ExpQ Transient Response Rev0

ExpQ Transient Response rev0.docx

CARLETON UNIVERSITY

Department of Mechanical and Aerospace Engineering

MAAE 3901 EXPERIMENT NUMBER: Q TITLE: Transient Response of Temp. Sensors and Introduction

to Data Acquisition.


1.0 Purpose The purpose of this laboratory exercise is to introduce the concepts of transient response, time constants and convective heat transfer by using thermocouple temperature probes as practical examples. Theory will be experimentally substantiated with respect to the effects of thermocouple probe size, thermocouple probe shape, and immersed gas stream velocity. 2.0 Introduction The signal from an ideal temperature sensor (such as a thermocouple, thermistor, fluid-filled bulb, etc.) would faithfully and instantaneously indicate any change in the temperature being measured. However, as the transfer of heat is a rate (or time dependent) process, there is an unavoidable lag inherent in any real sensor with finite dimensions and hence finite thermal capacity. Errors resulting from the thermal inertia of the sensor can be considerable as may be seen from Figure 1 below,

FIG. 1

which depicts temperatures downstream from the turbine blades of a large jet engine during start-up. Two features are necessary in the design of a temperature sensor to minimize this delay in response to system changes.

a) Low thermal capacity. b) Intimate thermal contact with the measured medium.

As one seldom has a wide choice in the selection of materials for the sensing element, the first aim is usually achieved by minimizing the size of concurrently. Thermally insulating the leads and probe support also serves to reduce the thermal capacity of the measuring element by reducing the amount of material that must undergo the temperature change. The second aim is normally achieved by exposing the active part of the sensing element directly to the measured medium. However, as the object here is to maximize the heat transfer coefficient


between the measured medium and the sensing element, one must bear in mind that the properties of the medium itself are highly significant. For instance, the convective heat transfer coefficient h between a flowing stream of gas and a thermocouple bead is strongly influenced by the velocity of the stream and the properties of the gas (k, , Cp, p, etc.). A typical construction for an exposed thermocouple probe is sketched below. The desirable features for rapid time response as described above are clearly present in this design.

In contrast, the enclosed thermocouple probe would have inferior time response characteristics but, because of its rugged construction, would undoubtedly enjoy a longer service life. The total temperature probe shown in the third sketch clearly incorporates some features from both the above types as well as stagnating the flow to give the total temperature reading 3.0 Analytical Prediction of Thermocouple Response Time A temperature sensor can only indicate its own temperature. Only when the sensor is at the temperature of the surrounding fluid, therefore, will the indicated temperature represent that of the fluid. When the sensor is not at the same temperature as its surroundings, heat is transferred between them until the temperatures are equal. For this reason, a heat transfer analysis is necessary to determine the time scale required to bring the two temperatures into close proximity. To estimate the time response characteristics of a thermocouple junction enclosed in a spherical or cylindrical probe, consider the sketches shown in Fig. 2.


FIG. 2

If we ignore the transfer of heat by radiation and conduction along leads from the probe, then the heat transfer can be described in terms of the convective coefficient h, or RateofHeatTransfertotheProbe (1) where h = Convective Heat Transfer Coefficient ( or

)

A = surface area of contact between the gas and the bead enclosure, (m2 or ft2) TG = temperature of the gas stream (C or F) TP = temperature of the probe including the junction

Strictly speaking, the temperature at the surface of the probe should appear in place of the probe temperature in Equation 1. Whenever the internal resistance to heat transfer is much less than the external resistance, (i.e. external being in the gas surrounding the probe) we ignore internal temperature gradients and treat the probe as though it is all at the same temperature. In this case we will make that assumption and check its validity later. From the first law of thermodynamics, we know that in the absence of work interactions, (2) where E is the energy of the system (Joule; Btu) is time (sec) is the rate of heat transfer to the system. In this case, only the internal energy is changed. The expression for the internal energy probe can be written as (3) where m is the mass of the probe (gm, lbm) c is the heat capacity (kJ/kgC, Btu/lbmF)


Tp is the probe temperature (C, F) Tr is a reference temperature at which E = 0. Combining Equations (1), (2) and (3) results in

(4) It should be noted that only when the probe temperature equals the gas temperature will the probe temperature be unchanging with time, i.e. steady. If the gas temperature is changing with time, then the solution of Equation (4) will help to explain by how much the probe temperature is lagging the gas temperature in its change. In this laboratory exercise, a step change in gas temperature is simulated by suddenly inserting the probe from the cool gas stream to a hot stream or vice versa. Solution of Equation (4) for that case should predict the degree to which the probe follows the step change. To aid in the solution, as long as the gas temperature is not changing with time, following the step change, (i.e. during the time that TP is changing) we can select the reference temperature Tr to equal the gas temperature, TG Equation (4) can then be re-arranged to the form (5) Or, for m and c constant with time, re-arrangement can be made further to give,

(6)

Solution of this ordinary differential equation for constant results in the familiar result. exp

(7)

where TPo is the temperature of the probe at the zero time when the step change in gas temperature occurred. The mass of the prove can be written in terms of the product of its density, , and volume, V, so that

(8)

The denominator of Equation (8) represents the thermal capacity of the prove and associated mass, and the numerator is the thermal conductance of the fluid layer surrounding the probe. That is Cth= cV Rth=1/hA


Or

(9)

where is the thermal time constant of the probe. There is a direct analog here with the charging or discharging of an electrical capacitance with a resistance in series, as depicted in Fig. 3.

FIG. 3

The time constant determines the rate of equilibrium with the driving potential, whether that potential is the difference in temperature, as in the thermal situation, or the difference in voltage as in electrical analogy. The smaller the time constant, the more quickly the circuit responds. From Equation (9), it can be seen that the time constant can be reduced by increasing the surface area to volume ration, by reducing the density or specific heat of the probe material or by increasing the convective coefficient. The surface to volume ratio is a purely geometric factor; the density and specific heats are material properties, and the convective coefficient is dependent on the geometry, temperature and gas flow rate. In this experimental apparatus, two geometries of probes are supplied; spherical and cylindrical. The surface area to volume ratio of each is as follows:

6

6


4

4

The convective coefficient is most conveniently described in terms of a functional relationship

with Reynolds number, in the form of semi-empirical relationships between Nusselt and Reynolds where and are the viscosity coefficient and thermal conductivity respectively at the film temperature, where the film temperature is the average between the gas and probe temperatures.

For spheres in gas streams, the following two equations may be used: a) For gases at Re from 25 to 100,000

0.37

.

0.37. (10) b) For gases at Re from 1.0 to 25

. . (11)

For cylinders in gas streams, an equation of the form of Equation 12 can be used.

(12)

with the constants given by

Range of C n

1-4 0.891 0.330 4-40 0.821 0.385

40-4,000 0.615 0.466 4,000-40,000 0.174 0.618

40,000-250,000 0.0239 0.805 To determine the time constant experimentally, one can insert the probe into a gas stream which is at different temperature and observe the response. When , the time constant, the thermocouple read-out will have reached 63.2% of the change in temperature that it will attain in total, as illustrated on Fig. 4.


FIG. 4

Consequently, for fast response, one wants as small as practically possible, which can be done by several different means, as pointed out. To demonstrate the calculation procedure, and to provide some quantitative assessment of the relative effect of the parameters discussed in the proceeding paragraphs, example calculations for a typical flow environment have been detailed in the following section. Example Computation Basic Data We will establish the dependence of response time on thermocouple bead size for the following condition: Fluid: Air

Temperature: 200F Velocity: 10 ft/sec Pressure: atmospheric

For each probe, the thermocouple junction is formed by encasing the dissimilar T/C leads in a spherical drop or cylindrical shell of high temperature silver solder. As set out in the theoretical development we shall neglect effects of conduction along the lead wires (and radiant heat transfer to or from the bead) on the understanding that probe design has minimized these errors. Properties of a Silver Solder Bead Composition Component Component (by weight) S.G.s Cps @ 68F 70% Silver 10.5 0.056 20% Copper 8.89 0.0915 10% Zinc 7.14 0.0918


S. G. 0.7 10.5 0.2 0.0915 0.1 0.0918 0.0579 Heat Transfer Properties of the Flow Field

10ft/s

14.7 14453.3 660 0.0601

145 10 sec K&KGASTABLES

10 0.0601

12

145 10 sec 3455

where D is in inches 3455

so, for spherical probes, a) for D>0.00725, 25 and Equation (10) holds b) for 0.00725>D>0.00029, 1 25 and Equation (11) holds

a) in the case of the larger bead sizes.

0.37

.

eqn. 10

0.0181 (Macadams Table A-25)

0.37 .

. with D in inches

0.37 0.0181 .. 12. .

10.6. .. where D is in inches (and D>0.00725)

D 0.010 in 0.020 in 0.050 in 0.100 in

D0.4 0.1585 0.2095 0.3015 0.3985


65.9 50.6 35.2 26.6

Table I. for larger T/C beads b) in the case of smaller bead sizes. .

. Eqn. 11

@200 0.241 (Krieth Table A-3)

0241 10 3600 0.0601 2.23455 0.48

3455

where D is in inches

521.0 0.637 10

8.18 10

Table II shows the results of this calculation for various bead sizes. D 0.0005 0.001 0.002 0.005

0.02235 0.0316 0.0447 0.0707

855 467 261 126.5

Table II - for smaller T/C beads It can be noted from Table I and II that increases with decreasing probe diameter. A similar comparison for a given probe size, but at different velocities would show that is much greater for higher velocity flows. Time Constants

From the theoretical development, the time constant is given by

For the spherical probe, this reduces to

For this example,


615 0.0579 Then

615 0.05796 12 3600

This equation yields in seconds for D in inches and h in Btu/hr ft2 oF. Table III gives results of combining previous values of and with this equation for the time constant.

D In 0.001 0.002 0.005 0.010 0.020 0.050 0.100 h

467 261 126.5 65.9 50.6 35.2 26.6

sec 0.0038 0.0136 0.0724 0.27 0.704 2.53 6.69 Table III Time Constants for Various Sized Beads for the Conditions in the Example.

4.0 Experimental Procedure The experimental apparatus is shown in Fig. 6.


FIG. 5 Schematic Diagram of Thermocouple Response Rate Rig.

This rig facilitates rapid transfer of a probe (or probes) from cold to hot air streams and vice-versa. Instrumentation has been fitted to measure the steady temperature of the hot and cold streams and the velocity of the cold stream. The data acquisition system records the outputs form the thermocouples. The PC computes the time constants and presents the results in both graphical and numerical form. The experimental procedure will investigate the effect upon transient response and time constant due to the following:

a) effect of probe mass; b) effect of probe shape c) effect of stream velocity;

Four parameters, namely thermocouple size, thermocouple shape, air-stream velocity and the direction of temperature transition (ie; hot to cold or cold to hot) will be varied to produce the cases that will be experimentally recorded. Five thermocouples, numbered one through five, will be subjected to as many as three different flow rates in the two temperature directions. The specific cases are listed as follows and wills result in 22 cases, each of which will produce a corresponding graphical record (the TA may adjust the number of cases). The Laboratory Q involves the substantiation of thermocouple theory with respect to the changing of four variables. These four variables are thermocouple size, thermocouple shape, rate


of flow of air over the thermocouple, and the temperature transition direction. This substantiation is the result of a comparison between subsets of the 22 figures that are generated during the lab session. The figures must be generated using the following procedural steps:

1) Open the data acquisition program utilizing LabVIEW software. 2) Prepare for the specific test case by adjusting the heater and airflow controls (important-

start the airflow first!) to achieve the desired temperature and airflow (they are inter-related), insert desired thermocouple into test setup, position the sliding prove carrier for the proper direction of temperature transition, and allow the system to reach steady state values)

3) Initiate the data acquisition software and then quickly slide the thermocouple carrier to achieve a fast transition between airstreams, thereby producing the desired graphical recording

4) Repeat above steps to produce graphical recordings for all desired cases. Presentation of Results

1) Perform theoretical calculations to determine the time constants for all cases. 2) Graphically determine the time constants directly from the graphs supplied by the data

acquisition system and compare these with the numbers computed by the data acquisition system and the theoretical calculations.

This laboratory experiment uses five different thermocouples of various masses and shape (all are brass). Each thermocouple is designated a number as follows 1-small sphere, 2- medium sphere, 3- large sphere, 4- small cylinder, 5- large cylinder. The dimensions are on page XX. Three different air flow rates are used to demonstrate the effect of flow-rate on the thermocouple time constant. These flow rates are measured using a manometer and are ordered 1 for lowest velocity and 3 for highest. The three flow rates must be recorded and converted from delta h values into appropriate flow rate units. Suggested values will be provided by the TA. A transient response of a thermocouple will be produced when the temperature sensor is quickly moved from one environment into another having a different temperature. In the following list, C-H implies a cold to hot transition and H-C implies a hot to cold transition. List of Cases

Graph Case Number

Thermocouple Number

Flow rate Number

Transition Direction

1 1 1 C-H 2 1 1 H-C 3 2 1 C-H 4 2 1 H-C 5 3 1 C-H 6 3 1 H-C 7 4 1 C-H


8 4 1 H-C 9 5 1 C-H 10 5 1 H-C 11 1 2 C-H 12 1 2 H-C 13 2 2 C-H 14 2 2 H-C 15 3 2 C-H 16 3 2 H-C 17 4 2 C-H 18 4 2 H-C 19 5 2 C-H 20 5 2 H-C 21 1 3 C-H 22 1 3 H-C

Important Constants

Air Viscosity 126.5 10 Air Density 0.07734

Specific Heat 0.24983 Specific Heat 0.090045

Brass Density 530.587 Thermal Conductivity 0.0181

Thermocouple wire: Type T Copper Constantan, 24 A.W.G. Probe # 1: 1/8 dia. Probe # 2: 3/16 dia. Probe # 3: 1/4" dia. Probe # 4: 1/8 dia. 1 long Probe # 5: 1/4" dia. 1 long Ref: Holman, J.P., Heat Transfer, 5th Edition, McGraw-Hill (1981).

ExpQ Transient Response Rev0

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