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Action on Structures Exposed to Fire
— Heat Transfer in Multiple Layers
Introduction
This is the third verification example from Ref. 1 which is part of the European Standard EN-1991-1-2:2010-12, Eurocode 1: Actions on structures - Part 1-2: General actions - Actions on structures exposed to fire. It computes heat transfer through layers with different material properties.
Model Definition
The modeled geometry is depicted in Figure 1. The total side length is 0.1 m.
Figure 1: Model geometry and setup. The size of the outer layer in this image is larger for better visibility.
The thickness of the steel layer is 0.5 mm.
The material properties for the filling are listed in Table 1
Table 1: Material properties.
Property
Value
Thermal conductivity
0.05 W/(m·K)
Density
50 kg/m3
Heat capacity
1000 J/(kg·K)
The material properties for steel are given in Ref. 2. The heat capacity is a highly nonlinear function of the temperature (Figure 2).
Figure 2: Temperature-dependent heat capacity.
The thermal conductivity depends linearly on the temperature according to
The model computes the heat transfer over 180 min starting from an initial temperature of 0°C. All outer boundaries are cooled by convective and radiative heat flux
with the heat transfer coefficient h = 10 W/(m²⋅K) and Text = 0°C.
Results and Discussion
The temperature distribution after 180 min is shown in Figure 3.
Figure 3: Temperature distribution after 180 min.
The reference and computed temperatures are compared in Figure 4.
Figure 4: Reference (blue) and calculated temperature (green) over time.
Table 2 lists the reference and calculated temperatures as well as the absolute and relative errors.
Table 2: Results.
Time (s)
Reference
temperature(°C)
Calculated
temperature(°C)
Absolute
error (K)
Relative
error (%)
30
340.5
340.6
0.1
0.03
60
717.1
723.0
5.9
0.8
90
881.6
885.7
4.1
0.5
120
950.6
952.9
2.3
0.2
150
979.3
980.6
1.3
0.1
180
991.7
992.0
0.3
0.03
To fulfill the norm the maximum deviation from the reference values must not exceed a relative error of 1% and an absolute error of 5 K. The norm is not fulfilled by means of absolute error in the range of temperature where the heat capacity is strongly nonlinear. The reference values are defined as the average temperature calculated by different software packages, whereas one value was out of range itself and hence the temperatures are too low in this area.
References
1. DIN EN 1991-1-2/NA National Annex - Nationally determined parameters - Eurocode 1: Actions on structures - Part 1-2: General actions - Actions on structures exposed to fire.
2. DIN EN 1993-1-2 Eurocode 3: Design of steel structures - Part 1-2: General rules - Structural fire design; German version EN 1993-1-2:2005 + AC:2009.
Application Library path: Heat_Transfer_Module/Verification_Examples/fire_effects_multiple_layers
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D.
2
In the Select Physics tree, select Heat Transfer > Heat Transfer in Solids (ht).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Time Dependent.
6
Click  Done.
Global Definitions
Create an interpolation function for the reference temperatures to compare the simulation results with these data.
Reference temperature
1
In the Home toolbar, click  Functions and choose Global > Interpolation.
2
In the Settings window for Interpolation, locate the Definition section.
3
From the Data source list, choose File.
4
Click  Browse.
5
Browse to the model’s Application Libraries folder and double-click the file fire_effects_multiple_layers_Tref.txt.
6
Click  Import.
7
In the Label text field, type Reference temperature.
8
Locate the Definition section. In the Function name text field, type Tref.
9
Locate the Units section. In the Function table, enter the following settings:
Function
Unit
Tref
degC
10
In the Argument table, enter the following settings:
Argument
Unit
t
min
Geometry 1
Square 1 (sq1)
1
In the Geometry toolbar, click  Square.
2
In the Settings window for Square, locate the Size section.
3
In the Side length text field, type 0.201.
4
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
0.0005
5
Click  Build All Objects.
6
Select the Layers to the left checkbox.
7
Select the Layers to the right checkbox.
8
Select the Layers on top checkbox.
9
Click  Build All Objects.
Materials
Steel
1
In the Model Builder window, under Component 1 (comp1) right-click Materials and choose Blank Material.
2
In the Settings window for Material, type Steel in the Label text field.
Heat Capacity
1
In the Model Builder window, expand the Component 1 (comp1) > Materials > Steel (mat1) node.
2
Right-click Component 1 (comp1) > Materials > Steel (mat1) > Basic (def) and choose Functions > Piecewise.
3
In the Settings window for Piecewise, type Heat Capacity in the Label text field.
4
In the Function name text field, type Cp.
5
Locate the Definition section. In the Argument text field, type T.
6
Find the Intervals subsection. In the table, enter the following settings:
Start
End
Function
20
600
425+7.73e-1*T-1.69e-3*T^2+2.22e-6*T^3
600
735
666+13002/(738-T)
735
900
545+17820/(T-731)
900
1200
650
7
Locate the Units section. In the Arguments text field, type degC.
8
In the Function text field, type J/(kg*K).
Plot the function and compare with Figure 2.
9
Click  Plot.
Thermal conductivity
1
In the Home toolbar, click  Functions and choose Global > Piecewise.
2
In the Settings window for Piecewise, type Thermal conductivity in the Label text field.
3
In the Function name text field, type k.
4
Locate the Definition section. In the Argument text field, type T.
5
Find the Intervals subsection. In the table, enter the following settings:
Start
End
Function
20
800
54-3.33e-2*T
800
1200
27.3
6
Locate the Units section. In the Arguments text field, type degC.
7
In the Function text field, type W/m/K.
Steel (mat1)
1
In the Model Builder window, under Component 1 (comp1) > Materials click Steel (mat1).
2
In the Settings window for Material, locate the Material Contents section.
3
In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Thermal conductivity
k_iso ; kii = k_iso, kij = 0
k(T)
W/(m·K)
Basic
Density
rho
7850
kg/m³
Basic
Heat capacity at constant pressure
Cp
Cp(T)
J/(kg·K)
Basic
Filling
1
In the Model Builder window, right-click Materials and choose Blank Material.
2
Select Domain 5 only.
3
In the Settings window for Material, type Filling in the Label text field.
4
Locate the Material Contents section. In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Thermal conductivity
k_iso ; kii = k_iso, kij = 0
0.05
W/(m·K)
Basic
Density
rho
50
kg/m³
Basic
Heat capacity at constant pressure
Cp
1000
J/(kg·K)
Basic
Definitions
Ambient Properties 1 (ampr1)
1
In the Physics toolbar, click  Shared Properties and choose Ambient Properties.
2
In the Settings window for Ambient Properties, locate the Ambient Conditions section.
3
In the Tamb text field, type 1000[degC].
Heat Transfer in Solids (ht)
Initial Values 1
1
In the Model Builder window, under Component 1 (comp1) > Heat Transfer in Solids (ht) click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values section.
3
In the T text field, type Tref(0).
Heat Flux 1
1
In the Physics toolbar, click  Boundaries and choose Heat Flux.
2
In the Settings window for Heat Flux, locate the Boundary Selection section.
3
From the Selection list, choose All boundaries.
4
Locate the Heat Flux section. From the Flux type list, choose Convective heat flux.
5
In the h text field, type 10.
6
From the Text list, choose Ambient temperature (ampr1).
7
Locate the Boundary Selection section. Click  Copy Selection.
Surface-to-Ambient Radiation 1
1
In the Physics toolbar, click  Boundaries and choose Surface-to-Ambient Radiation.
2
In the Settings window for Surface-to-Ambient Radiation, locate the Boundary Selection section.
3
Click  Paste Selection.
4
In the Paste Selection dialog, Press Shift+V on your keyboard.
5
Make sure all outer boundaries are in the Boundary Selection list.
6
click OK.
7
In the Settings window for Surface-to-Ambient Radiation, locate the Surface-to-Ambient Radiation section.
8
From the Tamb list, choose Ambient temperature (ampr1).
9
From the ε list, choose User defined. In the associated text field, type 0.8.
Mesh 1
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
From the Element size list, choose Extra fine.
4
Locate the Sequence Type section. From the list, choose User-controlled mesh.
Free Triangular 1
1
In the Model Builder window, under Component 1 (comp1) > Mesh 1 click Free Triangular 1.
2
In the Settings window for Free Triangular, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 5 only.
Mapped 1
In the Mesh toolbar, click  Mapped.
Size 1
1
Right-click Mapped 1 and choose Size.
2
In the Settings window for Size, locate the Element Size section.
3
Click the Custom button.
4
Locate the Element Size Parameters section.
5
Select the Maximum element size checkbox. In the associated text field, type 0.0002.
This way, you ensure to resolve the thin outer layer properly.
6
Click  Build All.
Study 1
Step 1: Time Dependent
1
In the Model Builder window, under Study 1 click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
From the Time unit list, choose min.
4
In the Output times text field, type 0 30 60 90 120 150 180.
Solution 1 (sol1)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 1 (sol1) node, then click Time-Dependent Solver 1.
3
In the Settings window for Time-Dependent Solver, click to expand the Time Stepping section.
4
From the Maximum step constraint list, choose Constant.
5
In the Maximum step text field, type 1.
This setting prevents the solver from taking too large time steps when the overall convergence is good. Otherwise, it could happen that the solver overestimates the time step size, which is required to resolve the transient behavior properly.
6
Click  Run.
The solver takes about 30 seconds to compute the solution.
Results
Change the unit of the temperature results to degrees Celsius.
Preferred Units 1
1
In the Results toolbar, click  Configurations and choose Preferred Units.
2
In the Settings window for Preferred Units, locate the Units section.
3
Click  Add Physical Quantity.
4
In the Physical Quantity dialog, select General > Temperature (K) in the tree.
5
Click OK.
6
In the Settings window for Preferred Units, locate the Units section.
7
In the table, enter the following settings:
Quantity
Unit
Preferred unit
Temperature
K
°C
8
Click  Apply.
Surface 1
1
In the Model Builder window, expand the Results > Temperature (ht) node, then click Surface 1.
2
In the Temperature (ht) toolbar, click  Plot.
Create a Cut Point dataset that is used to evaluate the temperature and compare it to the values in the norm.
Cut Point 2D 1
1
In the Results toolbar, click  Cut Point 2D.
2
In the Settings window for Cut Point 2D, locate the Point Data section.
3
In the X text field, type 0.1.
4
In the Y text field, type 0.1.
Point Evaluation 1
1
In the Results toolbar, click  Point Evaluation.
2
In the Settings window for Point Evaluation, locate the Data section.
3
From the Dataset list, choose Cut Point 2D 1.
4
Locate the Expressions section. In the table, enter the following settings:
Expression
Unit
Description
T
°C
Temperature
Tref(t)
°C
Reference temperature
5
Click  Evaluate.
Table 1
1
Go to the Table 1 window.
2
Click the Table Graph button in the window toolbar.
Results
Table Graph 1
Compare with Table 2.
1
In the Settings window for Table Graph, click to expand the Legends section.
2
Select the Show legends checkbox.
3
From the Legends list, choose Manual.
4
In the table, enter the following settings:
Legends
Temperature (°C)
Reference temperature (°C)
Temperature
1
In the Model Builder window, under Results click 1D Plot Group 2.
2
In the Settings window for 1D Plot Group, type Temperature in the Label text field.
3
Locate the Legend section. From the Position list, choose Lower right.
4
Locate the Plot Settings section.
5
Select the y-axis label checkbox. In the associated text field, type Temperature (°C).
6
In the Temperature toolbar, click  Plot.
Compare with Figure 4.
Finally, evaluate the absolute and relative errors.
Point Evaluation 2
1
In the Results toolbar, click  Point Evaluation.
2
In the Settings window for Point Evaluation, locate the Data section.
3
From the Time selection list, choose Manual.
4
In the Time indices (1-7) text field, type 2 3 4 5 6 7.
5
From the Dataset list, choose Cut Point 2D 1.
6
Locate the Expressions section. In the table, enter the following settings:
Expression
Unit
Description
abs(T-Tref(t))
K
Absolute error
abs(T-Tref(t))/(Tref(t)-273.15[K])
%
Relative error
7
Click  Evaluate.
Table 2
Go to the Table 2 window.
Results
1
In the Model Builder window, click Point Evaluation 2.
2
In the Settings window for Point Evaluation, clicknext to  Evaluate, then choose New Table.
Compare with Table 2.