Een niet-stationaire naaldmethode (warmtegeleiding, warmtecapaciteit, contactweerstand)

Precise numerical data on the thermal properties of granular materials are significant not only for fundamental research but also for a variety of applications in agriculture, science and technology. In agricultural research, the thermal conductivity and the heat capacity of the soil are relevant quantities in calculation of the heat balance at the soil surface. Thermal properties and differences in them wider various conditions can provide useful information too on the freezing process and structure of frozen soil.

To investigate these thermal properties of granular materials, the research reported in this thesis dealt with the development of a proper measuring device for thermal conductivity and heat capacity of these substances. The well known non-stationary needle-method was re-examined to improve theoretical treatment, construction of the measuring probe (the needle) and instrumentation. The study also provided a better physical understanding of contact resistance between the needle surface and the surrounding medium. The research achieved simultaneous measurement of thermal conductivity (λ), volumetric heat capacity (ρc) and contact resistance (Γ). Soil contains a variable amount of water. During measurement, migration of water might seriously disturb results. Thermal conductivity and heat capacity are defined for steady state, where possible heat effects resulting from migration of water (heat transfer by fluid flow or by transport of latent heat) are absent. To meet this complication, the total measuring time must be very short (maximum 300 s) and the final temperature rise very small (maximum 0,5°C). With these precautions, any transport of moisture was sufficiently suppressed to make the resulting effect on the steady-state thermal parameters negligible.

We distinguished the approach where volumetric heat capacity (pc) was pre viously known , and the approach where this quantity was previously unknown. In most laboratory measurements, the first approach applies with reasonable accuracy. In many outdoor measurements (especially in agricultural research), pc is previously unknown. The experiments dealt with well defined model materials, i.e. dilute agar gels, dry and wet glass beads and dry and wet silver sand. Agar gels were chosen because their thermal conductivity and volumetric heat capacity were almost equal to those of pure water. Moreover the contact resistance between the needle surface and the agar gel can be assumed to he zero.
An outline of the experimental set-up and of the measuring sensor are given in 3.2. The needle-shaped sensor consisted of a double-fold constantan heating wire and the heat junction of a manganin-constantan thermocouple which were carefully fitted into a stainless-steel envelope. To fix the position of the heating wire and the thermocouple in the cylindrical envelope and to prevent electrical contact, the remaining space was filled with silicon rubber. The diameter of the needle was 1 mm and its length 210 mm. As reference element the cold junction of the thermocouple was fitted into another stainless-steel cylinder, completely equal to the sensor needle (but without the heating wire). The measuring sensor and the reference element were fitted into the axes of the measuring cylinder and the reference cylinder, respectively. The dimensions of these cylinders were equal (both of length 240 mm and of diameter 120 mm). The two cylinders were filled with the material to be examined and were placed in a constant-temperature liquid bath. The electrical current for heating was provided by a direct-current source. During the measuring time of several minutes, the temperature in the needle was recorded once per second. Through a very sensitive amplifier, the data were recorded on paper tape by a data logger. The conditions that the measuring time must be of the order of 300 s and that the total temperature increase may not exceed 0,5°C, implies that the temperature rate near the end of the measuring time will be less than 0,001°Cs ^-1 . The measuring device must meet very high requirements to detect such small temperature increases . Chapter 6 describes the total set-up of the measuring device in detail and the equipment used to prepare a well defined dry or wetted granular structure.

Chapter 2 reviews literature on the non-stationary wire method, the non- stationary needle method and the various types of needle sensors. The simple line-source analysis (reproduced in 2.3) is still often used for theoretical description of the temperature-time curve in the measuring needle and in the surrounding medium. However this system of analysis is not sufficiently accurate to allow simultaneous measurement of thermal conductivity λand volumetric heat capacity ρc, and it is even impossible to measure contact resistance Γin this way. More accurate results may he obtained from the theoretical approach of Blackwell (1954), Jaeger (1956) and de Vries & Peck (1958a), who started from a cylindrical model of the measuring needle (2.4), which accounted for the different thermal properties of needle and surroundings. Size of the needle and its components, and contact resistance are also introduced in that model. The approach of the three groups of workers is very similar mathematically. There were differences in the composition and the thermal properties of the cylinder and in the position of the heating wire and of the temperature sensor.

Chapter 3 develops an adapted model for the measuring needle, based on that of Jaeger and called the modified Jaeger model. It fitted well into the aims of our study and was developed in two steps. First, we introduced some simplifying
assumptions on the measuring needle. The double-fold heating wire was replaced in our model by a cylindrical core; around that core the silicon rubber was placed as a coaxial cylinder enveloped in turn by the outer steel cylinder. The hot junction of the thermocouple was assumed to be infinitely small and its position P was given by the effective radial distance R to the axis of the system. Second, we identified our measuring needle with the solid, perfectly conducting cylinder of the original Jaeger model and so defined the modified Jaeger model (3.4). Introduction of the radial distance R facilitated estimation of the real contact
resistance Γbetween the needle surface and the surroundings. The internal resistance Wℓ,i between P and the wall of the needle could then be subtracted from the measured total heat resistance Γℓ,o reliable value for Wℓ,i was obtained with agar gels, for which real contact resistance Γwas zero (chapter 7).

In chapter 4 numerical mathematical analysis of the observed temperature-time curve is described. A time correction t ₀ had to be introduced for definite determination of λ, ρc and Γ. This quantity t ₀ primarily depended on the resulting measurements. Different methods of calculating t ₀ are discussed in 4.4. One proved best in giving much more precise and reliable results than the others. Thermal conductivity λand contact resistance Γwere then evaluated for "ρc known" (4.5). In 4.6 the method for "ρc unknown" was developed and allowed for simultaneous estimation of λ, ρc and Γ. This simultaneous estimation was approached by repeating the procedure of "ρc known" for different values of ρc and for different time intervals. The correct value of ρc was that for which λ(or Γ) was independent of the chosen time interval.

To verify the modified Jaeger model, the quantities λand Γestimated with this model, needed to be checked. The first check was on the effective distance R as calculated from the results of the agar gel measurements. The second was on the quantities λ, Γand ρc as measured for glass beads and silver sand. The means of checking was called the four-regions model (chapter 5). Characteristic for the model was the operation with the Laplacian transforms of the measuring results, seen as the solutions of the Laplacian-transformed differential equations of heat conduction. In the four-regions model, the thermal properties and the size of the needle were better expressed than in the modified Jaeger model. Another advantage of the four-regions model was the way in which the temperature curve was used.
The modified Jaeger model does not use the first part of the temperature-time curve, whereas the four-regions model uses the complete curve (from t = 0). So the two models are more or less independent of each other. In the four-regions model too, the effective distance R can be calculated from test measurements with the agar gels. With this value of R we can derive an equation for λ, ρc and Γ, characteristic for each needle. This equation allows testing of the quantities λand Γfrom the modified Jaeger model and (ρc) _s of the bulk test material for their mutual consistency.
In chapter 7 test measurements with the agar gels are discussed. The values of λwere compared with those of Powell et al. (1966), who measured the thermal conductivity of pure water. The differences were less than 1%. The differences in the values for λ, measured with several needles were not more than 1% either, and the reproducibility of the measurements was within 0,5%. In conclusion, the values of λprobably accurate to within 1%. The test measurements can also be used to calculate the internal heat resistance Wℓ,i and the effective distance R. For each element Wℓ,i and R are characteristic and were measurable to an accuracy of 3 and 0,3%, respectively. For each element, the average relative difference between the value of R from the four-regions model (7.4) and that from the modified Jaeger model (7.3) proved to be less than 0,5%, a remarkably low value that justified continuation of work on the modified Jaeger model. As a result of the slight in fluence of pc on λ(7.5) a rough estimate of ρc was sufficient to calculate an accurate value of λ.

Chapter 8 finally, describes measurements with the glass beads and the silver sand. For thermal conductivity λ, the results corresponded to those for agar gels (so far as comparable). For contact resistance Γ, average reproducibility was about 5% and accuracy was within an average of 10-20%. The influence of diameter of the grains of a dry granular material on the value of Γwas distinct and systematic: Γincreased almost rectilinearly with diameter. The relation of Γto packing density, moisture content and temperature was less clear. Measurements with glass beads saturated with water showed that (within the accuracy) Γwas zero. The simultaneous estimation of λand ρc for these materials resulted in a value for pc accurate to within an average of 5-10%. The four-regions model used the "measured" λand Γto obtain a volumetric heat capacity (ρc) ₄ . The difference between (ρc) ₄ and the original value (ρc) _s averaged not more than 3%. This result provides confirmation of the usefulness of the modified Jaeger model. Deviations were found when the diameter of the grains was of the same magnitude as the diameter of the needle and when the measured system was unsaturated moist glass beads. Simultaneous estimation of λ, ρc and Γaveraged over a group of measurements gave a ρc accurate to within 10-20%. In general, such accuracy is more than adequate for accurate estimation of λand Γ.