Stirling engines, or heat pumps, are systems that are able to work on incredibly low temperature differences. In fact, some types of Stirling engines only need human body heat in order to operate. Here, we explore the dynamics of this interesting machine that you can build at home and demonstrate how to model it using COMSOL Multiphysics.
Let’s begin by taking a step back in the history of the Stirling engine. Denominated the “engine of the future,” the Stirling engine was first developed by Robert Stirling nearly 200 years ago in 1816. While the technology never really came out on top, this type of heat engine has found extensive use in many modern applications. For example, the solar Stirling engine is used to directly transform solar heat into mechanical energy, which is in turn applied to powering a generator and producing electrical power. Additionally, analogous approaches exist that are based on geothermal energy or use industrial waste heat. The most astonishing modern application of the Stirling engine may be its operation in Swedish submarines — the absence of air is not an issue for a Stirling’s propulsion.
We’ve touched upon some of the applications of Stirling engines, but how does this machine operate? In a Stirling engine, thermal heat is converted into mechanical work (or the other way around) in a cyclic process. This can be realized in different ways, but the principle remains the same: The engine cycles through the four processes of cooling, compression, heating, and expansion. A gas is used to transport the heat from a permanent hot side to a cold side. The efficiency of the engine is restricted by the efficiency of the Carnot cycle.
In contrast to conventional engines, Stirling engines do not need to reach high temperatures to operate. Some Stirling engines only need a small Kelvin temperature difference between the hot side and the cold side. Furthermore, the sound level and subsequent energy losses are very low because there are no explosions and no exhaust. However, Stirling engines are most suitable for applications where constant power is required, as controlling dynamic power regulation would be an extensive task. This is likely the most prominent reason as to why there are still no cars powered by Stirling engines.
A Stirling engine operates using the heat from a human hand. (“A Stirling engine that works solely with the energy taken from the temperature-difference from the surrounding air and the palm of the hand” by Arsdell — Own work. Licensed under Creative Commons Attribution-Share Alike 3.0 via Wikimedia Commons).
For those of you with some experience in handcrafting, it is possible to build your own Stirling engine at home, without the need for professional tools or experience. Several video tutorials are available on YouTube that include instructions designed to guide you through the building process. The easiest of these examples is probably the version consisting of a cola can and some domestic odds and ends.
While easy to build, this model of a Stirling engine is likely not optimized from an efficiency point of view. Creating a numerical model of the engine offers a better solution.
With a numerical model of a Stirling engine, we are able to find and test sets of materials as well as parameter adjustments. The involved physics are heat transfer and fluid flow, and the mechanical process can be simplified by solving the equation of motion as an additional ODE.
The 2D-axisymmetric model consists of a main cylinder that includes the working fluid (air) and the displacer. The small cylinder on top contains the power piston. Both the displacer and the power piston are connected in parallel and joined 90º out of phase on a crankshaft, which is not featured as part of the model. The whole setup corresponds to a gamma type Stirling engine.
A model of a Stirling heat pump.
Here, the heat transport within the working gas is solved. The mechanical part is realized via a moving mesh (ALE) approach: The displacer and the power piston are free to move in the z-direction. The displacement is prescribed here, corresponding to a heat pump. Mechanical work is used to move thermal energy opposite to the direction of spontaneous heat flow. The other way around — a Stirling engine — can also be modeled by applying a heat source and solving for the resulting pressure forces at the power piston and the displacer. In any case, the system runs through a certain chain of processes that can be divided into the four steps of the Carnot cycle:
Thermodynamic processes that are acting on the working fluid.
While its efficiency is nowhere near the theoretical Carnot cycle diagram, the resulting pressure/volume graph shown below does correspond to experimental results.
A pressure/volume graph of the Stirling cycle.
The real advantage of the model is that we can analyze the physics within the heat pump. For example, the animation below indicates velocity distribution during the operation of the heat pump.
Velocity distribution during the heat pump’s operation.
Since the piston brings mechanical energy needed to pump heat, we can also investigate the dynamic temperature distribution within the heat pump during operation.
An animation depicting temperature distribution.
If you are looking to improve the efficiency of the Stirling engine, the goal is to maximize the enclosed area in the pressure/volume graph. This area corresponds to the work that is done by the engine. The engine’s overall efficiency can be improved in multiple ways. If we choose a working gas that features a high specific gas constant (i.e., a low molar mass), this will maximize the isothermal expansion (and therefore the work ability) of the engine. This is why hydrogen and helium are preferred as working gases. Another possibility is to maximize the heat that is transported by the displacer through using a porous medium as a regenerating displacer (see this paper).
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The development of drilling techniques during the last century provides the capability of spreading wells into several branches, among other investigations. This enhanced method allows a much wider reservoir area to be covered, either for oil and gas or for geothermal energy production.
Natural resource recovery: Oil is produced from a natural reservoir via mechanical pumps that raise the oil to the surface. A side-branch of the drilling reaches into an adjacent reservoir by a junction.
However, branching junctions are also less stable than single boreholes and have to be stabilized with a liner or casing quite often. Even if the borehole would stay open by itself after a drilling procedure is completed, it undergoes additional stress when the production process is engaged by starting pumping through the wellhole. Borehole casing is very expensive, it adds significantly to the overall cost for a drilling project. If possible, one would rather avoid the deployment of additional mechanical stabilization.
The question is: Is the open borehole stable without casing?
A chain is only as strong as its weakest link. To answer the question of stability, it is sufficient to reduce the investigation to the most critical point: the branching junction (or one of the junctions, if there are more branches). If the well is stable here, we may assume that it will hold up everywhere else. As a first approach, we assume only elastic deformation. Thus, all we need to build the model is the Subsurface Flow Module and the Solid Mechanics physics interface in COMSOL Multiphysics.
There is no need to take the fluid flow within the well into account. All we need to know is the pressure level at the well boundaries. This is prescribed by the pump and the depth of the junction. Also, the local symmetry of the part of interest allows us to reduce the geometry to one half and apply symmetry boundary conditions.
Model geometry of the multilateral well branching junction. Due to symmetry, only one half of the junction needs to be considered.
The subsurface flow within a porous media reservoir can be described by a combination of Darcy’s Law and a continuity equation:
with the permeability, \kappa, the dynamic viscosity, \mu, and the oil pore pressure, p_f. The pressure in the pores will change when the production process starts; a lower pressure at the walls of the well due to pumping leads to flow in the pores and a pore pressure drop.
Changes in the pressure load effect a deformation because the Cauchy stress tensor, \sigma, is related to p_f:
with the elasticity matrix \boldsymbol{C}=\boldsymbol{C}(E,\nu), which depends on Young’s modulus, E, and Poisson’s ratio, \nu, the strain tensor, \epsilon and the Biot-Willis coefficient, \alpha_B. This relation couples the fluid flow with the equation for the quasi-static deformation
where \boldsymbol{F} denotes any external body forces.
The only exit for the oil is the well. Thus, the boundaries perpendicular to the well axis (top and bottom) are assumed to provide No Flow conditions. As I mentioned above, the symmetry face provides symmetry conditions in both the flow and mechanical interfaces. The model constrains movement at all external boundaries, while the well opening is free to deform.
Variable | Description |
Value |
---|---|---|
\rho_f | Fluid density |
0.0361 lb/in^{3} |
\rho_s | Solids density |
0.0861 lb/in^{3} |
\mu | Fluid dynamic viscosity |
1·10^{-7} psi·s |
\kappa | Permeability |
1·10^{-13} in^{2} |
E | Young’s modulus |
0.43·10^{6} psi |
\nu | Poisson’s ratio |
0.16 |
p_r | Reservoir pressure |
122.45 psi |
p_w | Well pressure |
0 psi |
Model properties.
To get an impression of the results, let’s have a look at a re-mirrored result plot showing the total displacement due to the pressure drop and the resulting velocity field. So far, we know that the most critical part seems to be the area right above the branching. But we still do not know if the well is stable.
Surface displacement (color plot) and velocity field (arrow plot) of the branching junction.
The results of the model show how the pressure change affects the stress and strain distribution around the well. To find out if the hole is stable or not, we need a failure criterion. The 3D Coulomb criterion (a linearized form of the Lade criterion) reproduces the laboratory data of rock failure experiments well, but note that it requires calibration data.
The criterion relates rock failure, the three principal stresses (\sigma_1, \sigma_2, and \sigma_3), and the fluid pressures as follows:
where
with the Coulomb cohesion, S_0, and the Coulomb friction angle, \phi. For different results of the function, various scenarios can be expected: fail > 0 predicts stability; fail = 0 indicates the onset of rock failure; and fail < 0 denotes catastrophic failure.
Evaluation of the fail function (equation anchor: failure). The highest potential for failure is right above the branching.
When the fail values become increasingly negative, the potential for failure is higher. As expected, the fail function estimates show the greatest potential for failure just above the branching in the well. The failure potential can be reduced by adding a support force, e.g., a casing.
Below, you can see the results of a parametric sweep study. With increasing well support force, the failure potential decreases.
Evaluation of the fail function for different well support forces from a parameter study (support force from left to right: 5e^{4}, 7.5e^{4}, 1e^{5}, 2.5 N/m^2).
Here, we have showed you how to model a branching junction of an open-hole multilateral well. As the stability of open wells is an essential piece of information, this model is a good example of how numerical simulations can significantly save money. The important estimation of stability can be clarified in the early project phase.
The presented approach focuses on elastic deformation, while related analyses can also be conducted for elasto-plastic materials, for example. These are automated in the Geomechanics Module and in the Structural Mechanics Module. This model could also easily be expanded to study the influence of temperature by adding thermal expansion. This would be especially of interest for studies concerning geothermal energy approaches.
Modern building construction requires efficient climate control, preferably using sustainable energy resources. Such resources can be solar collector systems for warm-water support or ambient air or geothermal-based heat pump systems that can be used for cooling in the summer and heating in the winter.
The geothermal applications mostly consist of pipe installations with water or brine as fluid to provide the heat exchange between the subsurface and a heat pump. There are two main types of these so-called closed-loop systems: vertical borehole heat exchangers (BHEs), which are installed in boreholes (see part 1 of this series); and horizontal heat collectors, which are deposited on large areas and in very shallow depths of only one or two meters.
While the vertical BHEs are often simplified as infinite thermal line sources or sinks, the horizontal systems are more complicated from a modeling point of view. This is because they have to be arranged in patterns to cover large surface areas, and the use of this simplification does not match anymore. The total length of the pipes can easily reach 100 meters or more. The flow regime is preferably turbulent because of the resulting lower thermal resistivity between the fluid and the subsurface. You can imagine that a CFD simulation approach of systems with these dimensions is a challenge (despite the computational performance of modern HPC clusters). Fortunately, there is a solution for this kind of task.
Let’s have a look at how the Pipe Flow Module can be used in the growing sector of geothermal heat collector systems.
Thankfully, we do not have to care about the computational difficulties that may arise when simulating long pipe systems. The Pipe Flow Module provides all necessary capabilities to calculate pressure drops, velocities, and foremost — the heat transport between the pipe fluid and the subsurface, using prescribed 3D curved-edge correlation functions. All we need to do is to define the inlet conditions (temperature, velocity) and the pipe, fluid, and subsurface properties. Therefore, we can completely focus on questions like, “What do you think is the best horizontal pipe arrangement underneath your garden?”
Let’s find out.
The heat transfer example, provided in a recent blog post, shows how a prescribed heat extraction rate can be realized in the form of a numerically-calculated inlet temperature boundary condition. By doing so, any desired heat extraction rate can be given and the corresponding inlet temperature can be computed. Assuming that the fluid properties are temperature-independent, a simplified version of the inlet temperature can be written as:
(1)
Thus, it is a function of the outlet temperature (T_{out}), density (\rho_f), heat capacity (C_{p,f}), and volumetric flow rate (\dot{V}) of the working fluid in the pipes. The temperature difference between the inlet and outlet temperature is controlled via the heat extracted by the heat pump, P(t), which is a function of time because heat pumps (for house heating purposes) usually do not run all day. A typical heating rate of a one-family house heat pump is 8 kW, where 2 kW of electric power is needed to run it and 6 kW are to be extracted from the subsurface. To reach a daily heat demand of, say, 48 kWh during the winter season in Germany or North America, the heat pump needs to run six hours per day.
6 kW of heat extraction for six hours per day, in a three-day cycle.
The number of possibilities of how the heat collector pipes can be arranged in the subsurface is endless. Here, we will have a look at three randomly chosen patterns. Let’s call them snake, snail, and meander designs. The heat collector is embedded in the subsurface with typical thermal properties that you could find in the uppermost soil layer in your garden. The subsurface temperature distribution corresponds to that of the temperature in Germany during the month of January.
Three different layer designs for a geothermal heat collector system: 1) snake, 2) snail, and 3) meander design.
The COMSOL Multiphysics feature, geometry subsequence, allows us to create all of the three different pipe geometries within the same model. It also allows us to perform a parametric study that solves for each particular sub-geometry. We can compare the different designs by having a look at the resulting outlet temperature as a function of time.
Comparison of the outflow temperatures of the three collector designs.
The three collectors each show a different thermal behavior. This is due to the varying distances between the pipes, their different lengths, and thereby the surface available for heat transfer. Even though the freezing point of the water-glycol mixture inside the pipes is at about -13°C, the outflow temperature is — for environmental reasons — legally limited to stay above -5°C in Germany.
The snake design tends to go below this point only after a few days and would, thus, not be the preferred design here. However, a geothermal engineer could now proceed to optimize a design (modify the pipe length, pipe diameter, or the covered area to reach a maximum outlet temperature, for example.) One could, of course, also easily perform long time runs to study the impact of seasonal temperature variations or even use the pipes for cooling purposes in the summer.
Once the optimal design of the geothermal heat collector in your garden is found, why not go on and model the modern underfloor heating system in the house? The procedure is straight-forward. There is only a change of sign because heat is injected here.
Combined model of a garden heat collector together with an underfloor house heating system. The collector extracts heat from the subsurface at 6 kW, while the house heating system injects heat into the floor of the house at 30°C.
One of the greatest challenges in geothermal energy production is minimizing the prospecting risk. How can you be sure that the desired production site is appropriate for, let’s say, 30 years of heat extraction? Usually, only very little information is available about the local subsurface properties and it is typically afflicted with large uncertainties.
Over the last decades, numerical models became an important tool to estimate risks by performing parametric studies within reasonable ranges of uncertainty. Today, I will give a brief introduction to the mathematical description of the coupled subsurface flow and heat transport problem that needs to be solved in many geothermal applications. I will also show you how to use COMSOL software as an appropriate tool for studying and forecasting the performance of (hydro-) geothermal systems.
The heat transport in the subsurface is described by the heat transport equation:
(1)
Heat is balanced by conduction and convection processes and can be generated or lost through defining this in the source term, Q. A special feature of the Heat Transfer in Porous Media interface is the implemented Geothermal Heating feature, represented as a domain condition: Q_{geo}.
There is also another feature that makes the life of a geothermal energy modeler a little easier. It’s possible to implement an averaged representation of the thermal parameters, composed from the rock matrix and the groundwater using the matrix volume fraction, \theta, as a weighting factor. You may choose between volume and power law averaging for several immobile solids and fluids.
In the case of volume averaging, the volumetric heat capacity in the heat transport equation becomes:
(2)
and the thermal conductivity becomes:
(3)
Solving the heat transport properly requires incorporating the flow field. Generally, there can be various situations in the subsurface requiring different approaches to describe the flow mathematically. If the focus is on the micro scale and you want to resolve the flow in the pore space, you need to solve the creeping flow or Stokes flow equations. In partially saturated zones, you would solve Richards’ equation, as it is often done in studies concerning environmental pollution (see our past Simulating Pesticide Runoff, the Effects of Aldicarb blog post, for instance).
However, the fully-saturated and mainly pressure-driven flows in deep geothermal strata are sufficiently described by Darcy’s law:
(4)
where the velocity field, \mathbf{u}, depends on the permeability, \kappa, the fluid’s dynamic viscosity, \mu, and is driven by a pressure gradient, p. Darcy’s law is then combined with the continuity equation:
(5)
If your scenario concerns long geothermal time scales, the time dependence due to storage effects in the flow is negligible. Therefore, the first term on the left-hand side of the equation above vanishes because the density, \rho, and the porosity, \epsilon_p, can be assumed to be constant. Usually, the temperature dependencies of the hydraulic properties are negligible. Thus, the (stationary) flow equations are independent of the (time-dependent) heat transfer equations. In some cases, especially if the number of degrees of freedom is large, it can make sense to utilize the independence by splitting the problem into one stationary and one time-dependent study step.
Fracture flow may locally dominate the flow regime in geothermal systems, such as in karst aquifer systems. The Subsurface Flow Module offers the Fracture Flow interface for a 2D representation of the Darcy flow field in fractures and cracks.
Hydrothermal heat extraction systems usually consist of one or more injection and production wells. Those are in many cases realized as separate boreholes, but the modern approach is to create one (or more) multilateral wells. There are even tactics that consist of single boreholes with separate injection and production zones.
Note that artificial pressure changes due to water injection and extraction can influence the structure of the porous medium and produce hydraulic fracturing. To take these effects into account, you can perform poroelastic analyses, but we will not consider these here.
It is easy to set up a COMSOL Multiphysics model that features long time predictions for a hydro-geothermal application.
The model region contains three geologic layers with different thermal and hydraulic properties in a box with a volume V≈500 [m³]. The box represents a section of a geothermal production site that is ranged by a large fault zone. The layer elevations are interpolation functions from an external data set. The concerned aquifer is fully saturated and confined on top and bottom by aquitards (impermeable beds). The temperature distribution is generally a factor of uncertainty, but a good guess is to assume a geothermal gradient of 0.03 [°C/m], leading to an initial temperature distribution T_{0}(z)=10 [°C] – z·0.03 [°C/m].
Hydrothermal doublet system in a layered subsurface domain, ranged by a fault zone. The edge is about 500 meters long. The left drilling is the injection well, the production well is on the right. The lateral distance between the wells is about 120 meters.
COMSOL Multiphysics creates a mesh that is perfectly fine for this approach, except for one detail — the mesh on the wells is refined to resolve the expected high gradients in that area.
Now, let’s crank the heat up! Geothermal groundwater is pumped (produced) through the production well on the right at a rate of 50 [l/s]. The well is implemented as a cylinder that was cut out of the geometry to allow inlet and outlet boundary conditions for the flow. The extracted water is, after using it for heat or power generation, re-injected by the left well at the same rate, but with a lower temperature (in this case 5 [°C]).
The resulting flow field and temperature distribution after 30 years of heat production are displayed below:
Result after 30 years of heat production: Hydraulic connection between the production and injection zones and temperature distribution along the flow paths. Note that only the injection and production zones of the boreholes are considered. The rest of the boreholes are not implemented, in order to reduce the meshing effort.
The model is a suitable tool for estimating the development of a geothermal site under varied conditions. For example, how is the production temperature affected by the lateral distance of the wells? Is it worthwhile to reach a large spread or is a moderate distance sufficient?
This can be studied by performing a parametric study by varying the well distance:
Flow paths and temperature distribution between the wells for different lateral distances. The graph shows the production temperature after reaching stationary conditions as a function of the lateral distance.
With this model, different borehole systems can easily be realized just by changing the positions of the injection/production cylinders. For example, here are the results of a single-borehole system:
Results of a single-borehole approach after 30 years of heat production. The vertical distance between the injection (up) and production (down) zones is 130 meters.
So far, we have only looked at aquifers without ambient groundwater movement. What happens if there is a hydraulic gradient that leads to groundwater flow?
The following figure shows the same situation as the figure above, except that now there is a hydraulic head gradient of \nablaH=0.01 [m/m], leading to a superposed flow field:
Single borehole after 30 years of heat production and overlapping groundwater flow due to a horizontal pressure gradient.
During my time as a PhD student, I had to deal with different kinds of geothermal energy extraction techniques. The focus of my research was on shallow geothermal borehole heat exchangers, but within my team, I also came in contact with open borehole and deep geothermal approaches. To my surprise, I found that almost all of the processes involved in the various geothermal applications could be modeled with the Subsurface Flow Module.
Renewable energies are a growing industry and the geothermal energy branch is a hot topic of active research. Over the past few decades, different techniques were established to extract geothermal heat from shallow to deep subsurface levels. The closed-loop borehole heat exchanger (BHE) is a standard approach for lower- and mid-depth applications.
In a BHE, fluid circulates through pipes that are located inside of boreholes, resulting in indirect thermal contact between the fluid and the subsurface. This method was restricted by the closed nature of the heat exchanger with the effective heat exchange area limited by the equipment involved. This could could be significantly increased by extracting groundwater from aquifers, gaining thermal energy through the re-injection of hot fluid a distance from the injection point through hydrothermal doublets.
Enhanced geothermal systems (EGS) were developed to exploit geothermal resources in dry and impermeable rocks or hard dry rocks (HDR), by hydraulic fracturing, or fracking. This procedure involves pumping high-pressure water into the desired layer to establish new cracks and widen existing cracks and fractures. Injected water can then travel through the fractures, heat up, and be extracted at a second borehole.
Geothermal heat extraction methods.
Heat transport in the subsurface proceeds by convection, dispersion, and conduction. You need to know the thermal properties of the geological layers in order to run simulations. However, we can usually only obtain vague estimations based on geological maps and core samples. Convective heat transport, either naturally driven by buoyancy forces or artificially established through the wells, can play a role in — and even dominate — the processes.
Depending on the local geology, the subsurface flow regime may be situated in fully or partially saturated porous media, or it can proceed in fractures. Although the different techniques for geothermal heat production differ fundamentally, the Subsurface Flow Module provides the required features to simulate the subsurface thermal development. You can easily couple heat transport with the velocity field.
In some situations, the coupling must be applied bi-directionally. If the temperature gradients are high, the temperature dependencies of involved parameters (hydraulic conductivity, for example) are no longer negligible and must be taken into account. Also, poroelastic processes may play a role in some cases, especially when it comes to fracking.
Let’s have a look at an example to demonstrate some of the features that are necessary to simulate geothermal processes. The following model solves for the heat transport around a shallow geothermal installation embedded in a geological domain. The domain is separated into various parts, representing different geological layers with their particular properties. The thermal influence of seasonal temperature changes at the surface is taken into account.
A three-by-three borehole heat exchanger (BHE) array 135-meter deep, located in layered bedrock. Each BHE extracts heat at 20 W/mK throughout the year. Between 60-70 meters is an aquifer where groundwater flow occurs, causing horizontal convective heat transport. Shown on the right are the borehole wall temperatures of the three BHEs in the middle of the array. Due to the thermal interaction between the heat sinks, the temperature of middle BHE (green line) is lower than the other two. In the region of the aquifer, the BHE further downstream than the other two (red line) is affected by the heat exchange occurring upstream of it, resulting in a lower temperature.
Simulation is necessary for the prediction of long-term impacts on the BHEs in order to identify whether the pipe will avoid freezing. The most straightforward way to quickly model the BHEs is to neglect the glow and heat transfer within the borehole and apply an appropriate heat flux boundary condition at the walls. In doing so, the borehole becomes a local heat sink and heat will travel towards it. If more than one BHE is installed at a local site, the heat exchangers might start to interact after some time. In particular, if groundwater flow in an aquifer is present, the boreholes will connect thermally. This thermal interaction can cause significant efficiency losses of the whole geothermal system. On the other hand, groundwater flow also increases the thermal recovery rate. Reliable predictions can only be made if the geological data set is accurate enough.
The heat transfer through the BHEs over a period of time. The top surface temperature changes seasonally between 0-20°C, representing a city in central Europe or the east coast in the US. Notice how the heat pattern elongates in the direction of flow in the region of the aquifer.
Today, we have introduced how you can use multiphysics simulation in geothermal energy applications. Next up in this blog series we will present a more advanced application, involving coupling heat transfer in heat pipe arrays with subsurface porous media flow. Stay tuned!