Available online at www.sciencedirect.com
ScienceDirect Procedia - Social and Behavioral Sciences 182 (2015) 629 – 636
4th WORLD CONFERENCE ON EDUCATIONAL TECHNOLOGY RESEARCHES, WCETR2014
Simulation modelling of results of measuring surface temperature under electric voltage A. Bekbaev a*, V.P. Sherysheva, Y.A. Sarsenbayeva a
Kazakh National Technical University named after K. I. Satpayev, Satpayev Street, 22, Almaty 050013, Kazakhstan
Abstract Based on the concept of concentrated capacity and method of thermally thin layer the mathematical model of heat transfer in the system of metal frame-paraffin is created. The reduction of the mathematical model is made, the calculated formula for implementation in microcontroller of smart sensor of temperature for inaccessible surface (SS TIS) is obtained. On base of simulation modeling of measurement information with use of explicit solution of model problem it is became obvious that the absolute error of the surface temperature measurement under electric voltage can be reduced up to 1 ° C. © 2015The TheAuthors. Authors.Published Published Elsevier © 2015 by by Elsevier Ltd.Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of Academic World Research and Education Center. Peer-review under responsibility of Academic World Research and Education Center. Keywords: Temperature, smart sensor, inaccessible surface, simulation modeling, thermally thin layer, the process of heat transfer;
1. Introduction Measurement of surface temperature under high electric voltage of conduct bus with use of thermocouples [1] is often complicated due to the presence of noise that significantly exceeds the useful signal level - thermocouple EMF. In order to overcome this difficulty in [2] is proposed a method for measuring of surface temperature of the conduct bus using heating resistor which is placed in a metal bucket filled with paraffin, at the distance from the bottom of the bucket, ensuring the dielectric strength. The main disadvantage of this method that reduce the accuracy of measurement, is unavailability to measure the falling temperature of paraffin layer separating the heating resistor from the bottom of the metal bucket, attached close to the busbar. If we treat the surface of busbar as the surface that is inaccessible for direct measurements of
*A. Bekbaev. Tel.: +7-705-660-4459. E-mail address:
[email protected]
1877-0428 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of Academic World Research and Education Center. doi:10.1016/j.sbspro.2015.04.798
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temperature, then for improving of measurement accuracy, smart sensor of temperature for inaccessible surface (SS TIS) [3] can be used. This paper deals with the development of a mathematical model for heat transfer process in the system of metal frame-paraffin, the simplicity of which and the numerical implementation obtained by simplifications of calculation formulas permit to make verification of SS TIS. 2. Temperature measuring device Temperature measurement of the surface under electric voltage can be made with the help of the device shown in Figure 1 [2]. The devise consists of a metal frame 1 of a bucket type, tightly connected to a busbar 2, which is closed on the top by the insulator 3; frame 1 is filled with molten paraffin before filling with paraffin 4. Along the axis of a metallic bucket the heat resistive sensor element (heating resistor) 5 is placed. One part of lateral surface of the frame, which is located between the bottom of the bucket and the heating resistor, is separated from the environment by the cylindrical thermal insulator layer 6. The other part of the lateral surface of the frame 7 with rectangular slits (not shown in the figure) is designed for intensification of heat exchange with the environment. Heating resistor 5 is placed at the position so that the distance from it to the bucket walls and bottom provides the necessary electrical strength of paraffin. Outputs of heating resistor are connected to the input of the conditioning amplifier 8. The output of amplifier 8 is connected to the input of the analog-to-digital converter (ADC) 9. ADC output 9 is connected to the input of the microcontroller (MC) 10.
Fig. 1. Device for surface temperature measurement under high electric voltage of conduct bus: 1- metallic frame; 2- busbar; 3- insulator; 4- paraffin; 5- heating resistor; 6- cylindrical thermal insulator layer; 7- heat transfer part of the cylindrical lateral surface of the frame; 8- conditioning amplifier; 9- ADC; 10- microcontroller.
Heat, generated by the busbar 2 while being led by electric current, is transferred to the bottom part of the metal frame 1, that have a good thermal conductivity, and further through the walls of the frame to the environment (the field of the outer surface of the metallic frame 7) and to the paraffin 4. Through the cylindrical paraffin layer with the thickness G , providing dielectric strength, heat is transferred to the heating resistor 5. The sensor element of the heating resistor fixes the temperature of the surrounding paraffin at the point of its placement. The mathematical model of the thermal process in the system of the metallic bucket -paraffin is creating for modeling of the temperature of paraffin at the point of the sensor element placing. 3. Mathematical model Due to the axial symmetry of the temperature field, the temperature is calculated by spatial two-dimensional function T
t, r, z , defined in the right side of the system (Fig.2). At zero time t
0 the whole system is uniformly
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heated to the environment temperature T0 , i.e. for all r and z in the define area T 0, r , z
T0 . When r
0 on
the symmetry axis and on the heat-insulated part of outer surface of the metal frame r
r2 , and also on horizontal heat insulated surface of the space z 0 filled with paraffin, the conditions of absence of heat flow wT / wr 0 and wT / wz 0 are fulfilled (Fig. 2).
0
r1 r2 2
1
r
G
z z z z
Fig. 2. Calculating field and heating scheme
const , that heats the metal frame-paraffin system, according to the concept of the concentrated capacity [4] and the method of thermally thin layer [5] if the condition of G ¢¢r1 is fulfilled, on the border of the z z 2 , the following equation is made: With the constant heat flux from the busbar
Gc2J 2
wT wT O1 wt wz
and along the border
Gc2J 2
r
wT wt
q
q ,
(1)
r1 one-dimensional heat transfer is performed in accordance with the equation:
GO 2
w 2T wT O2 2 wr wz
O2 r r2
wT wr
,
(2)
r r1
where G - is the thickness of the bottom and the lateral wall of the metallic bucket; c - mass specific heat; J density; O - thermal conductivity coefficient (index 1 refers the respective value to the paraffin, and 2 - to metal). Taking into account the conditions of ideal thermal contact at the interface between the inner surface of the metal frame - paraffin:
O2
wT wr
O1 r r1 0
wT wr
,T r r1 0
r r1 0
T
r r1 0
,
(3)
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at the place of the outer surface of the frame with convective heat transfer with the environment equation (2) takes the form of:
Gc2J 2 Where
wT wt
GO 2
w 2T wT O1 2 wr wz
D T T0 , r
r1 ;0 z z1 ; t ! 0
z
z1 the
(4)
r r1
D is the coefficient of heat transfer, and at heat-insulated area of the surface the condition is fulfilled:
Gc2J 2
wT wt
GO 2
w 2T wT O1 2 wr wz
,r
r1 ; z1 d z z 2 ; t ! 0 .
(5)
r r1
Thus, the mathematical model of this thermal process can be represented by the following initial-boundary value problem of heat conduction:
c1J 1
wT wt
O1
w 2T w 2T O1 wT ,0 r r1 ;0 z z1 ; t ! 0 , O 1 wz 2 wr 2 r wr
(6)
wT wr
0, r
0;0 z z1 ; t ! 0 ,
(7)
wT wz
0, z
0;0 d r d r1 ; t ! 0 ,
(8)
Gc2J 2
wT wT O1 wt wz
Gc2J 2
wT wt
wT Gc2J 2 wt T
q, z
z 2 ;0 d r d r1 ; t ! 0 ,
w 2T wT O1 2 wr wz
r r1
w 2T wT GO 2 2 O1 wr wz
r r1
GO 2
T0 ,0 d r d r1 ;0 d z d z1 ; t
(9)
D T T0 , r
,r
r1 ;0 z z1 ; t ! 0 ,
r1 ; z1 d z z 2 ; t ! 0 ,
0 .
(10)
(11)
(12)
Solution of the initial-boundary value problem (6) - (12), and the dependence on the temperature time at the
r
0; z
z
1 , can be obtained by the finite difference method point of placing of the heating resistor sensor r, z: [6]. However, the implementation of implicit scheme of finite difference method with the use of computer is rather complicated and the explicit scheme imposes significant limits on the ratio of the grid spacing on time and space variable. That is why, it is reasonable to simplify the scheme of the measuring process and to adapt the mathematical model of the thermal process in order to develop an analytical method for the problem solving.
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4. Simulation of measuring information If to prolong the cylindrical layer of the heat insulator 6 until it touches the cover (insulator 3, Fig. 1) and to eliminate from consideration the bottom of the metal bucket, we can get a simplified diagram of the measurement process (Fig. 3), and subsequently the calculated formula for the surface temperature calculation under electric voltage can be obtained. It is important that the temperature measurement under electric voltage of the surface can be made with the help of use of the device shown in Figure 3. The device consists of paraffin made round bar 3, thermally insulated from the lateral surface and the upper end by the heat insulator 2. On the contact of the bar with the bus and at a distance δ from the contact surface, providing an electric resistance, temperature sensor elements 4 and 5 are placed.
Fig. 3. Scheme of modeled measuring process: 1- busbar under electric voltage; 2- heat insulating cylindrical shell; 3- space fulfilled with paraffin; 4,5- place of sensor elements for temperature detecting; 6- computer; 7- microcontroller
The simulated temperature data of the sensor 4 is calculated by computer 6 and by the USB bus is transmitted to the microcontroller 7. By the calculated formula, in the microcontroller the modeled temperature of the sensor 4 is converted to the temperature of the element 5 and transferred again by the USB bus into the computer's 6 memory. Due to the low thermal conductivity of paraffin the temperature T 1 of the sensor 5 may differ significantly from the temperature T 2 of the element 4, which leads to the need of recalculation. To determine the relationship between these values, according to [7], relative to the nondimensional temperature of the middle surface of the paraffin layer
0,25
T
T1 T 2
dT T dt
2
it is necessary to solve differential equation:
0,5 T 2 , T (0)
0. ;
where T 0 - the initial value of the dimensionless temperature of the middle surface; conductivity of the paraffin; q - heat flow density, flowing from the bus into the paraffin.
(13)
O - coefficient of thermal
Dimensionless temperature T 2 W is a continuous function. Therefore, after amplifying by the conditioning amplifier and before processing of the received signal, on the microcontroller is used the analog-to-digital converter.
T W
, and with it the At the output of the analog-to-digital converter the signal, proportional to the temperature 2 right part of the equation (13) is a step function of time. Therefore, at each step of the constancy of the right part the
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solution of the differential equation (13) has the form:
T W
Where
k
>0,5 T W @ k
2
k
W ª º 0 , 25 « », k 1,2, , n , u 1 e «¬ »¼
(14)
k k - the number of steps of a step function; W k Ot 2 - dimensionless time; c is the volumetric specific
cG
heat of paraffin; G - the thickness of the paraffin layer, which provides electrical strength. Introducing into the obtained solution correction coefficients k1 and k2, that compensates systematic error of k and, going to the dimensional temperature by the formula: polynomial of mth degree Pm W
T1 t k
q1G
O
T1 W k T0 ,
t k
cG 2W k , O
(15)
G
we obtain the calculated formula for the recalculating of the dimensionless temperature of the sensor 4 T 2 W , q with density value of heat flow and thermal resistance of the paraffin layer O , into actual temperature of the busbar surface, under electric voltage:
T t
T1 t
k
k
1
ª k G °° k1 q ®®2 u >0,5 T 2 W k @u «1 e O °° «
W k 2
0, 25
¬
¯¯
º » T 2 W k »¼
k
½ ° k ¾ Pm W ° ¿
½ ° ¾ T0 , k ° ¿
1,2,, n
(16) Computational formula (16) is implemented on the microcontroller [3]. In case of second-degree polynomial according to the simulation modeling of measurement information 5 regime parameters - two correction coefficients and three coefficients of second-degree polynomial are to be defined. To verify the correctness of the calculated formula (16), which is in case of second-degree interpolation polynomial and five correction coefficients k1=0,415, k2=4, a0=0,0394, a1=-1,21925, a2=5,962567 the simulation modeling of measurement information is conducted. The recalculation of dimensionless model temperature of the sensor 4 into the temperature of the sensor element 5, using the data of the Table 1, which shows thermal characteristics of the paraffin, by the formula (16), is made. Table 1 – Basic data Symbols
Name
Value
Dimension
δ
Half of the height of the paraffin bus
0.006362
m
λ
Thermal Conductivity
0,26
W / (m · K)
c
Specific heat capacity
1,93E + 06
J / (kg · K)
q
Specific heat flux
3,30E + 03
W/m2
T0
The initial temperature
2,00E + 01
° C.
The results of simulation modeling made in Excel XML Format, are presented in Figure 4. The exact nondimensional solution of the thermal conductivity problem and the dimensionless model
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temperature of the sensor 4 are calculated according to the formula given in [8] (formula (1.5.5)):
1 2 3 S2
T1 W W
T1 W W
f
1
¦n
1 2 2 24 S
n 1
f
2
1
¦n n 1
e n S W cos nS ,
2
2
2
e n S W cos nS . 2
2
(17)
(18)
Recalculation of dimensionless values obtained into the dimensional temperature, expressed in Celsius degrees, is made by the formulas:
T1 t k T2 t
k
qG
T1 W k T0 ° ° O ¾. qG k T 2 W T0 °° O ¿ ½
(19)
Fig. 4. The results of simulation modeling of measuring information: ∆ – explicit solution of heat conductivity problem; ◊ – temperature of the sensor 5, calculated by the formula (16); □ – modeling temperature of the sensor 4; –סdifference module of the exact and modeled temperature of the sensor 5.
The results of simulation modeling of measurement information, presented in Figure 4 show that the use of the sensor data with the sensing element 4 located in the paraffin at a distance of 6 mm from the busbar, as the indicating sensor 5, may lead to the absolute error measurement about 20 ° C. If the method of temperature measurement under electrical voltage of the surface is based on the use of SS TIS, implementing the calculated formula (16) when the current is 3.3 kWh / m 2 that leads to the specific heat flux from the busbar surface of can be achieved the absolute measurement error not exceeding 1°C. 5. Conclusions Based on the concept of concentrated capacity and method of thermally thin layer a mathematical model of the thermal process in the system of metal frame-paraffin, which is the main structural element of the device for measuring of the surface temperature under electrical voltage of conductor busbar, which is the initial boundary
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value problem of heat conduction. Solution of the initial-boundary value problem can be obtained by the method of finite differences on the rectangular grid. Simulation model of the temperature, measured by the heating resistor located on the axis of the metallic bucketparaffin system at the distance from the walls of the metal frame and the bottom of the bucket, providing electric strength, is the paraffin temperature of points r, z: r 0; z z1 calculated by computer. Use of SS TIS for the surface temperature measurement under electrical voltage makes it possible to achieve values of absolute measurement error no more than 1 ° C. References G. Andreev, А. Toloknov, B. Arpentyev et al. (1975). Measuring device of the surface temperature. Author’s certificate. No. 513271, MKI G 01 К 7/02. - № 2093989/27. USSR. Published on April 30, 1977, Bulletin 16. – 3 pp. Antonenkov A.N., Bessonov, A.I., Gertsman L.Y., Melnikov V. A., Pintiushenko A.D. (2006). Device for measuring of surface temperature, under electric voltage. Patent No. 227226, MPK7 G01K13/00, G01K1/08, G01K1/14, dated from 20.03.2006. Military Engineering Technical University. Russian Federation. A.B. Bekbaev, Y. Zhalmukhamed, R. M. Utebaev, N.A. Koltun (2013). Laboratory experimental device for testing of intelligent temperature sensor for inaccessible surface. Vestnik of Satpayev Kazakh National Technical University, 6 (100), p. 3642. Yerzhanov R. Zh, Matsevity Y.M., Sultangazin U.M., Sheryshev V.P. (1992). Concentrated capacity in problems of thermal physics and microelectronics. Kiev: Naukova Dumka, 295 pp. Bazhanov A.A., Chizhov V.N., Sheryshev V.P. (2005). Method of thermally thin layer in problems of modeling and identification of thermal processes. Almaty: Evero, 186 pp. Godunov S. K., Riabenkii V. S. (1962) Introduction to the theory of difference schemes. Moscow: Fizmatgiz. Bekbaev A.B., Karbozova A.M., Sheryshev V. P. (2012). Control of thermal condition of electrical contact. Topical Issue: Problems of automated electrical drive. Theory and Practice. Scientific and technical journal of Kremenchug. Kremenchug: KrNU, 3/2012 (19), pp. 575578. Beck J., B. Blackwell, Ch. St. (1989). Clair Incorrect inverse problem of heat conduction / Trans. from English.- Moscow: MIR.- 312 pp.
