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Analytical Solution of the Heat Pulse Method in a Parallelepiped Sample Space

^a Laboratory for Plant–Soil Interaction Processes, Ministry of Education, College of Resources and Environment, China Agricultural Univ., No. 2 Yuanmingyuan Xi Lu, Beijing 100094, P.R. China
^b Dep. of Agronomy, Iowa State Univ., Ames, IA 50011

View larger version (42K): [in this window] [in a new window]   Fig. 1. Schematic of the parallelepiped and the heat-pulse needles used in the model: (a) three-dimensional view [the heater needle is at (x, y) = (a/2, b/2), the sensor is at (x, y) = (a/2 + rcos[], b/2 + rsin[])] and (b) top view in polar coordinate [the heater needle is at (0, 0), the sensor is at (r, )]. For illustration purpose, the radius of heater probe and temperature probe are exaggerated and are not to scale.

View larger version (14K): [in this window] [in a new window]   Fig. 2. (a) Relative errors in volumetric heat capacity (C, filled symbols) and thermal diffusivity (k, hollow symbols) that occur when the model of Bristow et al. (1994) (Eq. [19–20]) is used to compute k and C rather than the model with zero surface temperature (ZST) boundary conditions (Eq. [15]). (b) Enlargement of the highlighted rectangular area of (a). Probe spacings are r = 2, 3, 4, 5, and 6 mm.

View larger version (32K): [in this window] [in a new window]   Fig. 3. The relative errors of temperature change in a parallelepiped (5 by 5 by 5 cm3) introduced by approximating the infinite sum with a truncated series (which contains a finite number of terms, Nseries) at time t = (a, b, and c) 60 s and (d, e, and f) 16 s.

View larger version (20K): [in this window] [in a new window]   Fig. 4. Calculated long-time temperature (T) curves (a) at the thermocouple location for a pulsed infinite line source (dash-dot line), a pulsed infinite cylindrical source (long dash line), a pulsed finite line source (short dash line), and a pulsed finite line source in a parallelepiped (5 by 5 by 5 cm3) with zero surface temperature (ZST, solid line) and adiabatic boundary condition (ABC, dotted line); and (b) for a pulsed finite line source in the same parallelepiped.

View larger version (38K): [in this window] [in a new window]   Fig. 5. The relative errors of temperature distribution (Eq. [23]) as a function of angle in a parallelepiped (5 by 5 by 5 cm3) with specified needle spacing for (a, b, and c) an adiabatic boundary condition (ABC) and (d, e, and f) a zero surface temperature (ZST) boundary condition with time t = 50, 100, 200, and 300 s and probe spacing r = 6, 10, 15, and 20 mm.

View larger version (51K): [in this window] [in a new window]   Fig. 6. Three-dimensional surface graphs of temperature (T) distribution at time t = 300 s and z = 2 cm for (a1) zero surface temperature (ZST) and (b1) adiabatic boundary conditions (ABC); (a2) and (b2) are the corresponding isothermal plots of the temperature curves of (a1) and (b1), respectively.

View larger version (30K): [in this window] [in a new window]   Fig. 7. The temperature (T) distribution along the direction of the sensing needle in a parallelepiped (5 by 5 by 5 cm3) with time t = 60 s for zero surface temperature (ZST, dotted lines) and adiabatic boundary conditions (ABC, solid lines). The numbers on the curves are the values of (a) needle spacing with fixed probe length (c0 = 4 cm) and (b) probe length with fixed needle spacing (r = 6 mm). The point TILS on the curve denotes the temperature of an infinite line source.

View larger version (39K): [in this window] [in a new window]   Fig. 8. Relative errors in (a) volumetric heat capacity (C) and (b) thermal diffusivity (k) that occur when the model of Bristow et al. (1994) (Eq. [19–20]) is used to compute k and C rather than the model with zero surface temperature (ZST) boundary conditions (Eq. [15]). Relative errors are shown for sample heights of c = 4, 4.5, 5, 6, 8, 10, 12, 15, and 20 cm.

View larger version (35K): [in this window] [in a new window]   Fig. 9. Relative errors in (a) volumetric heat capacity (C) and (b) thermal diffusivity (k) that occur when the model of Bristow et al. (1994) (Eq. [19–20]) is used to compute k and C rather than the model with adiabatic boundary conditions (ABC, Eq. [18]). Relative errors are shown for sample heights of c = 4, 4.5, 5, 6, 8, 10, 12, 15, and 20 cm.

View larger version (18K): [in this window] [in a new window]   Fig. 10. Relative errors in (a) volumetric heat capacity (C) and (b) thermal diffusivity (k) that occur when the model of Bristow et al. (1994) (Eq. [19–20]) is used to compute k and C rather than the model with zero surface temperature (ZST) boundary conditions (Eq. [15]). Relative errors are shown for sample lengths (a) and widths (b) of 25, 28, 31, 35, 40, and 60 mm, probe length (c0) of 4 cm, and sample height (c) of 5 cm.

View larger version (20K): [in this window] [in a new window]   Fig. 11. Relative errors in (a) volumetric heat capacity (C) and (b) thermal diffusivity (k) that occur when the model of Bristow et al. (1994) (Eq. [19–20]) is used to compute k and C rather than the model with adiabatic boundary conditions (ABC, Eq. [18]). Relative errors are shown for sample lengths (a) and widths (b) of 25, 28, 31, 35, 40, and 60 mm, probe length (c0) of 4 cm, and sample height (c) of 5 cm.

View larger version (20K): [in this window] [in a new window]   Fig. 12. Relative errors in (a) volumetric heat capacity (C) and (b) thermal diffusivity (k) that occur when the model of Bristow et al. (1994) (Eq. [19–2]) is used to compute k and C rather than the model with zero surface temperature (ZST) boundary conditions (Eq. [15]). Relative errors are shown for sample heights (c) of 3, 3.2, 3.4, 12, and 21 cm and probe length (c0) of 3 cm.

View larger version (27K): [in this window] [in a new window]   Fig. 13. Relative errors in (a) volumetric heat capacity (C) and (b) thermal diffusivity (k) that occur when the model of Bristow et al. (1994) (Eq. [19] and [20]) is used to compute k and C rather than the model with ABC (Eq. [18]). Relative errors are shown for sample heights (c) of 3, 6, 12, 15, and 21 cm and probe length (c0) of 3 cm.

View larger version (20K): [in this window] [in a new window]   Fig. 14. The temperature (T) distribution along the axis of the sensing needle in a parallelepiped (length a = width b = 5 cm for lines, and a = b = 10 cm for circle symbol C) for (a) adiabatic boundary conditions (ABC) and probe length of 3 cm, and (b) zero surface temperature (ZST) boundary condition and probe length/sample height ratios of 0.1, 0.2, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1 at time t = 60 s. Needle spacing was fixed (r = 6 mm). Arrows indicate the midpoint position of the sensing needle, which is parallel along the z axis.

View larger version (37K): [in this window] [in a new window]   Fig. 15. Relative errors in volumetric heat capacity (C) and thermal diffusivity (k) that occur when the model of Bristow et al. (1994) (Eq. [19–20]) is used to compute k and C rather than the model with zero surface temperature (ZST, Eq. [15]) and adiabatic boundary conditions (ABC, Eq. [18]). Relative errors are shown for different (a) quantities of heat liberated (q') and (b) pulsed heat source duration (t0) and the enlargements of the highlighted rectangular areas of (a) and (b).