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Analysis of various design and operating parameters of the thermal conductivity probe
./011171N/ of /+od Enginwring 30 (1046) 2(1’)-225 Copyright 0 lY96 Elsevicr Scicncc L.imited Printed in Great Britain. All rights resewed 0260-8774!Yh/$ ISN +I).00 0260-877J(96)00010-6
Fl.SF.VIER
Analysis of Various Design and Operating Parameters of the Thermal Conductivity Probe* E. G. Murakami,” V. E. Sweat,h S. K. Sash-y’ & E. Kolbe” “US Food and Drug Administration/National Center for Food Safety and Technology, 6502 S. Archer Ave., Summit-Argo, IL 60501, USA “Texas A & M University, Department of Agricultural Engineering, Rm 201, Scoates Hall I, College Station, TX 77843-2117, USA ’ The Ohio State University, Department of Agricultural Engineering, 590 Woody Hayes Dr., Columbus, OH 43210, USA “Oregon State University, Departments of Bioresource Engineering and Food Science and Technology. Corvallis, OR Y7331-3906, USA (Received
21 February
1995; revised 27 Novcmbcr
lYY5; accepted 5 February
lYY6)
ABSTRACT The thermal conductivity (k) probe is derived from an idealized transient heat transfer model; therefore, there are inevitable differences between the real probe and the theoretical model. However; the k probe is still an accurate and practical measurement device with wide-ranging applications if designed and used properly. Users of a thermal conductivity probe must be aware of Its limitations and the possible errors that can develop in its application. This paper includes a theoretical derivation sf the k probe equation and some experimental and theoretical simulations of parameters that can cause errors in the application of the k probe. An explanation is given of the significance of certain design and operating parameters. Some of the findings in the study are that the time-correction factor is not required and contact resistance does not affect accuracy. The calibration factor is necessary since it compensates for the difference in the thermal masses of the probe and the sample. Errors due to edge effects and convection can be avoided by limiting data analysis to the linear section of the time-temperature plot. The truncation error is minimized by making /jas small as possible. Copyright 0 1996 Elsevier Science Limited
“‘The contents of this paper arc solely the opinions of the authors, represent the official views of the US Food and Drug Administration. 2OY
and do not necessarily
210
E. G. Murakami et al.
NOTATION C,
&2 D E,, k L M e ; s t, to, Lf t-T T, T,
X,Y& CY ZT, AT,,
Euler’s constant (0.5 7721) specific heat (Jfkg”C) parameters of the heated-rod equation diameter (m) truncation error thermal conductivity (WIm”K) length (m) ratio of thermal masses of specimen and probe power input (Wlm) radius (m) contact resistance between the probe and the medium thermal mass per unit length of the heater (JImOK)
time, final time, initial time and time-correction time-temperature plot in semilog axes temperature and reference temperature (“C) Cartesian coordinates thermal diffusivity (m’/s) density (kg/m3) temperature rises at times t and to (“C)
(m “KIW)
factor (s)
INTRODUCTION The thermal conductivity (k) probe apparatus is a versatile and simple device that has gained popularity in food engineering applications. It has been used on a wide variety of food materials including powders, liquids, bulk grains and whole fruits, and owes its popularity to its low cost, ease of operation and simple design. However, its theory is based on the line-heat source technique, which is derived from an idealized transient heat conduction model; thus, there are differences between the real probe and its theoretical model. The main difference is that the real probe has finite thermal mass, whereas the theoretical model assumes it to be negligible. Moreover, published applications of the k probe have differed considerably from the theoretical assumptions. In the thermal conductivity measurements of rocks in situ, the contact resistance between the probe and the rock, which is theoretically assumed to be negligible, is finite because the procedure involves a loose-fit probe dropped into diamond-drilled holes that are crooked (Blackwell, 1954). In liquids and gases, radiation, convection, variable thermal properties and Knudsen effect, which are all assumed to be theoretically negligible, could become significant in some experimental situations (Healy et al., 1976; Nieto de Castro et al., 1988). In food engineering applications, the above problems are compounded by specimens with finite size and anisotropic properties. For instance, the thermal conductivity probe had been used to measure small items like corn kernels (Kusterman et al., 1981), radially asymmetric materials like tobacco (Casada & Walton, 1989) and tortilla chips (Moreira et al., 1992). These deviations can lead to unavoidable errors; however, their effects can be minimized and sometimes compensated through careful attention to the design and operating parameters and data analysis. To minimize the various design errors,
Design und operating parameters of the thermal conductivity probe
211
Murakami et al. (1995) developed a guideline for designing and fabricating k probes for non-frozen food materials. Researchers in the past accounted for these errors in various ways. Nieto de Castro et al. (1988) treated each error as a temperature deviation and algebraically added it to the measured temperature. Van der Held and Van Drunen (1949) as well as many other subsequent users, theorized that the finite size of the heater caused a delay in heating the products; they therefore subtracted a time-correction factor from the time value. Another often-used correction factor has been the calibration factor. It is recommended that k probes be calibrated with materials of known thermal conductivity, usually either water--agar gel or glycerin, before each use. The ratio between the published and the measured thermal conductivity values of the calibration material becomes the calibration factor. This value is then multiplied by the measured thermal conductivity values of the materials. For isotropic materials, Carslaw & Jaeger (1959) and Davis (1984) suggested using the geometric mean of the k at the two dimensions perpendicular to the axis of the k probe. The objectives of this paper are to make probe users familiar with the theoretical assumptions and limitations of the k probes and to provide them with recommendations for maximizing probe performance. Both theoretical and experimental t-T plots are used to aid in discussing the significance of various factors (truncation error, thermal mass ratio of specimen to probe, contact resistance, convection, edge effect, initial time, time-correction factor and calibration factor). In addition, the various line-heat source equations are compared to the more realistic but complicated heated-rod equation.
THEORETICAL
CONSIDERATION
Line-heat source technique The line-heat source technique measures thermal conductivity, assuming unsteadystate heat conduction from an infinitely long line-heat source immersed in an infinite and homogeneous medium. Although its working equation is widely known, the theoretical derivation is not. To be aware of the limitations and requisites for its effective application, the user must be familiar with the theoretical assumptions and derivations. Considering a string of instantaneous point sources (Q,) which is continuously generated over a period of time at a constant rate; then the temperature rise at r due to conduction in a medium with constant thermophysical properties is equal to (Carslaw & Jaeger, 1959):
where: ri
(14
L1 = 42
(t -
to)
212
E. G. Murakami et al.
(lb) Q = QIPC,
(lc)
The radius r is the distance between the heater and the point of interest, gap between them is filled with the medium. The analytical solution to eqn (1) is as follows (Carslaw & Jaeger, 1959):
The exponential integral E,(/12) is an infinite series (Abramowitz and therefore the expansion of eqn (2) is: AT=
Q - $
2rck
B2 _ 1)” 2(1!) 4(2!)
-In@)+
I
& Stegun, 1964)
1
*”
and the
(3)
This is the complete or nontruncated equation of the line-heat source technique. However, it is not used for calculation of thermal conductivity because it requires the thermal diffusivity of the material. Moreover, this equation is numerically too complicated for many applications and it must be simplified for practical purposes. In practice, the variables r and t in the fi parameter are referred to as the heater size and the test duration, respectively; thus b is both a design and an operating parameter. If the k probe apparatus is either designed or operated such that b is kept small, p” and the higher order terms can be truncated from eqn (3): AT=
-&-[-
$-
-In(b)]
(4)
This is the truncated equation of the line-heat source technique. Its main feature is that the temperature rise is linear with the natural logarithm of p, and thus the thermal conductivity can be derived from the slope. However, for calculation, eqn (4) requires the thermal diffusivity of the specimen, which is more likely to be unavailable. To avoid this complication, it is evaluated within a time interval (t-t,,). The resulting expression is the working equation of the line-heat source technique or the probe method (van der Held & van Drunen, 1949): AT-ATo=
Q
4nk
Heated rod The theoretical assumptions in the line-heat source technique are idealized and impractical. A real line-heat source apparatus is inevitably different from the theoretical model because it has finite diameter and mass. Moreover? in solids, the contact resistance on the probe surface, which was not considered m the line-heat source technique, may not be negligible.
qf the thermal conductivity probe
Design and operating parametets
213
Carslaw & Jaeger (1959) derived the temperature rise in an infinitely long heated probe with finite radius (r), thermal mass (S), and surface or contact resistance (K). For any value of /L the temperature rise (Aq in a heated rod is as follows: 2p’(cz - 2)
/?(4C, -C2)
T=
c
In([Lc)+...
(6)
C,
7
I
where, C, = 2nRk
(ha)
C1 = 2nr’pCJS
(C)b)
C = exp(C,)
(hc)
Note that if the contact resistance is negligibly small, the thermal masses of the heater and the material are equal (Cz = 2), and if the probe radius is relatively small, eqns (4) and (6) are equal.
DISCUSSION Truncation error The working probe equation [eqn (5)] has inherent truncation error (E,,) as a result of mathematical simplification from the nontruncated [eqn (3)] to the truncated form [eqn (4)]. By eliminating the infinite series, the predicted temperature rise using the truncated equation (ATT) is always smaller than that calculated with the nontruncated equation (AT,). However, the difference decreases as the value of [i decreases (Fig. 1). The truncation error, which is the ratio of that difference to the calculated temperature rise using eqn (3), is:
5
4
I
Truncation “\\
60
i
‘i‘_ ‘:,.,!/’
t 4
i
Truncated
_ _ 0.10
50
i i
\ ,* Nontruncated ‘\
I
0 lmL-~.----__ 0.01
error
\
G‘3 e 52 ;2
70
I
‘\.\
“ >yj,
_,’ --~--- ’ L_.I__--__+
30:s
40 g :
I,0 \ L ‘\
10 IJ I .oo
Beta
Fig. 1. Simulated temperature rise in water, calculated using eqns (10) and (11) and the resulting truncation errors for steel probes with r = 0.3 15 mm.
214
E. G. Murakami et al. 10
T
r = 0.315 mm
-2t Time (s)
Fig. 2. Simulated
temperature
rise in water, calculated using eqns (10) and (11) for probes with radii of 0.315 and 1.0 mm.
ATb-AT7 &, =
AT,
(7)
The truncation error is an error for each data point. Its highest value is at lo and it decreases thereafter with time. It is minimized by calculating the thermal conductivity from the linear portion of the t-T plot (Fig. 2). Vos (1955) recommended setting the truncation error to ~0.59%; Murakami et al. (1995) suggested minimizing it by making the k probe as small as possible. In non-food applications, researchers prefer a truncation error of < 0.2% (Richmond et al., 1984). Thermal mass ratio of specimen to probe In theory, the ratio of the thermal mass of the specimen to that of the k probe (M) is not considered in the line-heat source equation. In reality, the M changes the slope of t-T plots, affecting the calculated k value. Another effect of M is that it makes the t-T plot nonlinear, and consequently the required to increases. The k probe generally consists of a heater, temperature sensor, and tubing to make the assembly sturdy. By assuming a solid structure, its average thermal mass can be calculated from the thermal properties of the individual components. The thermal mass (S) per unit length of k probe is as follows: S = ~r2pCplprohct Thus, the parameter
(8)
C2 in eqn (6) becomes: C2=2M
(8a)
where M is the ratio of the thermal masses of the specimen and the probe, that is:
(9)
Design and operating parameters of the thermal conductivity probe
215
If M = 1, the effect of thermal mass in eqn (6) is eliminated. For reference, the M for water and a solid-steel probe is 1.07; for glycerin with the same probe, M = 0.79. Effect on Slope
A mathematical simulation was conducted to evaluate the effect of M on the slope of the t-T plot. The t-T plots of the heated-rod equation [eqn (6)j at various M values and of the nontruncated line-heat source equation, which represents an ideal situation (M = 1 and R = 0), are shown in Fig. 3. The line-heat source plot is chosen as the reference because it represents an ideal condition, viz negligible thermal mass of the k probe. At the linear section, as M increases, the slope of the t-T plot decreases and the deviation of the slope from the reference plot increases. At M values 50.6 and 22.5, the slope changes by as much as 25% per unit of 44. The deviation of the slope from the reference is remedied by using a calibration factor, which is directly proportional to the degree of the deviation (see calibration factor section) and the filling material for the probe tube. Efj&t on t,,
Another effect of M is in the nonlinear portion of the t-T plot (Fig. 3). This portion curves upward at low values of M (MS 1) and then becomes more linear as ,izI gets closer to 1. The number of data points in the nonlinear portion increases as the M decreases and consequently the to increases (see initial time section). I’ructical
Implications
Figure 3 shows that the calibration material can be chosen such that the slopes of the t-T plot of the calibration material and the specimen are as close as possible. Some researchers prefer fillers in the k probe for various reasons. Because tillers can change the thermal mass of the probe and also because of other considerations (e.g. thermal stability), they should be chosen in the same manner as the calibration material.
6.~
M=0.6 ,,/
z ._2 a c
xi_ 4
:/ M =
-9’
,,7
, ‘,” ,I
2.5 , -<,;s ,-:’ ,H
. . ..a’ / Nontruncated 0. I
1
10
100
Time (s)
Fig. 3. Simulated AT (heated-rod eqn, water with a steel k probe of r = 0.315 mm) showing the effect of M and comparison with the ideal situation (M = 1, R = 0)as represented hy the non-truncated eqn.
E. G. Murakami et al.
216
Contact resistance The contact resistance at the probe-specimen interface is assumed to be negligibly small in the derivation of the line-heat source equation. In reality, contact resistance could be significant, especially in powders or when the hole where the probe is inserted into the specimen becomes enlarged through repeated use. Contact resistance causes the t-T plot to shift, but it does not change the slope. Thus, it does not affect the probe accuracy. Its other effect is that it changes the nonlinear portion of the t-T plot. The equation [eqn (6)] derived by Carslaw & Jaeger (1959) for a heated rod is plotted to evaluate the effect of contact resistance (Fig. 4). In this figure, the plots for the heated-rod equation are simulations of a solid-steel k probe immersed in water. In contrast to the idealized plot of the line-heat source equation [eqn (3)], Fig. 4 shows that contact resistance shifts the linear portion of the t-T plot without changing the slope. Thus, the effect of contact resistance can be eliminated by limiting data analysis within the linear portion of the t-T plot. This finding is relevant in calibrating k probes for porous specimens (see ). Contact resistance also affects the nonlinear portion of the t-T plot. In Fig. 4, the curvilinear portion gets bigger and moves upward as the contact resistance decreases and consequently the to increases. Note that the plots of the heated rod with R = 0 and the nontruncated line-heat source equation are not equal because the heated rod was calculated with finite thermal mass ratio. Convection The k probe is based on pure conduction. The added effect of convection increases the rate of heat transfer through the specimen. Experimental t-T plots for glycerin and water at an extended test time of 110 s and with power input of 11 W/m are shown in Fig. 5. No convection-deterrent material was added to either specimen. This figure shows that in water convection started after 10 s, whereas in glycerin the
8-
R=0.15 /’
/’
R = 0.05
6
Nontruncated I
IO
/ 100
Time (s)
Fig. 4. Effect of contact resistance on the t-T plot of a heated rod and comparison line-heat source, using the non-truncated eqn.
with the
Design at~d operating parameters
qf the thermal conductivity probe
217
Glycerin
Water
.+_.+_
+_tt__-.----c IO
If-.-
+~--t----t-r--tc-ntl 100
1000
Time (s)
Fig. 5. Effect of convection on the experiment t-T
plots of water and glycerin (t,, = 1.0 s.
Q = I I W/m).
onset of convection occurred much later, at around ‘70 s. The obvious change in the slope indicates that convection effects can be easily eliminated by limiting data analysis to the linear portion of the t-T plot. Thus with Q = 11 W/m, water and more viscous liquids can be used as either test or calibration materials without adding convection-deterrent materials. Figure 5 also shows that convection shortens the linear portion of the t-‘T plot and may render test results useless because of insufficient data points for analysis. Convection can be avoided by using low power input, reducing test duration, and adding convection-deterrent materials like agar and glass wool. The appropriate values for power input and test times are determined by trial-and-error during the calibration. Agar may be added to liquid test samples to turn them into gels. Sweat (1986) recommended adding 0.5% agar to water. Murakami (1993, 1994) found that a water-agar gel with 0.6% agar was an excellent calibration material that could be kept for several days. It was easily reconstituted by reheating when the probe holes became enlarged and too numerous because of repeated use. For oil and water specimens at high temperature, Sweat (1986) recommended placing glass wool in the specimens to mechanically inhibit convection. Edge effect Edge effect is caused by heat transfer at specimen edges, which either enhances or diminishes conduction through the specimen. The influence on edge effect of specimen environment, operating parameters, and specimen radius is discussed. T/U?Role of Specimen Environment The thermal conductivity of the environment compared to that of the specimen affects the development of edge effect by making the t-T plot nonlinear toward the end of the test. An experiment was conducted to illustrate edge effect due to different types of environment. A potato cylinder with k = 0.56 W/m “K (Mur-
E. G. Murakami et al.
218
akami, 1993) and D = 7 x lo-” m was exposed thermal conductivity environments.
to relatively
lower
and higher
Environment with Low k An environment with relatively low k was simulated with still air, which has a value of k = 0.0244 W/m “K (ASHRAE, 1976). The potato cylinder was placed in a beaker loosely filled with paper towels to prevent convection. Results showed that in the still-air environment, the t-T plot moved upward as soon as edge effect developed (Fig. 6). This result indicated that when the heat reached the specimen edges, the conduction through the specimen slowed. The air, whose thermal conductivity is one order of magnitude lower than that of water, acted as an insulator and this decreased the temperature gradient in the specimen. It can be observed from Fig. 6 that edge effect started at ~~40 s, and at this time the calculated temperature rise due to convection was 0.7”C. If a temperature change of 0.7”C is to be considered significant by the probe user, the edge effect can be eliminated by discarding the data points at t240 s. If edge effect is not noted and if only the nonlinear section at the initial portion of the t-T plot is eliminated in the analysis, edge effect due to environments with k lower than that of the specimen would lower the calculated thermal conductivity. Environment with High k The environment with relatively high k was simulated by using stirred water (k + 0.6 W/m “K). In this setup, the specimen was immersed in a beaker filled with water and the water was continuously stirred with a magnetic stirrer. The result showed that the edge effect caused the t-T plot to bend downward (Fig. 6). The high rate of heat transfer through the edges enhanced the heat conduction in the specimen by keeping the temperature gradient. If the data are analysed in a similar manner as discussed above, the calculated thermal conductivity would be higher than without the edge effect.
Still air / Potato cylinder
6
/’
/,’
(D = 7 mm)
& ._2 4-p! C-. 2 --
0
i-t::; 1
Fig. 6. Evidence
10
100
of edge effect in the experimental temperature rise of potato cylinders still-air and stirred-water environments (Q = 11 W/m).
in
319
Design and operating parameters of the thermal conductivity probe
The Role of Operating Parameters
The effects on the occurrence of edge effect of operating time (t) and power input (Q) were also evaluated.
parameters
such as test
Power Input
The effect of power input on edge effect was evaluated by calculating the temperature distribution in water with the nontruncated line-heat source equation. The slope of the temperature rise across the specimen, which can be approximated from the truncated equation [eqn (4)], is calculated as follows:
d(AT)
( 10)
dr
Based on this equation, the temperature rise (AT) decreases toward the specimen edges, and its slope is affected by power input, thermal conductivity, and distance from the heater. The heat from the probe causes an advancing hot front (ATrO.Ol”C) to move toward the specimen edges. A simulation of water at t = 10 s shows that if the power input Q is varied between 5 and 50 W/m, the distance of the hot front is within 4.25.5x10P’ m (Fig. 7). This indicates that the distance of the hot front from the heater is found to be relatively insensitive to the power input, because a 900% increase in Q caused the r to increase by only 31%. Test Dura tiorl
The effect of test duration on edge effect was studied with the following numerical simulation. With the power input fixed at 5 W/m, the temperature profile in water was calculated for t = 5, 20 and 50 s (Fig. 8). From t = 5 s to t = 50 s, the hot front moved from r = 3 x lop3 m to r = 8 x lo-” m. The distance of the hot front from the probe increased by 167% when the test time was increased by 900%. Thus, the hot front is comparatively more sensitive to the test time than to the power input.
x
pjib -t-
-y-
‘\ -*\
Q = 5 W/m Q=
10 W/m
Q = 20 W/m
Q = 50 W/m
\ ‘k
_~ 0.1
~_.__~
-a-*xw+ ---tcM I
IO
Radius (mm)
Fig. 7. Simulated temperature distribution in water at a test time of 10 s.
E. G. Murakami et al.
220
I t
01 0.1
1
10
Radius (mm)
Fig. 8. Simulated temperature profile in water at a power input of 5 W/m.
Minimum Specimen Radius
To avoid edge effect, researchers have been using the following equation, mended by Vos (1955) for calculating the minimum specimen radius: 1 <0.6 B’
recom-
(11)
However, this equation does not include the power input and requires knowledge of the thermal diffusivity of the specimen for calculation. Using eqn (11) and a value of t = 10 s, the minimum radius for s ecimens with high moisture content (those with 2% 1.44 x lop7 m’/s) is 3.1 x IO-- P m. For an infinite medium, the calculated AT [eqn (3)] at this point is 0.06”C if Q = 5 W/m, and 0.6”C if Q is increased to 50 W/m. The heat transfer from the edges with AT = 0.6”C may be negligibly small but could be significant at AT = 0.6”C. Thus, eqn (11) is effective only at low power input levels. Probes users should use this equation as a first estimate of specimen radius and then examine the t-T plot for edge effect. Combination of power input, test duration, and specimen radius
In this study and previous studies by Murakami (1993, 1994), a power input of Q = 11 W/m was found to be sufficient for high moisture food materials (k = 0.50.6 W/m “K). Simulations using eqn (3) with a specimen having thermal properties close to water and Q = 11 W/m, showed that if the specimen r is 2 1 x lo-’ m, tests can be performed at up to 50 s without edge effect. Although convection develops in liquid water within 10 s of test time (Fig. 5) longer test times can be beneficial with solid food materials such as fresh potatoes and scallops. Conversely, theoretical simulations have shown that if the specimen type and power input are the same and if the test time is limited to 10 s, the specimen can be as small as r = 5 x lo-” m without edge effect. For specimens smaller than r = 5 x lo-’ m, edge effect can be eliminated by first decreasing the test time and then the power input.
Design und operuting purameters of the thermul conductivity probe
221
Initial time The primary purpose for the initial time (to) in the working equation [eqn (5)l is to eliminate the need for the specimen thermal diffusivity in the calculation. However, the t,, has another practical benefit. If chosen properly, it can be used to eliminate the initial nonlinear portion of t-T plots. Because its value does not affect the slope of the linear section of the t-T plot, data analysis can be simplified by having a single t,, value for all specimens. The to is affected by the probe radius, the ratio of the thermal masses of the specimen and the probe, and the contact resistance. The to can be used to linearize t-T plots if it is set equal to at least the time when the nonlinear portion of the t-T plot ends. In Fig. 9, the plots of water1 and glycerin1 include all the collected data, and it can be observed that because of differences in their thermal mass ratios, the t-T plot of water is linear, whereas that of the glycerin is initially nonlinear and then becomes linear after 2 s. By setting to = 2 s, the t-T plots of both water (water2) and glycerin (glycerin2) are linearized. Time-correction factor The actual t-T plots of some materials, i.e. glycerin, are nonlinear at the initial section. Because the working line-heat source equation is linear, early users like van der Held & van Drunen (1949), Woodside (1959), and Underwood and Taggart (1960) used the time-correction factor (tCf) to make the actual t-T plots straight. They modified the working equation to the following: AT-
AT,, =
(12)
Many justifications have been given for using the time-correction factor. According to van der Held and van Drunen (1949), the t,, compensated for the initial time
‘2
I
Effect of initial time on the experimental t-T plots of water and glycerin [note: for and glycerin1 , t* = T(t) - T(t = 0); for water2 and glycerin2, t” = t/2 and T* = T(t) - T(t = 2)].
222
E. G. Murakami et al.
10
I
(t - t,fMb - Lf) Fig. 10.
Effect of time-correction factor on t-T plots (to = 2.0 s, t,f = 1.0 s).
spent in heating the heater itself and for uncertainty in fixing the initial time, to. However, some researchers like D’Eustacio and Schreiner (1952) determined that the tCfwas unnecessary. Sweat (1986) found this value equal to zero. The tCf has no theoretical basis and, in fact, it can make a linear t-T plot curvilinear. As shown earlier, the t-T plot of glycerin was made linear by using an appropriate value of to (Fig. 9). However, when tCf= 1.0 s was subtracted from the time values while to = 2.0 s was maintained, the t-T plots of both glycerin and water became nonlinear (Fig. 10). Although a value of fCf can be found that can make those t-T plots linear, e.g. tCf= 0.5 s, there is no added benefit for using it. Thus, the use of tCf is not recommended. Calibration factor The calibration factor is used to increase the precision of the k probe. It is determined by performing a test run at the same operating parameters, e.g. power input and test time, with a material (called calibration or reference material) of known thermal conductivity. It is calculated from the ratio of the published values to the measured thermal conductivity values. The calibration factor is essential because of various factors that affect the slope of the linear portion of t-T plots. These factors are the thermal mass ratio of specimen to probe (see ) and the inaccuracy in the measurement of the power input. Effect of M As discussed in , the t-T plot of a k probe with thermal mass deviates from the ideal probe, -which has negligible thermal mass. The difference in the slopes of their t-T plots (at the linear section) is corrected in practice by using the calibration factor. The calibration factor is affected by the thermal mass of the calibration material. Thus, for a stainless-steel k probe (D = 7.1 x lop4 m, L = 22.8 x lo-” m) the calibration factor is 0.895 with water as the calibration material, and 0.835 with glycerin. Another factor that affects the thermal mass of the probe is the k probe filler, which
Design and operating parameters ofthe thermal conductivity probe has been used by researchers 1993) for various reasons.
(Morley,
1966; Karwe ef al., 1991; Gratzek
223 & Toledo,
Effect of Power Input The slope of the working equation increases with increasing power input. Thus, any inaccuracies in the determination of the power input directly affect the calculated slope of the t-T plot. The power input is calculated from the resistance of the heater wire and the electric current flowing through it. Although the resistivity equations for some commonly used heater wires are available in the literature, they have low accuracies because of variations in heater wire composition. The accuracy of digital ammeters is usually sufficient for application in k probes. Moreover, sufficiently stable DC power sources are available for powering the probe heater. However, instrument drift can occur and the calibration factor can be used to detect it. If the k probe is calibrated before and after a series of tests, the t-test can be used to determine if there is a statistical difference between the two calibration factors (Ott, 1977). Selection qf calibration material In theory, the thermal masses of the specimen and the calibration material must be equal, but this is impractical in actual use. In practice, the calibration material must he chosen such that the slope of its t-T plot is as close as possible to that of the specimen. Figure 11 shows the uncorrected t-T plots of various materials. The water specimen was made from 0.6% water-agar gel, and the food materials were all fresh (unprocessed) and had high moisture content (75% w.b.). All the t-T plots are almost parallel to each other, except for glycerin. Thus, water is a more appropriate calibration material than glycerin for these materials. If glycerin is chosen as the calibration material, the measured thermal conductivity values for the food materials in this example would have been lower by 6.7%. The most common calibration materials for nonfrozen foods are water and glycerin. If the test time is I 10 s, water does not need any convection deterrent materials in it (see the discussion on Presence of Convection). For frozen food and
0.1
I
-+-
Potato
-i-
Carrot
IO
Time (s)
Fig. 11.
Experimental t-T plots of various food and calibration
materials
(& = 11 W/m).
224
E. G. Murakami et al.
other high thermal conductivity materials (k >0.6 W/m “K), Tong et al. (1993) developed reference materials by blending copper powder with bentonite paste. CONCLUSIONS Although the k probe is based on an idealized model, accuracy can be maintained if it is carefully designed and operated and if the data are properly analysed. The errors caused by the various design and operating parameters can be determined from the t-T plot and can be corrected by limiting the data analysis within the linear region. The effects of the various design and operating parameters are as follows: (1) The truncation error can be minimized by making the k probe as small as the application and fabrication procedures allow. (2) The thermal mass ratio of specimen to heater (M) affects the slope of the t-T plot; thus its value should be maintained between calibration and actual application of the k probe. (3) The contact resistance does not affect the accuracy of the probe because it shifts the t-T plot without affecting the slope. (4) Convection affects the accuracy of the probe apparatus by changing the slope of the t-T plot. However, its effect can be eliminated by limiting data analysis to the linear portion. (5) Edge effect causes problems similar to that of convection and can be corrected in a similar manner. It can be avoided by either increasing the specimen diameter or decreasing the test time; decreasing the power input has minimal influence. (6) The initial time (to) should be chosen such that it is equal to at least the time when the t-T plot becomes linear. (7) The time-correction factor (tcf) is not recommended because it has no theoretical basis, and because its function for linearizing the t-T plot can be achieved with proper selection of the to. (8) The calibration factor is essential because the slope of the actual t-T plot is affected by the thermal mass of the specimen, accuracy in determining the power input and instrument drift. The reference materials should be chosen such that the slope of its t-T plot is close to that of the specimen. REFERENCES Abramowitz, M. 6t Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. NBS Applied Mathematics Series 55. U.S. Department of Commerce, Washington, DC. ASHRAE (1976). Thermophysical Properties of Refrigerants. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, GA [In ASHRAE 1982 Fundamentals].
Blackwell, J. H. (1954). A transient-flow method for determination of thermal constants of insulating materials in bulk. J. Appl. Phys. 25, 137-144. Carslaw, H. S. & Jaeger, J. C. (1959). Conduction of Heat in Solids. 2nd edn. Clarendon Press, Oxford, pp. 258-345. Casada, M. E. & Walton, L. R. (1989). Thermal conductivity of baled barley tobacco. Trans. ASAE
32 (3), 977.
Design and operating parameters
of thethermal conductivity probe
225
D’Eustachio. D. & Schreiner, R. E. (1952). A study of a transient method for measuring thermal conductivity. ASHI/E Trans. 58, 33 1. Davis, W. R. (1984). Hot-wire method for the measurement of the thermal conductivity of refractory materials. In Compendium of Thermophysical Property Measurement Methods, Vol. 1, eds K. Maglic, A. Cezairliyan & V. Peletsky. Plenum Press, New York, pp. 23 I-54. Gratzek, J. P. & Toledo, R. T. (1993). Solid food thermal conductivity determination at high temperatures. /. Food Sci. 58 (4), 908-913. Healy, J. J., de Groot, J. J. & Kestin, J. (1976). The theory of the transient hot-wire method for measuring thermal conductivity. Physica 82C, 392-408. Karwe, M. K., Tong, C. H. & Lund, D. B. (1991). Modeling of temperature distribution in a thermal conductivity probe with different tilling materials. Paper presented at the 1991 IFT Annual Meeting, Dallas, TX, 1-5 June Kusterman, M., Scherer, R. & Kutzbach, H.D. (1981). Thermal conductivity and diffusivity of shelled corn and grain. J. Food Process Engng 4 (3). 137-153. Moreira, M. G.. Palau, J. & Sweat, V. E. (1992). Thermal properties of tortilla chips during deep-fat frying. ASAE Paper No. 92-6595. Morley, M. J. (1966). Thermal conductivities of muscles, fats and bones. J. Food Technol. 1, 303-311. Murakami, E. G. (1993). The thermal properties of potatoes and carrots as affected by thermal processing. Paper presented at the 1993 IFT Annual Meeting, Chicago, IL, IO-14 July. Murakami, E. G. (1994). Thermal processing affects properties of commercial shrimp and scallops. /. Food Sci., 59(2), 237-24 1. Murakami, E. G., Sweat, V. E.. Sastry, S. K., Kolbe, E., Hayakawa, K. & Datta, A. (lY95). Recommended design parameters for thermal conductivity probes for nonfrozen food materials. J. Food Engng. 27, 10%123. Nieto de Castro, C.A., Taxis, B., Rodcr, H. M. & Wakeham, W. ‘4. (198X). Thermal diffusivity measurement by the transient hot-wire technique: a reappraisal. Int. J. Thermnphys. 9 (3), 293-315. Ott, L. ( 1977). An Introduction to Statistical Methods and Data Anafysis. Duxbury Press. North Scituate, MA, pp. 112-X. Richmond, J., Nilsson, 0. & Sandberg, 0. (1984). Thermal properties of some lubricants under high pressure. J. Appf. Phys. 56 (7), 2065-2067. Sweat, V. E. (1986). Thermal properties. In Engineering Properties qf Foods, eds M. A. Rao iic S. S. H. Rizvi. Marcel Dekker, Inc., New York, pp. 49-87. Tong, C. H., Sheen, S., Shah, K. K., Huang, V. T. blr Lund, D. B. (1993). Reference materials for calibrating probes used for measuring thermal conductivity of frozen foods. J. Food Sci., 58, 186- 189, 192. van der Held, E. M. F. Kr van Drunen, F. G. (1949). A method of measuring thermal conductivity of liquids. Physica 15, 865. Vos. B. (1955). Measurement of thermal conductivity by a non-steady state method. &~I. Sci. Res. A5, 425-438. Underwood, W. M. & Taggart, R. B. (1960). The thermal conductivity of several plastics. measured by an unsteady-state method. Heat Transfer (Storrs) Chem. Engng. Pro,q .Sympox Ser: 56 (30), 261. Woodsidc, W. (1959). Probe for the thermal conductivity measurement of dry and moist materials. ASHRAE 65, 291-3 IO.
Fl.SF.VIER
Analysis of Various Design and Operating Parameters of the Thermal Conductivity Probe* E. G. Murakami,” V. E. Sweat,h S. K. Sash-y’ & E. Kolbe” “US Food and Drug Administration/National Center for Food Safety and Technology, 6502 S. Archer Ave., Summit-Argo, IL 60501, USA “Texas A & M University, Department of Agricultural Engineering, Rm 201, Scoates Hall I, College Station, TX 77843-2117, USA ’ The Ohio State University, Department of Agricultural Engineering, 590 Woody Hayes Dr., Columbus, OH 43210, USA “Oregon State University, Departments of Bioresource Engineering and Food Science and Technology. Corvallis, OR Y7331-3906, USA (Received
21 February
1995; revised 27 Novcmbcr
lYY5; accepted 5 February
lYY6)
ABSTRACT The thermal conductivity (k) probe is derived from an idealized transient heat transfer model; therefore, there are inevitable differences between the real probe and the theoretical model. However; the k probe is still an accurate and practical measurement device with wide-ranging applications if designed and used properly. Users of a thermal conductivity probe must be aware of Its limitations and the possible errors that can develop in its application. This paper includes a theoretical derivation sf the k probe equation and some experimental and theoretical simulations of parameters that can cause errors in the application of the k probe. An explanation is given of the significance of certain design and operating parameters. Some of the findings in the study are that the time-correction factor is not required and contact resistance does not affect accuracy. The calibration factor is necessary since it compensates for the difference in the thermal masses of the probe and the sample. Errors due to edge effects and convection can be avoided by limiting data analysis to the linear section of the time-temperature plot. The truncation error is minimized by making /jas small as possible. Copyright 0 1996 Elsevier Science Limited
“‘The contents of this paper arc solely the opinions of the authors, represent the official views of the US Food and Drug Administration. 2OY
and do not necessarily
210
E. G. Murakami et al.
NOTATION C,
&2 D E,, k L M e ; s t, to, Lf t-T T, T,
X,Y& CY ZT, AT,,
Euler’s constant (0.5 7721) specific heat (Jfkg”C) parameters of the heated-rod equation diameter (m) truncation error thermal conductivity (WIm”K) length (m) ratio of thermal masses of specimen and probe power input (Wlm) radius (m) contact resistance between the probe and the medium thermal mass per unit length of the heater (JImOK)
time, final time, initial time and time-correction time-temperature plot in semilog axes temperature and reference temperature (“C) Cartesian coordinates thermal diffusivity (m’/s) density (kg/m3) temperature rises at times t and to (“C)
(m “KIW)
factor (s)
INTRODUCTION The thermal conductivity (k) probe apparatus is a versatile and simple device that has gained popularity in food engineering applications. It has been used on a wide variety of food materials including powders, liquids, bulk grains and whole fruits, and owes its popularity to its low cost, ease of operation and simple design. However, its theory is based on the line-heat source technique, which is derived from an idealized transient heat conduction model; thus, there are differences between the real probe and its theoretical model. The main difference is that the real probe has finite thermal mass, whereas the theoretical model assumes it to be negligible. Moreover, published applications of the k probe have differed considerably from the theoretical assumptions. In the thermal conductivity measurements of rocks in situ, the contact resistance between the probe and the rock, which is theoretically assumed to be negligible, is finite because the procedure involves a loose-fit probe dropped into diamond-drilled holes that are crooked (Blackwell, 1954). In liquids and gases, radiation, convection, variable thermal properties and Knudsen effect, which are all assumed to be theoretically negligible, could become significant in some experimental situations (Healy et al., 1976; Nieto de Castro et al., 1988). In food engineering applications, the above problems are compounded by specimens with finite size and anisotropic properties. For instance, the thermal conductivity probe had been used to measure small items like corn kernels (Kusterman et al., 1981), radially asymmetric materials like tobacco (Casada & Walton, 1989) and tortilla chips (Moreira et al., 1992). These deviations can lead to unavoidable errors; however, their effects can be minimized and sometimes compensated through careful attention to the design and operating parameters and data analysis. To minimize the various design errors,
Design und operating parameters of the thermal conductivity probe
211
Murakami et al. (1995) developed a guideline for designing and fabricating k probes for non-frozen food materials. Researchers in the past accounted for these errors in various ways. Nieto de Castro et al. (1988) treated each error as a temperature deviation and algebraically added it to the measured temperature. Van der Held and Van Drunen (1949) as well as many other subsequent users, theorized that the finite size of the heater caused a delay in heating the products; they therefore subtracted a time-correction factor from the time value. Another often-used correction factor has been the calibration factor. It is recommended that k probes be calibrated with materials of known thermal conductivity, usually either water--agar gel or glycerin, before each use. The ratio between the published and the measured thermal conductivity values of the calibration material becomes the calibration factor. This value is then multiplied by the measured thermal conductivity values of the materials. For isotropic materials, Carslaw & Jaeger (1959) and Davis (1984) suggested using the geometric mean of the k at the two dimensions perpendicular to the axis of the k probe. The objectives of this paper are to make probe users familiar with the theoretical assumptions and limitations of the k probes and to provide them with recommendations for maximizing probe performance. Both theoretical and experimental t-T plots are used to aid in discussing the significance of various factors (truncation error, thermal mass ratio of specimen to probe, contact resistance, convection, edge effect, initial time, time-correction factor and calibration factor). In addition, the various line-heat source equations are compared to the more realistic but complicated heated-rod equation.
THEORETICAL
CONSIDERATION
Line-heat source technique The line-heat source technique measures thermal conductivity, assuming unsteadystate heat conduction from an infinitely long line-heat source immersed in an infinite and homogeneous medium. Although its working equation is widely known, the theoretical derivation is not. To be aware of the limitations and requisites for its effective application, the user must be familiar with the theoretical assumptions and derivations. Considering a string of instantaneous point sources (Q,) which is continuously generated over a period of time at a constant rate; then the temperature rise at r due to conduction in a medium with constant thermophysical properties is equal to (Carslaw & Jaeger, 1959):
where: ri
(14
L1 = 42
(t -
to)
212
E. G. Murakami et al.
(lb) Q = QIPC,
(lc)
The radius r is the distance between the heater and the point of interest, gap between them is filled with the medium. The analytical solution to eqn (1) is as follows (Carslaw & Jaeger, 1959):
The exponential integral E,(/12) is an infinite series (Abramowitz and therefore the expansion of eqn (2) is: AT=
Q - $
2rck
B2 _ 1)” 2(1!) 4(2!)
-In@)+
I
& Stegun, 1964)
1
*”
and the
(3)
This is the complete or nontruncated equation of the line-heat source technique. However, it is not used for calculation of thermal conductivity because it requires the thermal diffusivity of the material. Moreover, this equation is numerically too complicated for many applications and it must be simplified for practical purposes. In practice, the variables r and t in the fi parameter are referred to as the heater size and the test duration, respectively; thus b is both a design and an operating parameter. If the k probe apparatus is either designed or operated such that b is kept small, p” and the higher order terms can be truncated from eqn (3): AT=
-&-[-
$-
-In(b)]
(4)
This is the truncated equation of the line-heat source technique. Its main feature is that the temperature rise is linear with the natural logarithm of p, and thus the thermal conductivity can be derived from the slope. However, for calculation, eqn (4) requires the thermal diffusivity of the specimen, which is more likely to be unavailable. To avoid this complication, it is evaluated within a time interval (t-t,,). The resulting expression is the working equation of the line-heat source technique or the probe method (van der Held & van Drunen, 1949): AT-ATo=
Q
4nk
Heated rod The theoretical assumptions in the line-heat source technique are idealized and impractical. A real line-heat source apparatus is inevitably different from the theoretical model because it has finite diameter and mass. Moreover? in solids, the contact resistance on the probe surface, which was not considered m the line-heat source technique, may not be negligible.
qf the thermal conductivity probe
Design and operating parametets
213
Carslaw & Jaeger (1959) derived the temperature rise in an infinitely long heated probe with finite radius (r), thermal mass (S), and surface or contact resistance (K). For any value of /L the temperature rise (Aq in a heated rod is as follows: 2p’(cz - 2)
/?(4C, -C2)
T=
c
In([Lc)+...
(6)
C,
7
I
where, C, = 2nRk
(ha)
C1 = 2nr’pCJS
(C)b)
C = exp(C,)
(hc)
Note that if the contact resistance is negligibly small, the thermal masses of the heater and the material are equal (Cz = 2), and if the probe radius is relatively small, eqns (4) and (6) are equal.
DISCUSSION Truncation error The working probe equation [eqn (5)] has inherent truncation error (E,,) as a result of mathematical simplification from the nontruncated [eqn (3)] to the truncated form [eqn (4)]. By eliminating the infinite series, the predicted temperature rise using the truncated equation (ATT) is always smaller than that calculated with the nontruncated equation (AT,). However, the difference decreases as the value of [i decreases (Fig. 1). The truncation error, which is the ratio of that difference to the calculated temperature rise using eqn (3), is:
5
4
I
Truncation “\\
60
i
‘i‘_ ‘:,.,!/’
t 4
i
Truncated
_ _ 0.10
50
i i
\ ,* Nontruncated ‘\
I
0 lmL-~.----__ 0.01
error
\
G‘3 e 52 ;2
70
I
‘\.\
“ >yj,
_,’ --~--- ’ L_.I__--__+
30:s
40 g :
I,0 \ L ‘\
10 IJ I .oo
Beta
Fig. 1. Simulated temperature rise in water, calculated using eqns (10) and (11) and the resulting truncation errors for steel probes with r = 0.3 15 mm.
214
E. G. Murakami et al. 10
T
r = 0.315 mm
-2t Time (s)
Fig. 2. Simulated
temperature
rise in water, calculated using eqns (10) and (11) for probes with radii of 0.315 and 1.0 mm.
ATb-AT7 &, =
AT,
(7)
The truncation error is an error for each data point. Its highest value is at lo and it decreases thereafter with time. It is minimized by calculating the thermal conductivity from the linear portion of the t-T plot (Fig. 2). Vos (1955) recommended setting the truncation error to ~0.59%; Murakami et al. (1995) suggested minimizing it by making the k probe as small as possible. In non-food applications, researchers prefer a truncation error of < 0.2% (Richmond et al., 1984). Thermal mass ratio of specimen to probe In theory, the ratio of the thermal mass of the specimen to that of the k probe (M) is not considered in the line-heat source equation. In reality, the M changes the slope of t-T plots, affecting the calculated k value. Another effect of M is that it makes the t-T plot nonlinear, and consequently the required to increases. The k probe generally consists of a heater, temperature sensor, and tubing to make the assembly sturdy. By assuming a solid structure, its average thermal mass can be calculated from the thermal properties of the individual components. The thermal mass (S) per unit length of k probe is as follows: S = ~r2pCplprohct Thus, the parameter
(8)
C2 in eqn (6) becomes: C2=2M
(8a)
where M is the ratio of the thermal masses of the specimen and the probe, that is:
(9)
Design and operating parameters of the thermal conductivity probe
215
If M = 1, the effect of thermal mass in eqn (6) is eliminated. For reference, the M for water and a solid-steel probe is 1.07; for glycerin with the same probe, M = 0.79. Effect on Slope
A mathematical simulation was conducted to evaluate the effect of M on the slope of the t-T plot. The t-T plots of the heated-rod equation [eqn (6)j at various M values and of the nontruncated line-heat source equation, which represents an ideal situation (M = 1 and R = 0), are shown in Fig. 3. The line-heat source plot is chosen as the reference because it represents an ideal condition, viz negligible thermal mass of the k probe. At the linear section, as M increases, the slope of the t-T plot decreases and the deviation of the slope from the reference plot increases. At M values 50.6 and 22.5, the slope changes by as much as 25% per unit of 44. The deviation of the slope from the reference is remedied by using a calibration factor, which is directly proportional to the degree of the deviation (see calibration factor section) and the filling material for the probe tube. Efj&t on t,,
Another effect of M is in the nonlinear portion of the t-T plot (Fig. 3). This portion curves upward at low values of M (MS 1) and then becomes more linear as ,izI gets closer to 1. The number of data points in the nonlinear portion increases as the M decreases and consequently the to increases (see initial time section). I’ructical
Implications
Figure 3 shows that the calibration material can be chosen such that the slopes of the t-T plot of the calibration material and the specimen are as close as possible. Some researchers prefer fillers in the k probe for various reasons. Because tillers can change the thermal mass of the probe and also because of other considerations (e.g. thermal stability), they should be chosen in the same manner as the calibration material.
6.~
M=0.6 ,,/
z ._2 a c
xi_ 4
:/ M =
-9’
,,7
, ‘,” ,I
2.5 , -<,;s ,-:’ ,H
. . ..a’ / Nontruncated 0. I
1
10
100
Time (s)
Fig. 3. Simulated AT (heated-rod eqn, water with a steel k probe of r = 0.315 mm) showing the effect of M and comparison with the ideal situation (M = 1, R = 0)as represented hy the non-truncated eqn.
E. G. Murakami et al.
216
Contact resistance The contact resistance at the probe-specimen interface is assumed to be negligibly small in the derivation of the line-heat source equation. In reality, contact resistance could be significant, especially in powders or when the hole where the probe is inserted into the specimen becomes enlarged through repeated use. Contact resistance causes the t-T plot to shift, but it does not change the slope. Thus, it does not affect the probe accuracy. Its other effect is that it changes the nonlinear portion of the t-T plot. The equation [eqn (6)] derived by Carslaw & Jaeger (1959) for a heated rod is plotted to evaluate the effect of contact resistance (Fig. 4). In this figure, the plots for the heated-rod equation are simulations of a solid-steel k probe immersed in water. In contrast to the idealized plot of the line-heat source equation [eqn (3)], Fig. 4 shows that contact resistance shifts the linear portion of the t-T plot without changing the slope. Thus, the effect of contact resistance can be eliminated by limiting data analysis within the linear portion of the t-T plot. This finding is relevant in calibrating k probes for porous specimens (see ). Contact resistance also affects the nonlinear portion of the t-T plot. In Fig. 4, the curvilinear portion gets bigger and moves upward as the contact resistance decreases and consequently the to increases. Note that the plots of the heated rod with R = 0 and the nontruncated line-heat source equation are not equal because the heated rod was calculated with finite thermal mass ratio. Convection The k probe is based on pure conduction. The added effect of convection increases the rate of heat transfer through the specimen. Experimental t-T plots for glycerin and water at an extended test time of 110 s and with power input of 11 W/m are shown in Fig. 5. No convection-deterrent material was added to either specimen. This figure shows that in water convection started after 10 s, whereas in glycerin the
8-
R=0.15 /’
/’
R = 0.05
6
Nontruncated I
IO
/ 100
Time (s)
Fig. 4. Effect of contact resistance on the t-T plot of a heated rod and comparison line-heat source, using the non-truncated eqn.
with the
Design at~d operating parameters
qf the thermal conductivity probe
217
Glycerin
Water
.+_.+_
+_tt__-.----c IO
If-.-
+~--t----t-r--tc-ntl 100
1000
Time (s)
Fig. 5. Effect of convection on the experiment t-T
plots of water and glycerin (t,, = 1.0 s.
Q = I I W/m).
onset of convection occurred much later, at around ‘70 s. The obvious change in the slope indicates that convection effects can be easily eliminated by limiting data analysis to the linear portion of the t-T plot. Thus with Q = 11 W/m, water and more viscous liquids can be used as either test or calibration materials without adding convection-deterrent materials. Figure 5 also shows that convection shortens the linear portion of the t-‘T plot and may render test results useless because of insufficient data points for analysis. Convection can be avoided by using low power input, reducing test duration, and adding convection-deterrent materials like agar and glass wool. The appropriate values for power input and test times are determined by trial-and-error during the calibration. Agar may be added to liquid test samples to turn them into gels. Sweat (1986) recommended adding 0.5% agar to water. Murakami (1993, 1994) found that a water-agar gel with 0.6% agar was an excellent calibration material that could be kept for several days. It was easily reconstituted by reheating when the probe holes became enlarged and too numerous because of repeated use. For oil and water specimens at high temperature, Sweat (1986) recommended placing glass wool in the specimens to mechanically inhibit convection. Edge effect Edge effect is caused by heat transfer at specimen edges, which either enhances or diminishes conduction through the specimen. The influence on edge effect of specimen environment, operating parameters, and specimen radius is discussed. T/U?Role of Specimen Environment The thermal conductivity of the environment compared to that of the specimen affects the development of edge effect by making the t-T plot nonlinear toward the end of the test. An experiment was conducted to illustrate edge effect due to different types of environment. A potato cylinder with k = 0.56 W/m “K (Mur-
E. G. Murakami et al.
218
akami, 1993) and D = 7 x lo-” m was exposed thermal conductivity environments.
to relatively
lower
and higher
Environment with Low k An environment with relatively low k was simulated with still air, which has a value of k = 0.0244 W/m “K (ASHRAE, 1976). The potato cylinder was placed in a beaker loosely filled with paper towels to prevent convection. Results showed that in the still-air environment, the t-T plot moved upward as soon as edge effect developed (Fig. 6). This result indicated that when the heat reached the specimen edges, the conduction through the specimen slowed. The air, whose thermal conductivity is one order of magnitude lower than that of water, acted as an insulator and this decreased the temperature gradient in the specimen. It can be observed from Fig. 6 that edge effect started at ~~40 s, and at this time the calculated temperature rise due to convection was 0.7”C. If a temperature change of 0.7”C is to be considered significant by the probe user, the edge effect can be eliminated by discarding the data points at t240 s. If edge effect is not noted and if only the nonlinear section at the initial portion of the t-T plot is eliminated in the analysis, edge effect due to environments with k lower than that of the specimen would lower the calculated thermal conductivity. Environment with High k The environment with relatively high k was simulated by using stirred water (k + 0.6 W/m “K). In this setup, the specimen was immersed in a beaker filled with water and the water was continuously stirred with a magnetic stirrer. The result showed that the edge effect caused the t-T plot to bend downward (Fig. 6). The high rate of heat transfer through the edges enhanced the heat conduction in the specimen by keeping the temperature gradient. If the data are analysed in a similar manner as discussed above, the calculated thermal conductivity would be higher than without the edge effect.
Still air / Potato cylinder
6
/’
/,’
(D = 7 mm)
& ._2 4-p! C-. 2 --
0
i-t::; 1
Fig. 6. Evidence
10
100
of edge effect in the experimental temperature rise of potato cylinders still-air and stirred-water environments (Q = 11 W/m).
in
319
Design and operating parameters of the thermal conductivity probe
The Role of Operating Parameters
The effects on the occurrence of edge effect of operating time (t) and power input (Q) were also evaluated.
parameters
such as test
Power Input
The effect of power input on edge effect was evaluated by calculating the temperature distribution in water with the nontruncated line-heat source equation. The slope of the temperature rise across the specimen, which can be approximated from the truncated equation [eqn (4)], is calculated as follows:
d(AT)
( 10)
dr
Based on this equation, the temperature rise (AT) decreases toward the specimen edges, and its slope is affected by power input, thermal conductivity, and distance from the heater. The heat from the probe causes an advancing hot front (ATrO.Ol”C) to move toward the specimen edges. A simulation of water at t = 10 s shows that if the power input Q is varied between 5 and 50 W/m, the distance of the hot front is within 4.2
The effect of test duration on edge effect was studied with the following numerical simulation. With the power input fixed at 5 W/m, the temperature profile in water was calculated for t = 5, 20 and 50 s (Fig. 8). From t = 5 s to t = 50 s, the hot front moved from r = 3 x lop3 m to r = 8 x lo-” m. The distance of the hot front from the probe increased by 167% when the test time was increased by 900%. Thus, the hot front is comparatively more sensitive to the test time than to the power input.
x
pjib -t-
-y-
‘\ -*\
Q = 5 W/m Q=
10 W/m
Q = 20 W/m
Q = 50 W/m
\ ‘k
_~ 0.1
~_.__~
-a-*xw+ ---tcM I
IO
Radius (mm)
Fig. 7. Simulated temperature distribution in water at a test time of 10 s.
E. G. Murakami et al.
220
I t
01 0.1
1
10
Radius (mm)
Fig. 8. Simulated temperature profile in water at a power input of 5 W/m.
Minimum Specimen Radius
To avoid edge effect, researchers have been using the following equation, mended by Vos (1955) for calculating the minimum specimen radius: 1 <0.6 B’
recom-
(11)
However, this equation does not include the power input and requires knowledge of the thermal diffusivity of the specimen for calculation. Using eqn (11) and a value of t = 10 s, the minimum radius for s ecimens with high moisture content (those with 2% 1.44 x lop7 m’/s) is 3.1 x IO-- P m. For an infinite medium, the calculated AT [eqn (3)] at this point is 0.06”C if Q = 5 W/m, and 0.6”C if Q is increased to 50 W/m. The heat transfer from the edges with AT = 0.6”C may be negligibly small but could be significant at AT = 0.6”C. Thus, eqn (11) is effective only at low power input levels. Probes users should use this equation as a first estimate of specimen radius and then examine the t-T plot for edge effect. Combination of power input, test duration, and specimen radius
In this study and previous studies by Murakami (1993, 1994), a power input of Q = 11 W/m was found to be sufficient for high moisture food materials (k = 0.50.6 W/m “K). Simulations using eqn (3) with a specimen having thermal properties close to water and Q = 11 W/m, showed that if the specimen r is 2 1 x lo-’ m, tests can be performed at up to 50 s without edge effect. Although convection develops in liquid water within 10 s of test time (Fig. 5) longer test times can be beneficial with solid food materials such as fresh potatoes and scallops. Conversely, theoretical simulations have shown that if the specimen type and power input are the same and if the test time is limited to 10 s, the specimen can be as small as r = 5 x lo-” m without edge effect. For specimens smaller than r = 5 x lo-’ m, edge effect can be eliminated by first decreasing the test time and then the power input.
Design und operuting purameters of the thermul conductivity probe
221
Initial time The primary purpose for the initial time (to) in the working equation [eqn (5)l is to eliminate the need for the specimen thermal diffusivity in the calculation. However, the t,, has another practical benefit. If chosen properly, it can be used to eliminate the initial nonlinear portion of t-T plots. Because its value does not affect the slope of the linear section of the t-T plot, data analysis can be simplified by having a single t,, value for all specimens. The to is affected by the probe radius, the ratio of the thermal masses of the specimen and the probe, and the contact resistance. The to can be used to linearize t-T plots if it is set equal to at least the time when the nonlinear portion of the t-T plot ends. In Fig. 9, the plots of water1 and glycerin1 include all the collected data, and it can be observed that because of differences in their thermal mass ratios, the t-T plot of water is linear, whereas that of the glycerin is initially nonlinear and then becomes linear after 2 s. By setting to = 2 s, the t-T plots of both water (water2) and glycerin (glycerin2) are linearized. Time-correction factor The actual t-T plots of some materials, i.e. glycerin, are nonlinear at the initial section. Because the working line-heat source equation is linear, early users like van der Held & van Drunen (1949), Woodside (1959), and Underwood and Taggart (1960) used the time-correction factor (tCf) to make the actual t-T plots straight. They modified the working equation to the following: AT-
AT,, =
(12)
Many justifications have been given for using the time-correction factor. According to van der Held and van Drunen (1949), the t,, compensated for the initial time
‘2
I
Effect of initial time on the experimental t-T plots of water and glycerin [note: for and glycerin1 , t* = T(t) - T(t = 0); for water2 and glycerin2, t” = t/2 and T* = T(t) - T(t = 2)].
222
E. G. Murakami et al.
10
I
(t - t,fMb - Lf) Fig. 10.
Effect of time-correction factor on t-T plots (to = 2.0 s, t,f = 1.0 s).
spent in heating the heater itself and for uncertainty in fixing the initial time, to. However, some researchers like D’Eustacio and Schreiner (1952) determined that the tCfwas unnecessary. Sweat (1986) found this value equal to zero. The tCf has no theoretical basis and, in fact, it can make a linear t-T plot curvilinear. As shown earlier, the t-T plot of glycerin was made linear by using an appropriate value of to (Fig. 9). However, when tCf= 1.0 s was subtracted from the time values while to = 2.0 s was maintained, the t-T plots of both glycerin and water became nonlinear (Fig. 10). Although a value of fCf can be found that can make those t-T plots linear, e.g. tCf= 0.5 s, there is no added benefit for using it. Thus, the use of tCf is not recommended. Calibration factor The calibration factor is used to increase the precision of the k probe. It is determined by performing a test run at the same operating parameters, e.g. power input and test time, with a material (called calibration or reference material) of known thermal conductivity. It is calculated from the ratio of the published values to the measured thermal conductivity values. The calibration factor is essential because of various factors that affect the slope of the linear portion of t-T plots. These factors are the thermal mass ratio of specimen to probe (see ) and the inaccuracy in the measurement of the power input. Effect of M As discussed in , the t-T plot of a k probe with thermal mass deviates from the ideal probe, -which has negligible thermal mass. The difference in the slopes of their t-T plots (at the linear section) is corrected in practice by using the calibration factor. The calibration factor is affected by the thermal mass of the calibration material. Thus, for a stainless-steel k probe (D = 7.1 x lop4 m, L = 22.8 x lo-” m) the calibration factor is 0.895 with water as the calibration material, and 0.835 with glycerin. Another factor that affects the thermal mass of the probe is the k probe filler, which
Design and operating parameters ofthe thermal conductivity probe has been used by researchers 1993) for various reasons.
(Morley,
1966; Karwe ef al., 1991; Gratzek
223 & Toledo,
Effect of Power Input The slope of the working equation increases with increasing power input. Thus, any inaccuracies in the determination of the power input directly affect the calculated slope of the t-T plot. The power input is calculated from the resistance of the heater wire and the electric current flowing through it. Although the resistivity equations for some commonly used heater wires are available in the literature, they have low accuracies because of variations in heater wire composition. The accuracy of digital ammeters is usually sufficient for application in k probes. Moreover, sufficiently stable DC power sources are available for powering the probe heater. However, instrument drift can occur and the calibration factor can be used to detect it. If the k probe is calibrated before and after a series of tests, the t-test can be used to determine if there is a statistical difference between the two calibration factors (Ott, 1977). Selection qf calibration material In theory, the thermal masses of the specimen and the calibration material must be equal, but this is impractical in actual use. In practice, the calibration material must he chosen such that the slope of its t-T plot is as close as possible to that of the specimen. Figure 11 shows the uncorrected t-T plots of various materials. The water specimen was made from 0.6% water-agar gel, and the food materials were all fresh (unprocessed) and had high moisture content (75% w.b.). All the t-T plots are almost parallel to each other, except for glycerin. Thus, water is a more appropriate calibration material than glycerin for these materials. If glycerin is chosen as the calibration material, the measured thermal conductivity values for the food materials in this example would have been lower by 6.7%. The most common calibration materials for nonfrozen foods are water and glycerin. If the test time is I 10 s, water does not need any convection deterrent materials in it (see the discussion on Presence of Convection). For frozen food and
0.1
I
-+-
Potato
-i-
Carrot
IO
Time (s)
Fig. 11.
Experimental t-T plots of various food and calibration
materials
(& = 11 W/m).
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E. G. Murakami et al.
other high thermal conductivity materials (k >0.6 W/m “K), Tong et al. (1993) developed reference materials by blending copper powder with bentonite paste. CONCLUSIONS Although the k probe is based on an idealized model, accuracy can be maintained if it is carefully designed and operated and if the data are properly analysed. The errors caused by the various design and operating parameters can be determined from the t-T plot and can be corrected by limiting the data analysis within the linear region. The effects of the various design and operating parameters are as follows: (1) The truncation error can be minimized by making the k probe as small as the application and fabrication procedures allow. (2) The thermal mass ratio of specimen to heater (M) affects the slope of the t-T plot; thus its value should be maintained between calibration and actual application of the k probe. (3) The contact resistance does not affect the accuracy of the probe because it shifts the t-T plot without affecting the slope. (4) Convection affects the accuracy of the probe apparatus by changing the slope of the t-T plot. However, its effect can be eliminated by limiting data analysis to the linear portion. (5) Edge effect causes problems similar to that of convection and can be corrected in a similar manner. It can be avoided by either increasing the specimen diameter or decreasing the test time; decreasing the power input has minimal influence. (6) The initial time (to) should be chosen such that it is equal to at least the time when the t-T plot becomes linear. (7) The time-correction factor (tcf) is not recommended because it has no theoretical basis, and because its function for linearizing the t-T plot can be achieved with proper selection of the to. (8) The calibration factor is essential because the slope of the actual t-T plot is affected by the thermal mass of the specimen, accuracy in determining the power input and instrument drift. The reference materials should be chosen such that the slope of its t-T plot is close to that of the specimen. REFERENCES Abramowitz, M. 6t Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. NBS Applied Mathematics Series 55. U.S. Department of Commerce, Washington, DC. ASHRAE (1976). Thermophysical Properties of Refrigerants. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, GA [In ASHRAE 1982 Fundamentals].
Blackwell, J. H. (1954). A transient-flow method for determination of thermal constants of insulating materials in bulk. J. Appl. Phys. 25, 137-144. Carslaw, H. S. & Jaeger, J. C. (1959). Conduction of Heat in Solids. 2nd edn. Clarendon Press, Oxford, pp. 258-345. Casada, M. E. & Walton, L. R. (1989). Thermal conductivity of baled barley tobacco. Trans. ASAE
32 (3), 977.
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of thethermal conductivity probe
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D’Eustachio. D. & Schreiner, R. E. (1952). A study of a transient method for measuring thermal conductivity. ASHI/E Trans. 58, 33 1. Davis, W. R. (1984). Hot-wire method for the measurement of the thermal conductivity of refractory materials. In Compendium of Thermophysical Property Measurement Methods, Vol. 1, eds K. Maglic, A. Cezairliyan & V. Peletsky. Plenum Press, New York, pp. 23 I-54. Gratzek, J. P. & Toledo, R. T. (1993). Solid food thermal conductivity determination at high temperatures. /. Food Sci. 58 (4), 908-913. Healy, J. J., de Groot, J. J. & Kestin, J. (1976). The theory of the transient hot-wire method for measuring thermal conductivity. Physica 82C, 392-408. Karwe, M. K., Tong, C. H. & Lund, D. B. (1991). Modeling of temperature distribution in a thermal conductivity probe with different tilling materials. Paper presented at the 1991 IFT Annual Meeting, Dallas, TX, 1-5 June Kusterman, M., Scherer, R. & Kutzbach, H.D. (1981). Thermal conductivity and diffusivity of shelled corn and grain. J. Food Process Engng 4 (3). 137-153. Moreira, M. G.. Palau, J. & Sweat, V. E. (1992). Thermal properties of tortilla chips during deep-fat frying. ASAE Paper No. 92-6595. Morley, M. J. (1966). Thermal conductivities of muscles, fats and bones. J. Food Technol. 1, 303-311. Murakami, E. G. (1993). The thermal properties of potatoes and carrots as affected by thermal processing. Paper presented at the 1993 IFT Annual Meeting, Chicago, IL, IO-14 July. Murakami, E. G. (1994). Thermal processing affects properties of commercial shrimp and scallops. /. Food Sci., 59(2), 237-24 1. Murakami, E. G., Sweat, V. E.. Sastry, S. K., Kolbe, E., Hayakawa, K. & Datta, A. (lY95). Recommended design parameters for thermal conductivity probes for nonfrozen food materials. J. Food Engng. 27, 10%123. Nieto de Castro, C.A., Taxis, B., Rodcr, H. M. & Wakeham, W. ‘4. (198X). Thermal diffusivity measurement by the transient hot-wire technique: a reappraisal. Int. J. Thermnphys. 9 (3), 293-315. Ott, L. ( 1977). An Introduction to Statistical Methods and Data Anafysis. Duxbury Press. North Scituate, MA, pp. 112-X. Richmond, J., Nilsson, 0. & Sandberg, 0. (1984). Thermal properties of some lubricants under high pressure. J. Appf. Phys. 56 (7), 2065-2067. Sweat, V. E. (1986). Thermal properties. In Engineering Properties qf Foods, eds M. A. Rao iic S. S. H. Rizvi. Marcel Dekker, Inc., New York, pp. 49-87. Tong, C. H., Sheen, S., Shah, K. K., Huang, V. T. blr Lund, D. B. (1993). Reference materials for calibrating probes used for measuring thermal conductivity of frozen foods. J. Food Sci., 58, 186- 189, 192. van der Held, E. M. F. Kr van Drunen, F. G. (1949). A method of measuring thermal conductivity of liquids. Physica 15, 865. Vos. B. (1955). Measurement of thermal conductivity by a non-steady state method. &~I. Sci. Res. A5, 425-438. Underwood, W. M. & Taggart, R. B. (1960). The thermal conductivity of several plastics. measured by an unsteady-state method. Heat Transfer (Storrs) Chem. Engng. Pro,q .Sympox Ser: 56 (30), 261. Woodsidc, W. (1959). Probe for the thermal conductivity measurement of dry and moist materials. ASHRAE 65, 291-3 IO.