Effusivity (thermal effusivity) is a thermal property present in materials of all formats—solid, liquid, paste, powder, and gas. Effusivity measures the rate at which a material can absorb heat and is defined as the square root of the product of the thermal conductivity, density, and specific heat capacity. Specifically, effusivity is the thermal property that dictates the interfacial temperature when two semi-infinite bodies meet (see Sample Thickness). [NOTE—See the final section of this chapter, Definition of Terms and Symbols, for the terms and symbols referred to in this chapter and commonly employed in effusivity technology.]

Every material has a specific thermal effusivity value. Effusivity is sensitive to composition because materials differ in effusivity value (Ws½/m2·K) from 5 for air to 1600 for water to several thousand for advanced composites. As a result, effusivity can differentiate between materials in solid, liquid, and powder formats. In the application of blend evaluation, the blending endpoint can then be identified when the effusivities of samples at different positions in the blender give minimum variation (see Definition of Terms and Symbols). The effusivity of a powder sample is driven by the properties of the particles and the interparticle material (typically air). Because effusivities of pharmaceutical solid materials (typically 150 to 800 Ws½/m2·K) are much higher than that of air, effusivity technology can also identify particle size distribution and density of powders. Because the effusivity of liquids (i.e., water is 1600 Ws½/m2·K) is so much greater than powder, effusivity technology can be used to monitor wet granulation and drying.

A big advantage of effusivity technology is that accurate measurements can be made rapidly and nondestructively. Furthermore, effusivity sensors only require brief contact with any materials investigated. As a result, effusivity technology allows in situ measurements and on-line monitoring of processes in challenging environments. If the material is in motion during the measurement but makes periodic contact with the sensor (as in blending), the measurement is taken only during the periods of contact.

A fast, nondestructive and accurate effusivity measurement can be performed using a transient heat interfacial reflectance MHW technique (see Definition of Terms and Symbols).

The sample is tested by placing the heating elements of the sensor of the apparatus against the surface of the sample (see Definition of Terms and Symbols). A known quantity of electrical current is passed through the heating elements of the sensor for a known time, which results in a temperature rise (less than 5 K) at the sensor/sample interface. The size of the temperature increase is measured, and because the temperature rise of the element is inversely proportional to the ability of the sample to transfer heat, a relationship between the change in temperature and material characteristics can be developed. This relationship is created through the use of a calibration equation using well-characterized samples (see Definition of Terms and Symbols). The value of effusivity can then be measured by inputting the appropriate sample test data into the calibration equation.

During the transient measurement, the heat flows into the sample. The longer the test duration, the further the heat wave penetrates into the sample based on the Einstein equation:

where:

d = distance traveled by 1% of the heat wave (m)

a = the thermal diffusivity of the material (m2/s)

t = the time (s)

Tests with a longer duration cause heat to travel further into the sample, and, therefore, the resulting effusivity measurement represents a larger volume of material, or a larger scale of scrutiny. It is important to balance the scale of scrutiny appropriately with the expectations of homogeneity and consistency in the material evaluated.

For powder samples, the effusivity varies with packing conditions because air has an effusivity of 5 Ws½/m2·K. As the material is compressed, air is removed and the effusivity value increases. After a user-defined nominal pressure of 15 Pa is applied for most materials, the effusivity measurement reaches a plateau when it is plotted against applied pressure. This is referred to as the threshold pressure and should be independently determined for materials that are tested offline. Alternately, if a sensor used for effusivity measurements weighs at least 500 g and has a surface area equal to or less than a 6.4-cm (2.5-inch) diameter, adequate pressure is supplied. During on-line testing, the column of powder above a sensor face provides adequate pressure if the sensor is at least 20 cm below the powder surface.

Although the scale of scrutiny is only 200 to 500 mg, in off-line blend evaluation, each sample should be 25 g or more to establish reproducible compression and consistency in sample treatment. It is recommended that samples be placed in a container whose inner diameter is in close proximity to the outer diameter of the sensor. This prevents additional movement and spreading of the sample and results in improved reproducibility. In the case of a 6.4-cm diameter sensor, a 250-mL beaker makes an excellent single, sample holder. On the other hand, the threshold pressure will not affect evaluation results in on-line blend evaluation as long as the sensors are placed at a level well below the fill line in order for the downward pressure of the powder above the sensor to generate the equivalent pressure (see Definition of Terms and Symbols).

Other Factors Affecting Effusivity Quantification

Moisture— Effusivity values are sensitive to the amount of moisture contained in a sample. It is, therefore, important to maintain consistent sample moisture during calibrations and sample testing, unless it is the moisture variations that are the desired measurement. Because water has an effusivity of 1600 Ws½/m2·K, the content of 1% water increases the effusivity of a typical powder by 3%.

Environmental Temperature— The variation of environmental temperature may jeopardize the target precision and accuracy (see Definition of Terms and Symbols). It is recommended that the variation of the environmental temperature should be less than ±1 K to achieve absolute effusivity measurements with accuracies of 1% to 2%. If the environmental temperature cannot be controlled, then it should be measured and recorded. The temperature data then provide a basis upon which a suitable correction for temperature effects can be formulated. Because the major application of effusivity is the relative effusivity or determination of effusivity plateaus rather than absolute effusivity (e.g., blend evaluation and drying), and each set of measurements is taken at the same environmental temperature, this factor typically has little or no affect on the final results.

Sample Thickness— Because heat is applied and detected on the same side of the material, if the sample is completely penetrated during a test, the material on the other side of the sample will be factored into the test results, producing a false representation of the effusivity value of the sample. Therefore, any off-line sample that is tested should be thicker than the depth of heat penetration during the test. In blend evaluation, a typical 2-second test penetrates into the powders approximately 0.5 mm, which is why a 25-g sample placed in a container with sides matching the sensor dimensions is recommended. In an on-line measurement, the thickness of sample has no significance, because the sample is semi-infinite and the heat never reaches the far side of the sample. The worst-case scenario for minimum sample thickness would be to use the longest test time of 10 seconds and a high effusivity material (crystalline material with high moisture). In this scenario, a minimum sample thickness of 5 mm would be required.

Sensor/Sample Interface Temperature— The differences in test results caused by variability in the starting temperature are not negligible. It is, therefore, critical to monitor the sensor/sample interface temperature and ensure that the temperature is the same at the start of each test. To achieve this, a cooling time may be used between two successive experiments (see Definition of Terms and Symbols). Otherwise, the temperature of the sensor/sample interface should be measured and recorded. The temperature data then provide a basis upon which a suitable correction for the effects of starting sensor/sample interface temperature can be formulated.

(In)homogeneity— Effusivity at the surface of a sample may or may not represent the effusivity of the bulk of the sample. The extent of regional differences in effusivity will depend on the homogeneity or inhomogeneity of the sample. As a result, differences in homogeneity or inhomogeneity can be detected by regional differences in effusivity values. In pharmaceutical analysis, this characteristic enables effusivity technology to be an efficient, effective tool for blend evaluation. The limiting factor is the range of effusivities of the components in the mixture. If all the materials had effusivities with a span of 10 Ws½/m2·K, the technique would not be able to differentiate between blended and unblended material because the relative standard deviation (RSD) between sensors at different locations in the powder bed would start at a maximum of 1.25%.

Power Supply— Accuracy and precision of the test results depend directly on the accuracy and precision of the power supply for heating elements. Therefore, it is desirable to use a high-precision, constant current and voltage power supply in the constant current and voltage configuration, respectively. Also, the power supply should be capable of delivering sufficient current to the heating elements such that the sensor generates a statistically significant temperature rise at the sensor/sample interface. The ideal temperature rise at the sensor/sample interface ranges from statistically significant to less than 5 K.

The sensor of the apparatus is based on the modified transient hot wire technique, which means that the heating elements are supported on a backing that provides mechanical support and electrica1 and thermal insulation (see Definition of Terms and Symbols). Such a modification eliminates the intrusive nature of hot wire technique, thereby allowing solids to be tested without melting, or otherwise modifying the sample to conform to the geometry of the test cell.

The sensor of this apparatus consists of three parallel wires mounted on a piece of thermally and electrically insulating backing. The parallel wires function as heating and guarding elements to generate a measurable rectangular one-dimensional heat flow. The constant current or voltage electrical source supplies power to the heating elements. The two outer wires are guard heaters, preventing the undesired lateral spread of heat from the heating elements and generating rectangular one-dimensional heat flow. The sensor of this apparatus functions by measuring the temperature rise at the sensor/sample interface. The temperature rise at the sensor/sample interface is measured as the voltage rises over time in a constant current configuration, or as the current rises over time in a constant voltage configuration. An ADC (analog/digital converter) with suitable input range and resolution should be connected to the sensor. The temperature rise is recorded using an appropriate data acquisition rate.

In on-line blend evaluation, the effusivity sensors can be retrofitted onto a blender. It is recommended to retrofit the sensors onto the covers of the blender for reasons of flatness and economics as well as for constant applied load of the material on top of the sensors. The measurement is taken while the blender is inverted. The number of sensors placed depends on the system and the size of the blenders, but typically it is four to eight sensors. For bin blenders with flat surfaces, the volume limitation on the lower end is related to the surface area of the bin so that no fewer than two sensor ferrules can be retrofitted on each side of the blender.

MHW technique is a comparative (secondary) method of measurement and the sensor must be calibrated with standards, which have known effusivity values. This calibration is conducted at the factory. The MHW instruments are factory calibrated with four calibration standards in triplicate. [NOTE —Calibration standards should be selected from or traceable to a recognized source of national standards.] During calibration, the test time and cooling time are established.

The highest/lowest effusivity of the sample must be less/greater than the highest/lowest effusivity of the calibration standard. The heating elements of the sensor must be completely covered by the standard. Handling the standard should be minimized. Avoid touching the sample/standard with bare hands or fingers because such touching can lead to significant disruption of thermal equilibrium.

Relatively stable thermal equilibrium must exist between the sensor and the sample in order for accurate and precise measurements to be made. Thermal equilibrium is disrupted during each test because the temperature of the sensor/sample interface increases during a test. As a result, it is desirable to employ a cooling time between two successive experiments to allow the system to (1) re-establish thermal equilibrium between the sensor and the sample and (2) ensure that the temperature of the sensor/sample interface is the same at the start of the each test. Depending on thermal diffusivity of the sample and sensor power, the cooling time needed may vary from a few seconds to several minutes or more. An acceptable cooling time should make temperature or voltage differences from the beginning to end of a test sequence less than 0.5 K or 0.9 mV. In blend evaluation, the typical cooling time is 1 to 3 minute(s). This is a factor that is set during factory calibration and is directly proportional to the test time. If the tests are conducted without cooling, the temperature of the sensor/sample interface should be recorded. The apparatus should be corrected for the difference in the initial sensor/sample interface temperature.

Test time may vary from less than 1 second to a few minutes, depending on the sample's thermal diffusivity (see Definition of Terms and Symbols). The lower the sample's thermal diffusivity, the shorter the test time, and vice versa. In blend evaluation, the typical test time is 1 to 10 seconds.

Data collection rates should be fast enough to provide statistical confidence in the test results. Shorter test times will require faster data collection rates. The data collection rate is determined such that at least 100 data points can be collected at each test. In blend evaluation, the typical data collection rate is 100 to 400 Hz.

The factory calibration is acceptable when the squared correlation coefficient resulting from the least-square correlation between raw instrument response and known thermal effusivities of the standards is greater than or equal to 0.990 (R2 ³ 0.990). Other correlation techniques should result in equivalent precision and bias in the tests after an acceptable calibration (see Definition of Terms and Symbols). Upon an acceptable calibration, both precision and bias estimates are less than ±3% RSD between sensors and less than ±1% RSD within a single sensor. Appropriateness and accuracy of the calibration are proven when a reference standard is tested and the effusivity value is within ±5%.

An effusivity reference material should be checked daily to ensure that the calibration has not drifted. The absolute effusivity value should be maintained within acceptance criteria at all times. Aside from that, the calibration should be carried out when the testing environment changes or when a need arises to change operating parameters such as test time and sampling frequency.

ABSOLUTE EFFUSIVITY is the numerical value of effusivity.

ACCURACY is the closeness of test results obtained to the true value.

BACKING is a piece of thermally and electrically insulating material to provide mechanical support for heating elements, to help generate one-dimensional heat flow, and to avoid a short circuit.

BIAS, also referred to as systematic error, is a fixed deviation that is inherent in every measurement.

BLEND ENDING POINT is a blend status when the blend uniformity is reached.

CALIBRATION is the process by which standard materials are used to determine the settings of instruments that correspond to particular values of voltage, current, frequency, or other output.

CALIBRATION EQUATION is an equation used to convert raw instrument response to effusivity values.

CALIBRATION STANDARD is a sample having known thermal effusivity value and selected from or traceable to a recognized library of national standards.

COOLING TIME is the time interval between two successive experiments, in seconds (s).

DATA ANALYSIS WINDOW is a calculation method to select part of data instead of whole data collected to determine the effusivity of samples tested.

DENSITY is the mass per unit volume of the material, in kg/m3.

ENDING TIME is the time point counted as the last valid data for the calculation of effusivity. It defines the ending point of the data analysis window.

HEATING ELEMENT is a thin material with high electrical resistance to generate a measurable temperature increase at the sensor/sample interface.

INTERFACIAL HEAT REFLECTANCE is a reflectance measurement technique where the sensors supply the heat source to samples and detect the heat flow reflected from the samples.

PRECISION is the degree of agreement among individual test results when the method is applied repeatedly to multiple samplings of a homogeneous sample. It is important to distinguish between within sensor and between sensor precisions in all reports.

RELATIVE EFFUSIVITY is the effusivity value of a specific sample relative to the mean effusivity of a group of such samples.

SAMPLING FREQUENCY is the frequency of testing during each set of test.

SPECIFIC HEAT CAPACITY is the quantity of heat required to raise the temperature of a unit of mass of a substance by a unit change in temperature, in J/(kg·K).

START TIME is the elapsed time from the beginning of the test to the time point counted as the first valid data for the calculation of effusivity. It defines the start point of the data analysis window, in s.

TEST TIME is the elapsed time from start to finish of passing current through the heating elements to perform a single test, in s.

THERMAL CONDUCTIVITY is the time rate of steady state heat flow through a unit area of a homogeneous material induced by a unit temperature gradient in a direction perpendicular to that unit area, in W/m·K.

THERMAL DIFFUSIVITY is the ratio of thermal conductivity of a substance to the product of its density and specific heat, in m2/s.

THERMAL EFFUSIVITY, e, also referred to as effusivity or thermal inertia, is the square root of the product of the thermal conductivity, density, and specific heat capacity, in Ws½/m2·K. In the MHW technique, e, is calculated using the following formula:

in which, DT is the temperature rise at the sensor/sample interface, and dt is the time period corresponding to this temperature rise.