Assuming one-dimensional heat flow through the FRM components of the specimen sandwich, a solution will be determined for the case where the temperature of the surfaces of the exposed specimens is increasing/decreasing at a constant rate. We consider a pair of FRM specimens, each of thickness l, with the initial condition that the temperature is constant through the thickness of the specimen and slug, i.e., T(z,0)=0. By symmetry, the mid-plane of the steel slug plate will be an adiabatic boundary, so that we need only consider one specimen and one half of the steel slug plate. Assuming constant properties, the temperature in the specimen must satisfy:
where α=k/C is the thermal diffusivity [m2/s], k is the thermal conductivity [W/(m·K)], C=ρcp is the volumetric heat capacity [J/(m3·K)], and ρ is density [kg/m3], all for the FRM specimen material. The steel slug plate is assumed to have a sufficiently high thermal conductivity that it can be considered to be isothermal at any given time. The "thermal capacity" of the slug plate, per unit area, is taken to be 2H = (areal density of the plate)x(heat capacity of the plate), with the factor of 2 arising from the fact that we need only consider one half of the steel slug plate. The thermal capacity, H, has units [J/(m2·K)].
The boundary condition at the exposed surface of the specimen, z=0, is:
| T(0,t) = Ft | (2) |
where F is the (constant) temperature increase rate having the units [K/s]. The boundary condition at the specimen surface, z=l, which is in contact with the steel slug plate is:
which follows from the fact that the heat conducted out of this face of the specimen must equal the heat absorbed by (one half of) the steel slug plate.
Assuming that the transient (exponentially decaying) terms in the solution, which depend on the thermal diffusivity of the FRM specimen and time, can be neglected, we arrive at a solution of the form:
The temperature difference across the specimen of the FRM, ΔT, is thus:
Finally, the thermal conductivity of the specimen can be computed as:
If the masses of the slug (MS) and FRM specimen (MFRM) are known, equation (6) can be rewritten as:
where A is the cross-sectional area of the slug (or specimen, 0.152 m by 0.152 m = 0.0232 m2 in our experimental setup), and cpS and cpFRM refer to the heat capacities of the steel plate and FRM specimen in units of [J/(kg·K)], respectively. A similar analysis can be performed for the cooling case (F<0, T(z,0)=constant), and it can be shown that equation (7) applies in this case as well.
Equation (7) was conveniently implemented in a spreadsheet program to determine the effective thermal conductivity from the acquired temperature-time data points. The measured temperature-versus-time series for the slug and for the exterior FRM surfaces were used to compute the instantaneous values of F (∂T/∂t) and ∆T for use in equation (7). Measured heat capacities of the 304 stainless steel (Figure 2) and FRMs (Figure 4) as a function of temperature and mass losses versus temperature for the FRMs (Figure 5) were used to further refine the parameters used in equation (7). The resulting values of k will be graphed against the mean specimen temperature ([T(0,t)+T(l,t)]/2) in the results that follow. While this paper presents preliminary results to indicate the feasibility of utilizing the slug calorimeter method to evaluate the thermal performance and specifically the effective thermal conductivity of FRMs, an expanded uncertainty analysis could be conducted based on equation (7) and the law of propagation of uncertainty.12 Assuming that the heat capacities of the steel slug and the FRM specimen are fairly well known and that lengths and masses can be measured with less than 1% uncertainty, the uncertainties in the thermocouple measurements at high temperatures, which are used to calculate both F and ∆T in equation (7), will be the most significant contributors to the overall uncertainty. Thus, the uncertainty can be reduced simply by applying equation (7) over a larger time interval. For example, assuming an optimistic uncertainty of 1 ºC for the thermocouple readings at high temperatures, changing the sampling frequency from 1 min to 5 min reduces the estimated uncertainty in the effective thermal conductivity from about 25 % to about 5 % for thermal conductivities computed in the temperature range of 400 ºC to 700 ºC during heating. During cooling, using a 25 min interval to calculate the effective thermal conductivity reduces its uncertainty to about 8 % from a value of about 40 % for 5 min intervals.