How to simulate a planting pot in winter

[Andrea Heil]

Hello,

I want to do a transient thermal simulation of wet soil with phase change in a planting pot for my master's thesis. I want to analyse when the temperature in the pot falls below -5 °C which is critical for the roots during winter. I also want to test how much difference it makes to wrap the pot in different insulation material.
It would be crucial that the software considers the energy which is needed to freeze the water (latent heat) and the changing heat capacity and thermal conductivity when there is ice instead of water in the soil. And as I have a cylindric pot, I would like to use rotation symmetry. Furthermore, wet soils do not freeze immediately but have a temperature difference between solidus- and liquidustemperatur of about 2 Kelvin, so I need to be able to define solidus- and liquidustemperature for the material. Can WUFI 2D do that and how?

You find attached a picture of the principal setup.

For validation issues, I want to proof the simulation with an experiment by putting a pot into a refrigerator. Is it possible to set an initialisation temperature (+20 °C for the whole pot) and a constant outdoor temperature as boundary condition (-18 degrees) to be able to compare simulation with experiment? For me it looks like if it is only possible to let the calculation run with a whole year's climate data.
For the comparison I need to visualize the temperature over time for single points in the pot, where I have my temperature sensors.

I would be very grateful for your help.

Thank you!

Andrea

[Thomas]

Hello,

I want to do a transient thermal simulation of wet soil with phase change in a planting pot for my master's thesis. I want to analyse when the temperature in the pot falls below -5 °C which is critical for the roots during winter. I also want to test how much difference it makes to wrap the pot in different insulation material.
It would be crucial that the software considers the energy which is needed to freeze the water (latent heat) and the changing heat capacity and thermal conductivity when there is ice instead of water in the soil. And as I have a cylindric pot, I would like to use rotation symmetry. Furthermore, wet soils do not freeze immediately but have a temperature difference between solidus- and liquidustemperatur of about 2 Kelvin, so I need to be able to define solidus- and liquidustemperature for the material. Can WUFI 2D do that and how?

Hi Andrea,

WUFI takes these things into account in principle, but in a simplified way which is sufficient for the majority of applications in building physics.

The release of latent heat upon freezing is taken into account (with the numerical value 333 kJ/kg).

The fact that the thermal capacity and the thermal conductivity of ice are different from those of water is not taken into account. (And I think that the thermal conductivity of a mixture of ice and liquid water in the freezing pores probably depends on the details of the interconnection structure of the pores, so you would probably need empirical thermal conductivities for different types of porous materials...)

WUFI can do calculations with rotation symmetry.

The fact that for temperatures below freezing some temperature-dependent percentage of the pore water remains liquid is taken into account, but without adjustable parameters. See the description in section 2.3.6 of Dr. Künzel's thesis (en, de). It is based on the description by Neiß (Neiß J.: Numerische Simulation des Wärme- und Feuchtetransports und der Eisbildung in Böden, VDI 1982) who in turn references calorimetric measurements on freezing soils. You'll have to see for yourself whether WUFI's freezing model is close enough to what you need.

For validation issues, I want to proof the simulation with an experiment by putting a pot into a refrigerator. Is it possible to set an initialisation temperature (+20 °C for the whole pot) and a constant outdoor temperature as boundary condition (-18 degrees) to be able to compare simulation with experiment? For me it looks like if it is only possible to let the calculation run with a whole year's climate data.

This would be extremely simple in WUFI:

Set the initial temperature of the simulated "component" to +20°C.

Choose "sine curves" for the climatic conditions, select "user-defined" curves, set the amplitude of the temperature sine curve to zero and the mean value to -18°C. This will apply constant temperature conditions to the "component" surface. Then run the simulation.

No file with climate data is needed in this case (but could be used in case you want to use one).

For the comparison I need to visualize the temperature over time for single points in the pot, where I have my temperature sensors.

After the calculation you can read out temperature curves for each grid cell of the computational grid.

Regards,
Thomas

[Andrea Heil]

Hello Thomas,

thank you very much!

The release of latent heat upon freezing is taken into account (with the numerical value 333 kJ/kg).

The fact that the thermal capacity and the thermal conductivity of ice are different from those of water is not taken into account. (And I think that the thermal conductivity of a mixture of ice and liquid water in the freezing pores probably depends on the details of the interconnection structure of the pores, so you would probably need empirical thermal conductivities for different types of porous materials...)

I would like to model the material in a simplified way with empirical values for the whole mixture soil-water-air. In WUFI it would be like a homogenous material. The latent heat would be taken into account according to the amount of water in the soil - see the formula attached. For this,I would need the possibility to define my own value for latent heat for a user-defined material.
The density, heat capacity and thermal conductivity would be modeled with empirical values like in the diagrams attached. In the literature I found -2°C as solidus temperature and 0°C as liquidus temperature because of the effects you mentioned.

Can I model the material like this in WUFI?

Best,
Andrea

[Andrea Heil]

somehow I can only attach 3 picures, so here is the 4th

[Thomas]

I would like to model the material in a simplified way with empirical values for the whole mixture soil-water-air. In WUFI it would be like a homogenous material. The latent heat would be taken into account according to the amount of water in the soil - see the formula attached. For this,I would need the possibility to define my own value for latent heat for a user-defined material.

So you are not interested in any redistribution or evaporation of the water? If it's a purely thermal problem with effective thermal parameters for the "mixture" of soil and water, I think it should be possible. The key is WUFI's ability to use tabulated values for the specific enthalpy of the material, so that any desired heat capacity and latent heat can be modeled. See the "Gypsum Board; PCM" in the catalog "Wooden Materials; Boards" in the material database for an example.

(1)
The heat capacity at constant pressure \(c_p\) is the derivative of the specific enthalpy \(h\) of the material with respect to temperature:

\(c_p = \left( \frac{\partial h}{\partial T} \right)_{p = const}\)

So the heat capacity \(c_p\) at any temperature is simply the slope of the tabulated enthalpy curve at this temperature, and by tabulating an appropriate curve you can describe any desired temperature-dependent heat capacity by choosing the corresponding slope at each temperature.

(2)
If we can neglect the occurrence of (absorbed or released) heat of absorption when water is absorbed in the soil, the total enthalpy of the pot is simply the sum of the enthalpies of the soil and the contained water:

\(H = H_s + H_w\)

The specific enthalpy of the pot, which is needed for the tabulation, is the total enthalpy divided by the total mass:

\(h = \frac{H_s + H_w}{m_s + m_w} = \frac{h_s \ m_s + h_w \ m_w}{m_s + m_w} = \frac{h_s \ \rho + h_w \ w}{\rho + w}\)

where \(\rho\) and \(w\) are the densities of the soil and the contained water, respectively (\(w\) is the current water content, not the density of bulk water). The specific enthalpies of the soil \(h_s\) and the water \(h_w\) can be computed by integrating over their known heat capacities. This allows you to compute the enthalpy curve for any given water content \(w\).

(3)
The specific latent heat of fusion is simply the difference of the specific enthalpies of the ice and the water at the ice point. It corresponds to a step with the appropriate height in the enthalpy curve. If you wish to take into account that in the soil freezing occurs over a certain temperature interval, you can tabulate a correspondingly "gradual" step. You can even adjust the height of the step to reflect any desired numerical value of the latent heat, in case the latent heat of fusion in the soil is different from the value in bulk water.

The latent heat is a contribution to the specific enthalpy of the water \(h_w\) (see above).

(4)
WUFI can use a temperature-dependent thermal conductivity \(\lambda\). If you have measured values for the thermal conductivity of the "mixed material" soil + water, you can simply tabulate them.

(5)
You should switch off "moisture transport" in WUFI's numerical settings and set the initial water content to zero because the effect of the water has already been taken into account in the effective material parameters of the "mixed material". Inclusion of any water in the calculation would account twice for the water.

Regards,
Thomas