A granular thermometer

Abstract

The behaviour of granular materials is known to be strongly affected by the granular temperature, which is a measure of the intensity of particle vibration. However, no experimental technique exists to measure the granular temperature and its variation inside arbitrary media during deformation. Here, we present a new experimental technique for measurement of this granular temperature field by tracking tracer particles over time, using X-ray radiography with two simultaneous orthogonal source/detector pairs. We show that errors in particle tracking lead to systematic measurement biases. The main sources of these biases are identified and methods for their quantification are developed so that unbiased results can be calculated. The new granular thermometry technique is validated against discrete element simulations. Finally, an experimental example is shown for the case of a vibro-fluidised system, where we find that the granular temperature is highest at the free surface.

Graphical Abstract

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary material 1 (DOCX 24045 kb)

Supporting information appendix (SI)

1.1 Appendix 1: Coarse-graining in cylindrical coordinates

Coarse-graining is an established technique for converting discrete particle measurements to continuum fields (see e.g.[45, 49, 50]). However, it is usually employed in Cartesian coordinate systems, whereas the experimental configuration of this paper is better suited to cylindrical coordinates $\varvec{r}=(r,\theta ,z)$, as shown in Fig. 7. We therefore outline below the coarse-graining process and the resulting granular temperature measurements in cylindrical coordinates.

The calculation is based on the mathematical formulation in [45, 49, 50]. We first introduce the coarse-graining function ${\mathcal {W}}$,

$$\begin{aligned} {\mathcal {W}}({\mathbf {r}}) = \frac{1}{2\pi \sigma _{r} \sigma _{z}}\exp {\left( -\frac{r^{2}}{2\sigma _{r}^{2}}-\frac{z^{2}}{2\sigma _{z}^{2}}\right) }, \end{aligned}$$

(11)

where $\sigma _{r}$ and $\sigma _z$ are the coarse-graining widths in the radial and vertical directions, respectively. Note the lack of $\theta$ dependence in (11), because we are interested in temperature fields only in the (r, z) half plane (and also independent of time). The normalisation is chosen to ensure that the coarse-graining function integrates to 1, i.e.

$$\begin{aligned} \iiint _{V} {\mathcal {W}}({\mathbf {r}})r\ dr\ dz\ d\theta = 1. \end{aligned}$$

(12)

It is then possible to define the coarse-grained solid volume $V(\varvec{r})$ as

$$\begin{aligned} V({\mathbf {r}})&= \sum _{k=1}^n\sum _{i \in S} \frac{\pi }{6} d_{i}^3 {\mathcal {W}}({\mathbf {r}}-{\mathbf {r}}_{i}(t_k)), \end{aligned}$$

(13)

where $\varvec{r}_{i}(t_k)$ represents the position of particle i at time $t_{k}$, and $d_{i}$ is the diameter of particle i. The sum $i\in S$ represents summing over all particles, and $k=1$ to n summing over n time steps.

We can then use this coarse-grained volume to obtain the steady-state average velocity field in the rz half-plane. For the radial component, the velocity $v_{r}(\varvec{r})$ is given by

$$\begin{aligned} v_{r} ({\mathbf {r}}) = \frac{1}{V(\varvec{r})}\displaystyle \sum _{k=1}^{n}\sum _{i \in S} \frac{\pi }{6} d_{i}^{3} {\mathcal {W}}({\mathbf {r}}-{\mathbf {r}}_{i}(t_k))v_{r,i}\left( t_{k}\right) , \end{aligned}$$

(14)

where $v_{r,i}(t_k)$ is particle i radial velocity at time $t_k$. This is calculated by converting the Cartesian positions from the tracking algorithm to cylindrical coordinates and then calculating the corresponding velocities. In a similar fashion, we can calculate an approximation of the square of the radial velocity as

$$\begin{aligned} v_{r}^{sq} ({\mathbf {r}}) = \frac{1}{V(\varvec{r})}\sum _{k=1}^{n}\sum _{i \in S} \frac{\pi }{6} d_{i}^{3} {\mathcal {W}}({\mathbf {r}}-{\mathbf {r}}_{i}(t_k))v_{r,i}^2(t_k). \end{aligned}$$

(15)

By definition, the granular temperature $T_{g}$ is the trace of the velocity variance tensor, in which the diagonal components are the variances of the average velocity of each component. $T_{g} = \frac{1}{3}(T_{g,r} + T_{g,\theta } + T_{g,z}) = \frac{1}{3}({\mathrm {Var}}(v_{r}) + {\mathrm {Var}}(v_\theta ) + {\mathrm {Var}}(v_z))$. Because the granular temperature of each component is the variance of the velocity field, we can define the coarse-grained granular temperature in the radial direction as

$$\begin{aligned} T_{g,r}({\mathbf {r}})= v_{r}^{sq} ({\mathbf {r}}) - (v_{r} ({\mathbf {r}}))^2. \end{aligned}$$

(16)

Analogously, components $T_{g,\theta }$ and $T_{g,z}$ are calculated by substituting the relevant velocity terms.

1.2 Appendix 2: Missed collisions

Sampling particle locations at discrete times necessarily means that not all collisions are observed. To account for this, we begin by considering the definition of the diffusivity, D, where if there is no net movement of the material, we have

$$\begin{aligned} D = \frac{{\mathbb {E}}[\Delta { x}^{2}]}{2\Delta { t}}, \end{aligned}$$

(17)

where ${\mathbb {E}}[\Delta x^2]$ is the mean square displacement of a large number of grains after a time $\Delta { t}$. For the case of uniform shear in the x direction, it has been shown [51] that the effect of the shear is to change this relationship to

$$\begin{aligned} D \propto \frac{{\mathbb {E}}[\Delta { x}^{2}]}{2\Delta { t}^{3/2}}. \end{aligned}$$

(18)

The constant of proportionality must therefore have the units of square root of time. For convenience, we assume that this time scales with the shear time, $|\dot{\gamma }|^{-1}$, so that we can write

$$\begin{aligned} D = \sqrt{a_{1}}\frac{{\mathbb {E}}[\Delta { x}^{2}]}{2\Delta { t}^{3/2}\sqrt{|\dot{\gamma }|}}, \end{aligned}$$

(19)

where $a_{1}$ is a non-dimensional fitting parameter. Recalling (6), we can define the measured granular temperature as

$$\begin{aligned} \widetilde{T}_{g} = \frac{1}{3}{\mathbb {E}}[||\varvec{v}'||^2] = \frac{1}{3}\frac{{\mathbb {E}}[\Delta x^2]}{\Delta { t}^2}. \end{aligned}$$

(20)

Substituting (19) in (20) gives

$$\begin{aligned} \widetilde{T}_{g} = \frac{2D}{3\sqrt{a_{1}\Delta { t}/|\dot{\gamma }|}}. \end{aligned}$$

(21)

Additionally, we can include some recent micromechanical measurements of $T_{g}$ [52] and D [4, 53], which give at steady state

$$\begin{aligned}&T_{g} \propto d|\dot{\gamma }|\sqrt{P/\rho }, \end{aligned}$$

(22)

$$\begin{aligned}&D \propto d^2|\dot{\gamma }|/\sqrt{I}, \end{aligned}$$

(23)

where d is the particle diameter, P the pressure, $\rho$ the particle density and $I=\dot{\gamma }d\sqrt{\rho /P}$ is the inertial number. From these, we can deduce that

$$\begin{aligned} \frac{T_{g}}{D} \propto \frac{|\dot{\gamma }|}{\sqrt{I}}. \end{aligned}$$

(24)

Substituting this relationship in (21) gives

$$\begin{aligned} \frac{T_{g}}{\widetilde{T}_{g}} \propto \frac{3}{2} \sqrt{a_{1}\frac{\Delta { t}}{t_{i}}}, \end{aligned}$$

(25)

where $t_{i}=d\sqrt{\rho /P}$ is the inertial time. We choose to replace the proportionality with an equality by assuming a translation of the curve in $\Delta { t}$ for the amount of $t_{0}$, rather than a strict proportionality, such that

$$\begin{aligned} \frac{T_{g}}{\widetilde{T}_{g}} = \frac{3}{2}\sqrt{\frac{a_{1}(\Delta { t} + t_{0})}{t_{i}}}. \end{aligned}$$

(26)

When $\Delta { t}=0$ we must have that $\widetilde{T}_{g} = T_{g}$, which yields $t_{0}$, and so by substituting $t_{0}$ this scaling becomes

$$\begin{aligned} \frac{T_{g}}{\widetilde{T}_{g}} = \alpha = \sqrt{a_{1}\frac{9}{4}\frac{\Delta { t}}{t_{i}} + 1}, \end{aligned}$$

(27)

In Fig. 5 we show a best fit of the parameter $a_{1}$ for the DEM data described in Sect. 4.

1.3 Appendix 3: Finite pixel size

Due to the finite size of the pixels on the detector panels, tracer positions cannot, even in principle, be measured perfectly, and thus every measured tracer position bears some randomness. The mathematical treatment of this randomness was developed in [48], and is reproduced here for clarity. The measured location of the particle centroid along the x axis, $\widetilde{x}$, differs from the true particle centroid x by a factor

$$\begin{aligned} \widetilde{x} = x + \tfrac{1}{2}p\xi , \end{aligned}$$

(28)

where p is the pixel size in meters, with $p=0$ representing perfect resolution, and $\xi$ is a random variable uniformly distributed in the domain $[-1, 1]$. The measured component of the tracer’s velocity in the x direction between two frames labelled 0 and 1 is then

$$\begin{aligned} \widetilde{v}_{x} = \frac{\widetilde{x}_{1} - \widetilde{x}_{0}}{\Delta { t}} = \frac{x_{1} - x_{0}}{\Delta { t}} + \frac{p(\xi _{1} - \xi _{0})}{2\Delta {t}} , \end{aligned}$$

(29)

where $v_{x}$ is the finite difference to the actual velocity and distribution of $\xi _{1} - \xi _{0}$ can be calculated by using the equation for compound probability density functions, as shown in [48]. Using this distribution the bias in the measured average velocity ${\mathbb {E}}[\widetilde{v}_{x}]$ is

$$\begin{aligned} {\mathbb {E}}[\widetilde{v}_{x}] = {\mathbb {E}}[v_{x}] + \frac{p}{2\Delta { t}}{\mathbb {E}}[\xi _{1} - \xi _{0}] = {\mathbb {E}}[v_{x}], \end{aligned}$$

(30)

since the expectation value of $\xi _{1} - \xi _{0}$ is 0, and hence no bias is introduced to the measurement of the average velocity. Nevertheless, $\xi _{1} - \xi _{0}$ introduces an uncertainty on the average velocity which is equal to one standard deviation of its distribution function, $\sigma _{\xi _{1} - \xi _{0}}$. Therefore the measured average velocity can be expressed as

$$\begin{aligned} {\mathbb {E}}[\widetilde{v}_x] = {\mathbb {E}}[v_x] \pm \frac{p}{2\Delta { t}}\sqrt{\tfrac{2}{3} - {\mathrm {Cov}}\left( \xi _{1}, \xi _{0} \right) }. \end{aligned}$$

(31)

The resulting error in the measured granular temperature component $\widetilde{T}_{g}$ in the x direction is

$$\begin{aligned} \widetilde{T}_{g_{xx}} = T_{g_{xx}} + \left( \frac{p}{2\Delta { t}}\right) ^{2}\left( {\mathrm {Var}}(\xi _{1}) + {\mathrm {Var}}(\xi _{0}) - {\mathrm {Cov}}\left( \xi _{1}, \xi _{0} \right) \right) , \end{aligned}$$

(32)

which, in the case that $\xi _{1}$ and $\xi _{0}$ are independent, simplifies to

$$\begin{aligned} \widetilde{T}_{g_{xx}} = T_{g_{xx}} + \frac{1}{6}\left( \frac{p}{\Delta { t}}\right) ^2. \end{aligned}$$

(33)

Because this bias is isotropic, we can additionally state that the overall granular temperature term can be expressed as

$$\begin{aligned}&\widetilde{T}_{g} = T_{g} + \Delta T_{g}^{\mathrm {res}}, \end{aligned}$$

(34)

$$\begin{aligned}&\Delta T_{g}^{\mathrm {res}} = \frac{1}{6}\left( \frac{p}{\Delta { t}}\right) ^{2}. \end{aligned}$$

(35)

1.4 Appendix 4: Absorption heterogeneity

The nature of granular materials is that they are generally heterogeneous at the particle scale. Because of this, radiographs of granular materials at this scale typically show heterogeneous absorption patterns, allowing us to observe the internal packing structure. This heterogeneity, however, introduces uncertainties in our measure of the particle centroids, as our definition of the tracer particle being the darkest disk in the image is only true for a homogeneous background.

Fig. 9

Example of a tracer in an artificial radiograph. The tracer is located at the centre of the image, and due to its high attenuation coefficient is the darkest region in the image. The heterogeneous background within intruder’s boundary causes skewing of intruder’s center towards the darker regions; right of intruder’s true centre in this case. The bulk particles are 15 cm (50 diameters) deep

In a dense granular packing, the effect of the heterogeneity on the accuracy of the tracer tracking algorithm depends on; (a) the particle size, (b) the number of bulk particles along the ray path, (c) the granular packing, and (d) the relative attenuation coefficient of the two materials. Here, we limit ourselves to tracers of the same size as the bulk particles, and to a single relative attenuation coefficient typical of a steel tracer in a glass medium. In this case, to approximate the error associated with the heterogeneity we have generated a series of high resolution (400 px/mm, chosen to eliminate detector resolution dependence from this analysis) artificial radiographs from DEM simulation data. Each radiograph represents a different thickness of glass beads along the path of the X-ray beam, and a single identical steel tracer is placed in front of each configuration. An example of these radiographs is shown in Fig. 9, with the tracer particle at the centre of the image.

For 25 different thicknesses, ranging from $n=H/d=1$ to 80 particle diameters deep, a tracer particle was placed at a known location and then the tracking algorithm was used to search for the particle location. This was repeated for 10,000 tracer locations uniformly distributed on a $100\times 100$ grid in each radiograph, and the error between the true location and the measured location was fitted with a zero mean Laplace distribution with non-dimensional width b (non-dimensionalised by the particle diameter). The Laplace probability density function (PDF) as a function of the fraction of the intruder diameter is given by

$$\begin{aligned} f(\chi ) = \frac{1}{2b}{\mathrm {e}}^{-\frac{|\chi |}{b}}, \end{aligned}$$

(36)

where $\chi$ is the normalized distance in the horizontal direction measured from the tracer’s center, and b is the non-dimensional width of the Laplace distribution calculated as a fitting parameter for each depth of bulk material. The dependence of this width on the packing depth is shown in Fig. 10 and can be well fitted by

$$\begin{aligned}&b\left( n\right) =a_2(1-{\mathrm {e}}^{-a_3\frac{H}{d}}), \end{aligned}$$

(37)

with coefficients $a_2=0.0048$ and $a_3=6.8\times 10^{-5}$. The width of the distribution and consequently the bias in the granular temperature asymptotically progresses towards a stable value with increasing bulk particle depth. This suggests that, in an ideal scenario, intruders are always visible even for a very large number of particles in front of them. In reality, however, with increasing depth of bulk particles it becomes more difficult to find the intruder due to the finite dynamic range of the detector panels, which prevent the detection of slight differences in the intensity of pixels.

Fig. 10

Estimating the effect of heterogeneity. Width of Laplace distribution b, in terms of intruder diameter fraction, plotted against number of particle layers in front of the tracer particle. Insets—histogram of error in measurement of position and line of fitted Laplace distribution for different bulk particle depths a $H/d=10/3$ and b $H/d=250/3$

To calculate the error caused by this heterogeneity we follow a similar logic to that in SI 1. The measured coordinate of the tracer can be expressed as the sum of the true value and a discrete random variable $\chi$ multiplied by the diameter of intruder d, as

$$\begin{aligned} \widetilde{x} = x + d\chi , \end{aligned}$$

(38)

where $\chi$ is distributed as in (36). It follows that the tracer’s x velocity component is given by

$$\begin{aligned} \widetilde{v}_x = \frac{\widetilde{x}_{1} - \widetilde{x}_{0}}{\Delta { t}} = \frac{x_{1} - x_{0}}{\Delta { t}} + d\frac{(\chi _{1} - \chi _{0})}{\Delta { t}} = v_{x} + \frac{d}{\Delta { t}}(\chi _{1} - \chi _{0}), \end{aligned}$$

(39)

The compound distribution function of the difference of $\chi _{1}$ and $\chi _{0}$ is symmetric around its zero mean, and does not introduce bias to the measured average velocity field, but introduces the uncertainty

$$\begin{aligned} {\mathbb {E}}[\widetilde{v}_{x}] = {\mathbb {E}}[v_{x}] \pm \frac{d}{\Delta { t}}\sqrt{(\mathrm {Var}(\chi _{1} - \chi _{0}) - 2\mathrm {Cov}(\chi _{1}, \chi _{0}))}. \end{aligned}$$

(40)

The expression for the x component of the measured granular temperature is

$$\begin{aligned}&\widetilde{T}_{xx} = T_{xx} + \Delta T_{xx}^{\mathrm {het}}\end{aligned}$$

(41)

$$\begin{aligned}&\Delta T_{xx}^{\mathrm {het}} = \left( \frac{d}{\Delta { t}}\right) ^2( {\mathrm {Var}}(\chi _{1} - \chi _{0}) - 2\mathrm {Cov}(\chi _{1}, \chi _{0})) \end{aligned}$$

(42)

The effect of this term on the measured temperature field is therefore strongly dependent on the independence of $\chi _{0}$ and $\chi _{1}$. In the case where they are independent (i.e. long sampling times relative to the particle displacement), the covariance approaches zero resulting in the maximum value of bias.

For short sampling times relative to the particle displacement, where the covariance of the two terms can be assumed to be close to the value of the variance, the bias caused by heterogeneity tends towards zero. In the cases reported here, we have assumed that the covariance and variance terms are equal in magnitude due to the short sampling times and similar background conditions in subsequent radiographs and have therefore ignored this error term.

1.5 Appendix 5: Finite sample size

Here we investigate the effect of using the finite sums (4) and (5) to approximate the true mean and variance (equivalently granular temperature) of the velocity field. For completeness, we redefine these estimators below

$$\begin{aligned} \widetilde{\mu }&= \frac{1}{N}\sum _{k=1}^{N} {v}_{k}, \end{aligned}$$

(43)

$$\begin{aligned} \widetilde{T}_{g}&= \frac{1}{N}\sum _{k=1}^{N} ({v}_{k}-\widetilde{\mu })^{2}, \end{aligned}$$

(44)

where it is now assumed that the true velocity measurements $v_{k}$ are used, so that we can study the finite sample size errors in isolation. Assuming these measurements are independent, it is straightforward to show that

$$\begin{aligned} {\mathbb {E}}[\widetilde{\mu }]&= {\mathbb {E}}[v], \end{aligned}$$

(45)

and

$$\begin{aligned} {\mathbb {E}}[\widetilde{T}_{g}]&= \frac{N-1}{N}\left( {\mathbb {E}}[v^{2}]-({\mathbb {E}}[v])^{2}\right) = T_{g} - \frac{1}{N}T_{g}. \end{aligned}$$

(46)

Hence the estimator for $T_{g}$ will, on average, underestimate the true granular temperature by $(1/N)T_{g}$.

Since the number of observations is typically large for the DEM simulation ($N>1000$), this bias can safely be neglected, although may start to become significant when fewer measurements are available.

1.6 Appendix 6: Measuring bulk temperature

Equipartition of energy is known to break down in agitated granular materials [54]. This means that we do not expect our tracer particle to have the same kinetic energy as the materials surrounding it. The kinetic energy in these systems is primarily dissipated through the contacts between grains. Any tracer particle which changes the local contact network may therefore affect the local temperature and bias any measurement. As a result, we do not expect the granular temperature of any tracer particles to be a true measure of the granular medium that would exist in the absence of the tracer particle. Whilst the relationship between the intruder and bulk temperatures are not known, and are the subject of intense research activity [1, 44, 55, 56], we here use DEM simulations to calculate the correction required for a system under simple shear, and use this value to correct our experimental data. In general, we expect this relationship to depend on at least the size, stiffness, density and friction ratios between the bulk and intruder particles.

The DEM simulation performed here is composed of a 3D Lees-Edwards boundary, of dimension $15\times 15\times 15$ mean bulk particle diameters, and sheared at constant volume within a 20% margin of the same bulk nominal values as in Table 1. For this case, we require that the contacts between grains are accounted for with realistic accuracy, and so choose to use a Hertz-Mindlin contact law [57, 58], with reduced glass bead and steel elastic modulii of 10 MPa and 30 MPa, particle densities of each material 2500 and 8000 kg/m$^3$, restitution coefficient of 0.95 and friction coefficient 0.5. The domain is filled with bulk particles of diameter $3\pm 0.3~\hbox {mm}$, with a single tracer particle, such that the total solid fraction is 0.596, which is chosen to achieve the target pressure. For this case, we find that the steel tracer particle has a lower temperature than the bulk particles by a factor of 1.5, and so the values reported in this paper could be divided by this value to recover the granular temperature of the glass beads.

1.7 Video 1

Caption: An overview of how X-ray radiography is used to measure the granular temperature for a vibro-fluidised bed.