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Time evaluation for the steady state condition of the weight before the measurements
Measurement 25 (1999) 299–306 www.elsevier.com / locate / measurement
Time evaluation for the steady state condition of the weight before the measurements ¨ Vahit C¸iftc¸i*, Sevda Kac¸maz, Umit Akc¸adag, Orhan Sakarya ¨ ¨ ¨ ( UME), P.K.21, 41407 Gebze, Kocaeli, Turkey , Ulusal Metroloji Enstitusu TUBITAK Received 12 September 1998; received in revised form 4 February 1999; accepted 24 March 1999
Abstract The temperature difference between the weight and its surrounding is an important parameter in the evaluation of the waiting time required by the weights to reach a steady state condition before starting a measurement, because the thermal gradients cause the thermal convection effect on the mass during the weighing process. In this study, investigation of both theoretical and experimental applications were carried out to evaluate the steady state condition of the weight before the calibration. Two kinds of unsteady state heat conduction equation were introduced, theoretical calculation results were compared to experimental results, and comparisons were very satisfactory for a 20 g weight. Then theoretical calculations were extended for different initial temperatures and for different weights. It was realized that the steady state condition of the waiting time was less dependent on initial temperature, but greatly dependent on the sizes of the weights. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Unsteady state heat conduction; Waiting time of the weight
1. Introduction It is always a question for the metrologists that after acceptance and cleaning of a weight, how long does it take for it to reach a steady state condition for an accurate calibration. There has been neither a common nor a recommendation procedure established yet to evaluate waiting time as a function of the initial temperature condition of the weights and their size. In the literature there are a few papers dealing with this subject; two papers were published by Glaser about ‘‘Response of apparent mass to thermal gradients’’ [1] and ‘‘Effect of free convection on the apparent mass of 1 kg mass standards’’ [2], another *Corresponding author. Tel.: 1262-646-6355; fax: 1262-6465914.
paper was from Plassa about ‘‘Variation in the mass of standard weights due to contact with water’’ [3]. However, none of the papers deals with the waiting time of the weights before calibration. For that reason, some weights were manufactured and cleaned, then initial conditions of the weights were changed by putting in the refrigerator or by cleaning in soxhlet apparatus to increase the initial temperature to about 808C, then measurements were done.
2. Brief description of the experimental work It has been decided to manufacture four weights at UME for the investigation of the waiting time of cleaned weights by applying various cleaning procedures and measuring, then comparing these ex-
0263-2241 / 99 / $ – see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S0263-2241( 99 )00015-9
300
V. C ¸ iftc¸i et al. / Measurement 25 (1999) 299 – 306
perimental results by theoretical modeling using the unsteady state heat conduction equation. The balance used for weighing was a single-pan, electro-magnetically compensated, mass comparator with a capacity of 50 g, readability of 1 mg and standard deviation of less then 2 mg for 20 g. Three weights, MK1 to MK3, were manufactured at UME, with a stainless steel having 18% chromium, 8% nickel content and with density 7900 kg m 23 which was determined by a hydrostatic weighing system. The roughness values (average peak to valley heights) of the polished weights were determined as 0.1 mm by UME Dimensional Laboratory. For comparison, one of the three weights, MK1, was used as a reference standard and stored on a turntable inside the balance. The temperature inside the balance chamber was monitored by PT-100 temperature sensors and was around 228C. Before cooling and cleaning, the apparent mass (d m) 221 (difference between MK2 and MK1) was 2 5323 mg at laboratory conditions and for MK2 cooled to about 108C, then weighed. Before cleaning the apparent mass (d m) 321 (difference between MK3 and MK1) was 2 12 347 mg at laboratory conditions and for the MK3 weight washed at about 808C in soxhlet apparatus and left until at 308C (because of the technical specifications of the balance) and then weighed. For each situation before the test, the surface temperature of the weight was measured by a PT-100 resistance thermometer with a digital temperature indicator. The variations of apparent mass difference d m(t) between the test and reference weight, as a function of time t, were measured every 10 hours. 3. Theoretical model Two kinds of theoretical Unsteady State Heat transfer model were introduced here related with Biot numbers, which are nondimensional and described by Eq. (11).
3.1. Transient heat conduction; lumped-parameter analysis with Bi ,0.1 A process designated ‘‘unsteady state’’ or ‘‘transient’’ is one that is time dependent.
Fig. 1. The properties of a weight and its surroundings.
Fig. 1 shows the properties of a weight and its surroundings. We assume the initial temperature of the material is uniform and that the temperature is a function of time only and thus the temperature is uniform throughout the material at any time. Heat is transferred between the material and its surroundings by convection only. When material is put in a cold or hot environment, the question is, what will be the temperature distribution for a given time. From the first law of Thermodynamics, Ref. [4]: dE / dt 5 Q
(1)
There is no work in or out, giving the following initial conditions: at the centre t 5 0 → T 5 T i
(2)
at the boundary Q 5 2 h 3 S 3 (T 2 T ` )
(3)
From the internal energy equation: dE / dt 5 r 3V 3 c 3 dT / dt
(4)
By applying Eqs. (3) and (4) into Eq. (1), then
r 3V 3 c 3 dT / dt 5 2 h 3 S 3 (T 2 T ` )
(5)
V. C ¸ iftc¸i et al. / Measurement 25 (1999) 299 – 306
301
Assuming r, V, c, h, S and T ` are constant and integrating, t 5t
T
E
dT /(T 2 T ` ) 5 2
T5T i
E h 3 S /( r 3V 3 c)
(6)
t50
So that (T 2 T ` ) /(T i 2 T ` ) 5 e 2(h3S / ( r 3c 3V ))3t
(7)
If we define a unit of length ‘‘L’’ as; L 5V/S
(8)
Fig. 2. The geometry for the case where Bi .0.1.
(8a)
The governing differential equation with the boundary condition is:
and for a cylinder L 5V/S 5 D 3 H /(4 3 (H 1 D/ 2))
where V is the volume and S is the surface area. For a cube, L 5V/S 5 B / 6 where S is the surface area, H is the height, and B is the side length of cube. The thermal diffusivity is defined as: a 5 k /r 3 c
≠ 2 T / ≠x 2 5 (1 /a) 3 ≠T / ≠t
(14)
With initial conditions: at t 5 0, T(x, 0) 5 T i
(15)
(9) and boundary conditions:
Thus Eq. (7) can be re-written as (T 2 T ` ) /(T i 2 T ` ) 5 e 2(h3L / k )3(a 3t / L 32)
(10)
The Biot number is defined as: Bi 5 h 3 L /k
(11)
The Fourier number is F0 5 a 3 t /L
2
(12)
So, Eq. (10) becomes: (T 2 T ` ) /(T i 2 T ` ) 5 e
at x 5 0, ≠T(0, t) / ≠x 5 0 at x 5 L, ≠T(L, t) / ≠x 5 2 (h /k) 3 (T 2 T ` )
J
The wall is initially at uniform temperature T i and surrounding temperature T ` , the convective surface coefficient is h. If we define u 5 T 2 T ` , ui 5 T i 2 T ` and the temperature distribution as
u (x, t) 5 X(x) 3 R(t) 2Bi 3F 0
(13)
A commonly used rule of thumb is that the error inherent in a lumped-parameter analysis will be less than 5% for a value of Bi less than 0.1.
(17)
then, using the boundary condition, Eq. (14) can be solved,
u (x,t) /ui 5 (T 2 T ` ) /(T 2 T ` )
Oe `
52
d n 2 3(a*t / L 2 )
3 (sin dn 3 cos dn
n 51
3.2. One-dimensional transient heat conduction; system with Bi .0.1 When the (applicable value of) the Biot value is greater than 0.1, a lumped parameter analysis will introduce more than 5% error, which is generally unacceptable [4]. The case to be considered is shown in Fig. 2. The wall thickness is 2L and the material extends to infinity.
(16)
3 x /L) /(dn 1 sin dn 3 cos dn ).
(18)
Hence, cot(k n 3 L) 5 k n 3 k /h dn 5 kn 3 L
n 5 0, 1, 2, 3,
J
(19)
Bi 5 h 3 L /k 5 dn 3 tan dn ; F0 5 a 3 t /L 2 . Thus, according to Eq. (18) the temperature at a given time and position in a plane wall is a function
302
V. C ¸ iftc¸i et al. / Measurement 25 (1999) 299 – 306
of Bi, F0 and a non-dimensional position parameter, x /L can be calculated. The above Eq. (18) is for a one dimension model of a plane wall. For infinite cylinders, a similar result to Eq. (18) is given in Ref. [4], and also a graphical form which was calculated and presented by Gurney, Heisler etc. was given in the same reference that can be seen by Eq. (20). Thus, by using those graphs [4] for the cylinder related parameters, the temperature distribution can be calculated (T 2 T ` ) /(T i 2 T ` ) 5 f(a 3 t /L 2 ,k /(h 3 L), r /r 0 (20)
(T 2 T ` ) /(T i 2 T ` ) 5 e 20.014344.5 ; T 5 66.888C.
3.3. Two and three-dimensional transient heat conduction; system with Bi .0.1 For the defined two dimensional cylinder, by using the Heisler graph, (T c 2 T ` ) /(T i 2 T ` ) cyl 5 (T c 2 T ` ) /(T i 2 T ` ) undefined cyl 3 (T c 2 T ` ) /(T i 2 T ` ) plane (21) can be obtained.
3.3.1. Example The properties of a 20 g mass are given in Fig. 3. The following holds for natural convection in air for a vertical cylinder [4]: h 5 A 3 (DT /H )b ; 10 4 , Gr L Pr , 10 9 ; A and b are constant and A51.42, b50.25; H5height of the above mass50.022 m; D5diameter of the mass50.013 m; T ` 5208C; T i 5 708C initially; T 5 T 2 T ` 570220550; h51.423 (DT / 0.022)0.25 ; h51.423(50 / 0.022)0.25 59.8 W/ m 2 ; at t51 min: F0 5 a 3 t /L 2 ; a 5 k /( r 3 c)5 0.45310 25 m 2 / s; L 5V/S 5 D 3 H /(4 3 (H 1 D/ 2)) (From Eq. (8a)); F0 5 44.5; Bi 5 h 3 L /k; Bi 5 0.0014; Bi < 0.1. Thus Eq. (13) can be applied and then the temperature after 1 minute will be
The Bi number is very small, therefore the temperature in the centre of the mass is very close to the surface temperature and we call it the Material Temperature, so that Theoretical Model 3.1 is applicable here. By using Eq. (13), steady state conditions for many different weights and temperatures were calculated. Those results are given in Table 1. As seen from the table, the Bi numbers are much less than 0.1 for all the calculations. Table 1 was obtained by extending the above example which shows the calculation of time evalua-
Fig. 3. The properties of a 20 g mass.
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303
Table 1 Time evaluation of steady state conditions at temperature initially from 708C to 208C for a 20 g mass Run No
Time (h)
Surrounding temperature 8C
Conduction coefficient W/ m K
Convection coefficient W/ m 2 K
Bi Number
F0 Number
Material Temper. 8C
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
0.0167 0.0833 0.2500 0.5000 0.7500 1.0000 1.2500 1.5000 1.7500 2.0000 2.2500 2.5000 2.7500 3.0000 3.2500 3.5000 3.7500 4.0000 4.2500 4.5000 4.7500 5.0000 5.2500 5.5000 5.7500 6.0000 6.2500 6.5000 6.7500 7.0000 7.2500 7.5000 7.7500 8.0000
20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20
17.0000 16.9688 16.8642 16.7049 16.6066 16.5698 16.5471 16.5344 16.5257 16.5200 16.5158 16.5128 16.5105 16.5088 16.5074 16.5063 16.5054 16.5047 16.5041 16.5036 16.5032 16.5028 16.5025 16.5023 16.5020 16.5019 16.5016 16.5016 16.5013 16.5014 16.5010 16.5014 16.5007 16.5016
9.8045 9.6479 9.0580 7.8442 6.6625 5.9932 5.4330 5.0223 4.6682 4.3847 4.1330 3.9229 3.7315 3.5681 3.4158 3.2848 3.1593 3.0521 2.9455 2.8569 2.7634 2.6908 2.6053 2.5485 2.4649 2.4272 2.3357 2.3277 2.2088 2.2573 2.0675 2.2396 1.8743 2.3443
0.0014 0.0014 0.0013 0.0012 0.0010 0.0009 0.0008 0.0008 0.0007 0.0007 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003 0.0003 0.0003 0.0003 0.0003 0.0004
44.4990 222.0867 662.1540 1311.7926 1956.1156 2602.3734 3248.5171 3895.2242 4542.0293 5189.1016 5836.2510 6483.5552 7130.9072 7778.3612 8425.8412 9073.3964 9720.9609 10 368.5880 11 016.2095 11 663.8907 12 311.5497 12 959.2756 13 606.9560 14 254.7244 14 902.4090 15 550.2274 16 197.8908 16 845.7841 17 493.3803 18 141.4118 18 788.8455 19 437.1756 20 084.2373 20 733.3301
66.88 56.42 40.49 30.66 26.98 24.71 23.44 22.57 22.00 21.58 21.28 21.05 20.88 20.74 20.63 20.54 20.47 20.41 20.36 20.32 20.28 20.25 20.23 20.20 20.19 20.16 20.16 20.13 20.14 20.10 20.14 20.07 20.16 20.03
tion for steady state conditions by the theoretical method at temperatures from 708C to 208C for a 20 g mass.
4. Comparison of the experimental and theoretical results Table 2 shows the comparison of the experimental results (measured temperature) with the theoretical model (calculated temperature) for the time evaluation of steady state conditions at temperature from 308C to 228C for a 20 g mass. Table 3 shows the
comparison of the experimental result with the theoretical model at temperature from 108C to 228C for the 20 g mass. The last column of Table 2 and Table 3 shows the measured mass differences d m(t) (apparent mass) between the test and the reference mass after cleaning. The total change in mass difference during the cooling down was about 72 mg and during the heating up was bout 11 mg. As can be seen from the tables, the calculated and measured temperature differences are about 18C which probably is part due to lumped–parameter analysis and part due to reading error of the surface temperature of the weight. Also the variations of d m(t) show that
V. C ¸ iftc¸i et al. / Measurement 25 (1999) 299 – 306
304
Table 2 Time evaluation of steady state condition at initial temperature 308C a Run No
Time (h)
Time (min)
Surrounding temperature 8C
Calculated temperature 8C
Measured temperature 8C
m(t) mg
1 2 3 4 5 6 7 8 9 10 11 12 13 14
0.0167 0.0833 0.2500 0.5000 0.7500 1.0000 2.0000 3.0000 4.0000 5.0000 6.0000 7.0000 8.0000 9.0000
1 5 15 30 45 60 120 180 240 300 360 420 480 540
22 22 22 22 22 22 22 22 22 22 22 22 22 22
29.68 28.54 26.48 24.78 23.96 23.43 22.57 22.29 22.17 22.11 22.08 22.05 22.01 22.01
30.00 – – – – – 21.50 21.03 21.01 21.08 21.07 20.90 20.96 21.07
239
a
27 118 133 122 10 112 239 239
d m(t) 5 (Mass) Test 2 (Mass) Reference 2 (d m) 321 .
Table 3 Time evaluation of steady state condition at initial temperature 108C a Run No
Time (h)
Time (min)
Surrounding temperature 8C
Calculated temperature 8C
Measured temperature 8C
d m(t) mg
1 2 3 4 5 6 7 8 9 10 11 12 13 14
0.0167 0.0833 0.2500 0.5000 0.7500 1.0000 2.0000 3.0000 4.0000 5.0000 6.0000 7.0000 8.0000 9.0000
1 5 15 30 45 60 120 180 240 300 360 420 480 450
22 22 22 22 22 22 22 22 22 22 22 22 22 22
10.53 12.40 15.67 18.21 19.38 20.11 21.28 21.64 21.79 21.87 21.91 21.94 21.97 21.97
10.00 – – – – 20.30 20.50 20.60 20.80 20.82 20.85 20.90 20.90 20.90
1 31.00
a
1 1 1 1 1 1 1 1 1
23.00 22.83 20.58 20.17 20.00 19.75 19.92 19.75 19.92
d m(t)5(Mass) Test 2(Mass) Reference2 (d m) 221 .
after sufficient time thermal equilibrium was achieved. Fig. 4 was obtained by theoretical calculations which were done for the 10, 20, 50, 100, 200, 500, 1000, 2000, 5000 and 10 000 g masses and shows the time evaluation of steady state condition at temperature from 708C to 208C. Fig. 5 shows the time evaluation of steady state condition at temperatures from initially 10, 30, 40, 70, and 1008C to 208C for a 20 g mass.
5. Discussion of the results and conclusions The apparent mass of a body is determined by a balance in air, depending on ambient conditions, such as the density or convective currents in the surrounding air. In high accurate mass comparisons, the temperature of the masses to be compared might be in thermal equilibrium with the weighing chamber of the balance. If the temperature of weight to be weighed differs from the weighing chamber of the
V. C ¸ iftc¸i et al. / Measurement 25 (1999) 299 – 306
305
Fig. 4. For different masses, the time evaluation of steady state condition at the temperature from 708C to 208C.
Fig. 5. For 20 g, the time evaluation of steady state condition from initially different temperatures.
balance, the apparent mass of the weight deviates by negative or positive values from the value at thermal equilibrium [1]. Also, the experimental results ob-
tained show that mass variations are small but negligible because the chemical composition of the stainless steel weight might be changed. Tables 2
306
V. C ¸ iftc¸i et al. / Measurement 25 (1999) 299 – 306
and 3 show that the comparison between theoretical calculation and experimental results demonstrate reasonably good agreement. Fig. 4 shows for 9 different masses the evaluation of steady state condition at the temperature from 708C to 208C. For each mass, the calculations were done and tables which are similar to Table 1 were prepared and all the ‘‘Biot numbers’’ were much less than 0.1. The results shown in Fig. 4 demonstrate that the steady state condition time of the weights directly depends on the sizes and the material of the weights. Fig. 5 shows that regardless of the initial temperature of the weight, all the lines collapse together in a few hours time. In other words, it is possible to reach the steady state condition in a few hours time from any initial temperature. Finally, the result of this study shows that the
steady state condition of the waiting time of the weight was less dependent on initial temperature but greatly dependent on the sizes and the material of the weights.
References [1] M. Glaser, Response of apparent mass to thermal gradients, Metrologia 27 (1990) 95–100. [2] M. Glaser, J.Y. Do, Effect of free convection on the apparent mass of 1 kg mass standards, Metrologia 30 (1993) 67–73. [3] M. Plassa, E. Angelini, P. Bianco, A. Torino, Variations in the mass of standard weights due to contact with water, in: XIII IMEKO World Congress, Torino, September 1994, Vol. 1, 1994, pp. 303–308. [4] J.R. Welty, in: Engineering Heat Transfer, John Wiley & Sons, 1978, pp. 114–174.
Time evaluation for the steady state condition of the weight before the measurements ¨ Vahit C¸iftc¸i*, Sevda Kac¸maz, Umit Akc¸adag, Orhan Sakarya ¨ ¨ ¨ ( UME), P.K.21, 41407 Gebze, Kocaeli, Turkey , Ulusal Metroloji Enstitusu TUBITAK Received 12 September 1998; received in revised form 4 February 1999; accepted 24 March 1999
Abstract The temperature difference between the weight and its surrounding is an important parameter in the evaluation of the waiting time required by the weights to reach a steady state condition before starting a measurement, because the thermal gradients cause the thermal convection effect on the mass during the weighing process. In this study, investigation of both theoretical and experimental applications were carried out to evaluate the steady state condition of the weight before the calibration. Two kinds of unsteady state heat conduction equation were introduced, theoretical calculation results were compared to experimental results, and comparisons were very satisfactory for a 20 g weight. Then theoretical calculations were extended for different initial temperatures and for different weights. It was realized that the steady state condition of the waiting time was less dependent on initial temperature, but greatly dependent on the sizes of the weights. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Unsteady state heat conduction; Waiting time of the weight
1. Introduction It is always a question for the metrologists that after acceptance and cleaning of a weight, how long does it take for it to reach a steady state condition for an accurate calibration. There has been neither a common nor a recommendation procedure established yet to evaluate waiting time as a function of the initial temperature condition of the weights and their size. In the literature there are a few papers dealing with this subject; two papers were published by Glaser about ‘‘Response of apparent mass to thermal gradients’’ [1] and ‘‘Effect of free convection on the apparent mass of 1 kg mass standards’’ [2], another *Corresponding author. Tel.: 1262-646-6355; fax: 1262-6465914.
paper was from Plassa about ‘‘Variation in the mass of standard weights due to contact with water’’ [3]. However, none of the papers deals with the waiting time of the weights before calibration. For that reason, some weights were manufactured and cleaned, then initial conditions of the weights were changed by putting in the refrigerator or by cleaning in soxhlet apparatus to increase the initial temperature to about 808C, then measurements were done.
2. Brief description of the experimental work It has been decided to manufacture four weights at UME for the investigation of the waiting time of cleaned weights by applying various cleaning procedures and measuring, then comparing these ex-
0263-2241 / 99 / $ – see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S0263-2241( 99 )00015-9
300
V. C ¸ iftc¸i et al. / Measurement 25 (1999) 299 – 306
perimental results by theoretical modeling using the unsteady state heat conduction equation. The balance used for weighing was a single-pan, electro-magnetically compensated, mass comparator with a capacity of 50 g, readability of 1 mg and standard deviation of less then 2 mg for 20 g. Three weights, MK1 to MK3, were manufactured at UME, with a stainless steel having 18% chromium, 8% nickel content and with density 7900 kg m 23 which was determined by a hydrostatic weighing system. The roughness values (average peak to valley heights) of the polished weights were determined as 0.1 mm by UME Dimensional Laboratory. For comparison, one of the three weights, MK1, was used as a reference standard and stored on a turntable inside the balance. The temperature inside the balance chamber was monitored by PT-100 temperature sensors and was around 228C. Before cooling and cleaning, the apparent mass (d m) 221 (difference between MK2 and MK1) was 2 5323 mg at laboratory conditions and for MK2 cooled to about 108C, then weighed. Before cleaning the apparent mass (d m) 321 (difference between MK3 and MK1) was 2 12 347 mg at laboratory conditions and for the MK3 weight washed at about 808C in soxhlet apparatus and left until at 308C (because of the technical specifications of the balance) and then weighed. For each situation before the test, the surface temperature of the weight was measured by a PT-100 resistance thermometer with a digital temperature indicator. The variations of apparent mass difference d m(t) between the test and reference weight, as a function of time t, were measured every 10 hours. 3. Theoretical model Two kinds of theoretical Unsteady State Heat transfer model were introduced here related with Biot numbers, which are nondimensional and described by Eq. (11).
3.1. Transient heat conduction; lumped-parameter analysis with Bi ,0.1 A process designated ‘‘unsteady state’’ or ‘‘transient’’ is one that is time dependent.
Fig. 1. The properties of a weight and its surroundings.
Fig. 1 shows the properties of a weight and its surroundings. We assume the initial temperature of the material is uniform and that the temperature is a function of time only and thus the temperature is uniform throughout the material at any time. Heat is transferred between the material and its surroundings by convection only. When material is put in a cold or hot environment, the question is, what will be the temperature distribution for a given time. From the first law of Thermodynamics, Ref. [4]: dE / dt 5 Q
(1)
There is no work in or out, giving the following initial conditions: at the centre t 5 0 → T 5 T i
(2)
at the boundary Q 5 2 h 3 S 3 (T 2 T ` )
(3)
From the internal energy equation: dE / dt 5 r 3V 3 c 3 dT / dt
(4)
By applying Eqs. (3) and (4) into Eq. (1), then
r 3V 3 c 3 dT / dt 5 2 h 3 S 3 (T 2 T ` )
(5)
V. C ¸ iftc¸i et al. / Measurement 25 (1999) 299 – 306
301
Assuming r, V, c, h, S and T ` are constant and integrating, t 5t
T
E
dT /(T 2 T ` ) 5 2
T5T i
E h 3 S /( r 3V 3 c)
(6)
t50
So that (T 2 T ` ) /(T i 2 T ` ) 5 e 2(h3S / ( r 3c 3V ))3t
(7)
If we define a unit of length ‘‘L’’ as; L 5V/S
(8)
Fig. 2. The geometry for the case where Bi .0.1.
(8a)
The governing differential equation with the boundary condition is:
and for a cylinder L 5V/S 5 D 3 H /(4 3 (H 1 D/ 2))
where V is the volume and S is the surface area. For a cube, L 5V/S 5 B / 6 where S is the surface area, H is the height, and B is the side length of cube. The thermal diffusivity is defined as: a 5 k /r 3 c
≠ 2 T / ≠x 2 5 (1 /a) 3 ≠T / ≠t
(14)
With initial conditions: at t 5 0, T(x, 0) 5 T i
(15)
(9) and boundary conditions:
Thus Eq. (7) can be re-written as (T 2 T ` ) /(T i 2 T ` ) 5 e 2(h3L / k )3(a 3t / L 32)
(10)
The Biot number is defined as: Bi 5 h 3 L /k
(11)
The Fourier number is F0 5 a 3 t /L
2
(12)
So, Eq. (10) becomes: (T 2 T ` ) /(T i 2 T ` ) 5 e
at x 5 0, ≠T(0, t) / ≠x 5 0 at x 5 L, ≠T(L, t) / ≠x 5 2 (h /k) 3 (T 2 T ` )
J
The wall is initially at uniform temperature T i and surrounding temperature T ` , the convective surface coefficient is h. If we define u 5 T 2 T ` , ui 5 T i 2 T ` and the temperature distribution as
u (x, t) 5 X(x) 3 R(t) 2Bi 3F 0
(13)
A commonly used rule of thumb is that the error inherent in a lumped-parameter analysis will be less than 5% for a value of Bi less than 0.1.
(17)
then, using the boundary condition, Eq. (14) can be solved,
u (x,t) /ui 5 (T 2 T ` ) /(T 2 T ` )
Oe `
52
d n 2 3(a*t / L 2 )
3 (sin dn 3 cos dn
n 51
3.2. One-dimensional transient heat conduction; system with Bi .0.1 When the (applicable value of) the Biot value is greater than 0.1, a lumped parameter analysis will introduce more than 5% error, which is generally unacceptable [4]. The case to be considered is shown in Fig. 2. The wall thickness is 2L and the material extends to infinity.
(16)
3 x /L) /(dn 1 sin dn 3 cos dn ).
(18)
Hence, cot(k n 3 L) 5 k n 3 k /h dn 5 kn 3 L
n 5 0, 1, 2, 3,
J
(19)
Bi 5 h 3 L /k 5 dn 3 tan dn ; F0 5 a 3 t /L 2 . Thus, according to Eq. (18) the temperature at a given time and position in a plane wall is a function
302
V. C ¸ iftc¸i et al. / Measurement 25 (1999) 299 – 306
of Bi, F0 and a non-dimensional position parameter, x /L can be calculated. The above Eq. (18) is for a one dimension model of a plane wall. For infinite cylinders, a similar result to Eq. (18) is given in Ref. [4], and also a graphical form which was calculated and presented by Gurney, Heisler etc. was given in the same reference that can be seen by Eq. (20). Thus, by using those graphs [4] for the cylinder related parameters, the temperature distribution can be calculated (T 2 T ` ) /(T i 2 T ` ) 5 f(a 3 t /L 2 ,k /(h 3 L), r /r 0 (20)
(T 2 T ` ) /(T i 2 T ` ) 5 e 20.014344.5 ; T 5 66.888C.
3.3. Two and three-dimensional transient heat conduction; system with Bi .0.1 For the defined two dimensional cylinder, by using the Heisler graph, (T c 2 T ` ) /(T i 2 T ` ) cyl 5 (T c 2 T ` ) /(T i 2 T ` ) undefined cyl 3 (T c 2 T ` ) /(T i 2 T ` ) plane (21) can be obtained.
3.3.1. Example The properties of a 20 g mass are given in Fig. 3. The following holds for natural convection in air for a vertical cylinder [4]: h 5 A 3 (DT /H )b ; 10 4 , Gr L Pr , 10 9 ; A and b are constant and A51.42, b50.25; H5height of the above mass50.022 m; D5diameter of the mass50.013 m; T ` 5208C; T i 5 708C initially; T 5 T 2 T ` 570220550; h51.423 (DT / 0.022)0.25 ; h51.423(50 / 0.022)0.25 59.8 W/ m 2 ; at t51 min: F0 5 a 3 t /L 2 ; a 5 k /( r 3 c)5 0.45310 25 m 2 / s; L 5V/S 5 D 3 H /(4 3 (H 1 D/ 2)) (From Eq. (8a)); F0 5 44.5; Bi 5 h 3 L /k; Bi 5 0.0014; Bi < 0.1. Thus Eq. (13) can be applied and then the temperature after 1 minute will be
The Bi number is very small, therefore the temperature in the centre of the mass is very close to the surface temperature and we call it the Material Temperature, so that Theoretical Model 3.1 is applicable here. By using Eq. (13), steady state conditions for many different weights and temperatures were calculated. Those results are given in Table 1. As seen from the table, the Bi numbers are much less than 0.1 for all the calculations. Table 1 was obtained by extending the above example which shows the calculation of time evalua-
Fig. 3. The properties of a 20 g mass.
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Table 1 Time evaluation of steady state conditions at temperature initially from 708C to 208C for a 20 g mass Run No
Time (h)
Surrounding temperature 8C
Conduction coefficient W/ m K
Convection coefficient W/ m 2 K
Bi Number
F0 Number
Material Temper. 8C
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
0.0167 0.0833 0.2500 0.5000 0.7500 1.0000 1.2500 1.5000 1.7500 2.0000 2.2500 2.5000 2.7500 3.0000 3.2500 3.5000 3.7500 4.0000 4.2500 4.5000 4.7500 5.0000 5.2500 5.5000 5.7500 6.0000 6.2500 6.5000 6.7500 7.0000 7.2500 7.5000 7.7500 8.0000
20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20
17.0000 16.9688 16.8642 16.7049 16.6066 16.5698 16.5471 16.5344 16.5257 16.5200 16.5158 16.5128 16.5105 16.5088 16.5074 16.5063 16.5054 16.5047 16.5041 16.5036 16.5032 16.5028 16.5025 16.5023 16.5020 16.5019 16.5016 16.5016 16.5013 16.5014 16.5010 16.5014 16.5007 16.5016
9.8045 9.6479 9.0580 7.8442 6.6625 5.9932 5.4330 5.0223 4.6682 4.3847 4.1330 3.9229 3.7315 3.5681 3.4158 3.2848 3.1593 3.0521 2.9455 2.8569 2.7634 2.6908 2.6053 2.5485 2.4649 2.4272 2.3357 2.3277 2.2088 2.2573 2.0675 2.2396 1.8743 2.3443
0.0014 0.0014 0.0013 0.0012 0.0010 0.0009 0.0008 0.0008 0.0007 0.0007 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003 0.0003 0.0003 0.0003 0.0003 0.0004
44.4990 222.0867 662.1540 1311.7926 1956.1156 2602.3734 3248.5171 3895.2242 4542.0293 5189.1016 5836.2510 6483.5552 7130.9072 7778.3612 8425.8412 9073.3964 9720.9609 10 368.5880 11 016.2095 11 663.8907 12 311.5497 12 959.2756 13 606.9560 14 254.7244 14 902.4090 15 550.2274 16 197.8908 16 845.7841 17 493.3803 18 141.4118 18 788.8455 19 437.1756 20 084.2373 20 733.3301
66.88 56.42 40.49 30.66 26.98 24.71 23.44 22.57 22.00 21.58 21.28 21.05 20.88 20.74 20.63 20.54 20.47 20.41 20.36 20.32 20.28 20.25 20.23 20.20 20.19 20.16 20.16 20.13 20.14 20.10 20.14 20.07 20.16 20.03
tion for steady state conditions by the theoretical method at temperatures from 708C to 208C for a 20 g mass.
4. Comparison of the experimental and theoretical results Table 2 shows the comparison of the experimental results (measured temperature) with the theoretical model (calculated temperature) for the time evaluation of steady state conditions at temperature from 308C to 228C for a 20 g mass. Table 3 shows the
comparison of the experimental result with the theoretical model at temperature from 108C to 228C for the 20 g mass. The last column of Table 2 and Table 3 shows the measured mass differences d m(t) (apparent mass) between the test and the reference mass after cleaning. The total change in mass difference during the cooling down was about 72 mg and during the heating up was bout 11 mg. As can be seen from the tables, the calculated and measured temperature differences are about 18C which probably is part due to lumped–parameter analysis and part due to reading error of the surface temperature of the weight. Also the variations of d m(t) show that
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Table 2 Time evaluation of steady state condition at initial temperature 308C a Run No
Time (h)
Time (min)
Surrounding temperature 8C
Calculated temperature 8C
Measured temperature 8C
m(t) mg
1 2 3 4 5 6 7 8 9 10 11 12 13 14
0.0167 0.0833 0.2500 0.5000 0.7500 1.0000 2.0000 3.0000 4.0000 5.0000 6.0000 7.0000 8.0000 9.0000
1 5 15 30 45 60 120 180 240 300 360 420 480 540
22 22 22 22 22 22 22 22 22 22 22 22 22 22
29.68 28.54 26.48 24.78 23.96 23.43 22.57 22.29 22.17 22.11 22.08 22.05 22.01 22.01
30.00 – – – – – 21.50 21.03 21.01 21.08 21.07 20.90 20.96 21.07
239
a
27 118 133 122 10 112 239 239
d m(t) 5 (Mass) Test 2 (Mass) Reference 2 (d m) 321 .
Table 3 Time evaluation of steady state condition at initial temperature 108C a Run No
Time (h)
Time (min)
Surrounding temperature 8C
Calculated temperature 8C
Measured temperature 8C
d m(t) mg
1 2 3 4 5 6 7 8 9 10 11 12 13 14
0.0167 0.0833 0.2500 0.5000 0.7500 1.0000 2.0000 3.0000 4.0000 5.0000 6.0000 7.0000 8.0000 9.0000
1 5 15 30 45 60 120 180 240 300 360 420 480 450
22 22 22 22 22 22 22 22 22 22 22 22 22 22
10.53 12.40 15.67 18.21 19.38 20.11 21.28 21.64 21.79 21.87 21.91 21.94 21.97 21.97
10.00 – – – – 20.30 20.50 20.60 20.80 20.82 20.85 20.90 20.90 20.90
1 31.00
a
1 1 1 1 1 1 1 1 1
23.00 22.83 20.58 20.17 20.00 19.75 19.92 19.75 19.92
d m(t)5(Mass) Test 2(Mass) Reference2 (d m) 221 .
after sufficient time thermal equilibrium was achieved. Fig. 4 was obtained by theoretical calculations which were done for the 10, 20, 50, 100, 200, 500, 1000, 2000, 5000 and 10 000 g masses and shows the time evaluation of steady state condition at temperature from 708C to 208C. Fig. 5 shows the time evaluation of steady state condition at temperatures from initially 10, 30, 40, 70, and 1008C to 208C for a 20 g mass.
5. Discussion of the results and conclusions The apparent mass of a body is determined by a balance in air, depending on ambient conditions, such as the density or convective currents in the surrounding air. In high accurate mass comparisons, the temperature of the masses to be compared might be in thermal equilibrium with the weighing chamber of the balance. If the temperature of weight to be weighed differs from the weighing chamber of the
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305
Fig. 4. For different masses, the time evaluation of steady state condition at the temperature from 708C to 208C.
Fig. 5. For 20 g, the time evaluation of steady state condition from initially different temperatures.
balance, the apparent mass of the weight deviates by negative or positive values from the value at thermal equilibrium [1]. Also, the experimental results ob-
tained show that mass variations are small but negligible because the chemical composition of the stainless steel weight might be changed. Tables 2
306
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and 3 show that the comparison between theoretical calculation and experimental results demonstrate reasonably good agreement. Fig. 4 shows for 9 different masses the evaluation of steady state condition at the temperature from 708C to 208C. For each mass, the calculations were done and tables which are similar to Table 1 were prepared and all the ‘‘Biot numbers’’ were much less than 0.1. The results shown in Fig. 4 demonstrate that the steady state condition time of the weights directly depends on the sizes and the material of the weights. Fig. 5 shows that regardless of the initial temperature of the weight, all the lines collapse together in a few hours time. In other words, it is possible to reach the steady state condition in a few hours time from any initial temperature. Finally, the result of this study shows that the
steady state condition of the waiting time of the weight was less dependent on initial temperature but greatly dependent on the sizes and the material of the weights.
References [1] M. Glaser, Response of apparent mass to thermal gradients, Metrologia 27 (1990) 95–100. [2] M. Glaser, J.Y. Do, Effect of free convection on the apparent mass of 1 kg mass standards, Metrologia 30 (1993) 67–73. [3] M. Plassa, E. Angelini, P. Bianco, A. Torino, Variations in the mass of standard weights due to contact with water, in: XIII IMEKO World Congress, Torino, September 1994, Vol. 1, 1994, pp. 303–308. [4] J.R. Welty, in: Engineering Heat Transfer, John Wiley & Sons, 1978, pp. 114–174.