Thermal Properties Part III Asst. Prof. Dr. Muanmai Apintanapong

Thermal diffusivity

Thermal Diffusivity Thermal diffusivity, a ratio involving thermal conductivity, density, and specific heat, is given as, The units of thermal diffusivity are m /s 2 Thermal diffusivity may be calculated by substituting values of thermal conductivity, density, and specific heat in this equation. Choi and Okos (1986) provided the following predictive equation, obtained by substituting the values of k, p, and c p in this equation.

Thermal Diffusivity Heating and cooling of agricultural materials involves the unsteady state or transient heat conduction. The temperature distribution in body for the the unsteady state condition is given by Fourier’s general law of heat conduction in the from of a partial differential equation as follows where t is the temperature at any point given by the coordinates x, y and z is time in hours

Measurement of Thermal Diffusivity Direct calculation from Use heating or cooling curves: Dickerson’s method by cylindrical object and time-temperature data.  The apparatus consisted of an agitated water bath in which a cylinder with high thermal conductivity containing the sample was immersed. Thermocouples were soldered to the outside surface of the cylinder monitoring the temperature the temperature of the sample at radius R. A thin thermocouple probe indicated the temperature at the center of the sample. The cylinder is inserted in the bottom cap, made of teflon with  = 4.17X10 -3 ft 2 /hr, and filled with the sample of known weight with a uniform rate of packing. Next, the top cap is put in place. Then the thermocouple tube is inserted to full immersion to insure proper radial positioning. The cylinder is placed in the agitated water bath and the time-temperature is recorded until a constant rate of temperature rise is obtained for both inner and outer thermocouples.

Chromium plated brass tube

(T a –T) (T a –T i ) N Fo =  t / D 2 1/N Bi

Method I for thermal diffusivity calculation Determine the time-temperature history of sample enclosed in a copper tube at its center when cooled in controlled temperature water bath. Based on Fourier’s method, assume that:  Infinite cylinder shape  h is very high (   ) then 1/Bi = 0 Determine  from time-temperature chart at 1/Bi = 0.

Method II for thermal diffusivity calculation From Newton’s law of cooling Assume h and then calculate for 

Method III Baird & Gaffney (1976) A simplified technique for the determination of  from transient cooling experiment. whenl = minimum distance within the object with maximum temperature difference during transient, m. G = geometric index (for sphere, infinite cylinder and infinite slab; G = 1.00, and 0.250, respectively) T = temperature within object (  C) T = temperature of surroundings (  C) t = time (s) For infinite cylinder, the length of tube is more than 4 times of the diameter hence radial conduction of heated is ensured.

Method IV Singh (1982) The condition where Bi is high can be utilized to determine  of the object with considerably high thermal conductivity. For situation where the sample is exposed for a long time, the equation applied for cylinder was:

Calculation by using time- temperature data

Cylinder object and time-temperature data Under the condition of constant temperature rise, the Fourier’s equation for the case when only radial temperature gradient exists can be given as: Letting A equals to the constant rate of temperature rise at all points in the cylinder, surface and center temperature be t s and t c ; therefore For infinite cylinder, the length of tube is more than 4 times of the diameter hence radial conduction of heated is ensured.

Example: If the temperature-time data given in Fig for a sample of liquid food. If the cylinder diameter is 2.24 inches, determine thermal diffusivity of the food material.

Spherical object and time-temperature data From the of temperature ratio and time yields a simple logarithmic curve:

Use of charts and graphical solution

(T a –T) (T a –T i ) N Fo =  t / D 2 1/N Bi

Example Q7 Final 2001 A spherical fruit having a radius of 7.0 cm is initially at uniform temperature of 30  C. The fruit is exposed to cold air at 5  C and its center temperature is recorded 7.5  C after 4.5 hours. If the following two points show a linear relationship between ln(TR) and Fo on the Fourier’s plot for the center temperature of a sphere, determine the thermal diffusivity of the fruit. TRFo

Example Q6 Final 2000 The following data were obtained for the center temperature of a long copper tube (dia = m) containing soybeans when immersed in a water bath maintained at 60  C. If the Fourier’s relationship for heating of a long cylinder can be represent by the equation ln(TR) = ln(1.73) – 5.82Fo, determine the thermal diffusivity of soybeans. Time (s) Temperature (C)

Time (s) Temperature (C)TRlnTR

lnTR vs time Ln TR Time (s)

Newtonian law of heating or cooling

Cooling rate (CR) or cooling coefficient

Half cooling time (z)