manual validation v42
TRANSCRIPT
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Validation ManualVersion 4.2
P+Z Engineering GmbH, Munich
www.theseus-fe.com
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THESEUS-FE 4.2 Validations Manual
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Legal Notices
Copyright 2012 P+Z Engineering GmbH. All Rights Reserved.
The information contained herein is the property of P+Z Engineering GmbH. Any use, copy, publication, distribution,
display, modification, or transmission of the information in whole or in part in any form or by any means without the priorexpress written permission of P+Z Engineering GmbH is strictly prohibited. Except when expressly provided by P+ZEngineering GmbH in writing, possession of this information shall not be construed to confer any license or rights underany of P+Z Engineering GmbHs intellectual property rights, whether by estoppel, implication, or otherwise.
ALL COPIES OF THE INFORMATION, IF ALLOWED BY P+Z ENGINEERING GMBH, MUST DISPLAY THIS NOTICEOF COPYRIGHT AND OWNERSHIP IN FULL.
THESEUS-FEis a copyright protected and registered trademark of P+Z Engineering GmbH.
All other brand and product names mentioned herein are the trademarks and registered trademarks of their respectiveowners.
Printed in Germany.June 2012
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About this manual
P+Z Engineering GmbH reserves the right to make changes or improvements to thesoftware product described in this document without notice. P+Z Engineering GmbHassumes no responsibility for any factual or typographical errors or omissions that may
have occurred. P+Z Engineering GmbH has however made every effort to ensure that theinformation contained in this Manual is accurate.
The idea of this manual is to demonstrate the high quality of our software THESEUS-FEby presenting a huge number of thermodynamic systems well validated with exact analyticresults, results from literature, or results achieved with our parent software tool INKA.
Additional background information can be found in the
GUI Manual
Keyword Manual
Tutorial Manual
Theory Manual
Transformer Manual
Oven Manual
also shipped with this release.
If you have any further questions, please contact:
P+Z Engineering GmbH
Anton-Ditt-Bogen 3
80939 MunichGermany
Phone: +49 89 31857 466
Fax: +49 89 31857 333
To see the latest THESUES-FE software and services, please visit our web site at:
http://www.theseus-fe.com
Questions about pricing, sales, availability and general issues should be directed to:
Technical and scientific support issues should be addressed to:
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Contents
CONTENTS .................................................................................................. IV
LIST OF FIGURES ...................................................................................... VII1 ANALYTIC VALIDATIONS ...................................................................... 1
1.1 Steady State Problems .......................................................................................... 21.1.1 1D wall with internal heat generation .................................................................... 31.1.2 1D composite wall with internal heat generation and convection .......................... 51.1.3 Conduction through a fin (bound. cond.: temp./temp.) .......................................... 71.1.4 Conduction through a fin (bound. cond.: temp./adiabatic) .................................... 91.1.5 2D conduction in rectangular plate ..................................................................... 111.1.6 2D conduction in a disk (bound. cond.: temp./temp./convec.) ............................ 131.1.7 2D conduction in a disk (bound. cond.: temp./adiabatic./convec.) ...................... 151.2 Transient Solutions .............................................................................................. 171.2.1 1D wall cooling .................................................................................................... 181.2.2 Sphere heating ................................................................................................... 201.2.3 Cylinder heating .................................................................................................. 221.2.4 Infinite body with internal heat impulse ............................................................... 251.3 Thermal Radiation Boundary Conditions .......................................................... 281.3.1 Numeric view factor integration ........................................................................... 291.3.2 Solar short wave radiation (from sun) ................................................................. 301.3.3 Thermal long wave radiation (closed cavity) ....................................................... 371.3.4 Thermal long wave radiation (opened cavity) ..................................................... 41
2 ADVANCED CONDUCTION .................................................................. 442.1 Heat Bridges in Buildings ................................................................................... 452.1.1 Heat bridges: Example 1 .................................................................................... 462.1.2 Heat bridges: Example 2 .................................................................................... 482.1.3 Heat bridges: Example 3 .................................................................................... 502.1.4 Heat bridges: Example 4 .................................................................................... 522.2 Anisotropic Conductivity .................................................................................... 542.3 Temperature Dependent Conductivity ............................................................... 562.4 Phase Change ...................................................................................................... 582.5 Cylinder Radiation (open grey body cavity) ...................................................... 602.6 Disk Radiation ...................................................................................................... 63
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2.7 Cylinder Radiation Coupled with 1D Flow ......................................................... 663 THESEUS-FE COMPARED WITH INKA ................................................ 70
3.1 Model a .................................................................................................................. 713.1.1 Linear convective airzone heating - without considering humidity (a 1) .............. 723.1.2 Linear convective airzone heating - considering humidity (a 2) .......................... 733.1.3 Linear convective airzone cooling - considering humidity (a 3) ........................... 743.1.4 Non-linear convective airzone heating - without considering humidity (a 4) ....... 763.1.5 Sun heating airzone - without considering humidity (a 5) ................................... 773.1.6 Sun heating airzone - without considering humidity (a 6) ................................... 783.1.7 Airzone heated by ventilation FVT (a 7) .............................................................. 793.1.8 Airzone heated by ventilation RVT (a 8) ............................................................. 813.1.9 Airzone cooled by ventilation RVT (a 9) .............................................................. 823.1.10
Airzone heated by ventilation RET (a 10) ........................................................... 84
3.1.11Airzone cooled by ventilation HRET (a 11) ......................................................... 853.1.12Airzone heated by inverse mode ventilation FAT (a 12) ..................................... 873.1.13Airzone cooled by inverse mode ventilation RAT (a 13) ..................................... 893.2 Model b ................................................................................................................. 923.2.1 Glass transmission without refraction (b 1) ......................................................... 933.2.2 Glass transmission with refraction (b 2) .............................................................. 953.3 Model c .................................................................................................................. 963.3.1 Box with glas roof (c1) ........................................................................................ 97
4 MANIKIN FIALA-FE VALIDATIONS .................................................... 994.1 Passive System Validation ................................................................................ 1004.1.1 Spherical body element .................................................................................... 1014.1.2 Cylindr. body element with metabolism and blood perfusion (1) ....................... 1044.1.3 Cylindr. body element with metabolism and blood perfusion (2) ....................... 1064.1.4 Dead man in a cold environment (10C) ........................................................... 1084.2 Thermal Neutrality Validation ........................................................................... 1104.2.1 Naked Manikin .................................................................................................. 1114.2.2 Clothed Manikin ................................................................................................ 1134.3 Active System Validation .................................................................................. 1144.3.1 Cooling at 5C (1) ............................................................................................. 1154.3.2 Cooling at 5C (2) ............................................................................................. 1184.3.3 Cool environment at 13C ................................................................................. 1224.3.4 Cool environment at 15C ................................................................................. 1244.3.5 Changing environment 28-18-28C .................................................................. 1264.3.6 Changing environment 28-33-28C .................................................................. 1294.3.7 Changing environment 18-42-18C .................................................................. 132
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4.3.8 Changing environment 28-48-28C .................................................................. 1344.3.9 Changing activity in a cold environment at 10C............................................... 1374.3.10 Stepwise changing activity in a warm environment at 30C .............................. 1394.3.11 Cool environment at 10C ................................................................................. 1414.3.12 Changing environment 43-17-43C .................................................................. 1454.3.13 Naked manikin - 1 hr exposure - wide range of environmental conditions ........ 1484.3.14 Clothed manikin - 3 hr exposure - wide range of environmental conditions ...... 1504.4 Thermal Comfort Validation .............................................................................. 1524.4.1 Thermal comfort at changing bound. cond.: neutral-cold-neutral ...................... 1534.4.2 Thermal comfort at changing bound. cond.: neutral-hot-neutral ....................... 154
BIBLIOGRAPHY ....................................................................................... 155
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List of figures
Fig. 1-1: Wall with heat generation system ....................................................................... 3Fig. 1-2: Wall with heat generation - results ......................................................................... 4Fig. 1-3: Composite wall with heat generation system ...................................................... 5Fig. 1-4: Composite wall with heat generation - results ....................................................... 6Fig. 1-5: Conduction through a fin with temp/temp BC system ......................................... 7Fig. 1-6: Conduction through a fin with temp/temp BC results .......................................... 8Fig. 1-7: Conduction through a fin with temp/adiabatic BC system ................................... 9Fig. 1-8: Conduction through a fin with temp/adiabatic BC - results .................................. 10Fig. 1-9: Rectangular plate conduction system ............................................................... 11Fig. 1-10: Rectangular plate conduction - results at x = 0.0833 and x = 0.5 ...................... 12Fig. 1-11: Rectangular plate conduction contour plot (temperature) ............................... 12Fig. 1-12: Conduction in a disk system ........................................................................... 13Fig. 1-13: Conduction in a disk temp. vs. Radius ............................................................ 14Fig. 1-14: Conduction in a disk contour plot (temperature) ............................................. 14Fig. 1-15: Conduction in a disk system ........................................................................... 15Fig. 1-16: Conduction in a disk temp. vs. Radius ............................................................ 16Fig. 1-17: Conduction in a disk contour plot (temperature) ............................................. 16Fig. 1-18: Cooling of wall system .................................................................................... 18
Fig. 1-19: Cooling of wall - FE model ................................................................................. 18Fig. 1-21: Results ............................................................................................................... 19Fig. 1-23: Sphere heating system ................................................................................... 20Fig. 1-24: Sphere heating time dep. results at center node ............................................ 21Fig. 1-25: Sphere heating contour plot (temperature) at t = 600s ................................... 21Fig. 1-26: Cylinder heating system ................................................................................. 22Fig. 1-27: Cylinder heating results for core & skin (t = 0..300s) ...................................... 23Fig. 1-28: Cylinder heating results for core & skin (t = 0..22500s) .................................. 23Fig. 1-29: Cylinder heating contour plot (temperatur) ..................................................... 24Fig. 1-30: Cylinder heating contour plot (convective heat flux density) ........................... 24Fig. 1-31: Internal heat impulse system .......................................................................... 25Fig. 1-32: Internal heat impulse results at t = 20 & 100s ................................................. 26Fig. 1-33: Internal heat impulse time dep. results at R = 0 .............................................. 26Fig. 1-34: Internal heat impulse contour plots (temperature) .......................................... 27Fig. 1-35: V21 system (closed cavity) ............................................................................. 28Fig. 1-36: V21viewfactors ............................................................................................... 29Fig. 1-37: V21 solar heating ............................................................................................ 30Fig. 1-38: THESEUS-FE shell group results stored in the hdf file ...................................... 31Fig. 1-39: V21 elementwise solar heatflux densities from THESEUS-FE (black body
cavity) ......................................................................................................................... 32
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Fig. 1-40: V21 interior diffuse transmittance and reflectance .......................................... 33Fig. 1-41: V21 results with and without interior diffuse transmittance and reflectance ... 33Fig. 1-42: V21 global energy balance for the BB-Cavity (Idf=0) ....................................... 34Fig. 1-43: V21 global energy balance for the PSDGB-Cavity (Idf=0) ............................... 34
Fig. 1-44: V21 global energy balance for the BB-Cavity (Idf=100W/m2) .......................... 35Fig. 1-45: V21 global energy balance for the PSDGB-Cavity (Idf=100W/m2) .................. 35Fig. 1-46: V21 global energy balance for the GB-Cavity (Idf=100W/m2) ......................... 36Fig. 1-47: V21 temperature boundary conditions for long wave radiation heat exchange
.................................................................................................................................... 37Fig. 1-48: THESEUS-FE shell group results stored in the hdf file ...................................... 38Fig. 1-49: V21 coarse mesh comparison between results (with and without reflection) .. 39Fig. 1-50: V21 fine mesh comparison between results (with and without reflection) ....... 40Fig. 1-51: V21 interior diffuse reflectance of long wave radiation .................................... 40Fig. 1-52: V21 temperature boundary conditions for long wave radiation heat exchange
.................................................................................................................................... 41Fig. 1-53: THESEUS-FE shell group results stored in the hdf file ...................................... 42Fig. 1-54: V21 fine mesh comparison between results (with and without reflection) ....... 43Fig. 1-55: V21 fine mesh view factor sum for elements of a closed cavity ...................... 43Fig. 2-1: T-shaped heat bridge with boundary conditions .................................................. 46Fig. 2-2: T-shaped heat bridge THESEUS-FE results ....................................................... 47Fig. 2-3: Right-angle heat bridge with boundary conditions ............................................... 48Fig. 2-4: Right-angle heat bridge THESEUS-FE results ................................................... 49Fig. 2-5: Right-angle heat bridge 2 with boundary conditions ............................................ 50Fig. 2-6: Right-angle heat bridge 2 THESEUS-FE results ................................................ 51Fig. 2-7: 3D heat bridge with boundary conditions. All units are specified in cm ................ 52Fig. 2-8: 3D heat bridge THESEUS-FE results ................................................................. 53Fig. 2-9: Anisotropic plate with boundary conditions .......................................................... 54Fig. 2-10: Anisotropic plate THESEUS-FE results compared with literature results.......... 55Fig. 2-11: Variable heat conductivity model with boundary conditions ............................... 56Fig. 2-12: THESEUS-FE results compared with isotropic lines from literature .................. 57Fig. 2-13: Phase Change model with boundary conditions ................................................ 58
Fig. 2-14: Table for assigning material property specific heat ............................................ 58Fig. 2-15: Phase change: THESEUS-FE results ............................................................... 59Fig. 2-16: Phase change: Comparison of FE results with analytical solution at x = 1 ........ 59Fig. 2-17: Cylinder Radiation example with boundary conditions ....................................... 60Fig. 2-18: Cylinder Radiation: THESEUS-FE results ........................................................ 61Fig. 2-19: Cylinder Radiation: THESEUS-FE results ........................................................... 62Fig. 2-20: Disk .................................................................................................................... 63Fig. 2-21: Fin Radiation: THESEUS-FE temperature results ............................................ 64Fig. 2-22: Fin Radiation: THESEUS-FEradiation per element results ................................. 65Fig. 2-23: Cylinder Radiation with 1D flow example: boundary conditions ......................... 66Fig. 2-24: Radiation Cylinder with 1D Flow: Theory discription .......................................... 67
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Fig. 2-25: Radiation Cylinder with 1D Flow: Comparison with results from literature ......... 68Fig. 2-26: Radiation Cylinder with 1D Flow: Field Results from Literature ......................... 69Fig. 3-1: Model a 1 system .............................................................................................. 72Fig. 3-2: Model a 1 - airzone temperature .......................................................................... 72
Fig. 3-3: Model a 2 system .............................................................................................. 73Fig. 3-4: Model a 2 - airzone temperature .......................................................................... 73Fig. 3-5: Model a 3 system .............................................................................................. 74Fig. 3-6: Model a 3 - airzone absolute humidity ................................................................. 74Fig. 3-7: Model a 3 - airzone temperature .......................................................................... 75Fig. 3-8: Model a 4 system .............................................................................................. 76Fig. 3-9: Model a 4 - airzone temperature .......................................................................... 76Fig. 3-10: Model a 5 system ............................................................................................ 77Fig. 3-11: Model a 5 - airzone temperature ........................................................................ 77Fig. 3-12: Model a 6 - system ............................................................................................ 78Fig. 3-13: Model a 6 - airzone temperature ........................................................................ 78Fig. 3-14: Model a 7 system ............................................................................................ 79Fig. 3-15: Model a 7 - airzone temperature ........................................................................ 79Fig. 3-16: Model a 7 - airzone absolute humidity ............................................................... 80Fig. 3-17: Model a 8 system ............................................................................................ 81Fig. 3-18: Model a 8 - airzone temperature ........................................................................ 81Fig. 3-19: Model a 9 system ............................................................................................ 82Fig. 3-20: Model a 9 - airzone absolute humidity ............................................................... 82Fig. 3-21: Model a 9 - airzone relative humidity ................................................................. 83Fig. 3-22: Model a 9 - airzone temperature ........................................................................ 83Fig. 3-23: Model a 10 - system .......................................................................................... 84Fig. 3-24: Model a 10 - airzone temperature ...................................................................... 84Fig. 3-25: Model a 11 - system .......................................................................................... 85Fig. 3-26: Model a 11 ventilation HRET .......................................................................... 85Fig. 3-27: Model a 11 airzone temperature ..................................................................... 86Fig. 3-28: Model a 11 - airzone relative humidity ............................................................... 86Fig. 3-29: Model a 12 system .......................................................................................... 87Fig. 3-30: Model a 12 - ventilation outlet temperature (Tout1) ............................................. 87Fig. 3-31: Airzone relative humidity .................................................................................... 88Fig. 3-32: Mode a 13 - system ........................................................................................... 89Fig. 3-33: Model a 13 - temperature at ventilation outlet .................................................... 89Fig. 3-34: Model a 14 abs. humidity at the ventilation outlet ........................................... 90Fig. 3-35: Model a 13 rel. humidity at the ventilation outlet ............................................. 90Fig. 3-36: Model a 13 airzone abs. humidity ................................................................... 90Fig. 3-37: Model a 13 airzone rel. Humidity .................................................................... 91Fig. 3-38: Model b 1 system ............................................................................................ 93Fig. 3-39: Model b 1 airzone temperature ....................................................................... 94
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Fig. 3-40: Model b 1 system ............................................................................................ 95Fig. 3-41: Model b 1 airzone temperature ....................................................................... 95Fig. 3-42: Model c 1 system ............................................................................................ 97Fig. 3-43: Model c 1 airzone temperature ....................................................................... 98
Fig. 3-44: Model c 1 FE temperatures (inside, t = 500s) ................................................. 98Fig. 4-1: Tissue temperature, radius 4.0 cm .................................................................... 102Fig. 4-2: Tissue temperature, radius 7.5 cm .................................................................... 102Fig. 4-3: Tissue temperature, radius 10.3 cm .................................................................. 103Fig. 4-4: Tissue temperature, radius 2.20 cm .................................................................. 105Fig. 4-5: Tissue temperature, radius 4.93 cm .................................................................. 105Fig. 4-6: Tissue temperature, radius 5.48 cm .................................................................. 105Fig. 4-7: Tissue temperature, radius 5.48 cm .................................................................. 107Fig. 4-8: Dead man rectal temp. vs time ....................................................................... 108Fig. 4-9: Dead man temperature distribution vs radius ................................................. 109Fig. 4-10: Thermo-neutral with KSU uniform .................................................................... 113Fig. 4-11: Mean skin temperature .................................................................................... 115Fig. 4-12: Metabolism ...................................................................................................... 116Fig. 4-13: Rectal temperature .......................................................................................... 116Fig. 4-14: Rectal temperature .......................................................................................... 117Fig. 4-15:Time dep. boundary cond.: ambient air temperature and relative humidity ...... 118Fig. 4-16: Mean skin temperature .................................................................................... 119Fig. 4-17: Metabolism ...................................................................................................... 119Fig. 4-18: Forehead temperature ..................................................................................... 119Fig. 4-19: Leg temperature .............................................................................................. 120Fig. 4-20: Chest temperature ........................................................................................... 120Fig. 4-21: Arm temperature .............................................................................................. 120Fig. 4-22: Rectal temperature .......................................................................................... 121Fig. 4-23: Mean skin temperature .................................................................................... 122Fig. 4-24: Metabolism ...................................................................................................... 123Fig. 4-25: Rectal temperature .......................................................................................... 123Fig. 4-26: Mean skin temperature .................................................................................... 125Fig. 4-27: Metabolism ...................................................................................................... 125Fig. 4-28: Rectal temperature .......................................................................................... 125Fig. 4-29: Time dependent boundary conditions: ambient air temperature ...................... 127Fig. 4-30: Mean skin temperature .................................................................................... 127Fig. 4-31: Metabolism ...................................................................................................... 128Fig. 4-32: Rectal temperature .......................................................................................... 128Fig. 4-33: Time dep. boundary cond.: ambient air temperature and relative humidity ..... 129Fig. 4-34: Mean skin temperature .................................................................................... 130Fig. 4-35: Evaporation heat loss ...................................................................................... 130Fig. 4-36: Rectal temperature .......................................................................................... 131
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Fig. 4-37: Time dependent boundary conditions: ambient air temperature ...................... 132Fig. 4-38: Mean skin temperature .................................................................................... 133Fig. 4-39: Evaporation heat loss ...................................................................................... 133Fig. 4-40: Rectal temperature .......................................................................................... 133
Fig. 4-41: Time dep. boundary cond.: ambient air temperature and relative humidity ..... 135Fig. 4-42: Mean skin temperature .................................................................................... 135Fig. 4-43: Evaporation heat loss ...................................................................................... 136Fig. 4-44: Rectal temperature .......................................................................................... 136Fig. 4-45: Time dependent boundary conditions: activity ................................................. 137Fig. 4-46: Mean skin temperature .................................................................................... 138Fig. 4-47: Weight loss ...................................................................................................... 138Fig. 4-48: Oesophageal temperature ............................................................................... 138Fig. 4-49: Time dependent boundary conditions: activity ................................................. 139Fig. 4-50: Mean skin temperature .................................................................................... 140Fig. 4-51: Skin evaporation .............................................................................................. 140Fig. 4-52: Rectal temperature .......................................................................................... 140Fig. 4-53: Time dependent boundary conditions: ambient air temperature ...................... 142Fig. 4-54: Mean skin temperature .................................................................................... 142Fig. 4-55: Metabolism ...................................................................................................... 143Fig. 4-56: Shoulder temperature ...................................................................................... 143Fig. 4-57: Arm temperature .............................................................................................. 143Fig. 4-58: Abdomen temperature ..................................................................................... 144Fig. 4-59: Deviation from inital value ................................................................................ 144Fig. 4-60: Time dependent boundary conditions: ambient air temperature ...................... 145Fig. 4-61: Mean skin temperature .................................................................................... 146Fig. 4-62: Metabolism ...................................................................................................... 146Fig. 4-63: Rectal temperature .......................................................................................... 146Fig. 4-64: Evaporation...................................................................................................... 147Fig. 4-65: Mean skin temperature, after 1hr exposure at different amb. Temperatures ... 148Fig. 4-66: Extra metabolism, after 1hr exposure at different ambient temperatures ........ 149Fig. 4-67: Rectal temperature, after 1hr exposure at different ambient temperatures ...... 149Fig. 4-68: Evaporation, after 1hr exposure at different ambient temperatures ................. 149Fig. 4-69: Skin blood flow, after 3hr expos. at diff. amb. temp. (rh = 85%) ...................... 150Fig. 4-70: Skin evaporation, after 3hr expos. at diff. amb. temp. (rh = 85%) .................... 151Fig. 4-71: Hypothal. temp., after 3hr expos. at diff. amb. temp. (rh = 85%) ..................... 151Fig. 4-72: Mean skin temperature, after 3hr expos. at diff. amb. temp. (rh = 85%) .......... 151Fig. 4-73: Comparison of comfort indices ........................................................................ 153Fig. 4-74: Comparison of comfort indices ........................................................................ 154
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1 Analytic Validations
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1.1 Steady State Problems
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1.1.1 1D wall with internal heat generation
System
Fig. 1-1: Wall with heat generation system
System and boundary conditions
Quantity Value Unit Description
Q 650 W / m Heat generation
k 0.79 W / m*K Conductivity
L 1 m Thickness
T1 = T2 20 C Temperature boundary condition
THESEUS-FE file Example_1_1_1.tfe
THESEUS-FE version 4.0
Problem description
Wall with internal heat generation Q has temperature boundary conditions T1 and T2. Thisis a steady state, 1D problem. The exact analytic solution is given in [14].
THESEUS-FE model
1 quad shell element (PSHELL3) with 1 layer and 3 discretization points.
T1
X
T2Q,k
L
System & boundary. cond.:
XL
THESEUS-FE model:
adiabatic
adiabatic
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Results
Fig. 1-2: Wall with heat generation - results
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1.1.2 1D composite wall with internal heat generation and convection
System
Fig. 1-3: Composite wall with heat generation system
System and boundary conditions
Quantity Value Unit Description
Q 1.5E6 W / m Heat generation
k1 75 W / m*K Conductivity in left layer
L1 0.05 m Thickness in left layer
k2 150 W / m*K Conductivity in right layer
L2 0.02 m Thickness in right layer
h 1000 W / m2
*K Convective heat transfer coefficient
T 30 C Ambient temperature
THESEUS-FE file Example_1_1_2.tfe
THESEUS-FE version 4.0
Problem descriptionA two layer composite wall with heat generation in the first layer is modelled. Thecomposite wall is adiabatic on one side and has convection boundary condition on theother.
The exact analytic solution for the temperature distribution is given in [14].
THESEUS-FE model
1 quad shell element (PSHELL3) with 2 layers and 3 discretization points per layer.
X
T,hQ1,k1
L1
k2
L2
adiabatic
X
L1 L2
System & boundary. cond.: THESEUS-FE model:
adiabatic
adiabatic
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Results
Fig. 1-4: Composite wall with heat generation - results
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1.1.3 Conduction through a fin (bound. cond.: temp./temp.)
System
X
T2T1k
L
w
X
1 2 3 4 5 6 7 8 9 10 11 12
System & boundary. cond.: THESEUS-FE model:
L
T, h
Fig. 1-5: Conduction through a fin with temp/temp BC system
System and boundary conditions
Quantity Value Unit Description
k 81 W / m*K Conductivity
L 0.833 m Length
w 0.083 m Width
h 100 W / m2
*K Convective heat transfer coefficient
T 20 C Ambient temperature
T1 100 C Left side temperature boundary condition
T2 20 C Right side temperature boundary condition
THESEUS-FE file Example_1_1_3.tfe
THESEUS-FE version 4.0
Problem description
A 1D fin is modeled with temperature boundary conditions at both ends and convection onthe top and bottom surfaces. The depth of the beam is 0.05m.The temperature distributionis plotted along the length of the beam.
The exact analytic solution is given in[14].
THESEUS-FE model
12 PSHELL3 elements are used to model the beam, each with 1 layer and 2 discretization
points in the depth direction to represent the top and bottom surfaces. The length of the FEmodel is longer than the actual length, to set boundary conditions on the 1st and 12thelement. Convection boundary condition is assigned to the top and bottom of elements 2through 11.
Comment
Convection that would take place on the sides of the fin (in the depth direction) was notaccounted for in the FE model and is also ignored in the analytic solution.
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Results
10,0
20,0
30,0
40,0
50,0
60,0
70,0
80,0
90,0
100,0
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8
Distance [m]
T[C]
THESEUS-FE
Analyt ical
Fig. 1-6: Conduction through a fin with temp/temp BC results
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1.1.5 2D conduction in rectangular plate
System
X
T2
T1k
L
Y
w
T1
T1
Group 4
Group 2
Gro
up3
Group1
Group 5
System & boundary. cond.: THESEUS-FE model:
L
w
Fig. 1-9: Rectangular plate conduction system
System and boundary conditions
Quantity Value Unit Description
k 81 W / m*K Conductivity
L 0.833 m Length
w 0.83 m Width
T1 100 C Temperature boundary condition
T2 20 C Temperature boundary condition
THESEUS-FE file Example_1_1_5.tfe
THESEUS-FE version 4.0
Problem description
A 2D rectangle is modelled with temperature boundary conditions on all 4 sides.
The exact analytic solution is given in[13].
THESEUS-FE model
12*12 PSHELL3 elements are used to model the rectangle; each with 1 layer and 2
discretization points in the depth direction to represent the top and bottom surfaces. The
length and width of the FE model are longer than the actual length, to set temperatureboundary condition on the faces of groups 1 through 4. Group 5 represents the domain inwhich temperature is calculated as a function of position.
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Results
20.0
30.0
40.0
50.0
60.0
70.0
80.0
90.0
100.0
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80
Distance [m]
T[C]
THESEUS-FE: x=0.0833
Analytical: x=0.0833
THESEUS-FE: x= 0.5
Analytical: x= 0.5
Fig. 1-10: Rectangular plate conduction - results at x = 0.0833 and x = 0.5
Fig. 1-11: Rectangular plate conduction contour plot (temperature)
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1.1.6 2D conduction in a disk (bound. cond.: temp./temp./convec.)
System
k
T1
Rin Rout0
T1
Group 1
Group 2
Group 3
System & boundary. cond.: THESEUS-FE model:
Rin Rout0
T
, h
Fig. 1-12: Conduction in a disk system
System and boundary conditions
Quantity Value Unit Description
k 81 W / m*K Conductivity
Rin 0.065 m Inner radius
Rout 0.185 m Outer radius
T1 50 C Temperature on inner/outer edge
h 100 W / m2
*K Convective heat transfer coefficient
T 20 C Ambiant temperature
THESEUS-FE file Example_1_1_6.tfe
THESEUS-FE version 4.0
Problem description
A 2D disk with a hole of radius 0.065m is modelled with temperature boundary conditionsspecified at the inner and outer edges. Convective boundary conditions hold for the rest ofthe disc.
The exact analytic solution is given in[13].
THESEUS-FE model
3 different groups are used to model the disk, each element is a PSHELL3 with 1 layer and2 discretization points in the depth direction to represent the top and bottom surfaces. Theinner and outer radius of the FE model are longer than the actual length, and are used to
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1.1.7 2D conduction in a disk (bound. cond.:temp./adiabatic./convec.)
System
k
T1
Rin Rout0
adiabatic
Group 1
Group 2
Group 3
System & boundary. cond.: THESEUS-FE model:
Rin Rout0
T, h
Fig. 1-15: Conduction in a disk system
System and boundary conditions
Quantity Value Unit Description
k 81 W / m*K Conductivity
Rin 0.065 m Inner radius
Rout 0.185 m Outer radius
T1 50 C Temperature on inner edge
h 100 W / m2
*K Convective heat transfer coefficient
T 20 C Ambiant temperature
THESEUS-FE file Example_1_1_7.tfe
THESEUS-FE version 4.0
Problem description
A 2D disk with a hole of radius 0.065m is modelled with temperature boundary conditionspecified on the inner edge, and adiabatic condition on the outer edge. Convectiveboundary conditions hold for the rest of the disc.
The exact analytic solution is given in[13].
THESEUS-FE model
3 different groups are used to model the disk, each element is a PSHELL3 with 1 layer and2 discretization points in the depth direction to represent the top and bottom surfaces. Theinner and outer radius of the FE model are shorter and longer than the actual lengths, and
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are used for applying the boundary conditions (group 1 and 3). Group 2 represents thedomain on which temperature is calculated as a function of position.
Results
35.0
40.0
45.0
50.0
0.06 0.08 0.10 0.12 0.14 0.16 0.18
Radius [m]
T[C]
THESEUS-FE
Analytical
Fig. 1-16: Conduction in a disk temp. vs. Radius
Fig. 1-17: Conduction in a disk contour plot (temperature)
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1.2 Transient Solutions
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1.2.2 Sphere heating
System
System & boundary. cond.: THESEUS-FE model:
R0
adiabaticT0
, k, c
adiabatic
R
T, h
T, h
Fig. 1-21: Sphere heating system
System and boundary conditions
Quantity Value Unit Description
k 50 W / m*K Conductivity
8000 kg / m Density
c 500 J / kg*K Heat capacitance
R 0.05 m Radius
h 100 W / m2 *K Heat transfer coefficient
T 100 C Ambient temperature
T0 20 C Initial temperature
THESEUS-FE file Example_1_2_2.tfe
THESEUS-FE version 4.0
Problem description
A sphere with initial temperature T0 is modeled as it warms to the convective ambienttemperature over time.
The exact analytic solution is given in[15].
THESEUS-FE model
Problem was modeled with 3 groups; group 1 is the center face (PSHELL3) where
adiabatic boundary condition is applied, group 2 is the outer shell (PSHELL3) where
convection is applied and group 3 is the inner solid comprised of HEXA elements.
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Results
20
25
30
35
40
45
50
0 100 200 300 400 500 600
time [sec]
T[C]
THESEUS-FE
Analytical
Fig. 1-22: Sphere heating time dep. results at center node
Fig. 1-23: Sphere heating contour plot (temperature) at t = 600s
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Results
20
25
30
35
40
45
50
55
60
65
70
0 50 100 150 200 250 300
time [sec]
T[C]
THESEUS-FE: Core Temp.
Analytical Solution: Core Temp
THESEUS-FE: Skin Temp
Analytical Solution: Skin Temp
Fig. 1-25: Cylinder heating results for core & skin (t = 0..300s)
0
100
200
300
400
500
600
700
0 5000 10000 15000 20000
time [sec]
T[C]
THESEUS-FE: Core Temp
Analytical Solution: Core Temp
THESEUS-FE: Skin Temp
Analytical Solution: Skin Temp
Fig. 1-26: Cylinder heating results for core & skin (t = 0..22500s)
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t = 200sec t = 300sec t = 600sect = 100sec t = 200sec t = 300sec t = 600sect = 100sec
Fig. 1-27: Cylinder heating contour plot (temperatur)
t = 200sec t = 300sec t = 600sect = 100sec t = 200sec t = 300sec t = 600sect = 100sec
Fig. 1-28: Cylinder heating contour plot (convective heat flux density)
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1.2.4 Infinite body with internal heat impulse
System
System & boundary. cond.: THESEUS-FE model:
L
R
T0, , k, c
E
System & boundary. cond.: THESEUS-FE model:
L
R
T0, , k, c
E
Fig. 1-29: Internal heat impulse system
System and boundary conditions
Quantity Value Unit Description
k 50 W / m*K Conductivity
10000 kg / m Density
c 500 J / kg*K Heat capacitance
L 0.6 m Length
R 0.3 m Radius
E 22643.38 J Initial energy input
T0 0 C Initial temperature
THESEUS-FE file Example_1_2_4.tfe
THESEUS-FE version 4.0
Problem descriptionA cylinder under a Dirac internal heat impulse is modeled.
The exact analytic solution is given in[15].
THESEUS-FE model
The problem was modeled with 2 groups. The heat impulse is applied on group 1, an
internal PSHELL3 mesh of area 2.26E-5 m2 over a time period of 1 second. Group 2 Is a
solid element mesh and serves as the body of the cylinder. The cylinder is large enoughas to represent an infinite solid for the point load.
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Results
0
5
10
15
20
25
30
35
40
0 0.02 0.04 0.06 0.08 0.1
Radius [m]
T[C]
Analytical (t=100sec)
Analytical (t=20sec)
THESEUS-FE (t=100sec)
THESEUS-FE (t=20sec)
Fig. 1-30: Internal heat impulse results at t = 20 & 100s
0
20
40
60
80
100
120
10 100 1000
time [sec]
T[C]
THESEUS-FE (R=0)
Analytical (R=0)
Fig. 1-31: Internal heat impulse time dep. results at R = 0
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t = 20sec t = 40sec
t = 60sec t = 80sec
t = 100sec t = 200sec
1C
0C
Tcore = 37.4C
Tcore = 7.0C
Tcore = 3.3C Tcore = 1.1C
Tcore = 4.5C
Tcore = 13.0C
t = 20sec t = 40sec
t = 60sec t = 80sec
t = 100sec t = 200sec
1C
0C
Tcore = 37.4C
Tcore = 7.0C
Tcore = 3.3C Tcore = 1.1C
Tcore = 4.5C
Tcore = 13.0C
Fig. 1-32: Internal heat impulse contour plots (temperature)
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1.3 Thermal Radiation Boundary Conditions
System
Fig. 1-33: V21 system (closed cavity)
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1.3.1 Numeric view factor integration
For the closed radiation cavity shown in Fig. 1-33 view factors from THESEUS-FE 3.0have been compared with some analytic results. Both results are shown in Fig. 1-34.
Fig. 1-34: V21viewfactors
THESEUS-FE file: Example_1_3_1.tfe
THESEUS-FE results: Example_1_3_1.rpt
In the tfe-file the Keyword VFCTRL that controls the view factor calculation is omitted.Thats why THESEUS-FE uses default settings:
#facets 10000 VF_MTH=S2S, INT_MTH=ADAPTIVE
The (adaptive) surface-to-surface method for view factor calculation does not deal withpartial shading. Partially shaded relations between facets in the matrix above get the viewfactor 0 as long as the element midpoint connection is interrupted by other elements. Toconsider the phenomenon of partial shading more realistic it is recommended to apply the
hemi-sphere (VF_MTH=HS) or hemi-cube method (VF_MTH=HC) together with a highrefinement level: e.g. RFL=5 and SUBELM=5.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 VFSUM:0,0000 0,1401 0,0155 0,0161 0,0686 0,0161 0,0000 0,0000 0,0213 0,0155 0,1401 0,0000 0,0441 0,0441 0,1018 0,0892 0,0000 0,0000 0,2794 0,0213 0,0140 1,02713
0,0000 0,1401 0,0155 0,0213 0,0155 0,1401 0,0442 0,0442 0,1018 0,0892 0,2793 0,0213 0,0140 1,00000
0,1752 0,0000 0,0000 0,0000 0,0271 0,0338 0,0000 0,0000 0,0385 0,0336 0,1001 0,0551 0,0000 0,0709 0,0551 0,0936 0,0000 0,0000 0,2749 0,0385 0,0122 1,00866
0,1751 0,0000 0,0386 0,0336 0,1001 0,0552 0,0709 0,0552 0,0936 0,2748 0,0386 0,0122 1,00000
0,0387 0,0000 0,0000 0,0000 0,0673 0,0901 0,0000 0,0000 0,2116 0,0563 0,0673 0,0286 0,0000 0,0497 0,0000 0,0000 0,0000 0,0000 0,0768 0,2116 0,0768 0,97474
0,0387 0,0000 0,2116 0,0562 0,0673 0,0497 0,0772 0,2116 0,0772 1,00000
0,0134 0,0000 0,0000 0,0000 0,2566 0,1374 0,1353 0,0740 0,0211 0,0300 0,0225 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0363 0,0130 0,0341 0,2198 0,99350
0,0000 0,2565 0,1374 0,1353 0,0741 0,0213 0,0363 0,0130 0,0341 0,2197 1,00000
0,0343 0,0109 0,0135 0,1540 0,0000 0,1540 0,1843 0,1315 0,0000 0,0135 0,0109 0,0000 0,0000 0,0000 0,0000 0,0000 0,0304 0,0304 0,0217 0,0269 0,1843 1,00022
0,1539 0,0000 0,1539 0,1842 0,1315 0,0852 0,0303 0,0303 0,0217 0,0269 0,1842 1,00000
0,0134 0,0225 0,0300 0,1374 0,2566 0,0000 0,1353 0,0740 0,0211 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0363 0,0000 0,0130 0,0341 0,2198 0,99350
0,1374 0,2565 0,0000 0,1353 0,0741 0,0211 0,0363 0,0130 0,0341 0,2197 1,00000
0,0000 0,0000 0,0000 0,1015 0,2304 0,1015 0,0000 0,1746 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0828 0,0828 0,0000 0,0585 0,1709 1,00296
0,1015 0,2303 0,1015 0,0000 0,1746 0,0828 0,0828 0,1709 1,00000
0,0000 0,0000 0,0000 0,0888 0,2630 0,0888 0,2794 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0954 0,0954 0,0000 0,0000 0,0892 1,00005
0,0889 0,2630 0,0889 0,2793 0,0000 0,0954 0,0954 0,0892 1,00000
0,0266 0,0385 0,1058 0,0316 0,0000 0,0316 0,0000 0,0000 0,0000 0,1058 0,0384 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,1686 0,2436 0,1686 0,95921
0,0266 0,0386 0,1058 0,0320 0,2130 0,0316 0,0000 0,1058 0,0386 0,1686 0,2436 0,1686 1,00000
0,0387 0,0673 0,0563 0,0901 0,0673 0,0000 0,0000 0,0000 0,2116 0,0000 0,0000 0,0286 0,0497 0,0000 0,0000 0,0000 0,0000 0,0000 0,0768 0,2116 0,0768 0,97470
0,0387 0,0673 0,0562 0,0000 0,0772 0,2116 0,0772 1,00000
0,1752 0,1001 0,0336 0,0338 0,0271 0,0000 0,0000 0,0000 0,0384 0,0000 0,0000 0,0551 0,0709 0,0000 0,0551 0,0936 0,0000 0,0000 0,2749 0,0385 0,0122 1,00854
0,1751 0,1001 0,0336 0,0386 0,0000 0,0552 0,0709 0,0552 0,0936 0,2748 0,0386 0,0122 1,00000
0,0000 0,0441 0,0114 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0114 0,0441 0,0000 0,1401 0,1401 0,1612 0,2794 0,0000 0,0000 0,0892 0,0325 0,0294 0,98299
0,0442 0,0442 0,0000 0,1401 0,1401 0,1612 0,2793 0,0892 0,0325 1,00000
0,0551 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0249 0,0709 0,1752 0,0000 0,1001 0,1752 0,2749 0,0000 0,0000 0,0936 0,0289 0,0000 0,99864
0,0552 0,0709 0,1751 0,0000 0,1001 0,1751 0,2748 0,0936 1,00000
0,0551 0,0709 0,0249 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,1752 0,1001 0,0000 0,1752 0,2749 0,0000 0,0000 0,0936 0,0289 0,0000 0,99864
0,0552 0,0709 0,0249 0,1751 0,1001 0,0000 0,1751 0,2748 0,0936 1,00000
0,1018 0,0441 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0441 0,1612 0,1401 0,1401 0,0000 0,2794 0,0000 0,0000 0,0892 0,0000 0,0000 1,00003
0,1018 0,0442 0,0442 0,1612 0,1401 0,1401 0,0000 0,2793 0,0892 1,00000
0,0558 0,0468 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0468 0,1746 0,1375 0,1375 0,1746 0,0000 0,0000 0,0000 0,1709 0,0585 0,0000 1,00293
0,0558 0,0468 0,0468 0,1746 0,1374 0,1374 0,1746 0,0000 0,1709 1,00000
0,0000 0,0000 0,0000 0,0000 0,1519 0,1088 0,3312 0,2385 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0540 0,0000 0,0326 0,0776 0,99450
0,1517 0,1088 0,3312 0,2384 0,0000 0,0540 0,0776 1,00000
0,0000 0,0000 0,0000 0,1088 0,1519 0,0000 0,3312 0,2385 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0540 0,0000 0,0000 0,0326 0,0776 0,99450
0,1088 0,1517 0,3312 0,2384 0,0540 0,0000 0,0776 1,00000
0,1746 0,1375 0,0192 0,0098 0,0271 0,0098 0,0000 0,0000 0,0843 0,0192 0,1375 0,0558 0,0468 0,0468 0,0558 0,1709 0,0000 0,0000 0,0000 0,0000 0,0000 0,99497
0,1746 0,1374 0,0193 0,0098 0,0271 0,0098 0,0000 1,00000
0,0266 0,0385 0,1058 0,0512 0,0673 0,0512 0,1170 0,0000 0,2436 0,1058 0,0385 0,0407 0,0289 0,0289 0,0000 0,1170 0,0163 0,0163 0,0000 0,0000 0,0000 1,09348
0,0266 0,0386 0,1058 0,0512 0,0673 0,0512 0,2436 0,1058 0,0386 0,0407 0,0000 1,00000
0,0087 0,0061 0,0192 0,1648 0,2304 0,1648 0,1709 0,0558 0,0843 0,0192 0,0061 0,0184 0,0000 0,0000 0,0000 0,0000 0,0194 0,0194 0,0000 0,0000 0,0000 0,98758
0,0087 0,0061 0,0193 0,1648 0,2303 0,1648 0,1709 0,0558 0,0843 0,0193 0,0061 0,0194 0,0194 0,0000 1,00000
shell elm. id:
-
shell elm. id:
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The solar heat hitting the ground:
2grnd
z
m/W63010058cos1000q
583290
The solar heat hitting the exterior glass roof:
W321005305.0Q
W1271005305.0Q
W1561005305.0Q
W315100)58cos(10005.0Q
396.0,496.07.590
405.0,495.058,0F,1F,5.0A
sky,effrf,sun
sky,efftr,sun
sky,effab,sun
bc,sun
sky,effsky,effsky,eff
grndsky
Fig. 1-36: THESEUS-FE shell group results stored in the hdf file
The solar heat hitting the exterior surface of element 5:
2bc,sunrf,sun
2bc,sunab,sun
2
bc,sun
0grndsky
m/W1861qq
m/W744qq
m/W9305.01.05545.0100)32cos(1000q
8.0,32,5.0F,5.0F
In a first step reflection and diffuse transmittance remain unconsidered inside the cavity:
then the solar heat hitting the interior surface of element 9 can be derived from
2bc,sunrf,sun
2
bc,sunab,sun
2
bc,sun
0
m/W431qqm/W172qq
m/W21558cos1000405.0q
8.0,58
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Now we add diffuse solar boundary conditions (Idf=100W/m2) and start with the Black Body
Approach:
77%
10%
13%
transmitting roof
opaque: interior surface
31.8W
156W
107.3W
85.9W
21.5W
77% system absorbed
=100%*(156+85.9)/315
10% system reflected
=100%*31.8/315
13% neglected
265W + 50W = 315W
107.3W + 19.8W =127.1W
Fig. 1-42: V21 global energy balance for the BB-Cavity (Idf=100W/m2)
Here not only diffuse reflection but also the diffuse transmitted solar heat flux remainsunconsidered within the Cavity. All together 13% of the solar heat will be neglected.
To considere diffuse energy portions we choose a Pseudo Grey Body Cavity:
87%
11%2%
transmitting roof
opaque: interior surface
31.8W
156W
142.3W
113.8W
28.5W
6.5W
3.2W
0.7W
2.6W
87% system absorbed
=100%*(156+113.8+3.2)/315
11% system reflected
=100%*(31.8+2.6)/315
2% neglected
265W + 50W = 315W
107.3W + 19.8W =127.1W
Fig. 1-43: V21 global energy balance for the PSDGB-Cavity (Idf=100W/m2)
The transmitted diffuse solar heat load of20W now hits the opque interior surfaces andleads to a rise from 122.2W (see Fig. 1-41) to 142.3W. A good approximation of theenergy balance with 2% heat loss results from a Pseudo Grey Body Approach that onceupdates the diffuse reflected solar heat (1 iteration).
The heat load of 6.493 W hitting the interior surface of the roof is divided into 3 parts:
3.218 W absorbed, 2.561 W transmitted and 0.714 W reflected. From this balance we canderive the absorbance of 0.496 and the transmittance of 0.394. Those values result froman effective incidence angle of 60 degree that is assumed for diffuse reflected solar heat.
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fulfil the energy balance: outgoingingoingabsorbed QQQ
As shown here differences for the absorbed long wave radiation heat flux strongly dependon the surface reflectance 1 . From that reasons a Grey Body Cavity is
recommended for100C.
Reflection neglected: Black Body (BB) Cavity Reflection considered: Grey Body (GB) Cavity
Absorbed long wave radiation heat flux density [W/m2]
Reflected rad. heat flux density [W/m2]
Fig. 1-47: V21 coarse mesh comparison between results (with and without reflection)
Reflection neglected: Black Body (BB) Cavity Reflection considered: Grey Body (GB) Cavity
Absorbed long wave radiation heat flux density: 0..22000 W/m2
from reflection !
Absorbed long wave radiation heat flux density 0..1000 W/m2
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Fig. 1-48: V21 fine mesh comparison between results (with and without reflection)
elm.:19 20 21
1
12
16
15
9
8
7
5
Fig. 1-49: V21 interior diffuse reflectance of long wave radiation
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1.3.4 Thermal long wave radiation (opened cavity)
Here we use the same THESEUS-FE model as in the chapter above, with 2 modifications:We delet the elements on the roof (19, 20, 21) and add the sky temperature -100C to thecavity definition in the tfe-file. This operation changes the closed cavity from the chapterabove to an opened cavity. Long wave radiation heat exchang takes now place not onlyinside the cavity, but also between elements inside and a so called background with auser-defined temperature.
0Celm:1
12
16
15
9
8
7
5
1000C
Tsky = -100C
Fig. 1-50: V21 temperature boundary conditions for long wave radiation heat exchange
System and boundary conditions
Quantity Value Unit Description
9T 1273.15 K
absolute temperature of the heated element
1810,81T 273.15 K
absolute temperature of all other elements
skyT 173.15 K absolute temperature of the background
5.67051E-8 W/(m2K
4) Stefan Bolzmann Constant
0.8 surface emissivity
= 1 - 0.2 surface reflectance
THESEUS-FE file V21_rad_BB_opencav.tfe
V21_fine2_rad_BB_opencav.tfe
V21_fine2_rad_GB_opencav.tfe
THESEUS-FE version 4.0
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2 Advanced Conduction
Several solved and experimental examples from literature have been taken and used for
further validation of THESEUS-FE.
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2.1 Heat Bridges in Buildings
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Results
THESEUS-FE: 19.92CLiterature: 17.92CTHESEUS-FE: 19.92CLiterature: 17.92C
Fig. 2-2: T-shaped heat bridge THESEUS-FEresults
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2.1.2 Heat bridges: Example 2
System
System & boundary. cond.: THESEUS-FE model:
L
L
k1
k2
Tout, hout
Tin, hin
Tout, hout
Adiabatic
Adiabatic
System & boundary. cond.: THESEUS-FE model:
L
L
k1
k2
Tout, hout
Tin, hin
Tout, hout
Adiabatic
Adiabatic
Fig. 2-3: Right-angle heat bridge with boundary conditions
System and boundary conditions
Quantity Value Unit Description
k1 0.21 W / m*K Conductivity
k2 0.56 W / m*K Conductivity
L 0.30 m Length
Tout 5 C Outside air temperature
hout 25 W / m2
*K Heat transfer coefficient
Tin 22 C Inside air temperature
hin 5 W / m2
*K Heat transfer coefficient
THESEUS-FE file Example_2_1_2.tfe
THESEUS-FE version 4.0
Problem description
Heat bridge example, with heat conduction through a composite wall. Convectionboundary conditions are assigned on the outer and inner surface of the wall. THESEUS-FEresults are compared with experimanal results at the corner location[19].
THESEUS-FE model
The problem was modeled with two groups of solid elements and two groups ofPSHELL3elements. Shell elements are placed on the inner and outer surface and are used forassigning boundary conditions.
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Results
THESEUS-FE: 17.02C
Literature: 17.07C
THESEUS-FE: 17.02C
Literature: 17.07C
Fig. 2-4: Right-angle heat bridge THESEUS-FEresults
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Results
THESEUS-FE: 18.40C
Literature: 17.92C
THESEUS-FE: 18.40C
Literature: 17.92C
Fig. 2-6: Right-angle heat bridge 2THESEUS-FEresults
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Problem description
Three dimansional heat bridge example, with heat conduction through a composite wall.Convection boundary conditions are assigned on the outer and inner surfaces of thewall.THESEUS-FE results are compared with experimanal results at several corner
locations and along edges[19].
THESEUS-FE model
The problem was modeled with eight groups of solid elements representing the different
parts of the composite wall and two groups of PSHELL3 elements. Shell elements are
placed on inner and outer surfaces and are used for assigning convection boundaryconditions. A uniform mesh with quad elements of approximate length 1cm was used witha total of 390,276 elements.
Results
THESEUS-FE: 18.9CLiterature: 18.4C
THESEUS-FE: 19.5C
Literature: 19.8C
THESEUS-FE: 19.8C
Literature: 19.6C
THESEUS-FE: 17.5C
Literature: 17.2C
THESEUS-FE: 17.7C
Literature: 17.9C
THESEUS-FE: 18.9CLiterature: 18.4C
THESEUS-FE: 19.5C
Literature: 19.8C
THESEUS-FE: 19.8C
Literature: 19.6C
THESEUS-FE: 17.5C
Literature: 17.2C
THESEUS-FE: 17.7C
Literature: 17.9C
Fig. 2-8: 3D heat bridge THESEUS-FEresults
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2.2 Anisotropic Conductivity
System
System & boundary. cond.: THESEUS-FE model:
L1
adiabatic
L1
T2T1
adiabatic
L2
L2
k1
k11
k22
System & boundary. cond.: THESEUS-FE model:
L1
adiabatic
L1
T2T1
adiabatic
L2
L2
k1
k11
k22
Fig. 2-9: Anisotropic plate with boundary conditions
System and boundary conditions
Quantity Value Unit Description
k1 0.56 W / m*K Conductivity
k11 0.056 W / m*K Conductivity
k22 0.0056 W / m*K Conductivity
T1 100 C Temperature boundary condition
T2 0 C Temperature boundary condition
L1 1 m Length
L2 0.8 m Length
45 Degrees Angle of rotation
THESEUS-FE file Example_2_2.tfe
THESEUS-FE version 4.0
Problem description
A planar square with a tilted square insert is used to demonstrate the effect of anisotropicconductivity. Referring to Fig. 2-9, the outer square is an isotropic material withconductivity k1. Constant temperature boundary conditions are imposed on the verticaledges of the square while the horizontal edges are insulated. The inner material isorthotropic with k11 = k1 / 10 and k22 = k1 / 100. The orientation of the material axes withrespect to the global coordinate axes is 45 degrees. Fig. 2-10 shows the temperature field.The distortion of the results due to the anisotropy is clearly visualized. The dotted lines areisothermal lines from literature for validation[20].
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THESEUS-FE model
The problem was modeled with four groups ofPSHELL3 elements. Temperature boundary
conditions are assigned on the left and right vertical groups. Isotropic material propertiesare assigned to the outer square while anisotorpic material properties are assigned to thelocal coordinates in tensor form of the inner square.
Results
Isothermal lines from literature.
Fig. 2-10: Anisotropic plate THESEUS-FEresults compared with literature results
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2.3 Temperature Dependent Conductivity
System
System & boundary. cond.: THESEUS-FE model:
L1
adiabatic
L3
adiabaticL2
R1
adiab
ati
c
adiab
ati
c
Tout, , hout
k1= Constant
k2= k1+ C1xTemp.
adiabatic
Tout, , hout
Q
Q
System & boundary. cond.: THESEUS-FE model:
L1
adiabatic
L3
adiabaticL2
R1
adiab
ati
c
adiab
ati
c
Tout, , hout
k1= Constant
k2= k1+ C1xTemp.
adiabatic
Tout, , hout
Q
Q
Fig. 2-11: Variable heat conductivity model with boundary conditions
System and boundary conditions
Quantity Value Unit Description
k1 1 W / m*K Conductivity
C1 0.01 Constant
Tout 0 C Temperature boundary condition
hout 0.875 W / m2
*K Heat transfer Coefficient
L1 3 m Length
L2 0.8 m Length
L3 1 m Length
R1 0.5 m Radius
Q 100 W / m2
Heat Flux
THESEUS-FE file Example_2_3.tfe
THESEUS-FE version 4.0
Problem description
This example illustrates the difference in results when conductivity variations are includedin the model. Fig. 2-11 contains a schematic and mesh for a simple planar geometry. Thetop and bottom halfes are occupied by different isotropic materials; the bottom materialhas constant conductivity, k1, while the top material has a conductivity that varies withtemperature as k= k1+C1*T. A constant heat flux is applied along the left edge of the thedomain. The right side boundaries are insulated. All other surfaces are convectively cooled
with a constant heat-transfer-coefficient and temperature[20].
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THESEUS-FE model
The problem was modeled with two groups of solid elements of arbitrary thickness and
four groups ofPSHELL3 elements. Heat flux and heat convection boundary conditions are
assigned on the shell elements. Transient 2nd order solver was used, due to the non-
liniearity of the problem, with an intial time step of 0.1 seconds and a run time of 25seconds. Variable conductivity is assigned to the second group of solid elements via theTemperature Table shown below:
Results
Isothermal lines from literature.Isothermal lines from literature.
Fig. 2-12: THESEUS-FEresults compared with isotropic lines from literature
Temperature[C]
Conductivity[W/mK]
0 1
100 2
200 3
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2.4 Phase Change
System
System & boundary. cond.: THESEUS-FE model:
T0, k1TBC
Adiabatic
Adiabatic
L1
Ad
iab
atic
x
Fig. 2-13: Phase Change model with boundary conditions
System and boundary conditions
Quantity Value Unit Description
k1 1.08 W / m*K Conductivity
1 kg / m3
Density
L 70.26 J / kg Latent heat
Tl -0.15 C Liqidus temperature
T0 0 C Initial temperature
TBC -45 C Temperature boundary condition
L1 4 m Length
tfinal 2 s Simulation time
THESEUS-FE file phase_change.tfe
THESEUS-FE version 4.0
Problem Description
A standard test problem for phase change is the one-dimensional Stefan problem. Theproblem consists of a material region, with length 4 meters, held initially at a unifromtemperature, T0 = 0C, that is greater than the liquidus temperature Tl = -0.15C . At timezero, the left face of the region, x = 0, is lowered to a temperature below the solidustemperature, to TBC= -45C, causing a solidification front to propagate into the material. All
other surfaces are insulated. The schematic of the problem is shown in Fig. 2-13[20].
0
10
20
30
40
50
-50 -40 -30 -20 -10 0 10Temperature [C]
SpecificHeat[J/KgK]
11
10
33.63-0.15
33.63-2.15
1-2.3
1-45
Specific Heat
[J/kgK]
T [C]
Latent Heat
70.26 J/kg
Fig. 2-14: Table for assigning material property specific heat
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THESEUS-FE model
The example was solved using 64 PSHELL3 elements in the domain. Temperature
boundary conditions are applied on the left hand side and the material specific heatproperty, was applied via Fig. 2-14 shown above. Computation was carried out for a phase
change temperature interval of 2 degrees, with solidification staring at -0.15 degrees.
Results
Time=0.0s
Time=0.5s
Time=1.0s
Time=1.5s
Time=2.0s
Solid Liquid
x=1
Time=0.0s
Time=0.5s
Time=1.0s
Time=1.5s
Time=2.0s
Solid Liquid
x=1
Fig. 2-15: Phase change: THESEUS-FEresults
In Fig. 2-16 THESEUS-FE results are compared with the analytical solution. We see adiscrepancy of approxametaly 1 degree, which is attributed to the numerical calculation ofthe Latent Heat addition. In nature, phase change or Latent Heat addition occursinstantaneously. Numerically, the dirac impulse function can not be modeled, and therefore we apply the Latent Heat addtion through a 2 degree temperature band. Thisintroduces a modeling error that results in an approximately 1 degree solution error.
-15
-13
-11
-9
-7
-5
-3
-1
1
0 0.5 1 1.5 2
Time [s]
Temperature[C]
Analytic
THESEUS FE Results
Fig. 2-16: Phase change: Comparison of FE results with analytical solution at x = 1
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2.5 Cylinder Radiation (open grey body cavity)
System
System & boundary. cond.: THESEUS-FE model:
D
Adiabatic
Adiabatic
L
k,
q
T0
System & boundary. cond.: THESEUS-FE model:
D
Adiabatic
Adiabatic
L
k,
q
T0
Fig. 2-17: Cylinder Radiation example with boundary conditions
System and boundary conditions
Quantity Value Unit Description
0.3 Surface emissivity
k1 1.0 W / m*K Conductivity
T0 0 K Surrounding temperature
L 4 m Length
D 1 m Diameter
q 200 W / m2 Applied heat flux
THESEUS-FE file cylinder_radiation.tfe
THESEUS-FE version 4.0
Problem Description
A simple heated enclosure is the circular tube, open at both ends and insulated on theoutside surface. For a uniform heat addition q = 200 W/m2 to the inside surface of the tubewall and a surrounding environment at 0K, we calculate the stead state temperaturedistribution along the inside surface[18].
THESEUS-FE model
The open ends of the tube are nonreflecting, and they are assumed to be black bodies atthe surrounding temperature 0 K. In THESEUS-FE, three major types of radiation heatexchange are realized; for the current problem the model used was open-cavity with graybody radiation. Gray body radiation considers reflection and absorption of each elementand performs a full view factor matrix calculation. On the inner surface of the cylinder wall,the boundary conditions applied are the background temperature and heat flux. This is apure thermal radiation and conduction problem; the material properties assigned are theemission coefficient and the conductivity k.
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Results
In Fig. 2-18 the THESEUS-FE results displayed are the view factor sum of each element,the radiation from each element, and the nodal temperature values.
The view factor is a quantity used to define the fraction of thermal power leaving oneelement and reaching another. The sum of the view factor on each element thenrepresents the fraction of thermal power leaving one elment and reaching all other elmentsin the domain. For enclosed bodies the view factor sum of each element is their fore one.For open cavitites, the value will be less then one, depending on how much thermal poweris lost to the environment. The radiation view factor depends strongly on distance and fromthe results we can see that thermal power is mainly lost near the cylinder ends, while verylittle power is lost from elements toward the center of the cylinder. The maximum andminimum view factor sums are 0.975 and 0.6.
The second diagram quantifies how much radiation heat is lost from each element. For
elements near the cylinder ends, it is of magnitude -55 W/m2
.In the last diagram, the temperature results are displayed and compared with the analyticsolution. There is a close agreement between THESEUS-FE results and the analyticresults. We can further see that there is a 75 degree temperature difference between thecenter of the cylinder, where little heat is lost compared with elements at the ends of thecylinder where most of the thermal power is lost via radiation.
View Factor Sum
Radiation [W/m2]
View Factor Sum
Radiation [W/m2]
Fig. 2-18: Cylinder Radiation: THESEUS-FEresults
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2.6 Disk Radiation
System
System & boundary. cond.: THESEUS-FE model:
0 Ri Ro
AdiabaticAdiabatic
k,
T0TBC
a
System & boundary. cond.: THESEUS-FE model:
0 Ri Ro
AdiabaticAdiabatic
k,
T0TBC
a
Fig. 2-20: Disk
System and boundary conditions
Quantity Value Unit Description
0.7 Surface emissivity
k 15.0 W / (m K) Conductivity
T0 -273.15 C Initial temperature
TBC 100 C Boundary condition
a 0.01 m Thickness
Rl 0.04 m Inner radius
R0 0.24 m Outer radius
THESEUS-FE file disk_radiation.tfe
THESEUS-FE version 4.0
Problem Description
An annular fin in vacuum is insulated on one face and aournd its outside edge. The diskhas thickness of 0.01m, inner radius 0.04m, outer radius 0.24m, and thermal conductivity15.0 W/mK. Energy is supplied to the inner edge from a pipe that fits the central hole andmaintains the inner edge at 100 C. The exposed annular surface, which is diffuse-graywith emissivity 0.7, radiates to the environment at 0 K to investigate performance in a coldenvironment. Results are shown for the temperature distribution as a function of radialposition along the disk, the radiation flux from each element and the radiation efficiency ofthe disk calculated from THESEUS-FE is compared with results from literature[18]. In Fig.2-22, Qrad-total is the total heat radiation from the disk to the environment taken from theTHESEUS-FE simulation and Qrad-boundary= A(T
4BC - T
40) is the calculated theoretical
value, if the disk was held at a constant temperature of TBC. Where is the Stefan-Boltzmann constant and A is the disk surface area
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Radiation heat loss
[W/m2]
Convection heat loss
Radiation heat loss
[W/m2]
Convection heat loss
Fig. 2-26: Radiation Cylinder with 1D Flow: Field Results from Literature
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3.1 Model aThis model is composed of a shell element and an airzone, both are thermally connected
via convection. An airzone is a mixture of dry air and steam with a homogeneoustemperature distribution. Besides temperature the humidity is the second degree offreedom of the airzone.
1m
1m
1m
alu plate
airzone
Tairz, h
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3.1.3 Linear convective airzone cooling - considering humidity (a 3)
System
airzone: V=1m3
alu plate: 1m*1m*1mm
1 quad, PSHELL1, disc=2
conductivity: k = 238 W/mK
spec. heat: c = 945 J/kgK
density: = 2700 kg/m3
(pos. side)
z
x,y
airzone: V=1m3
alu plate: 1m*1m*1mm
1 quad, PSHELL1, disc=2
conductivity: k = 238 W/mK
spec. heat: c = 945 J/kgK
density: = 2700 kg/m3
(pos. side)
z
x,y
z
x,y
Fig. 3-5: Model a 3 system
System and boundary conditions
Quantity Value Unit Description
T0 20 C Initial temperature for airzone and plate
RLF0 68.6 % Initial relative humidity
hpos = hneg 6.0 W / m2*C Convection coefficient at positive/negative side of
the plate
Tamb,pos = Tairz Ambient temp. at pos. side = airzone temp.
Tamb,neg -20 C Ambient temperature at negative side
tend 500 s End time of simulation
Solver file model_a3.tfe
THESEUS-FE version 4.0
Results
0
0.002
0.004
0.006
0.008
0.01
0.012
0 100 200 300 400 500
time [sec]
[kg/kg]
THESEUS-FE
INKA
Fig. 3-6: Model a 3 - airzone absolute humidity
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3.1.5 Sun heating airzone - without considering humidity (a 5)
System
airzone: V=1m3
(pos. side)
z
x,y
sun
alti = 60
alu plate: 1m*1m*1mm
1 quad, PSHELL1, disc=2
conductivity: k = 238 W/mK
spec. heat: c = 945 J/kgK
density: = 2700 kg/m3
airzone: V=1m3
(pos. side)
z
x,y
z
x,y
sun
alti = 60
alu plate: 1m*1m*1mm
1 quad, PSHELL1, disc=2
conductivity: k = 238 W/mK
spec. heat: c = 945 J/kgK
density: = 2700 kg/m3
Fig. 3-10: Model a 5 system
System and boundary conditions
Quantity Value Unit Description
T0 20 C Initial temperature for airzone and plate
RLF0 0 % Initial relative humidity
vpos = vneg 0 m / s Ambient air velocity
at positive/negative side of the plate (BC-FC)
Tamb,pos = Tairz Ambient temp. at pos. side = airzone temp.
Tamb,neg 20 C Ambient temp. at neg. side
qdr 1000 W / m2
Direct sun intensity
tend 500 s End time of simulation
Solver file model_a5.tfe
THESEUS-FE version 4.0
Results
0
10
20
30
40
50
60
70
0 100 200 300 400 500time [sec]
T[C]
THESEUS-FE
INKA
Fig. 3-11: Model a 5 - airzone temperature
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3.1.6 Sun heating airzone - without considering humidity (a 6)
System
airzone: V=1m3
(pos. side)
z
x,y
z
x,y
sun
alti = 20 alu plate: 1m*1m*1mm1 quad, PSHELL1, disc=2
conductivity: k = 238 W/mK
spec. heat: c = 945 J/kgK
density: = 2700 kg/m3
Fig. 3-12: Model a 6 - system
System and boundary conditions
Quantity Value Unit Description
T0 20 C Initial temperature for airzone and plate
RLF0 0 % Initial relative humidity
vpos = vneg 0 m / s Ambient air velocity
at positive/negative side of the plate (BC-FC)
Tamb,pos = Tairz Ambient temp. at pos. side = airzone temp.
Tamb,neg 20 C Ambient temperature at negative sideqdr 1000 W / m
2Direct sun intensity
tend 500 s End time of simulation
Solver file model_a6.tfe
THESEUS-FE version 4.0
Results
0
5
10
15
20
25
30
35
40
0 100 200 300 400 500time [sec]
T[C]
THESEUS-FE
INKA
Fig. 3-13: Model a 6 - airzone temperature
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3.1.9 Airzone cooled by ventilation RVT (a 9)
System
airzone: V=1m3
(pos. side)
z
x,y
z
x,y
VENTILT - RVT
dV/dt = 0.001 m3/sec
T_out = 0C
RLF_out = 100%
X_out =0.00374 (saturated)
alu plate: 1m*1m*1mm
1 quad, PSHELL1, disc=2
conductivity: k = 238 W/mK
spec. heat: c = 945 J/kgK
density: = 2700 kg/m3
Fig. 3-19: Model a 9 system
System and boundary conditions
Quantity Value Unit Description
T0 20 C Initial temperature for airzone and plate
RLF0 68.6 % Initial relative humidity
vpos = vneg 0 m / s Ambient air velocity
at positive/negative side of the plate (BC-FC)
Tamb,pos = Tairz Ambient temp. at pos. side = airzone temp.
Tamb,neg 20 C Ambient temperature at negative sidetend 500 s End time of simulation
Solver file model_a9.tfe
THESEUS-FE version 4.0
Results
0.007
0.008
0.009
0.010
0 100 200 300 400 500
time [sec]
[kg/kg]
THESEUS-FE
INKA
Fig. 3-20: Model a 9 - airzone absolute humidity
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3.1.10 Airzone heated by ventilation RET (a 10)
System
airzone: V=1m3
(pos. side)