Sublimation temperature sensor for temperature locale

The rudimentary measurement of temperature is arguably the most basic of needs when diagnosing, experimenting, cooking, manufacturing, etc. New devices to measure temperature have been created since at least the late sixteenth century; liquid-in-glass thermometers, thermocouples, thermistors, resistance temperature detectors (RTDs), and pyrometers, among others, have all been developed since that time. However, each of these devices measures the point-wise temperature of a specific location. An array of multiple sensors in a set pattern needs to be construed to provide a useful spatial understanding of a temperature gradient. Furthermore, a specific temperature of interest usually does not line up exactly with a spatial, point-wise sensor; this warrants the need to interpolate and/or extrapolate in between or beyond the range of sensors.

The current work presents the Sublime Temperature Sensor (STS, or “sublime sensor”), a novel approach to measuring both the temperature and specific location of that temperature in any heated gradient. Researchers at the Idaho National Laboratory’s (INL) Measurement Science Laboratory (MSL) see this as a paradigm shift in temperature metrology and are further developing this instrumentation. Rather than approaching the measurement of temperature in the traditional way, the novel sensor measures the specific location of a set or desired temperature.

The general build of an STS stages a material or compound with known properties inside a long narrow tube (i.e., L ≫ D) made of a gas tight, non-volatile material (e.g., quartz, alumina, and refractory metal). After evacuating the tube to a known pressure of high vacuum, the tube is sealed off. Upon completion of the seal, the individual sensor build is now set at a specific vacuum pressure, and therefore, the material inside will sublimate/deposit at a known temperature.¹ The sublimation process is paramount to the function of the sensor as the solid material stays stationary until the phase change occurs. At no time should the material exist in a liquid state. (A liquid could run down the length of the tube depending on the orientation of the sensor.) Furthermore, the solid state can form and be utilized as a deposited coating on the chamber’s inner surface. Upon heating the ensemble in a furnace, nuclear reactor, or other heated zone, the material will begin to off gas through sublimation and the gas will travel down the length of the tube, in vacuo, through random motion until deposition begins to occur on the inner wall; see Fig. 1. This creates a clear and defined line where the desired temperature measurement is located. If an opaque capsule is used, an x-ray or computed tomography (CT) scan of the STS can be performed as a post-examination step to determine whether the deposition has occurred. Through the selected means, this location can be equated with temperature at an exact location. If a symmetrical temperature profile is expected, the deposition locations are also symmetric in nature and will be auto-centered to the temperature profile on either extremity of the long narrow tube. Furthermore, individual tubes with unique materials per each can be bundled together to provide a complete temperature profile of the heated zone. The phrase, “printing temperatures,” is starting to be utilized for this process.

Currently, temperature gradients are measured by staggered thermocouples (or similar) followed by interpolating between any two thermocouples along a temperature gradient. Interpolating and/or extrapolating can lead to large uncertainties, especially if thermocouples are far enough apart from each other.² The sublime sensor helps resolve these potentially large uncertainties by pinpointing exactly where a temperature of interest is located along any temperature gradient. This is seen as a paradigm shift of how temperature is measured today, i.e., measuring locations of desired temperatures, rather than measuring temperatures at known locations. In short, the sublime sensor is essentially printing temperatures where they occur. In addition, the sublime sensor approach also provides greater detail to the validation and verification (V&V) processes needed in today’s modeling and simulation (e.g., MOOSE³).

Regardless of the simplicity of using the STS, there are a multitude of factors working in conjunction and must be discussed in greater detail: first, the phase changes of sublimation and deposition, followed by a broad discussion of the kinetic theory of gases⁴ and mean free path of free particles.⁵ 

The sublime sensor operates under the phase changes of sublimation and deposition. The two phenomena are exactly equal and opposite in principle. When holding pressure constant in deep vacuum, a material being heated below the pressure–temperature triple point, the material will begin to sublimate—a process of transitioning directly from a solid to a gas. The individual atoms of the material will then stay a gas until there is insufficient thermal energy to rebound off another item—either the material containment or another particle. The atoms in the gaseous state will then begin to reduce in number, coalesce together, and de-sublimate (i.e., deposition) back into a solid. This process is very predictable and consistent as the physical phenomenon is a natural occurrence in high vacuum.

The energy required to sublimate a solid into a gas follows Hess’s law; that is, in this case, the energy is directly equal to the amount of energy it takes to vaporize that same material at normal atmospheric conditions, as shown by

where ΔH is the change in enthalpy, either through thermal heating or bonding energy, of the particular mass and s, m, l, v, and g represent solid, melting, liquid, vaporization, and gas, respectively. The “sub/dep” subscript means sublimation and deposition, respectively, and shows that the process is the same in either direction. Finally, the subscript “therm” or “bond” shows whether the change in enthalpy is coming from thermal heating or bonding energy (i.e., latent heat).

There are defined varying degrees of vacuum pressures. In order to obtain a phase change indicative of sublimation, the material must be vacuumed to below the triple point of that material (e.g., see the pressure–temperature phase diagram of a pure element). Defined vacuum pressure ranges can be seen in Table I. Generally, the STS works at high, ultra, or extremely high vacuum ranges but is strongly material dependent.

Once below the triple point of the material, in terms of pressure, the temperature–pressure relationship generally follows the curve based on

or rather

where P is the absolute pressure of the system in mmHg, constants A and B are material dependent, and T is absolute temperature. Usually, constants A and B are found empirically and/or determined via published papers and tables. This equation can be used for pure elements and compounds alike.

If empirical data for sublimation of a solid are not available or the aforementioned curves are insufficient, a method outlined by Poling et al.⁶ shows the extrapolation method of obtaining a material’s vapor pressure and temperature ranges at which sublimation occurs. The method looks beyond a material or compound’s pressure and temperature triple point to estimate vapor pressure and associated temperature of a solid/gas interface (sublimation). The Clausius–Clapeyron equation can be utilized for both a solid and liquid,

where dP/dT is the slope of any standard pressure–temperature diagram. At the vapor pressure, P_vp, the ΔV can be approximated as wholly V_G as the specific volume of the gas is generally several orders of magnitude higher than V_S. Furthermore, V_G can be expressed as

from the ideal gas law. Utilizing both Clausius–Clapeyron equations for solid and liquid and substituting V_G in for ΔV, it can be shown that

where an abbreviated form of Eq. (1) was employed,

The enthalpy change in melting can be estimated from a linear relationship between the heat capacities of both the solid and liquid by

Upon inserting Eq. (5) into Eq. (4) and integrating them together, the result is as follows:

where

The rate at which a solid (or liquid) vaporizes into a gas is strongly dependent on the net outcome of the rate of evaporation vs the rate of condensation.⁷ Furthermore, it is conditionally dependent on the temperature, surface area, surface interface between gas and sublimating media, and rate of heat supplied. The governing equation for rate of vaporization is from Hertz–Knudsen,⁸ 

This gives the rate at which the solid material will vaporize through sublimation. As with any phase change, the temperature of the medium remains constant throughout the process.

Furthermore, on the macroscopic scale, the sublimation rate of change expressed as a mass, $ṁ$⁠, can be expressed by subtracting the rate of condensation from the rate of evaporation. Each is defined as follows:

where, overall, the rate of sublimation is in a state of equilibrium (i.e., $ṁ=0$⁠). This means that just as much material is sublimating as material de-sublimating down the length of the STS chamber.

Gases under vacuum move away from viscous or continuum flow (i.e., friction between particles) and tends toward free motion or the kinetic theory of gases.⁴ The particles (e.g., atoms or molecules) will travel in straight lines until acted upon (much like the game of billiards). The motion is considered constant and random. The distances between particles are considered to be much larger than the physical size of the particles themselves.

From the ideal gas law,

where P is the absolute pressure, V is volume, N is the number of particles, k_b is the Boltzmann constant, and T is the absolute temperature in kelvin, the temperature can be expressed in the form of kinetic energy of the particles,

where m and $v̄$ are the mass and average velocity of the particles, respectively. This form of the ideal gas law gives the average velocity of the particles as

In reality, the gas particles travel at a speed that is spread out over a wide spectrum called the Maxwell-Boltzmann distribution, and this average velocity, $v̄$⁠, is sometimes referred to as the most probable speed, v_p, meaning just as it sounds—the gas particles are moving most probably at this speed. However, it can be shown⁴ that

where v_rms is the root-mean-squared average velocity of the gas particles. The root-mean-squared molecular velocity most closely represents the speed of sound for that medium at any given temperature and is the velocity currently used in this present work.

The mean free path is the statistical average distance a particle travels before interacting with another.⁵ Under vacuum conditions, the mean free path of a gas—or, rather, the atoms and molecules that make up the gas—gets exponentially longer for smaller absolute pressure. Assuming a hard-sphere representation of the gas particles, the mean free path can be expressed as¹ 

where n is the number density of molecules per unit volume and the expression πd² is called the collision cross section. As an example, the mean free path of air at various vacuum pressures can be seen in Table I.

The mean free path can be used to compute the Knudsen number, Kn, which is defined as

where L_C is the characteristic length of the system, here, usually diameter or length of the STS. The Knudsen number ranges from continuum flow (Kn$<$ 0.01) up to free-molecular flow (Kn$>$ 10). For the STS, vacuum pressures are low enough such that 1000 $<$Kn$<$ 100 000 (i.e., Kn ≫ 10) or well into the free-molecular flow regime.

A representative model of the STS, in two dimensions, was modeled and simulated using MATLAB. The model follows the guidelines of the kinetic theory of gases and mean free path explained above. The governing equation is a time-stepping scheme as follows:

where T is the local temperature, v is the velocity of the particle dependent on local temperature from Eq. (12), Δt is the scheme time step, and σ is a relaxation or scaling factor. The bold typeset represents a vector of the individual positions being tracked per each particle.

The local temperature of each particle is then updated with

where a peak maximum temperature, T_m, is set at the center of the domain and ambient temperature, T_a, on either extremity. Any temperature gradient in the y-direction was deemed negligible, and the local x-temperature was assigned as the y-temperature [see Eqs. (19) and (20)]. If ever the local temperature of a particle, T(n), equates below the threshold temperature set by Eq. (2b), then the final position is recorded for that particle.

The domain and boundary conditions were defined as

and

such that there is a long, slender rectangular mesh (i.e., L × W, with L ≫ W) where simulated mass, m, could move about freely, but not leave the domain space (i.e., mass flux, ϕ, over time and area.) As shown in Eq. (18), a cosine squared temperature profile, T, was defined along the length of the mesh in the x direction. An appropriate pressure was selected (in the ranges of either high or ultra-high vacuum from Table I) to define the sublimation/deposition temperature, T_sub/dep, of the simulated material; the results were computed from Eq. (2b).

One thousand representative “particles” of gas (i.e., atoms or molecules) were released at random angles from 0° to 360° at the geometric center of the rectangular mesh. The inner walls of the STS chamber were assumed to be rough compared to the size of the particle. Thus, a diffuse reflection of the particles off the walls was used in the model and simulation. After each instance a particle would interact with the boundary of the mesh wall, a new trajectory was determined. Particle-to-particle interaction was not implemented into the model. This was considered negligible, as the mean free path was too large to merit this behavior at these low pressures. At any time particles hit the mesh boundary, the particle would physically turn around in an opposing and random direction by means of

where θ is from Eqs. (16) and (17). Simultaneously, the temperature [and therefore the velocity, see Eq. (12)] of each particle was updated. The temperature profile was introduced at the inception of the model start time. As mentioned previously, a sublimation/deposition temperature was set from the capsule’s absolute pressure; if at any time the particle encountered the wall below its deposition temperature, from Eq. (2b), the final position of the particle was recorded.

This simulation and model were applied using material properties of antimony (Sb) vapor at the conditions defined in Table II. The general objective of the model and simulation was to observe how the free particles behave in long slender tubes. The top of Fig. 2 shows the antimony atoms in the moment just after sublimation occurs. The bottom part of Fig. 2 shows the final state of all the particles after traveling down the length of the mesh sufficient enough to reach the deposition temperature defined in Eq. (2b). Figure 3 shows that, overall, the atoms behave as expected. A large proportion of the free particles solidify onto the wall of the rectangular mesh just beyond the threshold temperature, and a few particles traveling at oblique angles to the wall solidify further down. (A small uptick in count occurs on the extremities x = −L and L, as the atoms traveling at such an oblique angle to the wall naturally solidify on the end of the generated mesh.)

At pressures of 10⁻⁴ Pa [7.5(10⁻⁷) Torr], the antimony threshold temperature between sublimation and deposition is 277°C. From Eq. (18), T (0.42 m) = 277°C.

As a last note to this model and simulation effort, it should be pointed out that the auto-centering capability of the STS is unprecedented. The ability to find the exact “center” of a symmetrical thermal gradient is a significant capability and major selling point of this sensor, and the histogram in Fig. 3 shows just that.

For testing purposes, both a single core and multi-core (bundle) have been constructed and demonstrated.

A long quartz tube with 6 mm inside diameter was utilized in the build of the first STS. Pure zinc wire—with 99.99% purity—was placed at the lengthwise center of the tube. The absolute atmosphere inside was vacuumed down to 10⁻⁴ Pa [7.5(10⁻⁷) Torr] and was then sealed off. This assured that a temperature location of 320°C would be measured/located inside the furnace test run. The STS ensemble was then placed into a large tube furnace, open on both ends. In the top of Fig. 4, a small section of zinc wire can be seen before the STS was inserted into the furnace.

A furnace cycle was then run with a ramp to 550°C at 5°C/min and held there for 1 h, followed by an identical ramp rate back to room temperature. Upon completion, the STS was removed from the furnace, and zinc material deposits could be identified without difficulty on either end of the quartz tube—just as the model had predicted. These deposits of zinc defined the temperature locations of 320°C on either side of the symmetrical furnace profile when the furnace set point is 550°C. The bottom of Fig. 4 shows the left side of the material deposits in the STS sensor on the inside wall of the quartz tube. The sensor output is measured by reading the beginning of the material deposit coming outward from the center of the sensor (from right to left in this figure). The deposit begins at just above 26 cm (10.5 in.). The overall outcome of the material spread would naturally depend on the set point of the furnace; a wider spread means either a hotter furnace or a flatter temperature profile and vice versa for a narrow spread of material deposits.

This successful proof-of-concept of the STS showed that material properties could be used to measure temperature locations in a relatively easy, straightforward manner. The end user simply inserts the sensor into the thermal gradient of interest, runs a heat cycle, and measures the distance to their temperature of interest. In Sec. IV B, the STS will be shown to characterize the whole temperature profile.

The accuracy and precision of the STS were then measured extensively by running the test over again several more times. The results of these successive tests can be seen in Table III. Separate effects of tests of the temperature measurement location vs tube absolute pressure, inner diameter of the quartz tube, and repeatability (re-usability) of the sensor are shown. The parameters measured were the distance between symmetrical deposition locations. This test was then repeated multiple times and the results have been averaged, followed by a standard deviation of the distances. Finally, standard error was calculated to determine how good a repeatability is if these specific sensors could be used again and again. Each of the parameters is discussed in greater detail below.

The absolute pressure in each tube was carefully measured to the same value in four different similar builds of the STS. Problems with depressurizing chambers such as the STS capsule are real leaks, virtual leaks, pump back-streaming, permeation through the wall of the chamber, out gassing, diffusion, and desorption. These issues were all carefully monitored with precision instrumentation while vacuuming each sensor over long periods of time.

The time it takes to vacuum down each individual STS grows exponentially—though the volume is very small—as a long thin tube in such a high vacuum turns to molecular flow. The molecular volume flow rate, q, can be expressed by Knudsen’s equation⁴ 

where R is the universal gas constant, T is the absolute temperature, d is the inner diameter of the tube and l is the length, and ΔP is the pressure difference between any two points along the tube. The factor (2 − f)/f is representative of the number of atoms crossing over any cross section of the long thin tube, driven by the density gradient along the tube. The factor (2 − f)/f can also be seen as a ratio of the mean distance an atom traveled based on the last diffuse or specular reflection occurred.

However, the molecular conductance, C, of the atoms and molecules can be approximated closely with

where d and L are the inner diameter and overall length of the tube, respectively; both d and L are in units of cm. Note that the absolute pressure is not a factor in molecule conductance. This means that the random motion of the particles is unconcerned with the absolute pressure, and evacuating the tubes comes down to a statistical approach of the number of atoms removed from the system.

A turbo and roughing pump were utilized, in series, to evacuate the chamber of each STS. A vacuum gauge was placed on each end of the STS through valving and tees. As elevated temperatures have an exponential effect on baking out a chamber, a heat gun was run along the length of the STS to aid the desorption and diffusion of the inner walls of the STS. Care was taken to not elevate the temperature too high or quickly to avoid perturbation of the source material.

The final absolute pressure before sealing off the sensor was 10⁻⁴ Pa [7.5(10⁻⁷) Torr]. Each sensor was then run through a furnace cycle similar to that described above but in a much smaller tabletop furnace. This gave a temperature profile of 513 mm (20.2 in.) distance between 320°C on either side of the tube furnace. Standard deviation for the set and standard error for each sensor were 3.67 mm (0.144 in.) and 2.1 mm (0.083 in.), respectively.

The inner diameter of the quartz tubing was then varied between five different sensors by 1 mm (0.040 in.), 1.6 mm (0.063 in.), 3 mm (0.118 in.), 4 mm (0.157 in.), and 5.6 mm (0.220 in.), respectively. The same absolute pressure was then achieved as described above. Each sensor was then run through a furnace cycle. The outcome of the mean temperature location was 717.6 mm (28.25 in.), with standard deviation and standard error at 8.3 mm (0.328 in.) and 3.7 mm (0.147 in.), respectively. These results indicate that the inner diameter of the STS overall build has very little effect on the accuracy and precision of the reading.

The STS is reusable, where the same STS sensor can repeatedly be used, reinserted end for end to reset the material to the center, and then used again. A test to measure repeatability was performed on a single STS sensor. Zinc material was used as the core of an STS and vacuumed down to 10⁻⁴ Pa [7.5(10⁻⁷) Torr]. The sensor was then placed in the furnace and heated to a recorded temperature of 320°C. The end of the sensor tubing was then placed within the furnace just beyond halfway to drive the deposition location to the center. The tube was then flipped over—end for end—and the other end placed similarly within the furnace—driving the material at that deposition location to the center. The whole process was then started over again, each time recording the material spread where 320°C was located within the furnace isotherms. Repeatedly, the sensor would return to the same temperature location as before. The mean dimension between smears was 712.8 mm (28.0625 in.), with standard deviation and standard error calculated as 3.2 mm (0.125 in.) and 1.6 mm (0.0625 in.), respectively. This gives a standard error over a mean percentage of 0.22%.

The standard error as a percentage of the mean distance between deposition locations can be seen in Table III. This shows by simple relative means the absolute nature of specific temperatures being measured over large distances to a very accurate and precise location. Each test shows a relatively low variation in the outcome of the sensor. Further understanding and research of the sensor will only improve these results.

A multi-core build of the STS was designed to measure temperatures at different locations simultaneously. The concept is that one bundle of STS sensor cores could then be used to curve fit a complete temperature profile. By way of an example, a cartoon schematic in Fig. 5 shows four (4) sublime sensors bundled together to measure the maximum temperature profile of a nuclear reactor core. The original material location at the center of each tube and the final destination of each material are shown. The final position of the material can then be utilized to curve fit the temperature profile and find the absolute maximum temperature of the reactor core.

A test using zinc, tellurium, and antimony was carried out in a horizontal furnace at 700°C. This was achieved by vacuuming each material in its own quartz tubing down to an appropriate pressure {≈10⁻⁴ Pa [7.5(10⁻⁷) Torr] per tube}, coinciding with the desired temperature per material. The entire bundle of sensors was then co-located inside an open-ended, horizontal tube furnace. Upon completion of the test to 700°C, the sensor bundle was removed from the furnace. Figure 6 is a close-up view of the left side of the sensor, but the material deposits could be seen symmetrically on either side of the tubing—showing the symmetrical nature of the temperature profile. The final position of zinc, tellurium, and antimony was where the temperature of 320, 350, and 550°C was located, respectively.

Material interaction within the gaseous state is highly susceptible and should be largely mitigated. As mentioned above, using non-volatile materials for the housing—such as quartz or alumina—can greatly reduce material interactions. Using different STS materials inside the same tube can lead to gaseous mixtures and interaction that will behave erratically—losing the temperature location. It is recommended to use low enough vacuum pressures as to maximize mean free path or to use materials with sublimation temperatures greater than ∼500°C—although different construction methods may reduce this value, and it is a topic of further study for a later date.

The STS has shown to be a novel way of measuring temperature. The process of sublimation is well documented, but this is the first time to utilize sublimation as a means of measuring temperature. By looking for an actual temperature location rather than a varying temperature at a specific location helps characterize temperature profiles of hard to reach or defined places. A model was formed and validated with experimental work. From the experimental work, the accuracy of the temperature readings has shown to be within 1–3 mm over long ranges of 1–2 m. Further work is to improve the precision of the temperature readings, to correlate sublimation/deposition grains/granules with absolute maximum temperature and/or maximum temperature flux received, and to miniaturize the sensor through consolidating all materials into one tube or through advanced manufacturing processes, such as printing.

This work was supported by the U.S. Department of Energy under DOE Idaho Operations Office Contract No. DE-AC07-05ID14517. Accordingly, the U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript or allow others to do so, for U.S. Government purposes. This information was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. References herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.