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Intrigued by videos of erupting geysers, we wished to find out how these wonders of nature work. The questions we asked were: how does a geyser operate? What causes its periodicity? What are its different eruptive phases? To answer these questions, we built a model of a geyser of variable height, while respecting the main characteristics of natural geysers. Using the model, we collected pressure and temperature data with sensors and a data acquisition card. In particular, we discovered how the duration of an eruptive cycle varies, why there is overpressure at the beginning of an eruption, and why some eruptions begin normally but then shift to a continuous boiling regime without replenishment. We also provide models for depressurization and for replenishment.

Every year, Yellowstone national park attracts millions of visitors who come from all over the world to see a geyser erupting. They are willing to wait for days on end in order to admire one of the most impressive natural phenomena on Earth. This led us to wonder what physical process underpins these gushing jets of water, what causes their periodicity, and what the various eruptive phases are. The first studies on geysers date back to the nineteenth century: [

This study investigates the eruption phase of a geyser prototype and explains by mathematical models the overpressure at the beginning of the eruption as well as the depressurization and refilling phases that follow. We will first present the model geyser that we built, and then look at the different eruptive phases that we identified. We will then analyse the results, and in particular the duration of a cycle and its interpretation in a P-T diagram. Finally, we will present a model of the different phases of an eruption, before concluding.

It is currently very difficult for researchers from all over the world to collect data at real sites due to extreme temperature and pressure conditions, and because of the vulnerability of the sites, to which access is legally denied as at Yellowstone. Because of this, the studies presented in [

To gain a clear understanding of what happens below ground in the case of a real geyser, we decided to perform an experiment modelling a reduced-scale geyser, which enabled us to measure the pressure and temperature inside the experimental set-up and transpose the results to a real geyser. In the literature, we found Charles Lyell’s celebrated book [

1) Construction of a reduced-scale model. Options selected

Since we could not reproduce a geyser at its real size, we decided to build a reduced-scale model. However, the experiment could not be conducted on too small a scale since the hydrostatic overpressure needed to be sufficiently large. This is why we decided to reproduce the geyser process as faithfully as possible (with a permanent concern for safety). We therefore excluded small-scale geyser experiments on a lab bench: for instance, an experiment using conventional glassware (Erlenmeyer flask, tube, funnel, etc.) is relatively dangerous since glassware is not compatible with high pressures and temperatures due to its fragility. The different materials were chosen out according to the stresses they were to be subjected to:

・ The object modelling the cavity had to be resistant to the water pressure present in the vent. We therefore used an old 6-litre aluminium pressure cooker able to withstand a pressure in excess of 2 bar (

・ The heat- and pressure resistant vent was easier to model: to avoid having to enlarge the hole in the bar of the pressure cooker (and risk weakening it), we used copper plumbing pipes with a diameter of 14 mm able to withstand up to 60 bar and over 205˚C (

・ A temperature-resistant 20-litre basin to collect the water gushing out of the geyser, and to hold the water so that the pressure cooker could be replenished after the eruption (

・ A hotplate with a power of 1.5 kW. We had to alter the internal wiring in order to bypass the temperature regulation system: as a result, the heating power is constant but remains adjustable (0.83 kW or 1.5 kW) (

・ We used a pressure sensor (2 bar gauge) connected to the output provided for the safety valve by a hose made of transparent plastic (

・ A pointer pressure gauge provided a rapid indication of the pressure.

The above is the final list of the various components of the model, which developed along with our experiments and observations. For instance, we initially used a vent that was only 1.8 m high, but we quickly realised that the hydrostatic pressure was relatively low, which led us to lengthen the copper column to a height of 6 m. To keep the set-up stable, we attached it to the wall of the house to enable access to it from an upper-storey window (

2) Initial observations

The first trials were conclusive, and we observed some impressive eruptions (

We noted the striking resemblance with an experiment involving superheated water (

At each eruption, there is a large variation in pressure, while the temperature in the pressure cooker falls and then rises again linearly. The temperature of the basin increases in steps during the transitional regime, while in the permanent regime (

Later, we carried out this experiment on the 6 m high geyser (

t cycle 1 > t cycle 2 > t cycle 3 > t cycle 4 > t cycle 5 > t cycle 6 (1)

In order to reach periodic operation more quickly, we equipped the basin with two extra plastic pipes, one to supply it with cold water and the other for overflow. There are therefore three tubes leading to the basin, one of which is made of copper (

We will now take a detailed look at how a standard periodic eruptive cycle takes place.

a) The various eruptive phases of an experimental geyser.

During the experiment (after a transitional regime of a few cycles during which the air initially trapped inside the pressure cooker is expelled) the column is filled with water, as is the entire internal volume of the pressure cooker. For a 6 m column, the pressure inside the pressure cooker is approximately 1.6 bar, which corresponds to atmospheric pressure added to the pressure exerted on the pressure cooker by the water in the column (as in Pascal’s barrel experiment).

The hotplate heats the water to its boiling temperature in the pressure cooker (around 113˚C).

The steam formed builds up above the water in the pressure cooker: it cannot escape, since the tube is submerged fairly deeply inside the pressure cooker (

The pressure remains approximately 1.6 bar during the entire heating phase. The volume of steam increases and the level of the liquid water falls until the moment when a little steam begins to emerge from the bottom of the tube submerged inside the pressure cooker. The diagram in

Phase 1: From point E (end of the previous cycle) to point A. The liquid water (with mass M) filling the pressure cooker heats up until it reaches the vaporization temperature for the pressure conditions in the cooker (1.6 bar). The first bubbles of steam form in the pressure cooker (point A).

Phase 2: From point A to point B. The steam formed builds up beneath the lid; since the specific volume of the steam is much greater than that of water (by a factor of about 1000 at 1.6 bar), the level of the liquid water falls, and the water rises up the column initially producing a dome of water in the basin (

Phase 3: At point B, the pressure and temperature inside the pressure cooker are at a maximum. At this point, the steam contained in the pressure cooker can escape directly up the tube. At this precise moment, the steam is at its highest temperature during the whole experiment.

Phase 4: From point B to point C. Bubbles of steam start to rise up the column, producing explosive boiling or flash boiling. The pressure P steadily falls to atmospheric pressure, while at the same time the temperature of the water in the pressure cooker falls towards 100˚C. Because of the fall in pressure, the temperature of the water in the pressure cooker is higher than the boiling temperature (superheated water) corresponding to this pressure P. As a result, it boils violently and a large amount of the steam produced is expelled up the column (part of it condenses). In addition, although the hotplate continues to heat the pressure cooker, the temperature of the water inside the cooker falls: the violent boiling cools it all down (boiling is endothermic and absorbs the energy taken from the superheated water and the pressure cooker).

Phase 5: At point C. All the liquid water in the column has flowed out. The column is now filled with steam. The pressure in the pressure cooker is therefore that of atmospheric pressure plus the hydrostatic overpressure due to the height h_{basin}, and the boiling temperature is therefore close to 100˚C, the boiling temperature of water at atmospheric pressure. At this point, it can be considered that a mass of water ΔM has left the pressure cooker since the beginning of the eruption. This phase has a duration that depends on the heating power cf. §VIII. It ends with a slight increase in pressure.

Phase 6: From point C to point D. Part of the water in the basin flows back down into the pressure cooker (we can see the water flowing down the transparent plastic tube), which causes a slight initial increase in pressure; this colder water flowing down from above causes condensation of the steam present in the pressure cooker. The condensation causes a vacuum in the pressure cooker, which leads to a sudden drop in pressure from 1 bar to 0.7 bar. This depressurization can be seen in the basin, where a vortex is produced, together with the characteristic sound of suction (

Phase 7: At point D. The pressure reaches a minimum, and the depressurization is at its maximum.

Phase 8: From point D to point E, replenishment ends. At E, the pressure cooker and the column are entirely filled with water, which brings the pressure back up to 1.6 bar, while the temperature is at its minimum. Pressure oscillations are observed due to the violent water hammer that occurs when replenishment is complete (

b) The duration of an eruptive cycle: influencing parameters.

We attempted to model the thermal behaviour of our model geyser. We assume that heat loss is negligible since during phase 1 the temperature increase is linear. Moreover, we carried out eruptions with and without insulating the tube, and the experimental results were the same. We also ignore the pressure cooker’s heat capacity. Let P_{plate} be the power of the hotplate, M the mass of water to be heated in the pressure cooker, T the temperature of the water, and C_{water} the specific heat capacity of the water. During the heating phase, the hotplate transfers heat energy to the water in the pressure cooker in accordance with:

M C water d T d t = P plate (2)

By integrating this equation between point E and point A, and adding the duration of the eruption t_{eruption} we obtain the duration of a cycle t_{cycle}:

t cycle = M C water ( T ( A ) − T ( E ) ) P plate + t eruption (3)

In this expression, T(A) remains roughly constant (113˚C for 6 m), while T(E) depends on the temperature of the basin. A simplified calorimetric analysis is used to find in E:

T ( E ) ≈ ( M − Δ M ) ⋅ T ( C ) + Δ M ⋅ T basin M (4)

In Annex IB we will extend this model with a digital sequence, where index n is the number of moments of different successive replenishments. We can verify this model against the data in _{cycle}. Similarly, the lower the temperature of the basin T_{basin}, the lower is the minimum temperature reached T(E).

We verify experimentally that the duration of a cycle does indeed decrease in accordance with the heating power (

We note that the higher the column, the longer is the duration of the eruption t_{eruption}. An explanation will be provided in §VI for the duration of depressurization and in §VII for replenishment.

c) Interpretation in the water P-T diagram of an eruptive cycle.

i) First plot of an eruptive cycle with our experimental data.

During our experiment, we observed a cycle involving 8 successive phases that repeated over time. We place this cycle directly onto a P-T diagram (

During certain phases of the cycle of a geyser (phases 2 - 6 from A to D), there is an equilibrium in the pressure cooker between liquid water and steam. During these phases, the pressure is therefore the saturation vapour pressure, and the (P,T) coordinate points are located on the vaporization curve in the P-T diagram. Between points A and D, the measured points on the P-T diagram should be located on the vaporization curve, which is not the case, except for point C, where the pressure and temperature curves are almost stationary (

ii) Correction of data collected by the temperature sensor.

The temperature sensor only indicates variations in temperature after a certain response time. The experimental data are not therefore accurate, and the sensor output did not correspond to the temperature. It was therefore necessary to take into account the temperature sensor’s response time. To evaluate the sensor’s response time, we conducted a small experiment: we quickly moved the temperature probe from a water bath at room temperature (T = 20.85˚C) to a warmer water bath (T = 40˚C), recording the temperature as a function of time with Latis pro software. We thus obtained the response of the sensor to a temperature step-change. We obtained the recording shown in

time constant τ. The response of the temperature sensor is similar to the response of a first order differential equation system:

τ d S model d t + S model = T (5)

In this equation, τ is the time constant, and a first graphical estimate gives τ ≈ 7 s . To obtain a more accurate value, we solve the differential equation when T is a constant:

S model ( τ , t ) = S model ( t initial ) ⋅ e − ( t − t initial ) τ + T ⋅ ( 1 − e − ( t − t initial ) τ ) (6)

To find the most appropriate time constant, we use the least squares method.

We defined the quadratic error, which is a function of the time constant τ:

quadraticerror ( τ ) = ∑ k = 0 k = k final ( S model ( τ , k ⋅ T e c h ) − S sensor ( k ⋅ T e c h ) ) 2 (7)

k = 0 corresponds to the beginning of the step, and k = k_{final} corresponds to the end of the recording.

Using Matlab software, we obtained a plot of the quadratic error (

For the time constant found there is a good match between the model and the experimental result: henceforth, we will conflate the output from the model with the output from the sensor, that is S sensor ≈ S model . We can then estimate the temperature from the equation:

T ≈ τ d S sensor d t + S sensor (8)

which is feasible if we know d S sensor d t . To do this, we carry out the approxima- tion shown in

Then:

d S sensor d t ( k ⋅ T e c h ) ≈ S sensor ( ( k + 1 ) ⋅ T e c h ) − S sensor ( ( k − 1 ) ⋅ T e c h ) 2 T e c h for k ≥ 1 (9)

And:

T estimate ( k ⋅ T e c h ) ≈ τ S sensor ( ( k + 1 ) ⋅ T e c h ) − S sensor ( ( k − 1 ) ⋅ T e c h ) 2 T e c h + S sensor ( k ⋅ T e c h ) for k ≥ 1 (10)

The results are shown in

iii) Second plot of the eruptive cycle with corrected temperature data.

The cycle shown in

The black curve shows the pressure in the pressure cooker according to the output from the temperature sensor, and the pink curve shows the pressure in the pressure cooker according to the estimated temperature, which is closer to the real temperature. The temperature correction thus enables us to obtain a good match between the model and the experimental data: between the time the steam appears and its complete condensation, both phases are present, and the corresponding points on the P-T diagram are located on the vaporization curve.

We were thus able to understand the cyclical nature of the geyser and show this cycle in the P-T diagram.

d) Why is there sometimes overpressure at the beginning of an eruption?

In this section we will explain the pressure increase that takes place at the beginning of an eruption when the heating power is high. In

i) Demonstration of overpressure.

On

the On/Off switch, so as to have an eruption that is not heated for the duration of the eruption (once an eruption begins, explosive boiling can no longer be interrupted and it continues even without heating). By observing the formation of bubbles in the transparent tube at the pressure cooker outlet (

ii) Experiments performed.

We then measured the pressure in the pressure cooker when it was totally closed. We filled the pressure cooker up to the base of the column and plugged the column at its base. We then replaced the safety valve (which is normally triggered at 1.9 bar) and heated the pressure cooker at 1.5 kW. Only the pressure was recordable (

iii) Digital processing.

We then used digital processing on the data points. We shifted eruption 1 in time to make it begin at exactly the same moment as eruption 2 (at t = 2045 s). The pressure corresponding to eruptions 1 (in pink) and 2 (in red) are shown in

e) Depressurization model.

In this section we will model the depressurization of the pressure cooker corresponding to phase 4 in

d P d t = − k 1 ⋅ v (11)

where k_{1} is a proportionality factor that depends on the specific enthalpy of the outgoing steam, the mass of water contained in the pressure cooker during this phase, the specific enthalpy of water, the diameter of the tube and the density of the steam; v is the expulsion speed of the steam.

In addition, the pressure difference ΔP between the pressure cooker and the top of the tube introduces the notion of pressure drops:

Δ P = P − ρ ⋅ g ⋅ h basin − P atm = λ L D ρ steam v 2 2 (12)

where L is the length of the tube and D its diameter; λ is a coefficient obtained using a Moody chart, which in our situation where everything is fixed (L, D, viscosity of steam) only depends on v; it depends directly on the Reynolds number, which characterizes flow. Thus, at the beginning of phase 4, when the speed is at its highest, λ is greater than towards the end, when we approach the pressure at point C in

k 2 = k 1 ⋅ 2 ⋅ D λ ⋅ L ⋅ ρ steam (13)

d d t ( P − P atm − ρ ⋅ g ⋅ h basin ) = − k 2 ⋅ P − P atm − ρ ⋅ g ⋅ h basin (14)

When integrated, this differential equation simply gives:

P = P atm + ρ ⋅ g ⋅ h basin + α ⋅ ( t − t 0 ) 2 for t ≤ t 0 and α = k 2 2 4 (15)

It can be seen that the pressure has a parabolic shape locally (since k_{2}, like λ, depends on time). In

f) Refilling model.

In this section we will model the replenishment of the pressure cooker after an eruption. This corresponds to phase 6 in _{r} as the duration of phase 6, replenishment takes place steadily, and, ignoring water mixing dynamics, we can find an approximation to the instantaneous temperature of the pressure cooker on the basis of a heat balance (t is counted from the start of replenishment):

T ( t ) ≈ ( M − Δ M ) ⋅ T ( C ) + Δ M ⋅ T basin ⋅ t / τ r M − Δ M + Δ M ⋅ t / τ r (16)

In this relation, the temperature response depends on the temperature of the

basin. In _{basin} = 35 + 273 K, τ_{r} = 7.5 s; the model appears to be a good fit, which justifies our hypothesis. In

・ replenishment times are more or less the same,

・ the temperature of the basin affects the slope of the temperature.

In

・ replenishment time does not depend on the height of the geyser.

・ the previous T(t) model does not depend on the height of the geyser.

・ depressurization at the end of phase 4 for the 6 m geyser corresponds to depressurization for the 2.2 m geyser.

g) Why do some eruptions not lead to the replenishment of the pressure cooker?

In this section we will explain a phenomenon observed during certain eruptions. Looking at

an incomplete eruption: phase 5 in

P plate < L ⋅ π ⋅ R 2 2 ⋅ ρ ⋅ ρ steam ⋅ g ⋅ h basin (17)

We verified experimentally that for a given value of h_{basin}, a lower heating power (0.83 kW) is not accompanied by an incomplete eruption (

Thanks to this project, we have improved our understanding of geysers, but that isn’t all. We enjoyed working on this subject because it enabled us to connect our school work to our leisure activities. We love DIY, and this gave us the opportunity to build a model geyser, mainly with recycled materials. We then really enjoyed carrying out the experiments and collecting data so as to find answers to our questions about the phenomena we observed. The main difficulty was that initially we were not able to see anything, either in the column or in the pressure cooker. Thanks to the sensors, which “replaced” our eyes, we were able to collect accurate pressure and temperature data in the model geyser, using a computerized data acquisition system. This allowed us to get an even more detailed understanding of overpressure and certain special regimes, such as the fumarole regime.

We were able to model certain aspects of the geyser cycle, such as its period, depressurization and replenishment. We were lucky enough to be able to talk to researchers about our project and about the various problems we encountered throughout the project. They gave us valuable advice to help us refine our data and our interpretation of it. The project really gave us a desire to continue with our science studies and, indeed, to consider a career as a researcher.

1) Eruption without replenishment. Complement to paragraph VIII.

During a time interval Δ t , the energy supplied to the pressure cooker by the hotplate has the value Δ W = P plate ⋅ Δ t . This energy is used to vaporize Δ m = Δ W L where L is the latent heat of vaporization, which corresponds to a volume:

Δ V = Δ m ρ steam = P plate ⋅ Δ t L ⋅ ρ steam (18)

and Δ V = S ⋅ Δ l where S = π ⋅ R 2 is the cross-sectional area of the tube and Δ l the length of the tube occupied by the volume Δ V . This allows us to write:

P plate ⋅ Δ t L ⋅ ρ steam = S ⋅ Δ l (19)

At equilibrium, this steam produced will be expelled via the tube in a time interval Δ t . It will thus move Δ l in Δ t which corresponds to a speed:

v = Δ l Δ t = P plate L ⋅ ρ steam ⋅ π ⋅ R 2 (20)

This expulsion speed of steam via the tube produces a dynamic pressure P dyn = 1 2 ⋅ ρ steam ⋅ v 2 which can, if it is sufficient, compensate the hydrostatic pressure

P = ρ ⋅ g ⋅ h basin produced by the water of the basin above the top of the tube.

A continuous boiling regime (without replenishment of the pressure cooker) is produced when P dyn > ρ ⋅ g ⋅ h basin . We then need a heating power:

P plate > L ⋅ π ⋅ R 2 2 ⋅ ρ ⋅ ρ steam ⋅ g ⋅ h basin (21)

to prevent another replenishment from taking place.

2) Temperature as a sequence. Complement to paragraph III.

In this paragraph we will model the change in the temperature of the cooker along with the successive replenishments of the pressure cooker. In order to do this we use recursive sequences. Thus, each index n corresponds to the end of an eruption. By using the result of §III, we have:

( T ( E ) ) n + 1 ≈ ( M − Δ M ) ⋅ T ( C ) + Δ M ⋅ ( T basin ) n + 1 M (22)

The temperature of the basin at the end of an eruption depends of course on the temperature of the basin at the end of the previous eruption, of the water received at boiling temperature between T(A) and T(B), and also on the quantity of steam received by the basin during phase 4 (

{ ( T basin ) n + 1 ≈ a ⋅ ( T basin ) n + b ( T ( E ) ) n + 1 ≈ c ⋅ ( T basin ) n + 1 + d (23)

We simulated these two sequences with the initial conditions ( T ( E ) ) 0 = 71 + 273 and ( T basin ) 0 = 20.4 + 273 . The results of these two sequences are marked with stars in

In pink, the temperature of the pressure cooker with the blue stars corresponding to ( T ( E ) ) n , in green, the temperature of the basin and the red stars corresponding to ( T bassine ) n .

The sequence gives a result which is close to the recorded data. Although the model is quite simple, it needs to be improved for better precision.

Details of the experiment with the 6 m column.

Some images of the eruption of our geyser with a 6 m column.