Sizing with a thermal response test (TRT)

Thermal response tests (or TRTs) are measurements that can be used to fine-tune your geothermal borefield simulation to achieve the most accurate results. In this article, we’ll discuss what a TRT is and how it can be used to determine the undisturbed ground temperature, ground thermal conductivity, and effective borehole thermal resistance.

What is a thermal response test?

When it comes to the design of borefields, there are a few very important parameters: on the one hand, the ground thermal conductivity and undisturbed ground temperature, and on the other hand, the effective borehole thermal resistance. Although both can be estimated, there is always a difference between theory and practice.

For large projects, or when no accurate data is available, it is recommended to measure these parameters in situ.

Description of the test

A TRT is carried out on a sample borehole—a borehole at the specific project location that is representative of how the borefield will later be installed (same grout, same heat exchanger, same borehole depth). Once this borehole is installed, the grout is left for a few days to harden, allowing the borehole to return to thermal equilibrium with the surrounding ground.

After that, a thermal response test rig is brought to the site and connected to the borehole. This TRT rig includes a data logger to measure both inlet and outlet fluid temperatures, a circulation pump, and an electric heater. An illustration of the test site is shown in the figure below.

!Note
If you have not read our articles on the ground properties and the borehole effective thermal resistance, it may be helpful to check them out in order to fully understand this article.

Based on these temperature measurements, the TRT can determine the initial, undisturbed ground temperature (which is simply the original temperature of the fluid circulating through the test rig before the heating element is switched on), the ground thermal conductivity, and the effective borehole thermal resistance.

There are many different ways to estimate these parameters from the measurements, but in this article, we will focus on the most commonly used approach: the line source method.

!Note
There are different test setups for a thermal response test. The one described above is the most commonly used, with a constant flow rate and heat injection. Other setups exist, such as those with a constant inlet temperature and flow rate, or with both a constant inlet and outlet temperature. Each of these has its own advantages and disadvantages, but regardless of the setup, the type of results obtained is always the same.

Line source method

With the line source method, we approximate the geometry of the borehole as a line—meaning that the diameter is much smaller than the length. By doing so, we can theoretically express how the average temperature will change over time using the following equation:

$$\bar{T_f}(t)=\frac{Q}{H}\cdot\frac{1}{4\pi\lambda}\cdot ln(t)+\frac{Q}{H}\cdot\frac{1}{4\pi\lambda}\cdot\left[ ln \left(\frac{4\alpha}{r_0^2}\right)-y\right]+\frac{Q}{H}\cdot R_b^*+T_0$$

The different parameters of this equation are:

• $\bar{T_f}(t)$: the mean average fluid temperature [°C]
• $Q$: the injected power during the test [W]
• $H$: the length of the borehole [m]
• $\lambda$: the ground thermal conductivity [W/(mK)]
• $\alpha$: the ground thermal diffusivity [m²/s]
• $r_0$: the borehole radius [m]
• $y$: the constant of Euler (=0,5772)
• $R_b^*$: the effective borehole thermal resistance [mK/W]
• $T_0$: the undisturbed ground temperature

Although this equation might seem complicated at first, most of the parameters are constant or known in advance. The equation above can therefore be simplified as:

$$\bar{T_f}(t)-T_0=k\cdot ln(t)+m$$

The equation below shows a linear relationship between the difference in the average fluid temperature and the undisturbed ground temperature, and the logarithm of time. This is illustrated graphically in the figure below.

The figure on the left shows how the temperature is typically measured. After a sharp increase in temperature due to heat injection, the rate of increase in fluid temperature slows down, following a logarithmic behaviour. (This behaviour is similar to the curve of the g-function, which is discussed in this article). By changing the x-axis from a linear scale (with equal spacing between ticks) to a logarithmic scale (where each tick represents a multiple of 10), we can observe that the shape of the average fluid temperature curve changes.

After an initial steady period, the mean fluid temperature begins to rise, and after around 10 hours, it increases more or less linearly. This is the linear behaviour described in the formula above, which only becomes visible in this so-called semi-log plot. Based on the data points from hour 10 to 60, a logarithmic approximation can be drawn. The slope of this line determines the k-factor in the formula above, and the intersection of this line with the y-axis determines the m-factor. From the following two equations, the ground thermal conductivity and effective borehole thermal resistance can be derived.

$$\lambda = \frac{Q}{4\pi H k}$$ and $$R_b^*=\frac{H}{Q}\cdot(\bar{T_f}(t)-T_0)-\frac{1}{4\pi \lambda}\cdot \left[ ln(t)+ln \left(\frac{4\alpha}{r_0^2}\right)-0.5772\right]$$

!Note
There is one parameter, $\alpha$ (the thermal diffusivity), which is strictly speaking also unknown, as it depends on both the thermal conductivity and the volumetric heat capacity of the ground. This volumetric heat capacity can be estimated based on literature for the geological conditions of your project and typically has a smaller influence than the other parameters. If the effective borehole thermal resistance is well known, the equation above could instead be rearranged to determine the value of $\alpha$.

Example with GHEtool Cloud

The results from a TRT can be used to simulate your borefield with greater accuracy. All the measured parameters can be entered directly into the software. Imagine, for example, that we have the following measurement results:

• Undisturbed ground temperature: 11,78°C
• Ground thermal conductivity: 2,32 W/(mK)
• Effective borehole thermal resistance: 0,103 mk/W

The first two parameters can be entered under the ‘Ground’ tab like in the figure below.

!Note
Since the TRT is a measurement of the entire borehole, the parameters entered represent those of an equivalent homogeneous ground. The ground temperature should also be set to “measured” rather than “custom”, as no thermal gradient needs to be taken into account—the average temperature has already been measured.

The effective borehole thermal resistance can be entered in the ‘Borehole resistance’ tab by setting the resistance data to ‘measured’, as shown in the figure below.

!Caution
It is important to note that the measured effective borehole thermal resistance may not always be representative of the resistance you will have in your final project. Typically, a TRT is carried out without any antifreeze, or with a flow rate that may differ from your design flow rate. In addition, a TRT is usually performed under heat injection conditions, whereas the most critical resistance often occurs at the lowest temperature.

It is therefore recommended to always check the boundary conditions of the TRT to ensure they are applicable to your final design. If this is not the case, you can rely on the borehole thermal resistance calculated by GHEtool instead.

 

Conclusion

This article discussed the thermal response test, or TRT for short. This test can be used to obtain accurate measurements of both the ground properties (thermal conductivity and undisturbed ground temperature) as well as the effective borehole thermal resistance. In addition to the theoretical background, an example using GHEtool Cloud was presented. It was shown that the measurement of the effective borehole thermal resistance should be handled with caution, as it may not always be representative of the final project conditions. In such cases, it is preferable to calculate the effective borehole thermal resistance rather than relying solely on the measurement.

References

• Watch our video explanation over on our YouTube page by clicking here.