MHHII H’lllliHI'WIIJ I \ WIMilli”HHNIHUIWI 107 159 THS Date 0-7639 f LIBRARY Michigan State ' University \~________ I f- W This is to certify that the thesis entitled TRANSIENT SURFACE TEMPERATURE MEASUREMENTS USING NANMAC ERODING THERMOCOUPLES presented by JALAAL EBRAHIMZADEH has been accepted towards fulfillment of the requirements for M- 5. degree in MCCAAHI‘Cfi /Xr:yr Major professo MS U is an Affirmative Action/Equal Opportunity Institution __(EEE MSU RETURNING MATERIALS: Place in hock drop to remove this checkout from w your record. FINES will be charged if book is returned after the date stamped below. . .2314 ./ ‘L—«c‘ TRANSIENT SURFACE TEMPERATURE MEASUREMENTS USING NANMAC ERODING THERMOCOUPLES By JALAAL EBRAHIMZADEH A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Mechanical Engineering ~\ 1983 O/JV/UO ABSTRACT TRANSIENT SURFACE TEMPERATURE MEASUREMENTS USING NANMAC ERODING THERMOCOUPLES By Jalaal Ebrahimzadeh In this study, a new type of thermocouple, called the eroding thermocouple, is tested for its accuracy and precision in making trnasient temperature measurements. The eroding thermocouple is a product of NANMAC Corporation. The experiments involve transient surface temperature measurements of two identical specimens. Twelve eroding thermocouples with different design parameters are used. More than sixty experiments are performed, and the measure- ments produced by the eroding thermocouples are in good agreement with the expected values. The functional relationship between the measured surface temperatures of the specimens and several design parameters of eroding thermocouples are investigated using statistical F-test method. Comparisons between the responses of the eroding thermocouples are made and impor- tant design parameters are determined. Some numerical solutions of eroding thermocouple models are obtained using the finite difference technique. From these solutions, error analyses regarding the design characteristics of eroding thermocouples are made. ACKNOWLEDGMENTS The author wishes to express his deepest appreciation to Professor James V. Beck for his guidance, encouragement and patience. The author is grateful to Professor Mahlon C. Smith and R. W. Bartholemew for their guidance and suggestions. The author is thankful to Mr. Robert Rose and Mr. Vikas Sontakke of the Division of Engineering Research for their technical support. Gratitude is also extended to NANMAC Corporation for providing and installing the eroding thermocouples used for this research. Financial support for this research was provided by the National Science Foundation under grant number CME-79-20103 and the Division of Engineering Research, which is greatly appreciated. To his wife Lisa, special thanks for her patience and the many hours dedicated to the typing of this thesis. ii TABLE OF CONTENTS Page LIST OF TABLES . . . . . . . . . v LIST OF FIGURES . . .. . . . . . . vi LIST OF SYMBOLS . . . . . . . . . viii Chapter 1. INTRODUCTION . . . . . . . 1 1.1 Statement of the Problem . . . 2 1.2 Objective of this Investigation . 3 1.3 Literature Review . . . . . 3 2. DATA ACQUISITION SYSTEM . . . . 9 2.1 Experimental Apparatus . . . 9 2.2 Transient Temperature Measurement Using the PDP-11/03 Microcomputer . 15 2.3 NANMAC Eroding Thermocouples . . 17 3. EXPERIMENTAL RESULTS 1 . . . . 22 3.1 Experimental Procedure . . . 22 3.2 Results and Discussion . . . 25 3.3 Regression Analysis of Measured Peak Surface Temperature . . . . 31 3.4 Summary of Analysis . . . . 36 A. ERROR ANALYSIS . . . . . . . 43 4.1 Errors of the Measurement System . A3 iii Chapter Page 4.2 Errors due to Convective Heat Losses . . . . . . . . 44 4.3 Errors due to Nonuniform Heating . 45 4.4 Error due to NANMAC Eroding Thermocouples . . . . . . 50 5. SUMMARY AND CONCLUSION . . . . . 60 REFERENCES . . . . . . . . 64 iv Table 2.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 LIST OF TABLES Pin and cone diameters of the eroding thermocouples on specimen no.1 and no. 2 in inches . . . . . . . Measured peak temperature of specimen no.1 . . . . . . . . . . Measured peak temperature of specimen no.2? . . . . . . . . . . Numerical values of the independent variables . . . . . . . . Case 1: F-test table for specimen no.1 Independent variable is the pin diameter . Case 1: F-test table for specimen no.2? Independent variable is the clearance . . Case 3: F-test table for specimen no.1 Independent variable is the ratio of areas Case 4: F-test table for specimen no.2? Independent variable is the pin diameter . Case 5: F-test table for specimen no.2 Independent variable is the clearance . . Case 6: F-test table for specimen no.2? Independent variable is the ratio of areas Page 19 32 33 35 37 37 38 38 39 39 2.2 2.3 2.4 2.5 3.2 3.3 3.4 LIST OF FIGURES Geometry of an ideal intrinsic thermo- couple . . . . . . . . . Geometry of a beaded thermocouple . . Geometry of an indirect measurement method . . . . . . . . Picture of a specimen when the eroding thermocouples are assembled . . . . Schematic diagram of the testing assembly Picture of the Kapton electric heater Thermowell assembly of a NANMAC eroding thermocouple . . . . ,. . . . Test specimen and eroding thermocouples . A typical uncorrected transient surface ltemperature of specimen no.1 . . . . A typical corrected transient surface temperature of specimen no.1 . . . . A typical corrected transient surface temperature of specimen no.2? . . . A typical uncorrected transient surface temperature of specimen no.2 . . . . vi Page 10 11 13 18 2O 27 27 29 29 Figure 3.5 3.6 3.7 3.8 4.3 4.4 4.5 4.6 4.7 A typical corrected transient surface temperature of specimen no.1 using TC2, TC4, TCS and T06 . . . . . . . A typical corrected transient surface temperature of specimen no. 2 using TC2, TC4, TCS and T06 . . . . . . . A plot of peak temperature versus clearance for eroding thermocouples on specimen no.1 . . . . . . . . . . A plot of peak temperature versus clearance for eroding thermocouples on specimen no.2? . . . . . . . . . . Schematic of the electric heater problem . Effect of nonuniform heating on the surface of the specimens for t =0.01, 0.5 and 0.1 seconds . . . . . . . . . Cross-section of the thermowell assembly problem . . . . . . . . The finite difference nodal location for ‘ one half cross-section of the thermowell assembly problem . . . . . . . Steady state temperature disturbances for 1/8-inch diameter pin and hc:=0, 500 and 1000(W/m2-c). . . . . . . . Percent error on surface temperature rise after three seconds for hC=(J(W/m2-C) Percent of relative error in the surface temperature of the 1/8-inch diameter pin with respect to the undisturbed temperature distribution at the surface . . . . vii Page 30 30 41 41 46 49 52 54 56 57 58 erf(-) F(r') Radius Radius Linear Linear Radius Linear LIST OF SYMBOLS of the pin of the pin at xzo parameter parameter of the Ith thermocouple of the cone parameter Nonlinear parameter Count Degrees Celsius Error function Statistical F distribution Initial Green's Contact Thermal Length Number condition Function conductance coefficient conductivity along the Cartesian coordinate system of observations Regression function Number of parameters Heat flux Radial coordinate viii Residual sum of the squares Sum of the squares functions Number of continuous boundaries Surface element Time Temperature Temperature of the cone Average final temperature of the specimen Average final temperature of the Ith thermocouple Initial temperature Average initial temperature of the specimen Average measured temperature of the Ith thermocouple Corrected temperature of the Ith thermocouple Average initial temperature of the Ith thermocouple Temperature of the pin Cartesian coordinate system Clearance Independent variable Cartesian coordinate system Predicted peak temperature Measured peak temperature Independent variable Thermal diffusivity ix CHAPTER 1 INTRODUCTION Eroding thermocouples are ribbon type temperature sensors that have been developed to measure surface tempera- tures. They have been specifically constructed to follow extremely rapid temperature changes and to minimize the temperature errors due to the presence of the sensors. These thermocouples are a product of NANMAC Corporation and are frequently used in aerospace, chemical and automotive research. The purpose of this study is to investigate the accuracy of the NANMAC eroding thermocouples. This is accom- plished using both experimental and analytical approaches. This work is important because the eroding thermocouple is one of the few commercially available sensors for extremely rapid surface temperature measurements and its accuracy has not been previously investigated. Eroding thermocouples have been developed to meet the need for rugged and accurate temperature measurement devices in industery. The body in which the thermocouple ribbons are enclosed is called the thermowell assembly [1]. Itcxuibe made of any material with desired specifications to simulate the thermal properties of the medium. Eroding thermocouples can be used in a wide variety of applications. The applications have included the following: 1. temperature measurements involving high rate or 2 rapidly changing phenomena such as those occurring in a gun barrel, rocket nozzle, internal combustion engine and on the surface of a re-entry vehicle, and 2. determination of thermal porperties of low conduc- tivity materials such as Teflon, nylon and silica. 1.1 STATEMENT OF THE PROBLEM When thermocouples are used to measure surface temperature or heat flux distribution of a solid, signifi- cant errors in measurements can result if certairiprecautions are not taken. The sensor body (in this case, the thermo- well assembly) which replaces a part of the material should be designed and installed in such a way that its temperature rise due to heat input follows as closely as possible that of the surrounding material. If this condition is not met, large temperature gradients may occur at the early times of heating between the interface of the thermowell assembly and the material. The heat transfer between this interface can alter the local surface temperature distribution of the material whose temperature is being measured. In this investigation, the effect of design and installation method of the eroding thermocouples on the surface temperature distribution has been studied, experi- mentally and theoretically. Twelve eroding thermocouples with different thermowell specifications were installed by the NANMAC Co. on the surfaces of two specimens. 3 The experiments involved transient surface temperature measurements of the two specimens exposed to a constant heat flux. The finite difference technique is used for the analyses of the thermowell assembly problem. 1.2 OBJECTIVES OF THIS INVESTIGATION The objectives of this investigation are: 1. adaption of the previous transient measurement facility of the Heat Transfer Laboratory to a PDP/11-O3 microcomputer, 2. investigation of the performance of twelve eroding thermocouples when they are used to measure the surface temperature of two specimens subjected to a constant heat flux input, and 3. analysis of the possible sources of errors in experimental procedure. 1.3 LITERATURE REVIEW There have been numerous studies on the subject of thermocouple errors but, since the eroding thermocouples are fairly new, theoretical and experimental studies regarding the errors associated with them are not available. However, the same kinds of thermal considerations as for other types of thermocouple installation methods apply to the eroding thermocouples. One common method of transient surface temperature measurement is obtained by using an intrinsic thermocouple. it An intrinsic thermocouple is composed of one or two wires attached to the surface of an electrically conductive material , which is called the substrate. The substrate forms part of the thermocouple circuit [5]. Heat is trans- ferred from the substrate to the wire(s) due to temperature increase of the substrate. The heat transfer between the substrate and the intrinsic thermocouple can alter the local temperature distribution of the interface, and in so doing introduces errors in temperature or heat flux measurements. Burnett [2], Henning and Parker [3], Shewen [4], Keltner [5], Keltner and Beck [6], Litkouhi [7] and several others have investigated this problem experimentally and/or theoretically. Keltner [5] considered the classical model of an ideal intrinsic thermocouple (see Figure 1.1a) and developed a method for its response to a step change in substrate temperature. His studies using the finite difference technique and various analytical methods indicate that "at early times, there are very large temperature gradients near" the corners of the interface between the intrinsic thermocouple and the substrate. Litkouhi [7] further considered this case, which is important at early times. His results, which are in good agreement with Shewen [4] and Keltner [5], indicate that for a particular case of chromel substrate and an alumel wire the difference between the centerline and the corner of the interface at early times may be in excess of ten percent of the total step WIRE INSULATED 1.11 [11/11 SUBSTRATE Figure 1.1a Geometry of an ideal intrinsic thermocouple. THERMOCOUPLE WIRES BEAD SUBSTRATE Figure 1.1b Geometry of a beaded thermocouple. HEATED SURFACE THERMOCOUPLE\ CAVITY SUBSTRATE THERMOCOUPLE WIRES Figure 1.1c Geometry of an indirect measurement method. 6 temperature rise of the substrate. Keltner and Beck [6] have shown that "the difference between the undisturbed temperature and the thermocouple temperature" is maximum at zero time and decreases to zero at large times. The magnitude of the zero time error and the dimensionless time of the significant error are both dependent on the thermophysical properties of the wire and the substrate. Transient and steady-state temperature disturbances may result when beaded thermocouples are mounted on the surface of the substrate (refer to Figure 1.1b). The difference between the beaded thermocouples and intrinsic thermocouples is that the effective junction is displaced from the surface by the thickness of the bead [6]. The errors are the result of the thermal inertia of the bead and imperfect contact between the bead and the substrate [6, 8-10]. The effect of thermal inertia of the bead is a delay in the response of the thermocouple to the input heat flux at zero time and, in [6], "an increase in the error for all times but very late times." The effect of imperfect contact between the bead and the sub- strate is a slowing of the response at early times and a shift in steady-state temperature measurements [6, 10]. A direct transient surface temperature measurement is often difficult, such as when the surface is exposed to a very high heat flux or when the surface is in contact with a moving object. In these cases, measurements are performed by installing the thermocouples inside a cavity 7 which is drilled in the substrate (see Figure 1.10), and temperature or heat flux histories of the surface are] found by indirect methods. Therefore, it is important that the temperature measurements by an interior sensor be accurate. The errors are results of the very existence of the thermocouple and the cavity in the substrate [11.-15]. Beck [11, 12] and Chen [13-15] have found that the errors are inversely proportional to the ratios of the thermal conductivities and the heat capacities of the substrate to the thermocouple wire. If these ratios are less than one, then the errors are positive, which indi- cates overheating; errors are negative when these ratios are larger than one and the errors are maximum when the thermocouple is installed normal to the heated surface [11]. The presence of the cavity in the substrate would result in a hot spot at the heated surface of the substrate [12], and references [13] through [15] studied extensively the optimum cavity size and depth of the thermocouple installation from the heated surface to minimize these errors. NANMAC eroding thermocouples are designed to minimize the number of errors resulting from the application of thermocouples in temperature measurements, such as: 1. thermal disturbances due to thermOphysical properties of the thermocouple assembly, 2, errors due to thermal inertia of the junction, and 3. errors resulting from the unknown location of the junction. The thermocouple assembly of an eroding thermocouple (thermowell) can be made from the same material as of the substrate. This feature of eroding thermocouples elimi— nates the first error; however, temperature disturbances may occur as the result of the imperfect contact between the pins and the cone of the thermowell assembly and the taper of the pins (see Figure 2.4). The analyses regarding these errors are given in Chapter Four. The junction of an eroding thermocouple is effectively massless (see Section 2-3), and the junction is made flush with the heated surface of the substrate. When an eroding thermocouple is used to measure an interior location of a substrate, the cavity in which the thermowell is placed is eliminated by the use of the thermowell assembly. Also, in this case, the desired location of the thermocouple junction can be achieved within an accuracy of t .001 of an inch or better [1]. This is much more accurate than placing a thermocouple within a cavity such as those described in references [11] through [15]. CHAPTER 2 DATA ACQUISITION SYSTEM In this chapter the Data Acquisition System of the Heat Transfer Laboratory at Michigan State University is briefly described. Most of the components of this system are the same as those thoroughly described in [16]. The combined transient temperature measurement facility has been discussed, and only components which have been modi- fied cu" have been analysed more carefully are described here. This chapter is divided into three sections. The first section deals with the experimental set—up. The second section of this chapter describes the adaptation .of the previous experimental apparatus to a microcomputer system which was used to perform the transient surface temperature measurements. In the third section, construc- tion and components of NANMAC eroding thermocouples are described. 2.1 EXPERIMENTAL APPARATUS The combined experimental set-up consists of the following components: (1) hydraulic system, (2) testing unit, (3) AC power supply, (4) Kapton electric heater, (5) reference junction (6) computer signal conditioner, and (7) PDP-11/03 microcomputer. For a detailed descrip- tion of components (1), (3) and (6), refer to [16]. 9 (2) Testing Unit The testing unit consists of two specimens made of Armco iron and a load frame. One specimen is connected to the top of the load frame, which is stationary. The other specimen is connected to the piston of the hydraulic pump. The picture of one of the specimens and the schematic of the testing unit are shown in Figures 2.1 and 2.2. Before an experiment is performed, the specimens are brought and held together by moving the lower specimen with the hydraulic system. Figure 2.1 Picture of a specimen when the eroding thermo- couples are assembled. FASTENED TO THE LOAD FRAME \ \\ \\\\\\\\ -gQ--- 0) C2 '0 "U 0 50 0-3 H 2 C) "U H 2 (I) p---- -— 1.0" % SPECIMEN NO . 2 ELECTRIC HEATER HYDRAULIC \\\\\\ CYLINDER Figure 2.2 Schematic diagram of the testing assembly. 12 (4) Kapton Electric Heater The electric heater is made of a thin, flexible etched element, three inches in diameter, with Kapton film insula- tion material, approximately 0.015 inches thick (see Figure 2.3). The nominal heater resistance is ten ohms and the heater is designed to withstand a maximum temperature of 450°F [16]. The heater provides an approximately uniform heat flux. Detailed analysis of the departure from uniformity is given in Chapter Four. The temperature- resistance relationship for this heater was found to be a ~weak function of temperature [16]. In order to increase the thermal contact between the heater and the surface of the specimens, a thin layer (approximately 0.015 inches) of silicon heat sink compound was applied to the surface of both specimens with a special comb. Most of this compound squeezes out when the speci— mens are brought together, causing the effective thickness of the heat sink compound to be less than 0.005 inches on the heated surface of each specimen. The energy input was calculated by measurements of the input current and voltage. (5) Reference Junction The reference junction is made from a thick foam insulating box. All of the thermocouples' leads for each specimen are brought to terminal boards, one for the upper specimen and one for the lower specimen. Terminal boards Figure 2.3 Picture of the Kapton electric heater. 14 are placed in an insulating box to maintain the reference junction at approximately room temperature. A very reasonable assumption is that the reference junction's temperature stays constant for the duration of the experi- ments, which is ten seconds. (6) Computer Signal Conditioner The computer signal conditioner has been extensively described in [16]. Eight amplifiers of this unit were used to amplify the signals 1000 times. (7) PDP-11/03 Microcomputer The PDP-11/03 microcomputer (Plessy Peripheral Systems) used in the Heat Transfer Laboratory is based on DEC's popular PDP-11 architecture, under the RT-11 operating system. Important components interfaced for data acquisition purposes and discussed in this section are: analog to digital card no. DT2764 and real-time clock card no. DT2769, both manufactured by Data Translation, Inc. The DT2764 is a wide range input system with operating ranges from 1 10mv to t 10v. In this range the data acqui- sition module can operate at the maximum of 31 KHZ and a system accuracy of 0.03 percent for the operating input range is reported [19]. The DT2764 has eight differential inputs, which can be expanded to 32 differential inputs with the expander board DT2774. 15 The real-time clock (DT2769) interface board offers a full 16-bit event/interval counter that can be programmed to operate in any one of the following four modes: 1) single interval, 2) repeated interval, 3) external event timing, or 4) external event timing from zero base [17]. In this investigation the real-time clock was used as a pulse generator source, i.e., every time the real-clock generates a pulse, the analog to digital system will trigger one conversion. 2.2 TRANSIENT TEMPERATURE MEASUREMENT USING THE PDP-11/O3 MICROCOMPUTER Programs necessary to interface the DT2764 and DT2769 were developed by use of a software package called Real- Time Peripheral Support (Data Translation, Inc.). A library called DTLIB was generated from this software package, which contains numerous routines in assembly language to support different aspects of data acquisition operations. Several different programs were developed to test the optimum method of data collection. One method, which was used in program ATODO7, served the purpose of collecting more data than the internal memory capacity of the microcomputer. Operation of the program is described next. ATODO7 first calls routine (SETR) to set an operating rate and mode for the real-time clock. The operating rate and mode can be selected from any of those compatible with the DT2769. After the rate and mode are set, ATODO7 calls 16 the (RTS) routine to initiate and control'real-time sampling of the eight input channels which were used. The number of channels to be sampled is also arbitrary and cantxaincreased up to 32, the total number of existing channels of the DT2764. Once sampling is set up by the (RTS) routine, data is collected continously and the (RTS) stores the incoming data in a ring buffer which is divided into a number of sub-buffers. Each time a sub-buffer is filled, the (RTS) calls a completion routine to transmit the data from the subebuffer to a file on a floppy disk while sampling is in process. For more information on DTLIB and completion routines, see [18]. For eight thermocouples, ATODO7 can operate at a maximum sampling rate of 0.01 seconds. In this investi- gation, a sampling rate of 0.01 seconds was used. System Calibration The equipment which amplifies and transmits the thermocouples' signal to the microcomputer must be cali- brated. This requires calibration of the amplifiers and DT2764. Calibration of the amplifiers is discussed in [16]. The DT2764 has two calibration control knobs: one for the offset adjustment, and one for the range adjustment. Program SP0023 is provided by Data Translation, Inc. for testing of all the logics of the DT2764 and DT2769 inter- face boards and the calibration of them. It was found 17 that once the DT2764 was calibrated, it did not require recalibration. The PDP-11/03 requires a warm-up time of twenty minutes for stable operation of the DT2764. The DT2764 is a linear analog to digital converter, and converts input voltage from a thermocouple to corre- sponding counts. The relation between the counts, C, to output of a thermocouple and the corresponding temperature, T, was obtained from the knowledge of the thermocouple type and temperature range of the experiment. A second order polynomial, T=A+BC+DC2, was used to convert counts to temperature, where the coefficients A, B and D were obtained using the least square technique. 2.3 NANMAC ERODING THERMOCOUPLES In this investigation, twelve NANMAC eroding thermo- couples were used. A NANMAC eroding thermocouple is a special transducer designed for surface, inwall and immersion temperature measurements. The NANMAC eroding thermocouples which were used in this investigation consist of two thermocouple ribbons and a thermowell. The thermocouple ribbons are type "E" (chromel and constantan).' To reduce conduction errors and provide fast response, circular thermocouple wires have been flattened to form the ribbons (approximately 0.002 inches thick) in the immediate vicinity of the junction [1]. The junctions of the thermocouples are made by an abrasive action across the sensing surface [1], and are 18 MICA SHEETS TC RIBBONS NOTES: 1 - THERMOCOUPLE RIBBONS ARE ANSI TYPE "E" 2 - RIBBONS ARE 0.002 INCHES THICK 3 - MICA SHEETS ARE 0.001 INCHES THICK Figure 2.4 Thermowell assembly of a NANMAC eroding thermocouple. 19 positioned flush with the surface of the specimens. The thermowell assembly consists of two tapered pins and three mica sheets about 0.001 inches thick, and a cone. The thermocouple ribbons and mica sheets are placed between the two pins and then the assembly is press-fitted into the cone, The mica sheets provide electrical insulation between the thermocouple ribbons and the pins. The schematic diagram of a typical thermowell assembly is shown in Figure 2.4. To reduce temperature disturbances at the heated surface, the thermowell assemblies are made of the same material as the specimens, i.e., Armco iron. A set of six eroding thermocouples is installed on each specimen. Each set has six eroding thermocouples, which are different in the pin diameters and the cone diameters. Table 2.1 shows the prescribed dimensions of each thermocouple. In Figure 2.5, the location of each thermocouple on one of the specimens is shown. Table 2.1 Pin and cone diameters of the eroding thermo- couples on specimen no. 1 and no..2 in inches. TC. N0. PIN DIA. CONE DIA. 1 1/4 1/2 2 3/8 5/8 3 3/8 1/2 4 1/8 1/2 5 1/8 ’ 5/8 6 1/4 5/8 6-52-58 THD. 3/I6 055p- s HOLES REOD NOTES: 1 - ONLY ONE SPECIMEN NO.1 IS SHOWN. SPECIMEN NO. 2 IS DIMENSIONALLY THE SAME EXCEPT FOR THE TAPERED INSERTS 2 - ALL DIMENSIONS ARE IN INCHES 3 - SPECIMENS AND THERMOWELL ASSEMBLIES ARE MADE OF ARMCO IRON Figure 2.5 Test specimen and eroding thermocouples. 21 The first set of thermowells has 0.25 inches per foot tapered pins, which are fitted tightly into specimen no.1. The second set has 0.5 inches per foot tapered pins, which are installed into specimen no.2?. The selection of different pins, cone diameters and tapers was made to investigate the effects of various physical dimensions of thermowell assemblies on the heated surface temperature distribution of the two specimens. CHAPTER 3 EXPERIMENTAL RESULTS A primary objective of the experiments is to investi- gate the accuracy of the temperatures measured by the thermocouples. There are two identical specimens which are symmetrically heated. Each specimen has six thermo- couples at its heated surface. Hence, there are a total of twelve thermocouples measuring the "same" transient temperature history. Since the heat input is a known heat flux, the surface temperature histories of the speci- mens can be calculated. By comparing the measured temper- atures with the calculated values and with each other, the accuracy, reliability and precision of the NANMAC eroding thermocouples can be examined. More than sixty experiments were performed. Plots of temperature versus time for three experiments can be found in Section 3-2. A data analysis for results of twenty experiments is given in Section 3-3. 3.1 EXPERIMENTAL PROCEDURE In the preparation of each experiment, the specimens are carefully installed in the testing unit; they must be aligned and made parallel. A thin layer of silicon grease is applied to the heated surfaces of the specimens. Then a Kapton heating element is placed on the heated surface of the bottom specimen and the bottom specimen is raised to 22 23 come in contact with the heated side of the top specimen. After the specimens are mounted, the eight amplifiers of the signal conditioner unit are calibrated. The duration of each test is designed to be ten seconds and it is divided into three stages. In the first stage, the initial uniform temperature of the specimens is measured for approximately three seconds. In the second stage, the heater is activated by a hand switch for approxi- mately three seconds and transient surface temperature measurements of the two specimens are performed; In the third stage, the remaining time is used to measure the uniform final temperature of the specimens when the heater is deactivated and the specimens are approaching to a uniform temperature. The heat input for all of the experi- ments was approximately 900 watts. The average heat flux for each Specimen was 98500 W/m2. The sampling rate was selected to be 0.01 seconds. For this sampling rate the analog to digital card (DT2764) sweeps through eight input channels from the thermocouples. The times between reading the consecutive channels are negligible. In the prescribed testing period, 1000)(8::8000 data points are collected and stored on a floppy disk. Data Processing Method The collected data points are in counts. FORTRAN program PROCSS was developed to process the data. The sequence of the data processing performed by this program 24 is the following: 1. convert the data points from counts to degrees celsius using the relationship described in Chapter 2; 2. calculate average initial and final uniform temperature of the specimens (Ti, Tf); 3. calculate the average of five readings from each data set corresponding to each thermocouple input; 4. using the average initial and final uniform temperatures, calculate a set of normalization coefficients for each thermocouple input to correct the measured transient temperatures (see below), and 5. call subroutine RPLOT to write all of the data in a special format for plotting purposes. The averaging which was described in item 3 is done to reduce the effect of noise and unwanted characteristics of the measurement system. The correction method is described here. All of the input thermocouples should register nearly the same temper- ature for the initial and final uniform temperature of the specimens. ‘However, since the signals resulting from each thermocouple are amplified 1000 times and transmitted to the microcomputer separately, each thermocouple does not measure the same initial and final temperature. Hence, corrections are-necessary. Normalization coefficients are obtained from the simultaneous solution of ‘T T 1 AI * 131T1]: AI * BITfI f 25 where AI and BI are normalization coefficients. Ti and Tf are the initial and final average temperatures of the specimen measured by all of the thermocouples. The sub- script "I" is in reference to the thermocouple number. The temperatures Ti]: and TfI are the initial average temperature and final average temperature of the Ith thermocouple. The corrected temperatures are obtained by using the same relationships: at TI : AI-i-BITI it where TI is corrected temperature, and TI is the measured temperature of the Ith thermocouple. Since only the temperature rise is the interest of this study, the cor- rected temperatures were subtracted from the average initial temperature of each specimen. 3.2 RESULTS AND DISCUSSION In Section 2-1 it was mentioned that before the specimens were brought together, a thin layer of silicon grease was applied to the heated surfaces of the specimens. Experience with the use of silicon grease indicates that this coating can alter the surface temperature of the specimens if it is not applied correctly. For that reason, a numbering scheme is employed to distinguish between the batches of data taken from different silicon grease coatings. A number is assigned to each test to 26 indicate the test number, and the test number is followed by a letter indicating the silicon grease coating batCh. If the letters are different, it means that the specimens were disjointed and a new coat of silicon grease was applied to the heated surfaces of the two specimens. For example, experiment 5E has a different silicon grease coating from experiment 3C. All of the plots that will be shown are made using the program GENPLT at the Central Node in the Division of Engineering Research. Figure 3.1 shows the results of experiment is typical of tests for specimen no. 1. In this six thermocouples located on the heated surface no. 1were used. The data for Figure 3.1 is not 5E, which experiment, of specimen corrected according to the method described in Section 3-1. The plot of the corrected data for experiment SE is shown in Figure 3.2. Notice that the temperatures at the early and late times agree in Figure 3.2. Observation of Figure 3.2 indicates that TC1 and TC3 temperature responses of specimen no. 1do not have the same shape nor reach the same peak temperatures as the other four thermocouples on this specimen. The shapes and magni- tudes of the four clustered upper thermocouple responses agree with the predicted values. Hence, TC1 and TC3 of specimen no.1 are defective. The predicted temperature of both specimens after three seconds of heating is about 13.5 degrees Celsius. ,The temperature responses of TC2, 27 ' EXP 5E UNCOR 25.0 -: 20.0 150 100 IEHPIUSEKD [IllllllIIITHIPTITIHIIIIIlllllllllllilllllllllll 40 80 " &0 L00 TDEED Figure 3.1 A typical uncorrected transient surface tem- perature of specimen no.1. EXP 5E COR 20.0? 153—: 9. 1W: a .. m and g 150-: [— _. 0.0: __ ‘ 1133 “5.0 -lllllllilll'l'Illllll'lllllllllllll‘llllllllllllill as as 4.0 so an 109 TIMI-3:3) Figure 3.2 A typical corrected transient surface temper- ature of specimen no.1. 28 TC4, TC5 and TC6 on this specimen are very close together at the early times of heating and are different by about one degree Celsius at the end of the heating time, which is about 1 3.7 percent of the predicted value. Also, after the heating element is deactivated, the response of TC1 does not reach the same final uniform temperature as the other thermocouples (refer to Figure 3.1). The plots of temperature versus time for experiment 3C are given in Figures 3.3 and 3.4 for corrected and uncorrected data, respectively. These plots are for specimen no.2?. In experiment 30, during the early times of heating the thermocouples show slightly different responses to the input heat flux, and the spread between the recorded maximum temperatures of the thermocouples is about 1.5°C. It is interesting to observe that TC1 on this specimen is also defective. This was found for all of the experiments. One explanation for the behavior CM] the defective thermo- couples is that there might be more than one junction in their thermowell assemblies, and the measured temperatures are the average of the EMFS produced by each junction. 4 Many experiments were performed using TC2, TC4, TC5 and TC6 on the heated surfaces of both specimens. It was found that the difference between the responses of the eight eroding thermocouples at a given time is consistent in most of the cases. Results of one of these experiments, 22M, is shown in Figures 3.5 and 3.6 for specimens no.1 and no.2?,respectively. 29 EXP 3C COR 200 ' . 7] 15.0% I 3 10.01 E d E 5": 00 A AA ~ I “'53 nnlrrnlnnluuIllnluulnnllrnluulmrl 10 20 £0 80 as L00 TIMES) Figure 3.3 A typical corrected transient surface temper- ature of specimen no. 2. EXP 3C UNCOR flflf’RIflflE) '53 lTllllllllllllllllllllll]lllllllllllllllllllllllll as 2.0. w an an we tflfiflfi Figure 3.4 A typical uncorrected transient surface tem- perature of specimen no. 2. 30 EXP 22M COR 20.0—J :1 1534: g 103-: 'i’ I 0.0 :w ‘SJEWWWWPWW g E 4.0 6.0 8.0 1.0.0 TIPECS) Figure 3.5 A typical corrected transient surface temperature of specimen no.1 using T02, TC4, TC5 and TC6. EXP 22M COR 20.0 1.. 15.0 --: - TC a 1&9'2 .TC.2 E a ICES E 5" ‘ I— 2 an : w -53 ;memm on 2.9: w an an we macs; ' Figure 3.6 A typical corrected transient surfacetemperature of specimen no.2?using TC2, TC4, TC5 and TC6. 31 Results of twenty experiments using the same thermo- couples that were used in experiment 22M are tabulated ' for the specimens no. 1 and no. 2 in Tables 3.1 and 3.2. Tables 3.1 and 3.2 are the measured peak temperatures of the two specimens. As was mentioned previously, one of “the problems of interest is the difference between the responses of the thermocouples at any time. The time at the end of heating was chosen for data analysis which will be discussed in the next section of this chapter. 3.3 REGRESSION ANALYSIS OF MEASURED PEAK SURFACE TEMPERATURE In Section 3-2, two tables for the measured peak temperatures of the heated surfaces of the two specimens were presented, Tables 3.1 and 3.2. In this section, the statistical F-test method is used to investigate the possibility of a functional relationship between the measured peak temperatures and certain chosen independent variables. The independent variables will be introduced later in this section. The regression functions considered are as follows: Model 1, N BO 1 Model 2, Ni : B°+ 81X.j 2 Model 3, Ni - 130+ B1Xj 2 Model 5, N. l Bo+-B X.-+B Z. 13 23 32 'Table 3.1 Measured peak temperature of specimen no.1 Experiment T02 TC4 TC5 TC6 22M 14.3 14.8 13.1 15.2 23M 12.8 13.8 12.5 14.0 24M 13.4 14.1 12.3 14.4 25M 12.8 14.1 12.1 13.7 26M 13.6 14.4 12.2 14.5 27M 12.7 14.4 12.1 14.2 28M 13.1 14.4 12.4 14.6 29M 12.3 13.5 11.6 13.7 30M 12.3 13.1 11.6 14.0 31M 12.4 13.4 12.0 14.5 32M 12.3 13.6 11.7 13.9 33M 12.1 13.2 11.5 13.5 34M 12.7 .13.9 11.9 14.5 35M 413.4 14.2 12.6 14.6 36M 12.4 14.3 11.9 13.1 37M 12.9 13.8 12.1 15.0 38M 12.0 13.8 11.1 13.9 39M 13.3 13.7 12.3 14.5 40M 13.0 13.9 12.3 14.4 8 13.8 12.7 14.4 41M 13. 33 Table 3.2 Measured peak temperature of specimen no.2 Experiment TC2 TC4 TC5 TC6 22M 12.3 12.6 11.2 14.5 23M 11.7 13.5 11.7 13.6 24M . 12.0 13.7 11.6 11.8 25M 11.7 13.5 11.9 12.5 26M 12.1 13.1 11.9 13.5 27M 11.7 13.5 11.6 12.2 28M 11.8 14.2 11.7 12.4 29M 11.1 13.0 10.8 13.2 30M 12.0 13.8 11.3 12.8 31M 11.6 13.5 11.6 12.7 32M 11.4 13.2 10.9 12.8 33M 13.1 14.7 12.0 13.2 34M 12.2 14.2 11.7 13.0 35M 12.5 14.3 12.1 13.4 36M 12.4 14.3 11.9 13.1 37M 12.0 14.2 11.8 13.1 38M 13.1 13.8 12.0 13.5 39M 12.5 14.6 11.8 13.0 40M 12.1 14.2 11.3 13.2 41M 12.6 14.9 11.7 12.1 34 where B0, B1 and B2 are the parameters to be estimated independently for each model. Xj and Zj are the inde- pendent variables and are assumed to be errorless. N is the peak temperature predicted by the mathematical model. The subscript "1" corresponds to the number of measurements taken using one or more eroding thermocouples with the same and/or different particular independent variables. The subscript "j" refers to the number of independent variables. For any given data set and mathematical model, the least square technique is used to estimate the parameters. The parameters are estimated by minimizing the sum of the squares function,£3,defined by [20]: _ 2 ANS with respect to the parameters in the mathematical model, Ni' The number of observations is n and Yi is a measured temperature taken from Table 3.1 or 3.2. In order to make analysis using the sample variance, the following statistical assumptions regarding the temperature measurement's errors are assumed to be valid. It is assumed that errors are additive and uncorrelated, and have zero mean, constant variance and a normal distri- bution. The independent variables chosen are: pin diameter, clearance and ratio of the cross-sectional area. Values of the different pin diameters of thermowell assembly are shown in Table 2.1. Clearance is defined as the difference 35 between the pin radius and the cone radius of the well assemblies. The 2.1. Theretio of the the pin at the heated ’corresponding area of specimens. cone radius can be found in cross-sectional area is the surface of the specimens to the pin at the cold side of The numerical values corresponding to thermo- Table area of the the the various independent variables are shown in Table 3.3. Table 3.3 Numerical values of the independent variables. Variable Specimen TC2 TC4 TC5 TC6 pin diameter (cm) 1and2? 0.953 0.318 ' 0.318 0.635 clearance (cm) 1and2? 0.318 0.476 0.635 0.476 ratio of areas 1.121 1.440 1.440 1.190 ratio of areas 2.250 2.250 1.440 1.266 The optimum number of parameters necessary for a good fit is based on the statistical F distribution [20]. form of a statistic Ex is defined as: Fx=R(p-1)-R(p): R(p) /(n-p) AR £2 The where R(p) is the residual sum of the squares for p para- meters, and n is the number of observations. The ratios have one degree of freedom for the numerator and (n-p) degree(s) of freedom for the denominator. The rat i0 is a measure of how much an additional term in the regresssion function can improve the fit [21]. The correspond ing 56 values of F for five perCent probability and various of (n-p) are found in [21] and are shown in Tables 3.4 through 3.9. The criteria for the rejection of an additional term in the model is if Fx < F The other case is for Fx being larger than F; then one can be confident that the added parameter is needed [21]. Tables of the sum of the squares and F ratio for each case are given in Tables 3.4 through 3.9. There are a total of six cases: cases 1, 2 and 3 are for specimen no.1, considering the prescribed three independent variables in Table 3.3; cases 4, 5 and 6 are for specimen no.2?. The sum of the square value and Fx for model 5, which is a function of pin diameter and clearance, is only included in cases 1 and 4. 3.4 SUMMARY OF ANALYSIS To obtain the best model, the observed values of Fx should be compared with tabulated F values. When Fx is much larger than F, then the added parameter appears to be necessary in the model. Large differences in the F values indicate inn: residuals of the sum of the squares had a large decrease. The independent variable with the largest Fx value for both specimens is the clearance. This indi- cates that the clearance is the most important variable. There is some indication of possible correlation between the pin diameters and ratio of the areas, but they are Table 3.4 Case 1: 37 F-test table for specimen no.1 Independent variable is the pin diameter. Model p d.f. R SZ=R/d.f. AR Fx:AR/32 F 1 1 79 75.46 0.95 2 2 78. 75.26 0.96 0.20 0.212 3.96 3 2 78 75.42 0.97 0.04 0.04 3.96 4 3 77‘ 50.11 0.65 25.15 38.69 3.96 5 3 77 62.26 0.81 13.0 10.47 3.96 Table 3.5 Case 2: F-test table for specimen no.2 Independent variable is the clearance. Model p d.f. R 52: R/d.f. on Fx = AR/32 F 1 1 79 ' 75.46 0.95 2 2 78 69.28 0.96 6.18 6.40 3.96 3 2 78 64.57 0.83 10.90 13.15 3.96 4 3 77 18.72 0.24 45.85 188.68 3.96 38 Table 3.6 Case 3: F-test table for specimen no.1 -Independent variable is the ratio of areas. Model p d.f. R 32: R/d.f. AR Fx = Ala/32 F 1 1 79 75.46 ' 0.95 3.96 2 2 78 72.31 ‘o.93 3.15 3.40 3.96 3 2 78 68.30 0.88 7.16 8.18 3.96 4 3 77 55.93 0.70 16.38 23.40 3.96 Table 3.7 Case 4: F-test table for specimen no.2 Independent variable is the pin diameter. Model p d.f. R 52: R/d.f. AR Fx = “/32 F 1 1 79 86.36 1.09 2 2 78 82.72 1.06 3.63 3.43 3.96 3 2 , 78 81.41 1.04 4.95 4.76 3.96 4 3 77 76.43 0.99 5.29 5.34 .3.96 5 3 77 72.85 0.95 9.87 10.40 3.96 Table 3.8 Case 5: -39 F-test table for specimen no.2 [Independent variable is the clearance. Model p d.f. R 32: R/d.f. AR Fx = AR/82 F 1 1 79 86.36 1.09 2 2 ' 78 84.09 1.08 2.27 2.11 3.96 3 2 78 80.85 1.04 5.51 5.31 3.96 4 3 77 32.68 0.42 51.31 122.2 3.96 Table 3.9 Case 6: F-test table for specimen no.2 Independent variable is the ratio of areas. Model p d.f. R 32: R/d.f. AR szAR‘ISZ F 1 1 79 86.36 1.09 2 2 78 84.94 1.09 2.22 2.04 3.96 3 2 78 85.40 1.09 0.96 0.88 3.96 1 3 77 76.49 0.99 8.45 8.53 3.96 40 not as consistent and conclusive. The two accepted regression functions to predict surface temperature are, for specimen no.1, v: 0.93 + 57.63): -63.12x2 and for specimen no. 2, y = -o.24 + 59.07 X -63.63X2 where X is the clearance in (cm) and y is the predicted peak temperature in (C). The constantsfxn~X and X2 in both of these equations are very close. The plots of y versus clearance for specimen no.1 and specimen no.2?are shown in Figures 3.7 and 3.8. In these figures two sets of experimental data are shown for .476(cm) clearance. This is because there were two thermo- couples on each specimen with the same clearance. Obser- vation of these plots suggests that the thickness of the cone assemblies of the NANMAC eroding thermocouples should be limited to around 0.5cm to give a repeatable surface temperature measurement. The measured temperatures near this region are very close to the predicted temperature of the specimens, which is 13.5°C. It is also important to note that the variation in temperature measurements is minimum when the clearance is about 0.5(cm), which occurs at the peak of the curves. The extreme measured temper- atures of each specimen are also shown in Figures 3.7 and 3.8. An inspection of these extremes gives added validity of the regression models given above. 4.1 16 r . . . . 15 ’ ‘ 14 - I ' q 13 _ . 12 - . PEAKLTEMP (C) 11 . - v PEAK-TEMP (MATH MODEL) 0 AVERAGE PEAK-TEMP (EXPERIMENT) 9 1 1 1 1 L 0.2 0.3 0.4 0.5 0.6 0.7 0.8 CLEARANCE (CM) Figure 3.7 A plot of peak temperature versus clearance for eroding thermocouples on specimen no.1. 16 I I I r I 15- 1 1111. 12 I 11L PEAK-TEMP (C) V HflmETflfl’(MNfliMmfiflJ ..MERNEEPEMGTEWP(EUEmDflflfl) 9 1 1 1 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 CLEARANCE (CM) Figure 3.8 A plot of peak temperature versus clearance for eroding thermocouples on specimen no.2L 42 In conclusion, the experimental results and analyses do not show any large systematic errors produced by the thermowell components Of the NANMAC eroding thermocouples. The greatest statistical correlation is with the clearance. CHAPTER 4 ERROR ANALYSIS The analysis Of the errors is an essential part of a heat transfer experiment. The error analyses in this chapter are primarily done to provide an indepth under- standing Of the heat transfer consideration of the present measurement system in the Heat Transfer Laboratory and the thermocouples used in this study. The possible errors in the present experiment may include: 1. errors of the measurement system, 2. errors due to convection heat losses Of the specimens, 3. errors due to nonuniform heating method provided by the electric heater, and 4. errors due to thermal disturbances caused by NANMAC eroding thermocouples. Other errors may be involved, such as conduction heat losses through the supporting pins of the specimens, errors due to the presence of the thermocouple leads, and errors due to energy absorbed by the electric heater and heat sink compound (silicon grease), but they arerufi: considered in this analysis. 4.1 ERRORS OF THE MEASUREMENT SYSTEM The description of the data acquisition system and the accuracy Of its components are given in Chapter Two. Also, 43 44 a correction method of the errors due to signal amplifi- cation was introduced in Chapter Three (see Section 3-1). The accuracy of each component may not represent the performance of the measurement system in the presence of the environmental noises and noises generated by the microcomputer. Therefore, the accuracy of the measurement system was determined experimentally. An experiment was designed to provide a signal to one of the amplifiers, simulating a thermocouple signal, and then measurements were made. The error due to the measure- ment system was determined to be approximately i0.5°C. As was mentioned in Chapter Three, correction for this error was made by taking the average of the five readings from the thermocouple, and errors due to the measurement system were reduced to approximately 1 0.20°C. 4.2 ERRORS DUE TO CONVECTIVE HEAT LOSSES The value of h, the convection heat transfer coef- ficient, was determined experimentally. This value is 13.6(W7m2-C),which is in the mid-range Of natural convection heat transfer coefficients given in [22]. Using this value for the convection heat transfer coeffi- cient, the rate of heat loss of the specimens due to convection is found to be 1.451ior approximately 0.16 percent of the heat input. It is evident that this heat loss is negligible and insulation Of the specimens is not necessary during the brief transient experiment. 4.3 ERRORS DUE TO NONUNIFORM HEATING The electric heater used in the surface temperature measurements was commercially made by Minco of an etched flat heating element insulated with Kapton. The heating element has many strips 0.05 inches wide and 0.05 inches apart (see Figure 2.3). Heat transfer analysis in this section will show that the electric heater does not provide uniform heating throughout the flat surfaces Of the speci- mens. For convenience in the analysis, assume that there is perfect contact between the flat surface of a specimen and the electric heater; then, a portion of the flat surface Of the specimen (0.05 inch width) in contact with the electric heater can be modeled as a semi-infinite plate. The schematic geometry of the idealized heat transfer problem is shown in Figure 4.1. Referring to Figure 4.1, L =0.05 inches is the total width of a portion of the specimen in contact with the prescribed sections of the electric heater. It is reasonable to model the specified portion of the specimen as a semi-infinite plate extended in y direction, since the Fourier number (at/L2) associated with the characteristic length (L::0.05 inches) is 400 times larger than the Fourier number when the thickness of the specimens (L::1 inch) is taken as characteristic length. 46 90:0 r t 'Tlrlilll 0 L/2 L CONSTANT HEAT FLUX Figure 4.1 Schematic geometry of the electric heater problem. The transient two-dimensional differential equation of conduction for the prescribed geometry can be written as: 62T+62T=lfl (11.1) 8 2 2 a 6 x 6y where T is the temperature and a: k/pc is the thermal diffusivity. The boundary conditions and initial condition applied to this equation (see Figure 4.1) are: 47 i- -k%z| “30 inoixiL/z y y:0 ii-g-1l‘0 int/25x51. y yzO iii- .31‘ :0 atx:0andx=L X y=0~co iv - T(X’y’0)=Ti at 13:0 The solution for the nonhomogeneous differential equation (Equation 4.1) and its boundary conditions are Obtained by the method of Green's Functions. The general solution of a two-dimensional problem in terms Of Green's Functions is given in [23] as: T(r,t) I min-431,1) |1_0F(r')dV' (11.2) V _. t +§I er G(}_"_,h|}‘_",t) g(£',r)dV' 1::0 V ' t +9! d‘tZ I C(P,t)|r",1:)l 1- fids- 1:0 1 s 5""117 l i i wherellis the point of interest; V, F(rj) and dSi are the volume of the region, initial condition and differential area Of the surface element. The index, i::1, 2,...,s is for the boundaries and s is the number of continuous boundary paths of the region [23]. The appropriate Green's Function for the equation (4.2) can be determined by the product of Green's Functions resulting from the solutions of two one-dimensional homogeneous heat conduction 48 equations. The temperature distribution for the region of A is Obtained by substitution Of the appropriate Green's Function in equation (4.2). The solution after performing the integration for any time and initial temperature Of Ti :0 has the following form: 2 T(x,y,o> .3212! (1 -erf(dJE-_))+ “‘1? JE (1.e'y W“) 2 at k at +3332}; €23. 12 cos(m—:§-) sin (9%)” Pf (925:) (4.3) where qo is the input heat flux from the electric heater and L is the width of the specimen in perfect contact with the electric heater. The temperature distribution of interest is at the heated surface of the specimen when y::0 in equation (4.3). Therefore, T (x,0,t) = 234; a); 12 cos (-—)£) sin (T)er rf (Inn/at) k: In 1 m (4.4) The quasi-steady state maximum temperature difference between the x = 0 and x = L, temperatures at y = 0 is AT 20.916??- S-fc—L - (4.5) To obtain the temperature versus x at y :0, the properties of Armco iron, which are kz'T3W/m.C and a-2. 051(10'51n2/s [16] were used. The plot of temperature T(x,0,t)--Ti AT': 49 +1.0 1 _ t = O.‘l : -+O.5 " '3 '1-.=0.05 ‘ 0.0 - ’ t=0.01 [ «-O.5 " -1'.0 J 0 1/2L L POSITION, x Figure 4.2 Effect of nonuniform heating on the surface of the specimens for t :0.01, 0.5 and 0.1 seconds. 50 versus x for different times and heat input Of 900 watts can be seen in Figure 4.2. At the early time, t =0.01 seconds, the temperature difference between the heated section and the insulated section is less than 0.24°C, and for times about 0.05 seconds and larger, the heated surface of the specimen reaches quasi-steady state and the maximum difference between the two prescribed sections is about 1.28°C. The analyses in this section show that errors as large as : 0.64°C can be expected from nonuniform heating of the flat surface of the specimens. The width of the thermocouple ribbons (which is about pin diameter) is in all cases at least two times larger than L:=0.05 inches (refer to Figure 4.1). The thermocouple ribbons are also randomly aligned with respect to the heater strips. Therefore, errors of this magnitude are unlikely to occur. 4.4 ERRORS DUE TO NANMAC ERODING THERMOCOUPLES. In Chapter One, several potential sources of errors when making surface temperature measurements were described. The significant errors are results of the very existence Of the sensor altering the local surface temperature distri- bution. In this section, errors due to the installation method Of NANMAC eroding thermocouples are given. The installation method of the thermocouples was described in Chapter Two. It was shown (see Figure 2.4).that the thermocouple wires 51 I are a flat ribbon type and are embedded between the flat surfaces of two pins; the pins are tapered and are pressed-fit into a hollow cone. Heat transfer analyses in this section show that errors in the steady state and transient temperature measurements can be results of the taper of the pins and the imperfect contact between the pins and the cone surfaces. Figure 4.3 shows the cross-section Of the idealized thermowell assembly problem. At x:=0 where the heated surface is located, the surface of the thermowell assembly is exposed to constant heat flux and insulated on the other surfaces. With the assumption of constant thermal properties of Armco iron and constant contact coefficient (he) between the pins and the cone surfaces, the boundary conditions for the two-dimensional transient problem are given by: 6T 6T 1 - k(—a-3CE)| =k(-a?c-) =-qo whereT =f(x,r,t) X=O X=O p TC=f(X,r',t) 6T OTC “‘53 ‘53? -° X x=L X=L OT OT0 iii - R(—E) =-k(-—) =h (T -T ) 61" r=a 6P r=a C p 0 6T iv- k(_8—PE :0 r- b 6T V' a? =0 r=0 where Tp and Tc stand for the temperature of the pin and 52 ' CONSTANT HEAT FLUX X I '1 11 \‘\\t 3’ he, IMPERFECT d a a a a c P v / g/I”’_- CONTACT A | V a. ‘r/llv A V L I y I I A | ’ I V A I b __’ I '7’ A I V I V 1 ' ’ "_____-"Jrlwra'r :F77—7”' Figure 4.3 Cross-section of the thermowell assembly problem. 53 cone, respectively; r is the radius Of the pin normal to the circumference at any given x; b is the radius of the cone; and k is the thermal conductivity. Since the geometry is not "nice," the problem was solved by using finite differencesixlspace [24]. Due to the symmetry about the center line, annular elements are used. The locationscd‘the nodes for the problem are given in Figure 4.4. A total of 100 nodes were used, 25 in the axial direction and 4 in the radial direction. The resulting system of first order ordinary differential equations was solved with the use Of IMSL software package, which is compiled on the CDC 6500 computer. ~ The routine (DGEAR) from IMSL package was used. This routine solves the system of differential equations by some type of predictor-corrector method [25]. The temperature distributions for all of the cases that will be discussed were found using the prOperties of Armco iron given in [16] and an input heat flux of 31540 W/m2. This heat flux value corresponds to about 30 percent of the experimental heat flux used. This value, however, is not significant because relative errors only are important. Discussion of the Results' There are three parameters that can be varied for the present analysis. They are: 1. the contact conductance coefficient, hc’ between the pins and cone surfaces, 54 CONSTANT HEAT FLUX __‘_‘_'_‘_’ 111 1.— AP (4.1) \‘\\“‘\\\\ a a a A a 4 A I a a A A 4 \ (1,25) (4,25) Figure 4.4 The finite difference nodal location for one half cross-section of the thermowell assembly problem. 55 2. the pin and cone diameters, a and b, at the heated surface, x = 0, and 3. the taper Of the pins. Steady State Temperature Disturbances The steady state solution for this problem was simu- lated by solving the problem in which the nodes at x:=L were kept at a constant temperature. Some results of the steady state temperature distribution for the worst possible case, when the pin diameter is 1/8 inch and taper Of the pin is 1/2 inch per foot for different values of contact conductance coefficient (he) are shown in Figure 4.5. The undisturbed temperature distribution was found by setting hc:=a>. Figure 4.5 shows the effect of imperfect contact on the steady state temperature distribution Of the pins. It is evident that as the thermal contact resistance decreases, the magnitude Of the disturbances at the heated surface of the pin decreases; and if it is assumed that he = 1000 (W_/m2 A-C), then errors as large as 11 percent can be expected in steady state temperature measure- ments using this kind of installation method. This value of hC is probably much too low. Higher values of hC have smaller errors. It is expected that hc is at least as 2 large as 1000 (W/m - C). Transient Temperature Disturbances The transient heated surface temperatures for several different pin diameters and two different tapers of the 56 25 I I I I 20 — .— 15 — _ E-GH ' he : 1000 «E.» X E-‘Q H 10 — _ [3 110 : O 5 — UNDISTURBED TEMP, h :00 C 0 L L 1 I 0.2 0.4 0.6 0.8 1.0 X/L Figure 4.5 Steady state temperature disturbances for 1/8-inch diameter pin and he = 0, 500 and 1000 (W/m2- C). PERCENT ERROR IN TEMP RISE 57 ‘5 I ' 1 ' l ' P a 1 - 1/4 INCH PER FOOT TAPER ' 2 - 1/2 INCH PER FOOT TAPER ‘ 10 - l- 51- 1- l 1 l 1 l 1 O 1/4 1/2 3/4 1 PIN DIAMETER (INCHES) Figure 4.6 Percent error in surface temperature rise after three seconds Of heating for hc:(3(W/m2-C). 58 30 1 1 1 1 25 — d [:11 a) H a: g 20 - .— 21.. hc:0 2 1—1 ‘23 15 '- 8‘: £11 hc=500 S 8 1O _ \. \ 5 hc=1000 n. . 5 — _ 0 1 1 1 J 2 4 6 8 10 TIME (S) Figure 4.7 Percent Of relative error in the surface temper- ature of the 1/8-inch diameter pin with respect to the undisturbed temperature distribution at the surface. 59 pin were calculated using the difference procedure mentioned above. They are compared to the undisturbed transient temperature distribution at the heated surfaces. The percent errors in temperature rise after three seconds of heating for 1/2 and 1/4 inch per foot of taper and several different pin diameters are shown in Figure 4.6. These results are for the worst possible case of hc::0. Obser- vation Of Figure 4.6 indicates that maximum error of 13 percent can result from the combined effects of no contact (when hc:=0), and taper Of the pins. It is important to note that errors are much higher when the taperijsincreased. It is also significant to mention that even for the maximum possible contact resistance value, when he :0, the errors are less than 5 percent (refer to Figure 4.6) if the pin diameter is 1/2 inch or larger. Figure 4.7 shows the percent error in temperature rise of the heated surface of 1/8-inch diameter and 1/2 inch per foot taper of the pin when hc:=0, 500 and 1000 (W/mZ-C) with respect to the undisturbed surface temperature distribution for ten seconds of heating. As expected, the maximum temperature distortion occurs at an early time (t‘<8 seconds) and errors tend to decrease after that time. CHAPTER 5 SUMMARY AND CONCLUSION The primary objectives of this study were to investigate the accuracy and precision of the temperature measurements using NANMAC eroding thermocouples. To accomplish these objectives,zanumber of modifications in the previous data acquisition system were made. A new microcomputer system (PDP-11/03) was employed to do the real-time sampling and increase the power capacity Of the previous system. In Chapter One, several different measurementtechniques using thermocouples were described. The application Of each type of thermocouple and its associated problems were reviewed. In Chapter Two, the DTLIB software package by (Data Translation, Inc.) was introduced. Several data collection methods using this software were tried. FORTRAN program ATODO7 was developed to collect more data than the internal memory capacity of the microcomputer system. Eroding thermocouples were described. Twelve eroding thermocouples were installed by (NANMAC Co.) on the surfaces of two specimens made of Armco iron supplied by Michigan State University. A thin electric thermofoil was used to heat the surfaces of the specimens. In Chapter Three, the experimental procedure and data processing method were described. FORTRAN program PROCSS was developed to transform the responses of the 60 61 thermocouples from counts to degrees Celsius using the least square technique. To minimize the effect Of unwanted noise on the collected data, program PROCSS utilizes two correction methods. More than sixty experiments were performed to investi- gate the performance of the eroding thermocouples. It was found that 25 percent Of the eroding thermocouples were defective. The average measured peak surface temperature and expected value were found to be in good agreement. The results Of twenty experiments were tabulated in Section 3-2. A data analysis based on statistical F-test method using the measured peak temperatures of each specimen were performed. The data analysis showed a significant func- tional relationship between the measured peak temperatures and the clearance of the thermowell assemblies (see Section 3-3). An error analysis of the experimental method was given in Chapter Four. Combined accuracy of the measurement system after averaging the data was determined experi- mentally tO be about t 0.2 Celsius degrees. Errors due to convective heat losses were shown to be negligible. Possible nonuniform heat transfer distribution due to the design of the electric heater was investigated analytically. It was found that errors as large as t 0.64 Celsius degrees may occur due to nonuniform heating provided by the elec- tric heater. This result may be misleading since, as was described in Section 4-3 regarding alignment of heater 62 strips and the thermocouple ribbons, errors of this magni- tude are unlikely to occur. The numerical solution of the two-dimensional partial difference equation of conduction for the thermowell assembly was given in Section 4-4. In these analyses, the combined effect of taper and contact resistance of the pins and cone surfaces was considered. The following conclusions from the results Of the analyses were made: 1. Temperature rise errors are related to the taper of the pins and contact conductance coefficient, he, between the pins and cone surfaces. 2. Temperature rise errors are maximum at early times of heating and for the worst case (see Section 4-4) the magnitude of the errors is less than ten percent Of the expected values. Further work for improvement and in-depth understanding of the measurement system and performance of eroding thermocouples is needed. The following recommendations for future work are given: 1. The accuracy of the measurement system at present, before averaging is performed, is i 0.5 Celsius degrees. The present signal conditioning method can be improved to reduce the effect of noise in the transmission of the signals from the thermocouples to the PDP/11-03 micro- computer. 2. At present, the maximum sampling rate of the data collection program (ATODO7) is 0.01 seconds. A higher 63 sampling rate can be achieved with some modification in the operation of this program. 3. Computerized control of the power supply can Unprove the power transmission to the electric heater. 4. Heat transfer experiments and/or analyses are needed to provide adequate information about the heat capacity of the silicon heat sink compound and Kapton electric heater. 5. Heat transfer analysis of the errors due to conduction heat losses of the eroding thermocouples is needed. 6. Heat transfer analysis of temperature disturbances caused by the mica sheets in the thermowell assemblies is necessary for better understanding of the characteristics of eroding thermocouples. REFERENCES LIST OF REFERENCES NANMAC Temperature Handbook, 1981/82, NANMAC Corporation, 9-11 Mayhew Street, Framingham Center, Massachusettes. Burnett,l).R., "Transient Measurement Errors in Heated Slabs for Thermocouples Located at an Insulated Surface, " Journal of Heat Transfer, Trans. ASME, Series C, Vol. 83, 1961, pp. 505-506. Henning,(3.D. and Parker, R., "Transient Response Of an Intrinsic Thermocouple," Journal of Heat Transfer, Trans. ASME, Series C, Volume 397 1967, pp. 146-154. Shewen, E. C., "A Transient Numerical Analysis of Conduction between Contacting Circular Cylinders and Halfspaces Applied to a Biosensor," MS Thesis, Dept. of Mechanical Engineering, University of Waterloo, 197 . Keltner, N. R., "Heat Transfer in Intrinsic Thermo- couples Application to Transient Measurement Errors," Report SC-RR-72-0719, Sandia National Laboratories, Albuquerque, New Mexico, January 1973. Keltner, N. R. and Beck, J. V., "Surface Temperature Measurement Errors," to be published in Journal of Heat Transfer, 1983. Litkouhi, B., Surface Element Method in Transient Heat Conduction Problems," Ph.D dissertation, Dept. of Mechanical Engineering, Michigan State University, East Lansing, Michigan, 1982. Quandt, E. R. and Fink, E. W., "Experimental and Theoretical Analysis of the Transient Response of Surface Bonded Thermocouples," Beltis Technical Review, WAPD-BT-19, Reactor Technology, June 1960, p. 31. Wally, K. and Bickle, L. 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M., "Transient Temperature Distortion in a Slab due to Thermocouple Cavity," Journal of Heat Transfer, Vol. 14, July 1976, pp. 979-981. Chen, C. J. and Li, P., "Theoretical Error Analysis of Temperature Measurement by an Embedded Thermocouple," Letters in Heat and Mass Transfer, Vol. 1, 1974, pp. 171-180. Chen, D. J. and Li, P., "Minimization of Temperature Distortion in Thermocouple Cavities," Journal Of Heat Transfer, Vol. 15, June 1977, pp. 869-871. Khosrow, Farnia K. "Computer-Assisted Experimental and Analytical Study of Time/Temperature-Dependent Thermal Properties of the Aluminum Alloy 2024-T351," Ph.D. dissertation, Dept. of Mechanical Engineering, Michigan State University, East Lansing, Michigan, 1976. User Manual for Real-Time Clock, DT2769, Data Trans- lation Inc., 100 Locke Drive, Marlborough, Massachusettes, 1979. User Manual for DTLIB/RT V2.2, Real-Time Peripheral Support, SP-101-V02-02, Data Translation Inc., 100 Locke Drive, Marlborough, Massachusettes, 1981. User Manual for DEC Dual Height Analog Input Systems, DT2764, Data Translation Inc., 100 Locke Drive, Marlborough, Massachusettes, 1979. Beck, J. V. and Arnold, K. J., Paramater Estimation in Engineering and Science, John Wiley & Sons, Inc., 1977. Walpole, R. E. and Myers, R. H., Probability and Statistics for Engineers and Scientists, MacMillan, New York, 1972. Holman, J. P., Heat Transfer, 4th ed., McGraw-Hill,Inc., 1976. 23. 24. 25. 67 Ozisik, M. N. Heat Conduction, John Wiley & Sons, Inc., 1980. , Myers, G. E., Analytical Methods in Conduction Heat Transfer, McGraw-Hill, Inc., 1971. Gear, C. W., Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1971. HICHIGRN TATE NV. 111 11 111111111 312931M54591 RARIES MIN] 21