Periodic soil temperature fluctuations

1 Problem Specification In this study we want to find out how the soil temperature is influenced by the daily and yearly periodic air temperatures and solar radiation.

The goal of this study is to get an overview of periodic temperature profiles in the bottom with respect to the time and depth, for a typical Dutch climate.

Such information can be useful e.g. for dimensioning of vertical and horizontal soil collectors, which are used for heating/cooling of buildings.

In this study we limit the energy balance of the Earth surface to only sensible heat flux caused by solar radiation and outside air temperature. In the Theoretical Background section the whole energy balance of the Earth surface is given.

The energy transport equation in a one-dimensional setting is:

+

=

2

2+

Our problem can be approximated as a 1D transient diffusion problem with a source that represents the periodic solar radiation. So the governing equation can be reduced to:

=

2

2+

2

2 Theoretical background Soil characteristics and energy flows

The mean annual energy balance of the Earth is visualized in Fig. 1:

Fig. 1 The mean annual radiant energy and heat balance of the Earth. From Houghton et al., (1996:58), which used data from Kiehl and Trenberth (1996) [1].

As we mentioned before, this energy balance is more complex than we will include in our study. The energy flows are constrained by only sensible heat flows, caused by solar radiation and temperature differences between earth surface and outdoor air temperature.

The energy flows in the bottom are strongly dependent on the soil type and humidity. Fig. 2 shows us indicative thermal characteristics of typical soils. Minimum, maximum and typical values are given for these species, dependent on moisture storage.

Fig. 2 Indicative thermal characteristics of different soil textures (Novem., 2003).

3

Also the radiation reflection characteristics of different surfaces are greatly varying:

Fresh snow or ice: 80-95% Old melting snow: 40-70% Clouds: 40-90% Desert sand: 30-50% Earth ground (NL: grondaarde): 5-30% Tundra: 15-35% Grassland: 25-30% Forest: 10-20% Water: 10-60%

Soil temperature tool of Dr. D.L. Nofziger and Dr. J. Wu from

Dr. D.L. Nofziger and Dr. J. Wu from the Department of Plant and Soil Sciences on the Oklahoma State University have developed a free software tool, which uses a simple model to calculate the average soil temperatures and displays them at specific dates or soil depths. These calculations are based on the following input parameters:

Maximum and minimum surface temperatures Thermal Diffusivity Date Depth

This tool and background information can be found on the web: http://soilphysics.okstate.edu/software/SoilTemperature

3 Method In this study we are interested in the periodic temperature distribution into the soil. We will use the simulation software program COMSOL Multiphysics v3.5 to get this information. We use a model of three different soil types based on the indicative thermal characters out of the theoretical background section. Two soil types with extreme thermal diffusivities and one with a representative Dutch thermal diffusivity. These models are exposed to the mean Dutch air temperature and solar radiation with yearly and daily sinusoidal periods. After the simulation we create temperature plots at different depths, which gives us information such as penetration depths and time shifts.

We also use the tool of D.L. Nofziger and Dr. J. Wu to make some soil temperature plots.

4

4 The model Geometry and mesh properties The simulation model consists of six rectangles which each propose the upper 15 meter of the ground. For three different soil types we use one rectangle to simulate both yearly and daily periodic fluctuations and one with only daily periodic temperature variations. The other properties of the model are listed below.

Dimensions [m]:

Six rectangles (WxH) = 0,5x15 [m]

Mesh settings:

The mesh of the 6 rectangles together consists of 1836 elements.

Below the geometry and the mesh statistics of the model are shown:

Fig. 3 Left: Mesh geometry; right: Mesh statistics

5

Material Properties and Initial Condition

Initial temperature conditions = 11 [C]

Thermal conductivity k:

Soil with high thermal diffusivity = 5,2 [W/m.K]

Soil with low thermal diffusivity = 0,27 [W/m.K]

Dutch soil thermal diffusivity = 1,9 [W/m.K]

Density :

Soil with high thermal diffusivity = 1750 [kg/m3]

Soil with low thermal diffusivity = 1600 [kg/m3]

Dutch soil thermal diffusivity = 1700 [kg/m3]

Heat capacity c:

Soil with high thermal diffusivity = 1200 [J/kg.K]

Soil with low thermal diffusivity = 1000 [J/kg.K]

Dutch soil thermal diffusivity = 1200 [J/kg.K]

Thermal diffusivity:

Soil with high thermal diffusivity = 0,2065 [m2/day]

Soil with low thermal diffusivity = 0,0146 [m2/day]

Dutch soil thermal diffusivity = 0,0805 [m2/day]

6

Fig. 4 shows the simulation models with colors representing the thermal conductivity of each type of soil. Each of the three soil types was simulated with both yearly and daily temperature and solar periods (left side) and once with only the daily periodic conditions (right side).

Fig. 4 Visualization of the models, the colors represent the thermal conductivity of the 3 different soil types Boundary Conditions

Surface heat transfer coefficients = 25 W/m2.K (top boundaries)

Thermal insulation = left, right and bottom boundaries

On the top boundaries we implement the average transient climatologically quantities for in De Bilt (NL), which are represented by formulas. These formulas are given below.

7

Variable Air temperature on top boundaries:

- Simulation with both daily and yearly temperature variations (see Fig. 5):

9.41 7.35 cos 2(3600533)360024365

2.6 1.5 cos 2(360041)360024365

cos 2(36001.75)360024

[C]

- Simulation with only daily temperature variations (see Fig. 6):

9.41 2.6 1.5 cos 2(360041)360024365

cos 2(36001.75)360024

[C]

Fig. 5 Applied external temperature during a year with daily periodic variations included.

Fig. 6 Visualization of the splitted formula into yearly and daily periodic temperature curves

8

Variable Solar heat flux on top boundaries:

- Simulation with both daily and yearly temperature variations (see Fig. 7):

111.5 97.5 cos 2(3600222)360024365

166 127.4 cos 2(3600219)360024365

cos 2(360023.66)360024

[W/m2]

with:

a = 0,8 absorption coefficient.

- Simulation with only daily temperature variations (see Fig. 8):

111.5 166 127.4 cos 2(3600219)360024365

cos 2(360023.66)360024

[W/m2]

with:

a = 0,8 absorption coefficient.

Fig. 7 Applied solar radiation as incoming heat flux during a year with daily periodic variations included.

Fig. 8 Visualization of the partitioned formula into yearly and daily periodic solar radiation curves

9

Solver Settings

Type of analysis: Transient Heat Transfer by conduction

Linear system solver: Direct (UMFPACK)

Matrix symmetry: Automatic

Pivot threshold: 0,1

Memory allocation factor: 0,7

Time stepping: range(0,3600,5.5*365*24*3600) (5 year)

Relative tolerance: 0.001

Absolute tolerance: 0.0001

Times to store output: Specified times

Time steps taken by solver: Free

Maximum BDF order: 5

Minimum BDF order: 1

Singular mass matrix: Maybe

Consistent initialization of DAE systems: Backward Euler

Error estimation strategy: Include algebraic

10

5 Results

First we show plots of the whole 15 meter deep soil structure with its temperature distribution lines during the fifth simulation year. It represents both yearly and daily periodic variations. Y-coordinate: 0 represents the surface of the soil.

Fig. 9, Fig. 10 and Fig. 11 represent the soil with respectively high thermal diffusivity, low thermal diffusivity and a soil with common Dutch thermal diffusivity characteristics:

Fig. 9 Yearly temperature distribution lines through a soil with high thermal diffusivity

Soil with high thermal diffusivity:

a=0,2065 m2/day

11

Fig. 10 Yearly temperature distribution lines through a soil with low thermal diffusivity

Fig. 11 Yearly temperature distribution lines through a soil with common Dutch thermal diffusivity characteristics

Soil with low thermal diffusivity:

a=0,0146 m2/day

Soil with common Dutch thermal

diffusivity:

a=0,0805 m2/day

12

The following plots show us the temperature at different depths in the soil during the last two years of simulation time. Near the surface both daily and yearly periodic variations are visible. The plots are figured in the same soil type order: high, low and common Dutch thermal diffusivity.

Fig. 12 Two year temperature plot at different depths in a soil with high thermal diffusivity.

Soil with high thermal diffusivity:

a=0,2065 m2/day

13

Fig. 13 Two year temperature plot at different depths in a soil with low thermal diffusivity.

Fig. 14 Two year temperature plot at different depths in a soil with common Dutch thermal diffusivity.

Soil with low thermal diffusivity:

a=0,0146 m2/day

Soil with common Dutch thermal

diffusivity:

a=0,0805 m2/day

14

The daily periodic variations are visualized in the next temperature plots. These represent an average cycle of two days namely 8 and 9 July. The mean solar radiation on these days is shown in Fig. 16 and the outside air temperature in Fig. 16.

Fig. 16 Mean surface solar irradiation at 8 and 9 July

The influence of these daily variations on the three soil types is visualized in the following temperature plots at different depths just below the surfaces. The yearly periodic influence is still visible because the mean temperature at each point is decreasing with the depth:

Fig. 17 Two day (8-9 July) temperature plot at different depths in a soil with high thermal diffusivity.

Soil with high thermal diffusivity

a=0,2065 m2/day

Fig. 15 Mean outside air temperature at 8 and 9 July

15

Fig. 18 Two day (8-9 July) temperature plot at different depths in a low thermal diffusivity.

Fig. 19 Two day (8-9 July) temperature plot at different depths in a soil with common Dutch thermal diffusivity.

Soil with low thermal diffusivity

a=0,0146 m2/day

Soil with common Dutch thermal

diffusivity

a=0,0805 m2/day

16

The following plots are produced with the tool of D.L. Nofziger and Dr. J. Wu. The tool shows a plot of the yearly periodic temperature profile at a certain dept (left corner) and the temperature distribution through the whole soil structure at a certain day (right corner).

Temperature plots of the soil structure with high thermal diffusivity a=0, 2065:

Fig. 20 Plots of the temperature distribution in a soil with high thermal diffusivity

Temperature plots of the soil structure with low thermal diffusivity a=0,0146:

Fig. 21 Plots of the temperature distribution in a soil with low thermal diffusivity

17

Temperature plots of the soil structure with a common Dutch thermal diffusivity a=0, 0805:

Fig. 22 Plots of the temperature distribution in a soil with a common Dutch thermal diffusivity

18

6 Conclusion/Discussion

The tool of D.L. Nofziger and Dr. J. Wu is very easy to use with limitary input parameters as maximum-, minimum surface temperatures and thermal diffusivity of the soil. Besides, it is a free available for everyone. Comsol Multiphysics is a wider applicable powerful software tool with many more options concerning input parameters and result visualizations.

Analyzing the results we can conclude that the penetration depth of periodic temperature profiles varies widely between the soil types with extreme high and low thermal diffusivity. That is because the thermal diffusivity is about 14 times higher between these soiltypes. Including by the moisture content, the thermal conductivity in a common Dutch soil is quite high and therefore highly suitable for e.g. ground heat exchange systems. But is also means that the periodic temperature variations penetrates deeper into the ground.

The mean temperature and thus nearly stable temperature at a few meters below the surface is circa 10-11C in Dutch soil. Thats why we used 11C as initial temperature in the Comsol simulations. But the simulated temperature plots show us that the mean temperature is a couple degrees higher than 11C. Likely this is because we do not include all the natural energy flows in our model, such as the evaporation of latent heat.

Bibliography

1. milieu., N. N. (2003). Kwaliteitsrichtlijn Verticale Bodemwarmtewisselaars.

2. M.h. de Wit, [2009]: Heat, air and moisture in building envelopes. Course book Eindhoven University of Technology: blz 63

3. http://oceanworld.tamu.edu/resources/oceanography-book/radiationbalance.htm

1 Problem Specification2 Theoretical background3 Method4 The modelGeometry and mesh propertiesMaterial Properties and Initial ConditionBoundary ConditionsSolver Settings

5 Results6 Conclusion/DiscussionBibliography

/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False

/CreateJDFFile false /Description > /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ > /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles false /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /DocumentCMYK /PreserveEditing true /UntaggedCMYKHandling /LeaveUntagged /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ]>> setdistillerparams> setpagedevice