Improved Conduction Transfer Function Coefficients Generation In ...

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Proceedings of BS2013: 13th Conference of International Building Performance Simulation Association, ChambΓ©ry, France, August 26-28

π‘žΜ‡

π‘žΜ‡

π‘žΜ‡

,

= Β 

𝑏

𝑇

,

Β 

βˆ’

𝑐

 𝑇

𝑑

Β π‘žΜ‡

,

Β 

βˆ’

,

Β 

π‘žΜ‡

,

=

π‘Ž

 𝑇

βˆ’

,

𝑏

𝑇

𝑑

Β π‘žΜ‡

,

Β 

βˆ’

βˆ‘

Β 

βˆ‘

Β 

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π‘Ž 𝑑

= Β 

βˆ‘

Β 

βˆ‘

Β 

𝑏 𝑑

= Β 

,

βˆ‘

Β 

𝑐

βˆ‘

Β 

𝑑

=π‘ˆ

Proceedings of BS2013: 13th Conference of International Building Performance Simulation Association, ChambΓ©ry, France, August 26-28

βˆ†π‘₯ = Β 

𝛼 Β π›₯𝑑𝑏 Β  Β  Β  Β  Β (π‘€β„Žπ‘’π‘Ÿπ‘’ Β πΉπ‘œ = 200) πΉπ‘œ

π‘žΜ‡

π‘žΜ‡

𝑇 𝑑 … 𝑇 𝑇 𝑇 = [𝐴] … + [𝐡] 𝑇 𝑑𝑑 𝑇 π‘žΜ‡ π‘žΜ‡

𝐢 πΉπ‘œ =

𝛼=

𝛼 Β π›₯𝑑𝑏 𝐿

𝑇 𝑇 = [𝐢] … + [𝐷] 𝑇 𝑇

𝑑𝑇 = π‘ˆ (𝑇 𝑑𝑑

βˆ’ 𝑇 ) + π‘ˆ (𝑇

π‘žΜ‡ = π‘ˆ(𝑇 βˆ’ 𝑇 )

π‘˜ 𝜌  𝐢

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βˆ’π‘‡)

Proceedings of BS2013: 13th Conference of International Building Performance Simulation Association, ChambΓ©ry, France, August 26-28

𝐴(𝑛π‘₯, 𝑛π‘₯) 𝐡(𝑛π‘₯, 𝑛𝑒) π‘§π‘’π‘Ÿπ‘œπ‘ (𝑛π‘₯, 𝑛𝑒) ⎑ ⎀ 𝐼(𝑛𝑒) βŽ₯ 𝑀 = Β  βŽ’π‘§π‘’π‘Ÿπ‘œπ‘ (𝑛𝑒, 𝑛π‘₯) π‘§π‘’π‘Ÿπ‘œπ‘ (𝑛𝑒, 𝑛𝑒) π›₯𝑑𝑏 ⎒ βŽ₯ βŽ£π‘§π‘’π‘Ÿπ‘œπ‘ (𝑛𝑒, 𝑛π‘₯) π‘§π‘’π‘Ÿπ‘œπ‘ (𝑛𝑒, 𝑛𝑒) π‘§π‘’π‘Ÿπ‘œπ‘ (𝑛𝑒, 𝑛𝑒)⎦

𝛷=𝑒

π›₯𝑑𝑏

𝐹1 =  𝛷(1: 𝑛π‘₯, 𝑛π‘₯ + 1: 𝑛π‘₯ + 𝑛𝑒) 𝐹2 =  𝛷(1: 𝑛π‘₯, 𝑛π‘₯ + 𝑛𝑒 + 1: 𝑛π‘₯ + 2  𝑛𝑒) 𝐴 =  𝛷(1: 𝑛π‘₯, 1: 𝑛π‘₯) 𝐢 = 𝜌 Γ— 𝐢 Γ— 𝐿 = 2200 Γ— 840 Γ— 0.3 𝐽 = 554400 π‘šΒ²πΎ

𝐡 = 𝐹1 + 𝐴 βˆ— 𝐹2 βˆ’ 𝐹2 𝐢 =𝐢

554400 𝐽 = 138600 Β  4 π‘š 𝐾 554400 𝐽 𝐢2 = = 277200 2 π‘š 𝐾 1 π‘Š π‘ˆ1 = = β„Ž = 8.3 𝑅1 π‘š 𝐾 1 π‘˜ 1.7 π‘Š π‘ˆ2 = π‘ˆ3 = = = = 11.33 𝑅2 𝐿 0.15 π‘š 𝐾 2 1 π‘Š π‘ˆ4 = = β„Ž = 34.5 𝑅4 π‘š 𝐾

𝐷 = 𝐷 + 𝐢 βˆ— 𝐹2

𝐢1 = 𝐢3 =

𝑑𝑇 π‘ˆ1 π‘ˆ2 = Γ— (𝑇 βˆ’ 𝑇 ) + Γ— (𝑇 βˆ’ 𝑇 ) 𝑑𝑑 𝐢1 𝐢1 8.3 = Γ— (𝑇 βˆ’ 𝑇 ) 138600 11.33 + Γ— (𝑇 βˆ’ 𝑇 ) 138600

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Proceedings of BS2013: 13th Conference of International Building Performance Simulation Association, ChambΓ©ry, France, August 26-28

𝑑𝑇 π‘ˆ2 π‘ˆ3 = Γ— (𝑇 βˆ’ 𝑇 ) + Γ— (𝑇 βˆ’ 𝑇 ) 𝑑𝑑 𝐢2 𝐢2 11.33 = Γ— (𝑇 βˆ’ 𝑇 ) 277200 11.33 + Γ— (𝑇 βˆ’ 𝑇 ) 277200

𝐷 =  

7.537 βˆ’0.009171

βˆ’0.009171 23.68

𝑑𝑇 π‘ˆ3 π‘ˆ4 = Γ— (𝑇 βˆ’ 𝑇 ) + Γ— (𝑇 βˆ’ 𝑇 ) 𝑑𝑑 𝐢3 𝐢3 11.33 = Γ— (𝑇 βˆ’ 𝑇 ) 138600 34.5 + Γ— (𝑇 βˆ’ 𝑇 ) 138600

𝑑 𝑇 𝑇 𝑑𝑑 𝑇 11.33 138600 22.66 βˆ’ 277200 11.33 138600

19.63 βŽ‘βˆ’ 138600 ⎒ 11.33 =⎒ 277200 ⎒ ⎒ 0 ⎣ 8.3 ⎑ 0 ⎀ ⎒138600 βŽ₯ 𝑇 +⎒ 0 0 βŽ₯ Β  𝑇 34.5 βŽ₯ ⎒ 0 ⎣ 138600⎦

⎀ βŽ₯ 𝑇 11.33 βŽ₯ Β  𝑇 277200 βŽ₯ 𝑇 45.83 βŽ₯ βˆ’ 138600⎦ 0

π‘žΜ‡ = β„Ž (𝑇 βˆ’ 𝑇 ) = 8.3 Β (𝑇 βˆ’ 𝑇 ) π‘žΜ‡

= β„Ž (𝑇 βˆ’ 𝑇 ) = 34.5 Β (𝑇 βˆ’ 𝑇 ) Β  Β 

q̇ q̇

=

βˆ’8.3 Β  0 +

0 0

8.3 0

If TIMEB < 0 then useSsCtfMethod = .TRUE. TIMEB = -TIMEB Else useSsCtfMethod = .FALSE.

T 0 T βˆ’34.5 T T 0 34.5 T

Subroutine WALLS If useSsCtfMethod = .TRUE. Subroutine NodesDetermination + Subroutine CtfCoeffGen

0.6147 0.1999 0.01142 𝐴 = Β  0.09994 0.7725 0.07353 0.01142 0.1471 0.3133 0.1355 0.01029 𝐡 = 0.02049 0.06137 0.002475 0.3142 βˆ’8.3 0 0 𝐢 = 0 0 βˆ’34.5

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If useSsCtfMethod = .FALSE.

Subroutine TRANS

Proceedings of BS2013: 13th Conference of International Building Performance Simulation Association, ChambΓ©ry, France, August 26-28

23

22

Original - 1 h Modified - 12 min Reference

Temperature [Β°C]

21

20

19

18

17

16 4000

19 18.8

4005

4010

4015 Time [h]

4020

4025

4030

4019.5 Time [h]

4020

4020.5

4021

Original - 1 h Modified - 12 min Reference

18.6

Temperature [Β°C]

18.4 18.2 18 17.8 17.6 17.4 17.2 17 4018

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4018.5

4019

Proceedings of BS2013: 13th Conference of International Building Performance Simulation Association, ChambΓ©ry, France, August 26-28

21

Original - 3 h Modified - 1 h Reference

20.5

Temperature [Β°C]

20

19.5

19

18.5

18

17.5 4000

4005

4010

4015 Time [h]

4020

4025

4030

18.1

18

Temperature [Β°C]

17.9

17.8

17.7

17.6

17.5 4015.5

ICF DST SIPS Concrete Insulation Wood

Original - 3 h Modified - 1 h Reference 4016

4016.5 4017 Time [h]

4017.5

4018

Original timebase New timebase Number of nodes Calculation time (DRF method) (SS method) (SS method) (SS method) [h] [h] [-] [s] 1 0.2 255 2 0.25 0.1 323 3.4 0.15 0.05 350 3.7 0.25 0.2 268 2.2 0.5 0.1 333 3.4 3 1 347 3.5

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Proceedings of BS2013: 13th Conference of International Building Performance Simulation Association, ChambΓ©ry, France, August 26-28

22 21.5

π‘žΜ‡

DRF method (Timebase = 5h) SS method (Timebase = 1.5h)

21

Temperature [Β°C]

20.5 20 19.5

π›₯𝑑

19 18.5 18 17.5 17 4000

4005

4010

4015 Time [h]

4020

4025

4030

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Proceedings of BS2013: 13th Conference of International Building Performance Simulation Association, ChambΓ©ry, France, August 26-28

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