Determination of thermal conductivity of powders in different atmospheres by differential scanning calorimetry

Abstract

We have developed a new method to measure the thermal conductivity of powders by differential scanning calorimetry that works with masses in amounts as low as tens of mg. The method is based on that used by Camirand to determine the thermal conductivity of materials in the form of thin sheets but introducing a hemispherical pan to contain powders in such a way that the issue of heat transfer is reduced to a one-dimensional problem. The modification of the method was successfully validated on obtaining identical results in determining the thermal conductivity of a commercial silicone with both Camirand’s method and the modified method. We have also tested our method with materials that, in bulk, cover a wide range of thermal conductivities and have performed the experiments with several atmospheres and reference metals. The results are consistent with already published general trends in that they confirm that thermal conductivity of powders is mainly governed by thermal conduction through the surrounding gas.

Appendix: Slope of the DSC signal during melting

During melting, the metal reference remains at the melting temperature, T _M, whereas the DSC furnace is heated at a constant heating rate, β. Let us first demonstrate that any point r of the powder contained in the hemispherical holder (Fig. 1) will experience a constant heating rate according to the formula:

$$ T\left( {r,t} \right) = f\left( R \right) + \frac{{\beta_{\rm e} }}{{R_{\rm e} }}Rt, $$

(4)

where t is time, f is a time-independent function and R is the thermal resistance of the powder inside the radius r, i.e.:

$$ R = \frac{1}{2\pi \kappa }\left( {\frac{1}{{r_{i} }} - \frac{1}{r}} \right), $$

(5)

where κ is the powder thermal conductivity. Notice that the heating rate of the powder varies from zero at r _i to its maximum value β _e at r _e.

We must simply verify that T(r,t) of Eq. (4) satisfies the heat transport equation that, given the spherical symmetry of the system, can be written as:

$$ \frac{2}{r}\frac{\partial T}{\partial r} + \frac{{\partial^{2} T}}{{\partial r^{2} }} = \frac{\rho c}{\kappa }\frac{\partial T}{\partial t}, $$

(6)

or, substituting r by R in the partial derivatives:

$$ \frac{1}{2\pi }\frac{1}{{r^{4} }}\frac{{\partial^{2} T}}{{\partial R^{2} }} = \rho c\frac{\partial T}{\partial t}, $$

(7)

where ρ and c are the density and specific heat of the powder, respectively. This verification is straightforward after the introduction of Eq. (4) into (7), and is left for the reader.

Once demonstrated that the powder in contact with the pan (at r = r _e ) is heated at a constant rate, the value of β _e will be calculated with the assumption that we can neglect both the pan resistance and its contact resistance with the DSC sensor (i.e. T _S = T _e—see Fig. 1). We must simply impose a power measured by the DSC, $\dot{Q}_{{_{\text{DSC}} }}$ equal to the heat that enters into the powder through the pan’s inner surface, $\dot{Q}_{\text{e}}$. $\dot{Q}_{{_{\text{DSC}} }}$ can be easily calculated through the formula:

$$ \dot{Q}_{\text{DSC}} = \frac{{T_{\text{REF}} - T_{e} }}{R}, $$

(8)

where R is the thermal resistance of the DSC sensor, T _REF is the reference temperature that changes at the programmed heating rate:

$$ T_{\text{REF}} = T_{0} + \beta t, $$

(9)

and T _e is given by Eq. (4) at r = r _e. We obtain:

$$ \dot{Q}_{\text{DSC}} = A + \frac{{\beta - \beta_{\text{e}} }}{R}t, $$

(10)

where A is time-independent. On the other hand, $\dot{Q}_{\text{e}}$ is proportional to the temperature gradient at r _e :

$$ \dot{Q}_{\text{e}} = 2\pi r_{\text{e}}^{2} \kappa \left. {\frac{\partial T}{\partial r}} \right|_{\text{re}} . $$

(11)

Introduction of Eqs. (4) and (5) into Eq. (11) gives:

$$ \dot{Q}_{\text{e}} = B + \frac{{\beta_{\text{e}} }}{{R_{\text{e}} }}t, $$

(12)

where B is time-independent. Now, the condition $\dot{Q}_{\text{DSC}} = \dot{Q}_{\text{e}}$ delivers the value of β _e:

$$ \beta_{\text{e}} = \frac{{R_{\text{e}} }}{{R + R_{\text{e}} }}\beta . $$

(13)

And, finally, the slope of the DSC signal can be obtained after the substitution of β _e in Eq. (10):

$$ \frac{{{\text{d}}\dot{Q}_{\text{DSC}} }}{{{\text{d}}T_{\text{REF}} }} = \frac{{{\text{d}}\dot{Q}_{\text{DSC}} }}{{{\text{d}}(\beta t)}} = \frac{1}{{R_{\text{DSC}} + R_{\text{e}} }}. $$

(14)

If the contact resistance between the pan and the DSC sensor were not negligible, the result would be 1/R + R _c + R _e.