. fD is the Darcy friction factor, dh is the hydraulic diameter and u is the velocity mean value across a pipe cross section. F is a volume force term (SI unit: N/m3), in this case used to account for gravity. and A in Equation 2 are fluid density (kg/m3) and cross section area (m3), respectively. To find out more about these equations and variables, please refer to the section Theory for the Pipe Flow User Interface in the Pipe Flow Userss guide.

ETfa

F

Tture

wG TA N K 2 0 1 3 C O M S O L

xpressions for the Darcy Friction Factorhe Pipe Flow interface provides a library of built-in expressions for the Darcy friction ctor, fD.

igure 2: Select from different predefined Friction models in the Pipe Properties node.

his example uses the Churchill relation (Ref. 2) that is valid for laminar flow, rbulent flow, and the transitional region in between these regimes. The Churchill lation is:

(3)

here

(4)

(5)

fD 88

Re------

12A B+ -1.5+ 1 12=

A -2.457ln 7Re------

0.90.27 e d +

16=

B 37530Re

---------------- 16

=


2 0 1 3 C O M S O L

As seen from the equations above, the friction factor is a function of the surface roughness divided by diameter of the pipe. Surface roughness data can be selected from a predefined list in the Pipe Properties feature.

Fi

Tge

Tth

AInlo

chre

3 | D I S C H A R G I N G TA N K

gure 3: Specify the Pipe Shape and the Surface roughness in the Pipe Properties node.

he Churchill equation is also a function of the fluid properties and flow type, and ometry, through the Reynolds number:

(6)

he physical properties of water as function of temperature are directly available from e softwares built-in material library.

dditional Flow Resistances pipe networks, fittings, bends, valves, and so on, induce additional lumped pressure sses

(7)

aracterized by loss coefficients, Ki. The Pipe Flow interface can include such sistances through the following point features:

90 bend

45 bend

T-junction

Sudden contraction

Re ud-----------=

p 12---Kiu2=


4 | D I S C H A R G

Gradual contraction

Sudden expansion

Gradual expansion

Globe valve

Gate valve

Angle valve

Ball valve

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Fd

FI N G TA N K 2 0 1 3 C O M S O L

Butterfly valve

Swing check

his model uses two 90 bends and a Ball valve.

esults and Discussion

igure 4 shows the pressure drop over the pipe system, while Figure 5 shows the irection of flow and the fluid velocity.

igure 4: Pressure drop across the pipe system.


2 0 1 3 C O M S O L

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gure 5: The fluid velocity is constant at approximately 3.1 m/s.

he initial discharge rate is calculated to 54.5 m3/s. A check of the Reynolds number oduces Re = 4.6105 and indicates that the flow is well in the turbulent regime.

eferences

J.M. Coulson, J.F. Richardson, Chemical Engineering Vol. 1 4th Ed., Pergamon ress, 1990, 74-75.

S.W. Churchill, Friction factor equations span all fluid-flow regimes, Chem. Eng., l. 84, no. 24, p. 91, 1997.

odel Library path: Pipe_Flow_Module/Tutorial_Models/discharging_tank


6 | D I S C H A R G

Modeling Instructions

M O D E L W I Z A R D

1 Go to the Model Wizard window.

2 Click the 2D button.

3 Click Next.

4 In the Add physics tree, select Fluid Flow>Single-Phase Flow>Pipe Flow (pfl).

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Click Next.

Find the Studies subsection. In the tree, select Preset Studies>Stationary.

Click Finish.

E O M E T R Y 1

lygon 1In the Model Builder window, under Model 1 right-click Geometry 1 and choose Polygon.

In the Polygon settings window, locate the Coordinates section.

In the x edit field, type 0 30 30 90 90.1.

In the y edit field, type 0 0 15 15 15.

Locate the Object Type section. From the Type list, choose Open curve.

Now go to the Material Browser and select Water from the built-in Liquids and Gases library. The material properties will apply to the entire model domain by default.

A T E R I A L S

aterial BrowserIn the Model Builder window, under Model 1 right-click Materials and choose Open Material Browser.

In the Material Browser settings window, In the tree, select Built-In>Water, liquid.

Click Add Material to Model.

Click the Zoom Extents button on the Graphics toolbar.

Next, specify the dimensions and surface roughness of the pipe. Note that you can add multiple Pipe Properties features and assign them to different parts of a pipe network, should you have a system of made up of pipes with different characteristics.


2 0 1 3 C O M S O L

P I P E F L OW

Pipe Properties 11 In the Model Builder window, under Model 1>Pipe Flow click Pipe Properties 1.

2 In the Pipe Properties settings window, locate the Pipe Shape section.

3 From the list, choose Round.

4 In the di edit field, type 15[cm].

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4 7 | D I S C H A R G I N G TA N K

Locate the Flow Resistance section. From the Surface roughness list, choose Galvanized iron (0.15 mm).

Set boundary conditions for the inlet and outlet and use a Volume Force feature to take gravity effects into account.

essure 2In the Model Builder window, right-click Pipe Flow and choose Pressure.

Select Point 1 only.

In the Pressure settings window, locate the Boundary Pressure section.

In the p0 edit field, type 101325[Pa]+(25[m])*g_const*pfl.rho.

lume Force 1Right-click Pipe Flow and choose Volume Force.

Select Boundaries 13 only.

The default volume force is a gravity vector pointing in the negative y-direction.

Next, add a number of point features to include the pressure losses due to bends and the ball valve. The valve point may be difficult to select graphically with the mouse. Here, you can use the Selection list, as help to browse the points in a list.


end 1Right-click Pipe Flow and choose Bend.

Select Points 2 and 3 only.

lve 1Right-click Pipe Flow and choose Valve.


In the Valve settings window, locate the Valve Specification section.

From the Valve list, choose Ball valve (K = 4.5).

Now, mesh and solve the model.


8 | D I S C H A R G

M E S H 1

1 In the Model Builder window, under Model 1 click Mesh 1.

2 In the Mesh settings window, locate the Mesh Settings section.

3 From the Element size list, choose Extra fine.

4 Click the Build All button.

5 Right-click Model 1>Mesh 1 and choose Statistics.

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6I N G TA N K 2 0 1 3 C O M S O L

T U D Y 1

the Model Builder window, right-click Study 1 and choose Compute.

E S U L T S

locity (pfl)efault plots show the pressure drop in the pipe system, and the direction and velocity f the flow. Now select from predefined plot quantities to evaluate the volumetric flow te and the Reynolds number.

In the Model Builder window, expand the Velocity (pfl) node.

In the Model Builder window, expand the Results>Velocity (pfl)>Arrow Line 1.1 node, then click Color Expression 1.1.

In the Color Expression settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Pipe Flow>Mass flow rate magnitude (pfl.Qm).

Click the Plot button.

In the Color Expression settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Pipe Flow>Reynolds number (pfl.Re).



2 0 1 3 C O M S O L

Geo t h e rma l Hea t i n g f r om a Pond Loop

Introduction

Ponds and lakes can serve as thermal reservoirs in geothermal heating applications. In thsyNtelin

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is example, fluid circulates underwater through polyethylene piping in a closed stem. The pipes are coiled in a slinky shape and mounted onto sleds. The on-Isothermal Pipe Flow interface sets up and solves the equations for the mperature and fluid flow in the pipe system, where the geometry is represented by es in 3D.

gure 1: A sled carrying pipe coils shown before the system is submerged.

odel Definition

E O M E T R Y

igh density polyethylene pipe (20 mm diameter) is rolled into sixteen coils. Groups eight coils are mounted on two sleds. Each coil has a radius of 1 m and a length of proximately 75 m. The coil groups are connected to feed and return piping with a


2 | G E O T H E R M A

diameter of 50 mm (see Figure 2). The coil groups are 2.4 m in height an sit at the bottom of a pond that is 6 m deep. The total length of the piping is 1446 m.

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6 m9 m 0 mL H E A T I N G F R O M A P O N D L O O P 2 0 1 3 C O M S O L

igure 2: Polyethylene pipe system. Elevation above the pond bottom is indicated. Feed and turn piping (gray) is 50 mm in diameter while coils (black) are 20 mm in diameter.

he heat exchange between pond water and pipe fluid will depend on the temperature ifference between the two. A slow current in the pond will make the heat transfer ore effective than water at rest. The pond is warmer closer to the surface, as shown

y the temperature data in Table 1 below.

BLE 1: POND TEMPERATURE

LEVATION (m) TEMPERATURE (K)

284

288

291

293

2.4 m

2 m

9 m

2.4 m

2.4 m

0 m

feedreturn


2 0 1 3 C O M S O L

It is easy to set up a function in the software with linear interpolation between points so that the varying pond temperature can be taken into account in the simulation.

F L O W E Q U A T I O N S

The continuity and momentum equations below describe the stationary flow inside the pipe system:

(1)

AdeF

Gnea exprT

ETshR

Au 0= 3 | G E O T H E R M A L H E A T I N G F R O M A P O N D L O O P

(2)

bove, A (SI unit: m2) is the cross section area of the pipe, (SI unit: kg/m3) is the nsity, u (SI unit: m/s) is the fluid velocity, and p (SI unit: N/m2) is the pressure and

(SI unit: N/m3) is a volume force, like gravity.

ravity can be included explicitly in the model, but since the variation in density is gligible, and the model is not pressure driven, the only effect of including gravity is

change in the total pressure level. It is therefore common modeling practice to clude gravity by setting F=0 and interpret the pressure variable as the reduced essure , where z0 is the datum level of the free liquid surface. his reduces the model complexity and yields the same results.

xpressions for the Darcy Friction Factorhe right-hand side of Equation 2 describes the pressure drop due to internal viscous ear. The term contains the Darcy friction factor, fD, which is a function of the eynolds number and the surface roughness divided by the hydraulic pipe diameter,

0 p fD 2dh----------u u F+=

pr p g z0 z =


4 | G E O T H E R M

e/dh. The Non-Isothermal Pipe Flow interface provides a library of built-in expressions for the Darcy friction factor, fD.

F

Ttu

wA L H E A T I N G F R O M A P O N D L O O P 2 0 1 3 C O M S O L

igure 3: Select from different predefined Friction models in the Pipe Properties node.

his example uses the Churchill relation (Ref. 1) that is valid for laminar flow, rbulent flow, and the transitional region in between. The Churchill relation is:

(3)

here

(4)

(5)

fD 88

Re-------

12A B+ -1.5+ 1 12=

A -2.457ln 7Re-------

0.90.27 e d +

16=

B 37530Re

---------------- 16

=


2 0 1 3 C O M S O L

As seen from the equations above, the friction factor is a function of the surface roughness divided by diameter of the pipe. Surface roughness data can be selected from a predefined list in the Pipe Properties feature.

Fi

Tnu

TthRR

H

T

wtesediex 5 | G E O T H E R M A L H E A T I N G F R O M A P O N D L O O P

gure 4: Specify the Pipe Shape and the Surface roughness in the Pipe Properties node.

he Churchill equation is also a function of the fluid properties, through the Reynolds mber:

(6)

he physical properties of water as function of temperature are directly available from e softwares built-in material library. Inspection of Equation 3 reveals that for low eynolds number (at laminar flow), the friction factor is 64/Re, and for very high eynolds number, the friction factor is independent of Re.

E A T TR A N S F E R E Q U A T I O N S

he energy equation for the pipeline flow is:

(7)

here Cp (SI unit: J/(kgK)) is the heat capacity at constant pressure, T is the mperature (SI unit: K), and k (SI unit: W/(mK)) is the thermal conductivity. The cond term on the right-hand side of Equation 7 corresponds to friction heat ssipated due viscous shear. Qwall (SI unit: W/m) is a source/sink term due to heat change with the surroundings through the pipe wall:

Re ud-----------=

ACpu T Ak T fD+ 2dh---------- u 3 Qwall+=


6 | G E O T H E R M

(8)

Where Z (m) is the wetted perimeter of the pipe, h (W/(m2K)) an overall heat transfer coefficient and Text (K) the external temperature outside of the pipe.

The overall heat transfer coefficient includes contribution from internal film resistance, wall resistance, and external film resistance.

F

If

F

Qwall hZ Text T =A L H E A T I N G F R O M A P O N D L O O P 2 0 1 3 C O M S O L

igure 5: Temperature distribution across the pipe wall.

rn (m) is the outer radius of wall n

wrr0 (m) a wall coordinate, starting at the inner radius r0 wnrnrn1 (m) the wall thickness of wall nZn (m) is the outer perimeter of wall n

hint and hext are the film heat transfer coefficients on the inside and outside of the tube, respectively (W/(m2K)).

kn is the thermal conductivity (W/(mK)) of wall n

or a circular tube, an effective hZ in Equation 8 can be used such that

Layered pipe wall

1 2 NWall number:

r0r1

rN

T

r

TT0

T1T2

TNText

n

Tn

r0

wn


2 0 1 3 C O M S O L

(9)

The assumption is equal temperature around the circumference of the pipe, and that the heat transfer through the wall is quasi-static.

T

Tfo(R

T

ACan

w

P

TA

N

d

k

hZ eff 2

1r0hint--------------- 1

rNhext----------------

rnrn-1--------- lnkn

----------------------

n 1=

N

+ +------------------------------------------------------------------------------------= 7 | G E O T H E R M A L H E A T I N G F R O M A P O N D L O O P

he film resistance inside the pipe is given by:

. (10)

he internal Nusselt number is taken as 3.66 for the laminar flow regime (Ref. 2), and r the turbulent flow regime the Gnielinski correlation for internal pipe flow is used ef. 3):

(11)

he external film resistance around the pipe is:

(12)

slow current is present in the pond. For external forced convection around a pipe, OMSOL uses the Churchill and Bernstein (Ref. 4) relation for Nu, valid for all Re d for Pr > 0.2,:

(13)

here PrCpk.roperties of the pipe wall is given in the table below.

BLE 2: PIPE PROPERTIES

AME VALUE DESCRIPTION

wall 2 mm Pipe wall thickness

wall 0.46 W/(mK) Pipe wall thermal conductivity

hint Nuintkwater

d--------------=

NuintfD 8 Re 1000 Pr1 12.7 Pr2 3 1 +---------------------------------------------------------=

hext Nuextkwater

d--------------=

Nuext 0.30.62Re1 2 Pr1 3

1 0.4 Pr 2 3+ 1 4------------------------------------------------------- 1 Re 282000 5 8+ 4 5+=


8 | G E O T H E R M

Results and Discussion

Figure 6 shows the pressure (Pa) in the 1446 m pipe system assuming that water enters the system at a rate of 4 l/s.

FA L H E A T I N G F R O M A P O N D L O O P 2 0 1 3 C O M S O L

igure 6: Pressure drop over the pipe system.


2 0 1 3 C O M S O L

The plot below shows the temperature (K) distribution for the pipe fluid. It enters the pipe system at 5 C and exits with a temperature of approximately 11 C.

Fi 9 | G E O T H E R M A L H E A T I N G F R O M A P O N D L O O P

gure 7: Temperature of the pipe fluid.


10 | G E O T H E R

Turbulent flow conditions in the loop are important for good heat exchange between the pipes and the surroundings. A plot of the Reynolds number is shown in Figure 8, confirming that flow is turbulent (Re > 3000) throughout the system.

Ftu

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2ed

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igure 8: The Reynolds number in the pipe loop confirms that the flow conditions are rbulent.

ote: This model is also available in an extended version in the Introduction to Pipe low Module booklet.

eferences

. S.W. Churchill, Friction factor equations span all fluid-flow regimes, Chem. Eng., l. 84, no. 24, p. 91, 1997.

. F.P. Incropera and D.P. DeWitt, Fundamentals of Heat and Mass Transfer, 4th ., John Wiley & Sons, 1996. Eq 8.62 and Eq 9.34, respectively.

. V. Gnielinski, Int. Chem. Eng. vol. 16, p. 359, 1976.

. S. W. Churchill, M. Bernstein, J Heat Transfer, vol. 99, p. 300, 1977.


2 0 1 3 C O M S O L

Model Library path: Pipe_Flow_Module/Heat_Transfer/geothermal_heating



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6 11 | G E O T H E R M A L H E A T I N G F R O M A P O N D L O O P

Go to the Model Wizard window.

Click Next.

In the Add physics tree, select Fluid Flow>Non-Isothermal Flow>Non-Isothermal Pipe Flow (nipfl).

Click Next.

Find the Studies subsection. In the tree, select Preset Studies>Stationary.

Click Finish.

E O M E T R Y 1

art by creating the piping system geometry. You can simplify this by inserting a epared geometry sequence from file:

In the Model Builder window, under Model 1 right-click Geometry 1 and choose Insert Sequence from File.

Browse to the models Model Library folder and double-click the file geothermal_heating_geom_sequence.mph.

Click the Build All button.

you inserted the geometry sequence in the steps above, you can skip the geometry structions below and go directly to the section Definitions.

rametric Curve 1In the Model Builder window, under Model 1 right-click Geometry 1 and choose More Primitives>Parametric Curve.

In the Parametric Curve settings window, locate the Parameter section.

In the Maximum edit field, type 24.

Locate the Expressions section. In the x edit field, type cos(pi*s).

In the y edit field, type sin(pi*s).

In the z edit field, type 0.1*s.


12 | G E O T H E R

7 Click the Build Selected button.

Polygon 11 In the Model Builder window, right-click Geometry 1 and choose More

Primitives>Polygon.

2 In the Polygon settings window, locate the Coordinates section.

3 In the x edit field, type 1 1.1 1.1 1.1.

4 In the y edit field, type 0 0 0 1.5.

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In the z edit field, type 0 0 2.6 2.6.

Click the Build Selected button.

lygon 2Right-click Geometry 1 and choose More Primitives>Polygon.


In the x edit field, type 1 1.

In the y edit field, type 0 1.5.

In the z edit field, type 2.4 2.4.


irror 1Right-click Geometry 1 and choose Transforms>Mirror.

Click in the Graphics window, press Ctrl+A to highlight all objects, and then right-click to confirm the selection.

In the Mirror settings window, locate the Point on Plane of Reflection section.

In the y edit field, type 1.5.

Locate the Normal Vector to Plane of Reflection section. In the y edit field, type 1.

In the z edit field, type 0.

Locate the Input section. Select the Keep input objects check box.


rray 1Right-click Geometry 1 and choose Transforms>Array.


In the Array settings window, locate the Size section.

In the x size edit field, type 4.


2 0 1 3 C O M S O L

5 Locate the Displacement section. In the x edit field, type -3.


Polygon 31 Right-click Geometry 1 and choose More Primitives>Polygon.


3 In the x edit field, type 1 -15.

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In the y edit field, type 1.5 1.5.





In the x edit field, type 1.1 -15.

In the y edit field, type 1.5 1.5.



rray 2Right-click Geometry 1 and choose Transforms>Array.



In the y size edit field, type 2.

Locate the Displacement section. In the y edit field, type 10.




In the x edit field, type -15 -15 -28 -35 -45.

In the y edit field, type 1.5 11.5 11.5 11.5 9.

In the z edit field, type 2.4 2.4 6 10 10.



14 | G E O T H E R

Polygon 61 Right-click Geometry 1 and choose More Primitives>Polygon.


3 In the x edit field, type -15 -15 -28 -35 -45.

4 In the y edit field, type 1.5 11.5 11.5 11.5 9.

5 In the z edit field, type 2.6 2.6 6.2 10.2 10.2.


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otate 1Right-click Geometry 1 and choose Transforms>Rotate.

On the Graphics toolbar, first click Go to XY View and then Select Box.

In the Graphics window, draw a box around the upper coil group and the connecting pipe, and then right-click to select the objects.

In the Rotate settings window, locate the Rotation Angle section.

In the Rotation edit field, type 30.

Locate the Point on Axis of Rotation section. In the x edit field, type -15.

In the y edit field, type 11.5.


Click the Go to Default 3D View button on the Graphics toolbar.

E F I N I T I O N S

ow add some external data in the form of interpolation tables and variables.

terpolation 1In the Model Builder window, under Model 1 right-click Definitions and choose Functions>Interpolation.


2 0 1 3 C O M S O L

2 In the Interpolation settings window, locate the Definition section.

3 In the table, enter the following settings:

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t f(t)

0 284

2 288

4 291

6 293 15 | G E O T H E R M A L H E A T I N G F R O M A P O N D L O O P

Locate the Units section. In the Arguments edit field, type m.

In the Function edit field, type K.

riables 1In the Model Builder window, right-click Definitions and choose Variables.

In the Variables settings window, locate the Variables section.

Click Load from File.

Browse to the models Model Library folder and double-click the file geothermal_heating_variables.txt.

A T E R I A L S




O N - I S O T H E R M A L P I P E F L O W

pe Properties 1In the Model Builder window, under Model 1>Non-Isothermal Pipe Flow click Pipe Properties 1.

In the Pipe Properties settings window, locate the Pipe Shape section.

From the list, choose Round.

In the di edit field, type 20[mm].


16 | G E O T H E R

Temperature 11 In the Model Builder window, under Model 1>Non-Isothermal Pipe Flow click

Temperature 1.

2 In the Temperature settings window, locate the Temperature section.

3 In the Tin edit field, type 5[degC].

Pipe Properties 21 In the Model Builder window, right-click Non-Isothermal Pipe Flow and choose Pipe

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W1M A L H E A T I N G F R O M A P O N D L O O P 2 0 1 3 C O M S O L

Properties.

Select Edges 112, 15, 1721, 2729, 3335, 39, 4145, 5153, 5759, 63, 6569, 7577, 8183, 87, 8991, 97, 98, 102, and 103 only.

To make the selection, first click Go to XZ View and then Select Box. In the Graphics window, draw a box around the pipes, then right-click to select the objects. Click Go to Default View button. Alternatively, copy the entity numbers from the text, click in the selection box, and then press Ctrl+V.



In the di edit field, type 50[mm].

all Heat Transfer 1Right-click Non-Isothermal Pipe Flow and choose the edge condition Heat Transfer in Pipes>Wall Heat Transfer.


2 0 1 3 C O M S O L

2 Select Edges 7104 only.

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In1 17 | G E O T H E R M A L H E A T I N G F R O M A P O N D L O O P

In the Wall Heat Transfer settings window, locate the Heat Transfer Model section.

In the Text edit field, type T_pond.

Right-click Model 1>Non-Isothermal Pipe Flow>Wall Heat Transfer 1 and choose Internal Film Resistance.

all Layer 1Right-click Wall Heat Transfer 1 and choose Wall Layer.

In the Wall Layer settings window, locate the Specification section.

From the k list, choose User defined.

In the associated edit field, type k_wall.

From the w list, choose User defined.In the associated edit field, type d_wall.

xternal Film Resistance 1Right-click Wall Heat Transfer 1 and choose External Film Resistance.

In the External Film Resistance settings window, locate the Specification section.

From the External film heat transfer model list, choose External forced convection.

From the External material list, choose Water, liquid.

In the uext edit field, type 0.2[m/s].

The external slow flow of 0.2 m/s is the mild current in the pond. This is enough to consider it forced convection outside the tubes.

let 1In the Model Builder window, right-click Non-Isothermal Pipe Flow and choose the point condition Pipe Flow>Inlet.


18 | G E O T H E R

2 Select Point 1 only.

3 In the Inlet settings window, locate the Inlet Specification section.

4 From the Specification list, choose Volumetric flow rate.

5 In the qv,0 edit field, type 4[l/s].

Heat Outflow 11 Right-click Non-Isothermal Pipe Flow and choose the point condition Heat Transfer

in Pipes>Heat Outflow.

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E S H 1

In the Model Builder window, under Model 1 click Mesh 1.

In the Mesh settings window, locate the Mesh Settings section.

From the Element size list, choose Extremely fine.


Click the Go to Default 3D View button on the Graphics toolbar.

T U D Y 1

the Model Builder window, right-click Study 1 and choose Compute.

E S U L T S

ressure (nipfl)efault plot groups show the pressure (Figure 6), velocity, and temperature (Figure 7) the pipe system. To get a better view, do as follows:

Click the Zoom Box button on the Graphics toolbar.

Draw a box in the Graphics window to zoom in on the coils.

eproduce the Reynolds number plot in Figure 8 with the following steps.

D Plot Group 4In the Model Builder window, right-click Results and choose 3D Plot Group.

Right-click 3D Plot Group 4 and choose Line.

In the Line settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Non-Isothermal Pipe Flow>Reynolds number (nipfl.Re).

Locate the Coloring and Style section. From the Line type list, choose Tube.


2 0 1 3 C O M S O L

5 Click the Plot button. 19 | G E O T H E R M A L H E A T I N G F R O M A P O N D L O O P


20 | G E O T H E R M A L H E A T I N G F R O M A P O N D L O O P 2 0 1 3 C O M S O L


2 0 1 3 C O M S O L

Con v e c t i v e F l ow i n a Hea t E x c h ang e r P l a t e

Introduction

A well distributed flow is necessary to achieve optimal performance in plate heat exou

Mcothmsuthpr

FiS. 1 | C O N V E C T I V E F L O W I N A H E A T E X C H A N G E R P L A T E

changers. The flow distribution can be controlled through the design of the inlet and tlet manifolds that connect the plate channels.

odeling the detailed flow within the channels of a plate heat exchanger can be mputationally costly, or even prohibitive. However, it is often sufficient to describe e flow with a lumped pipe flow model. This way the computational time and emory requirements can be significantly reduced. The model presented here shows ch an approach where a Pipe Flow interface, solving for the velocity and pressure in e plate channels, is coupled to a Laminar Flow interface, that solves for the flow and essure in the inflow and an outflow manifolds.

gure 1: Plate heat exchanger assembly with stacked plates. Image courtesy of Varem p.a.


2 | C O N V E C T I V

Note: This model requires the Pipe Flow Module. Due to the use of the Laminar inflow boundary condition on inlets 16, either of the following modules are also required: the Batteries & Fuel Cells Module, the CFD Module, the Chemical Reaction Engineering Module, the Electrodeposition Module, the Heat Transfer Module, the Microfluidics Module, the Plasma Module, the Subsurface Flow Module, or the Corrosion module. To reproduce the model with only the Pipe Flow module, that inlet condition can be replaced with a simpler condition.

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odel Definition

igure 2 shows the model geometry. The inlet and outlet manifolds are 2 mm wide, mm high, and 85 mm long. The microchannels have a square cross section with a de of 1 mm. They are drawn as edges in the 3D geometry using a cosine parametric rve.

igure 2: Five microchannels, inlet and outlet manifold.

O M A I N E Q U A T I O N S

he fluid properties of water are used in both interfaces.

let and Outlet Manifoldshe laminar flow in the manifolds is described by the Navier-Stokes equation in 3D, t up by the Laminar Flow interface.

utlet Manifold

Inlet Manifold

Microchannels

Inlet

Outlet


2 0 1 3 C O M S O L

MicrochannelsThe 1D flow in the channels is modeled using the Pipe Flow interface, that calculates the pressure drop over a pipe and the continuity equation according to:

(1)

(2)

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wpe

Tshlibw

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InA

Oprin

p fD 2dh----------u u=

Au 0= 3 | C O N V E C T I V E F L O W I N A H E A T E X C H A N G E R P L A T E

here dh is the mean hydraulic diameter (m), given by:

(2-3)

here A is the pipe cross section area (m2) available for flow, and Z is the wetted rimeter (m).

he right hand side of Equation 1 describes the pressure drop due to internal viscous ear and contains the Darcy friction factor, fD. The Pipe Flow interface provides a rary of built-in expressions for fD covering laminar and turbulent flow regimes, as

ell as Newtonian and non-Newtonian fluids.

O U N D A R Y C O N D I T I O N S

oupling the 3D flow in the manifolds to the 1D flow in the channels requires some tra consideration.

let Manifold (Laminar Flow)n average velocity of 5 cm/s is set at the manifold inlet.

n the manifold outlets, facing the microchannel inlets, the pressure is set to the essure in the microchannels. The channel pressure at the connection is available only the point representing the channel inlet. Using an integration point operator makes

dh4AZ-------=


4 | C O N V E C T I

the pressure in the point available globally, and this way, the value can be applied to the outlet surface of the manifold.

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AverageV E F L O W I N A H E A T E X C H A N G E R P L A T E 2 0 1 3 C O M S O L

igure 4: Integration and Average operators are used to set up the interface couplings.

icrochannels (Pipe Flow)n a channel inlet, the flow velocity is set to the average velocity evaluated across the rresponding manifold outlet boundary. Average boundary operators are used for is purpose.

n the microchannel outlets the pressure is set to the pressure in the outlet manifold.

utlet Manifold (Laminar Flow)n the manifold inlets, facing the microchannel outlets, the average velocity in the aminar Inflow feature is set to the velocity in the microchannel. Once again the point lue of the channel outlet is made available to the manifold inlet surface through tegral point operators.

inally, atmospheric pressure is applied to the manifold outlet boundary.

Integration

operator

operator


2 0 1 3 C O M S O L


Figure 5 shows the pressure distribution in the model. The microchannels contribute to the major part of the pressure drop.

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Fchmch 5 | C O N V E C T I V E F L O W I N A H E A T E X C H A N G E R P L A T E

gure 5: The main pressure drop occurs the microchannels.

igure 6 shows the velocity in the microchannels. The velocity is lower in the central annels, reducing the heat exchange efficiency in this region. A change in the anifold design would be necessary for a more uniform flow distribution in the plate annels.


6 | C O N V E C T I

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igure 6: Flow velocity magnitude in the microchannels.

otes about the COMSOL implementation

the wavy pipes, the flow is so slow and viscid that you can assume that the viscous ear is the sole contribution to pressure drop. The pressure drop due to momentum ange in the bends is neglected. One could extend this model by adding a lumped

end point condition in the Pipe Flow interface at the inlets of the pipes and set the ss coefficient to a number that estimates all the bends together.

odel Library path: Pipe_Flow_Module/Heat_Transfer/eat_exchanger_plate

odeling Instructions

O D E L W I Z A R D



2 0 1 3 C O M S O L

2 Click Next.

3 In the Add physics tree, select Fluid Flow>Single-Phase Flow>Pipe Flow (pfl).

4 Click Add Selected.

5 In the Add physics tree, select Fluid Flow>Single-Phase Flow>Laminar Flow (spf).

6 Click Add Selected.

7 Click Next.

8 Find the Studies subsection. In the tree, select Preset Studies for Selected

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4 7 | C O N V E C T I V E F L O W I N A H E A T E X C H A N G E R P L A T E

Physics>Stationary.

Click Finish.

Start by drawing the geometry. Use millimeters as the default length unit.

E O M E T R Y 1

In the Model Builder window, under Model 1 click Geometry 1.

In the Geometry settings window, locate the Units section.

From the Length unit list, choose mm.

ote: To save time, you can insert a prepared geometry sequence from at_exchanger_plate_geom_sequence.mph now by right-clicking the Geometry 1 de, choosing Insert Sequence from File, and then browsing to the models Model

ibrary folder. If you do so, skip directly to the Materials section.

se a work plane to create the curves that define the channels in the Pipe Flow terface.

ork Plane 1Right-click Model 1>Geometry 1 and choose Work Plane.

In the Work Plane settings window, locate the Plane Definition section.

In the z-coordinate edit field, type 0.5.

rametric Curve 1In the Model Builder window, under Model 1>Geometry 1>Work Plane 1 right-click Plane Geometry and choose Parametric Curve.

In the Parametric Curve settings window, locate the Parameter section.

In the Maximum edit field, type 2*pi*25.

Locate the Expressions section. In the xw edit field, type s.


8 | C O N V E C T I

5 In the yw edit field, type cos(s)-1.

6 Locate the Position section. In the yw edit field, type 0.5.


8 Click the Zoom Extents button on the Graphics toolbar.

Array 1Create an array of the curve and enable Create selections to facilitate selection later on.

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4V E F L O W I N A H E A T E X C H A N G E R P L A T E 2 0 1 3 C O M S O L

In the Model Builder window, right-click Geometry 1 and choose Transforms>Array.

Select the object wp1 only.



Locate the Displacement section. In the y edit field, type 20.

Locate the Selections of Resulting Entities section. Select the Create selections check box.

lock 1reate the inlet and outlet manifolds using blocks.

Right-click Geometry 1 and choose Block.

In the Block settings window, locate the Size and Shape section.

In the Width edit field, type 2.

Locate the Position section. In the x edit field, type -2.



lock 2Right-click Model 1>Geometry 1>Block 1 and choose Duplicate.

In the Block settings window, locate the Position section.

In the x edit field, type 25*2*pi.


rray 2In the Model Builder window, right-click Geometry 1 and choose Transforms>Array.

Select the objects blk2 and blk1 only.




2 0 1 3 C O M S O L

5 Locate the Displacement section. In the y edit field, type 20.


Block 31 In the Model Builder window, under Model 1>Geometry 1 right-click Block 2 and

choose Duplicate.

2 In the Block settings window, locate the Size and Shape section.

3 In the Depth edit field, type 85.

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Locate the Position section. In the x edit field, type -4.


lock 4Right-click Model 1>Geometry 1>Block 3 and choose Duplicate.

In the Block settings window, locate the Position section.

In the x edit field, type 25*2*pi+2.

In the y edit field, type -4.


To simplify meshing, do as follows:

rm Composite Domains 1In the Model Builder window, right-click Geometry 1 and choose Virtual Operations>Form Composite Domains.

On the object fin, select Domains 112 only.


A T E R I A L S




In the Model Builder window, right-click Materials and choose Open Material Browser.

Because the material is sensitive to space dimension level, you need to add another instance of Water and assign it to the edge level, where the pipe flow equations are solved.


10 | C O N V E C T

5 In the Material Browser settings window, In the tree, select Built-In>Water, liquid.

6 Click Add Material to Model.

Water, liquid (2)1 In the Model Builder window, under Model 1>Materials click Water, liquid (2).

2 In the Material settings window, locate the Geometric Entity Selection section.

3 From the Geometric entity level list, choose Edge.

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2I V E F L O W I N A H E A T E X C H A N G E R P L A T E 2 0 1 3 C O M S O L

From the Selection list, choose Array 1.

The Laminar Flow interface uses a number of integration operators to couple to the pressure in the Pipe Flow interface. The integration operators make it possible to access the values for the pressure at the end points of the channel edge in the entire model.

Optionally, modify the default view to see the labels.

E F I N I T I O N S

iew 1In the Model Builder window, under Model 1>Definitions click View 1.

In the View settings window, locate the View section.

Select the Show geometry labels check box.

tegration 1ote that if you have problems finding certain points, you can always choose View > lection List.

In the Model Builder window, right-click Definitions and choose Model Couplings>Integration.

In the Integration settings window, locate the Source Selection section.

From the Geometric entity level list, choose Point.


tegration 2Right-click Model 1>Definitions>Integration 1 and choose Duplicate.





2 0 1 3 C O M S O L

Integration 41 Right-click Model 1>Definitions>Integration 3 and choose Duplicate.


Integration 51 Right-click Model 1>Definitions>Integration 4 and choose Duplicate.


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A1 11 | C O N V E C T I V E F L O W I N A H E A T E X C H A N G E R P L A T E











The Pipe Flow interface couples its inflow velocities to the 3D outflow velocity; average integration operators are needed for this coupling.

verage 1In the Model Builder window, right-click Definitions and choose Model Couplings>Average.

In the Average settings window, locate the Source Selection section.

From the Geometric entity level list, choose Boundary.

Select Boundary 25 only.

verage 2Right-click Model 1>Definitions>Average 1 and choose Duplicate.


12 | C O N V E C T

2 Select Boundary 30 only.

Average 31 Right-click Model 1>Definitions>Average 2 and choose Duplicate.


Average 41 Right-click Model 1>Definitions>Average 3 and choose Duplicate.

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verage 5Right-click Model 1>Definitions>Average 4 and choose Duplicate.


Now set up the Pipe Flow interface. Define it only on the curved edges representing the channels.

I P E F L O W

In the Model Builder window, under Model 1 click Pipe Flow.

In the Pipe Flow settings window, locate the Edge Selection section.

From the Selection list, choose Array 1.

uid Properties 1o settings are needed for the fluid properties because these are retrieved from the aterial.

ipe Properties 1In the Model Builder window, under Model 1>Pipe Flow click Pipe Properties 1.


From the list, choose Square.

In the wi edit field, type 1[mm].

let 1he inflow velocities into the pipe segments are taken from the Laminar Flow interface sing the average integration operators.

In the Model Builder window, right-click Pipe Flow and choose Inlet.


2 0 1 3 C O M S O L


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In the Inlet settings window, locate the Inlet Specification section.

From the Specification list, choose Velocity.

In the u0 edit field, type aveop1(-u2).

let 2Right-click Model 1>Pipe Flow>Inlet 1 and choose Duplicate.
















14 | C O N V E C T

4 In the u0 edit field, type aveop5(-u2).

The outlet pressures are set to the pressure calculated by the Laminar Flow interface. No coupling operator is needed here because the pressure in the Laminar Flow interface is defined in the edge points. Assume the pressure to be constant over the whole boundary in the Laminar Flow interface.

Pressure 11 In the Model Builder window, under Model 1>Pipe Flow click Pressure 1.

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In1I V E F L O W I N A H E A T E X C H A N G E R P L A T E 2 0 1 3 C O M S O L

In the Pressure settings window, locate the Boundary Pressure section.

In the p0 edit field, type p2.

In this case the constraint needs to be unidirectional for the coupling to work.

In the Model Builder windows toolbar, click the Show button and select Advanced Physics Options in the menu.

In the Pressure settings window, click to expand the Constraint Settings section.

From the Apply reaction terms on list, choose Individual dependent variables.

A M I N A R F L O W

ow set up the Laminar Flow interface. Again, the fluid properties are taken from the aterial. Set up the coupling of the pressure to the Pipe Flow interface. This is done the outlets of the inlet manifold.

let 1In the Model Builder window, right-click Laminar Flow and choose Inlet.


In the Inlet settings window, locate the Boundary Condition section.

From the Boundary condition list, choose Laminar inflow.

Locate the Laminar Inflow section. In the Uav edit field, type intop6(pfl.U).

In the Lentr edit field, type 1[cm].

let 2Right-click Model 1>Laminar Flow>Inlet 1 and choose Duplicate.


In the Inlet settings window, locate the Laminar Inflow section.

In the Uav edit field, type intop7(pfl.U).



2 0 1 3 C O M S O L


3 In the Inlet settings window, locate the Laminar Inflow section.

4 In the Uav edit field, type intop8(pfl.U).

Inlet 41 Right-click Model 1>Laminar Flow>Inlet 3 and choose Duplicate.


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O1 15 | C O N V E C T I V E F L O W I N A H E A T E X C H A N G E R P L A T E








This is the inlet to the manifold.



In the Uav edit field, type 5[cm/s].

utlet 1In the Model Builder window, right-click Laminar Flow and choose Outlet.


In the Outlet settings window, locate the Boundary Condition section.

In the p0 edit field, type intop1(p).

utlet 2Right-click Model 1>Laminar Flow>Outlet 1 and choose Duplicate.






16 | C O N V E C T


3 In the Outlet settings window, locate the Boundary Condition section.

4 In the p0 edit field, type intop3(p).

Outlet 41 Right-click Model 1>Laminar Flow>Outlet 3 and choose Duplicate.


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Right-click Model 1>Laminar Flow>Outlet 5 and choose Duplicate.

Set atmospheric pressure at the outlet of the manifold.

utlet 6Select Boundary 5 only.


In the p0 edit field, type 1[atm].

E S H 1

In the Model Builder window, under Model 1 click Mesh 1.

In the Mesh settings window, locate the Mesh Settings section.

From the Element size list, choose Finer.


To make the Model Builder tree easier to navigate in, you can now click the Collapse All button in the Model Builder window?s toolbar.

T U D Y 1

olver 1In the Model Builder window, right-click Study 1 and choose Show Default Solver.

Expand the Study 1>Solver Configurations>Solver 1 node.


2 0 1 3 C O M S O L

3 Right-click Stationary Solver 1 and choose Fully Coupled.

4 Right-click Study 1>Solver Configurations>Solver 1>Stationary Solver 1>Direct and choose Enable.

5 In the Model Builder window, click Study 1.

6 In the Study settings window, locate the Study Settings section.

7 Clear the Generate default plots check box.

8 Click the Compute button.

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Create the result plot shown in Figure 5 following these steps:

E S U L T S

D Plot Group 1In the Model Builder window, right-click Results and choose 3D Plot Group.

Right-click 3D Plot Group 1 and choose Volume.

In the Volume settings window, locate the Expression section.

In the Expression edit field, type p2-1[atm].

Click to expand the Range section. Click the Plot button.

Select the Manual color range check box.


In the Model Builder window, right-click 3D Plot Group 1 and choose Line.

In the Line settings window, locate the Expression section.

In the Expression edit field, type p-1[atm].

Click to expand the Range section. Click the Plot button.

Select the Manual color range check box.



In the Minimum edit field, type 0.


Locate the Coloring and Style section. Clear the Color legend check box.

From the Line type list, choose Tube.

In the Tube radius expression edit field, type 0.5.

Select the Radius scale factor check box.

From the Color table list, choose RainbowLight.


18 | C O N V E C T

22 Click the Plot button.

23 Click the Zoom Extents button on the Graphics toolbar.

3D Plot Group 21 Right-click 3D Plot Group 1 and choose Duplicate.

2 In the Model Builder window, expand the 3D Plot Group 2 node, then click Line 1.

3 In the Line settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Pipe Flow>Velocity magnitude (pfl.U).

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Locate the Range section. Clear the Manual color range check box.

Locate the Coloring and Style section. Select the Color legend check box.

In the Model Builder window, under Results>3D Plot Group 2 right-click Volume 1 and choose Disable.


2 0 1 3 C O M S O L

Coo l i n g o f an I n j e c t i o n Mo l d

Introduction

Cooling is an important process in the production of injection molded plastics. First of all, the cooling time may well represent more than half of the production cycle time. Second, a homogeneous cooling process is desired to avoid defects in the munan

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Fi 1 | C O O L I N G O F A N I N J E C T I O N M O L D

anufactured parts. If plastic materials in the injection molding die are cooled down iformly and slowly, residual stresses can be avoided, and thereby the risk of warps d cracks in the end product can be minimized.

s a consequence, the positioning and properties of the cooling channels become portant aspects when designing the mold.

he simulation of heat transfer in molds of relatively complex geometries requires a 3D presentation. Simulation of 3D flow and heat transfer inside the cooling channels are mputationally expensive. An efficient short-cut alternative is to model the flow and at transfer in the cooling channels with 1D pipe flow equations, and still model the rrounding mold and product in 3D.

his example shows how you can use the Non-Isothermal Pipe Flow interface together ith the Heat Transfer in Solids interface to model a mold cooling process. The uations describing the cooling channels are fully coupled to the heat transfer uations of the mold and the polyurethane part.

gure 1: The steering wheel of a car, made from polyurethane.


2 | C O O L I N G O

Model Definition

M O D E L G E O M E T R Y A N D P R O C E S S C O N D I T I O N S

The polyurethane material used for a steering wheel is produced by several different molds. The part considered in this model is the top half of the wheel grip, shown in gray in Figure 2.

FexF A N I N J E C T I O N M O L D 2 0 1 3 C O M S O L

igure 2: Polyurethane parts for a steering wheel. The top half of the grip is modeled in this ample.


2 0 1 3 C O M S O L

The mold consists of a 50-by-50-by-15 cm steel block. Two cooling channels, 1 cm in diameter, are machined into the block as illustrated in Figure 3.

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W 3 | C O O L I N G O F A N I N J E C T I O N M O L D

gure 3: Mold block and cooling channels.

he after injection of the polyurethane, the average temperature of the mold a the astic material is 473 K. Water at room temperature is used as cooling fluid and flows rough the channels at a rate of 10 liters/min. The model simulates a 10 min cooling ocess.

or numerical stability reasons, the model is set up with an initial water temperature of 3 K, which is ramped down to 288 K during the first few seconds.

I P E F L OW E Q U A T I O N S

he momentum and mass conservation equations below describe the flow in the oling channels:

(1)

(2)

ater inlets

ut------- p fD

2dh----------u u=

At----------- Au + 0=


4 | C O O L I N G

Above, u is the cross section averaged fluid velocity (m/s) along the tangent of the center line of a pipe. A (m2) is the cross section area of the pipe, (kg/m3) is the density, and p (N/m2) is the pressure. For more information, refer to the section Theory for the Pipe Flow User Interface in the Pipe Flow Module Users Guide.

Expressions for the Darcy Friction FactorThe second term on the right hand side of Equation 2 accounts for pressure drop due to viscous shear. The Pipe Flow physics uses the Churchill friction model (Ref. 1) to cabin

wO F A N I N J E C T I O N M O L D 2 0 1 3 C O M S O L

lculate fD. It is valid for laminar flow, turbulent flow, and the transitional region in etween. The Churchill friction model is predefined in the Non-Isothermal Pipe Flow terface and is given by:

(3)

here

(4)

(5)

fD 88

Re------

12A B+ -1.5+ 1 12=

A -2.457ln 7Re------

0.90.27 e d +

16=

B 37530Re

---------------- 16

=


2 0 1 3 C O M S O L

As seen from the equations above, the friction factor depends on the surface roughness divided by diameter of the pipe, e/d. Surface roughness values can be selected from a list in the Pipe Properties feature or be entered as user-defined values.

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wtethItth 5 | C O O L I N G O F A N I N J E C T I O N M O L D

gure 4: The Friction model and Surface roughness settings are found in the Pipe roperties feature.

the Churchill equation, fD is also a function of the fluid properties, flow velocity and ometry, through the Reynolds number:

(6)

he physical properties of water as function of temperature are directly available from e softwares built-in material library.

E A T TR A N S F E R E Q U A T I O N S

ooling Channelshe energy equation for the cooling water inside the pipe is:

(7)

here Cp (J/(kgK)) is the heat capacity at constant pressure, T is the cooling water mperature (K), and k (W/(mK)) is the thermal conductivity. The second term on e right hand side corresponds to heat dissipated due to internal friction in the fluid. is negligible for the short channels considered here. Qwall (W/m) is a source term at accounts for the heat exchange with the surrounding mold block.

Re ud-----------=

ACpTt------- ACpu T+ Ak T fD+A2dh---------- u 3 Qwall+=


6 | C O O L I N G

Mold Block and Polyurethane PartHeat transfer in the solid steel mold block as well as the molded polyurethane part is governed by conduction:

(8)

Above, T2 is the temperature in the solids. The source term Qwall comes into play for the heat balance in Equation 8 through a line heat source where the pipe is situated. TN

HTE

T

Incoa

Tfi

TcoausuIn

Intrasdpd

CpT2t---------- k T2=O F A N I N J E C T I O N M O L D 2 0 1 3 C O M S O L

his coupling is automatically done by the Wall Heat Transfer feature in the on-Isothermal Pipe Flow interface.

eat Exchangehe heat exchange term Qwall (W/m) couple the two energy balances given by quation 7 and Equation 8:

he heat transfer through the pipe wall is given by

(9)

Equation 9 Z (m) is the perimeter of the pipe, h (W/(m2K)) a heat transfer efficient and Text (K) the external temperature outside of the pipe. Qwall appears as source term in the pipe heat transfer equation.

he Wall heat transfer feature requires the external temperature and at least an internal lm resistance.

ext can be a constant, parameter, expression, or given by a temperature field mputed by another physics interface, typically a 3D Heat Transfer interface. h is tomatically calculated through film resistances and wall layers that are added as bnodes. For details, refer to the section Theory for the Heat Transfer in Pipes User terface in the Pipe Flow Module Users Guide.

this model example, Text is given as the temperature field computed by a 3D heat ansfer interface, and automatic heat transfer coupling is done to the 3D physics side a line source. The temperature coupling between the pipe and the surrounding omain is implemented as a line heat source in the 3D domain. The source strength is roportional to the temperature difference between the pipe fluid and the surrounding omain.

Qwall hZ Text T =


2 0 1 3 C O M S O L

The Wall Heat Transfer feature is added to the Non-Isothermal Pipe Flow interface, and the External temperature is set to the temperature of the Heat Transfer in solids interface.

Fite

Tna

wis

Cantuco

w

Nra

NTtu 7 | C O O L I N G O F A N I N J E C T I O N M O L D

gure 5: In the Wall Heat Transfer feature, set the External temperature to the mperature field calculated by the Heat Transfer in Solids interface.

he heat transfer coefficient, h, depends on the physical properties of water and the ture of the flow and is calculated from the Nusselt number:

(9-6)

here k is the thermal conductivity of the material, and Nu is the Nusselt number. dh the hydraulic diameter of the pipe.

OMSOL detects if the flow is laminar or turbulent. For the laminar flow regime, an alytic solution is available that gives Nu = 3.66 for circular tubes (Ref. 2). For rbulent flow inside channels of circular cross sections the following Nusselt rrelation is used (Ref. 3):

(10)

here Pr is the Prandtl number:

(11)

ote that Equation 10 is a function of the friction factor, fD, and therefore that the dial heat transfer will increase with the surface roughness of the channels.

ote: All the correlations discussed above are automatically used by the Wall Heat ransfer feature in the Pipe Flow Module, and it detects if the flow is laminar or rbulent for automatic selection of the correct correlation.

h Nu kdh------=

NuintfD 8 Re 1000 Pr

1 12.7 fD 8 1 2 Pr2 3 1 +------------------------------------------------------------------------------=

PrCp

k-----------=


8 | C O O L I N G


The steel mold and polyurethane part, initially at 473 K, are cooled for 10 minutes by water at room temperature. Figures below show sample results when flow rate of the cooling water is 10 liters/minute and the surface roughness of the channels is 46 m. After two minutes of cooling, the hottest and coldest parts of the polyurethane part differ by approximately 40 K (Figure 7).

FaO F A N I N J E C T I O N M O L D 2 0 1 3 C O M S O L

igure 7: Temperature distribution in the polyurethane part and the cooling channels fter 2 minutes of cooling.


2 0 1 3 C O M S O L

Figure 8 shows the temperature distribution in the steel mold after 2 minutes. The temperature footprint of the cooling channels is clearly visible.

Fi 9 | C O O L I N G O F A N I N J E C T I O N M O L D

gure 8: Temperature distribution in the steel mold block after 2 minutes of cooling.


10 | C O O L I N G

After 10 minutes of cooling, the temperature in the mold block is more uniform, with a temperature at the center of approximately 333 K (Figure 9). Still, the faces with cooling channel inlets and outlets are more than 20 K hotter.

F

TfuTwcob

TA

MM

S

S

S

A O F A N I N J E C T I O N M O L D 2 0 1 3 C O M S O L

igure 9: Temperature distribution in the steel mold block after 10 minutes of cooling.

he blue line in Figure 10 shows the average temperature of the polyurethane part as nction of the cooling time. The temperature is 333 K after 10 minutes of cooling. o evaluate the influence of factors affecting the cooling time, additional simulations ere run varying the flow rate of the cooling water, the surface roughness of the oling channels, and the mold material. The conditions are summarized in the table

elow.

BLE 1: COOLING CONDITIONS

OLD ATERIAL

WATER FLOW RATE (L/MIN)

SURFACE ROUGHNESS (MM)

AVERAGE T AFTER 10 MIN (K)

LINE COLOR

teel 10 0.046 333 Blue

teel 20 0.046 325 Green

teel 10 0.46 328 Red

luminium 10 0.046 301 Magenta


2 0 1 3 C O M S O L

Fico

CthAitinro

R

1.84

2.T

3. 11 | C O O L I N G O F A N I N J E C T I O N M O L D

gure 10: Average temperature of the polyurethane part as function of time and cooling nditions.

learly, the thermal conductivity of the mold material is the most important factor in is comparison, followed by flow rate and surface roughness of the cooling channels. ssuming that 340 K is an acceptable temperature at the end of the production cycle, can be found that changing the mold material reduces the cooling time by 67%, creasing the flow rate reduces the cooling time by 17%, and increasing surface ughness reduces the cooling time by 11%.

eferences

S.W. Churchill, Friction factor equations span all fluid-flow regimes, Chem. Eng., (24), 1997, 91.

Incropera, Frank P.; DeWitt, David P. (2002). Fundamentals of Heat and Mass ransfer (5th ed.). Hoboken: Wiley. pp. 486, 487. ISBN 0-471-38650-2.

V. Gnielinski, Int. Chem. Eng. 16, 1976, 359.


12 | C O O L I N G

Model Library path: Pipe_Flow_Module/Heat_Transfer/mold_cooling



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Click Next.

In the Add physics tree, select Fluid Flow>Non-Isothermal Flow>Non-Isothermal Pipe Flow (nipfl).

Click Add Selected.

In the Add physics tree, select Heat Transfer>Heat Transfer in Solids (ht).

Click Add Selected.

Click Next.

Find the Studies subsection. In the tree, select Preset Studies for Selected Physics>Time Dependent.

Click Finish.

L O B A L D E F I N I T I O N S

rametersIn the Model Builder window, right-click Global Definitions and choose Parameters.

In the Parameters settings window, locate the Parameters section.

In the table, enter the following settings:

tep 1reate a smooth step function to decrease the coolant temperature at the beginning of e process.

Right-click Global Definitions and choose Functions>Step.

In the Step settings window, locate the Parameters section.

ame Expression Description

_init_mold 473.15[K] Initial temperature, mold

_coolant 288.15[K] Steady-state inlet temperature, coolant


2 0 1 3 C O M S O L

3 In the Location edit field, type 2.5.

4 In the From edit field, type 1.

5 In the To edit field, type 0.

6 Click to expand the Smoothing section. In the Size of transition zone edit field, type 5.

Optionally, you can inspect the shape of the step function:


Va1

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F

Im1

Nam

T_in 13 | C O O L I N G O F A N I N J E C T I O N M O L D

riables 1Right-click Global Definitions and choose Variables.

In the Variables settings window, locate the Variables section.


E O M E T R Y 1

irst, import the steering wheel part from a CAD design file.

port 1In the Model Builder window, under Model 1 right-click Geometry 1 and choose Import.

e Expression Description

let T_coolant+(T_init_mold-T_coolant)*step1(t[1/s])

Ramped inlet temperature, coolant


14 | C O O L I N G

2 In the Import settings window, locate the Import section.

3 Click the Browse button.

4 Browse to the models Model Library folder and double-click the file mold_cooling_top.mphbin.

5 Click the Import button.

Second, draw the mold and cooling channels. To simplify this step, insert a prepared geometry sequence from file. After insertion you can study each geometry step in

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4 O F A N I N J E C T I O N M O L D 2 0 1 3 C O M S O L

the sequence.

In the Model Builder window, right-click Geometry 1 and choose Insert Sequence from File.

Browse to the models Model Library folder and double-click the file mold_cooling_geom_sequence.mph.



Click the Transparency button on the Graphics toolbar.

E F I N I T I O N S

reate the selections to simplify the model specification.

xplicit 1In the Model Builder window, under Model 1 right-click Definitions and choose Selections>Explicit.

In the Explicit settings window, locate the Input Entities section.

From the Geometric entity level list, choose Edge.

Select one segment of the upper channel and one segment of the lower.

Select Edges 6 and 7 only.


2 0 1 3 C O M S O L

5 Select the Group by continuous tangent check box to select entire channels.

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M

Tth

M1

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W1

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M1

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3 15 | C O O L I N G O F A N I N J E C T I O N M O L D

Right-click Model 1>Definitions>Explicit 1 and choose Rename.

Go to the Rename Explicit dialog box and type Cooling channels in the New name edit field.

Click OK.

A T E R I A L S

he next step is to specify material properties for the model. Select water and steel from e built-in materials database.




ater, liquidIn the Model Builder window, under Model 1>Materials click Water, liquid.

In the Material settings window, locate the Geometric Entity Selection section.

From the Geometric entity level list, choose Edge.

From the Selection list, choose Cooling channels.

aterial BrowserIn the Model Builder window, right-click Materials and choose Open Material Browser.

In the Material Browser settings window, In the tree, select Built-In>Steel AISI 4340.



16 | C O O L I N G

Steel AISI 43401 In the Model Builder window, under Model 1>Materials click Steel AISI 4340.

2 Select Domain 1 only.

Next, create a material with the properties of polyurethane.

Material 31 In the Model Builder window, right-click Materials and choose Material.

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H O F A N I N J E C T I O N M O L D 2 0 1 3 C O M S O L

Select Domain 2 only.

In the Material settings window, locate the Material Contents section.


Right-click Model 1>Materials>Material 3 and choose Rename.

Go to the Rename Material dialog box and type Polyurethane in the New name edit field.

Click OK.

O N - I S O T H E R M A L P I P E F L OW

In the Non-Isothermal Pipe Flow settings window, locate the Edge Selection section.


ipe Properties 1In the Model Builder window, under Model 1>Non-Isothermal Pipe Flow click Pipe Properties 1.



In the di edit field, type 1[cm].

Locate the Flow Resistance section. From the Surface roughness list, choose Commercial steel (0.046 mm).

mperature 1In the Model Builder window, under Model 1>Non-Isothermal Pipe Flow click Temperature 1.

roperty Name Value

hermal conductivity k 0.32

ensity rho 1250

eat capacity at constant pressure Cp 1540


2 0 1 3 C O M S O L

2 In the Temperature settings window, locate the Temperature section.

3 In the Tin edit field, type T_inlet.

Inlet 11 In the Model Builder window, right-click Non-Isothermal Pipe Flow and choose the

point condition Pipe Flow>Inlet.

2 Select Points 3 and 4 only.

3 In the Inlet settings window, locate the Inlet Specification section.

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From the Specification list, choose Volumetric flow rate.

In the qv,0 edit field, type 10[dm^3/min].

eat Outflow 1Right-click Non-Isothermal Pipe Flow and choose the point condition Heat Transfer in Pipes>Heat Outflow.

Select Points 269 and 270 only.

all Heat Transfer 1Right-click Non-Isothermal Pipe Flow and choose the edge condition Heat Transfer in Pipes>Wall Heat Transfer.

In the Wall Heat Transfer settings window, locate the Edge Selection section.


Locate the Heat Transfer Model section. From the Text list, choose Temperature (ht).

Right-click Model 1>Non-Isothermal Pipe Flow>Wall Heat Transfer 1 and choose Internal Film Resistance.

itial Values 1In the Model Builder window, under Model 1>Non-Isothermal Pipe Flow click Initial Values 1.

In the Initial Values settings window, locate the Initial Values section.

In the u edit field, type 0.1.

In the T edit field, type T_init_mold.

E A T TR A N S F E R I N S O L I D S

In the Model Builder window, expand the Model 1>Heat Transfer in Solids node, then click Initial Values 1.

In the Initial Values settings window, locate the Initial Values section.

In the T2 edit field, type T_init_mold.


18 | C O O L I N G

Heat Flux 11 In the Model Builder window, right-click Heat Transfer in Solids and choose Heat Flux.

2 In the Heat Flux settings window, locate the Boundary Selection section.

3 From the Selection list, choose All boundaries.

4 Locate the Heat Flux section. Click the Inward heat flux button.

5 In the h edit field, type 2.

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2 O F A N I N J E C T I O N M O L D 2 0 1 3 C O M S O L

E S H 1

dge 1In the Model Builder window, under Model 1 right-click Mesh 1 and choose More Operations>Edge.

In the Edge settings window, locate the Edge Selection section.


ize 1Right-click Model 1>Mesh 1>Edge 1 and choose Size.

In the Size settings window, locate the Element Size section.

From the Predefined list, choose Extra fine.

ee Tetrahedral 1 the Model Builder window, right-click Mesh 1 and choose Free Tetrahedral.

ize 1In the Model Builder window, under Model 1>Mesh 1 right-click Free Tetrahedral 1 and choose Size.

In the Size settings window, locate the Geometric Entity Selection section.

From the Geometric entity level list, choose Domain.


Locate the Element Size section. From the Predefined list, choose Fine.


T U D Y 1

tep 1: Time DependentIn the Model Builder window, under Study 1 click Step 1: Time Dependent.

In the Time Dependent settings window, locate the Study Settings section.


2 0 1 3 C O M S O L

3 In the Times edit field, type range(0,30,600).

4 In the Model Builder window, right-click Study 1 and choose Compute.

R E S U L T S

Data Sets1 In the Model Builder window, under Results right-click Data Sets and choose Solution.

2 Right-click Results>Data Sets>Solution 2 and choose Add Selection.

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In the Selection settings window, locate the Geometric Entity Selection section.

From the Geometric entity level list, choose Domain.


In the Model Builder window, right-click Data Sets and choose Solution.

Right-click Results>Data Sets>Solution 3 and choose Add Selection.

In the Selection settings window, locate the Geometric Entity Selection section.

From the Geometric entity level list, choose Boundary.

Select Boundaries 3 and 5 only.

mperature (nipfl)In the Model Builder window, expand the Results>Temperature (nipfl) node, then click Line 1.1.

In the Line settings window, locate the Coloring and Style section.

Select the Radius scale factor check box.

In the associated edit field, type 1.

Clear the Color legend check box.

Locate the Data section. From the Data set list, choose Solution 1.

From the Time list, choose 120.

In the Model Builder window, right-click Temperature (nipfl) and choose Surface.

In the Surface settings window, locate the Data section.

From the Data set list, choose Solution 2.

From the Time list, choose 120.


mperature (ht)In the Model Builder window, under Results click Temperature (ht).

In the 3D Plot Group settings window, locate the Data section.


20 | C O O L I N G

3 From the Time list, choose 120.


5 Click the Transparency button on the Graphics toolbar.

6 In the Model Builder window, expand the Temperature (ht) node, then click Surfac

pipeflowmodellibrarymanual.pdf

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