1 / 80

CHAPTER 13 Transient Temperature Measurement

CHAPTER 13 Transient Temperature Measurement. The True Meaning of a Term Is to Be Found by Observing What a Man Does With It, Not What He Says About It. 13. 1 GENERAL REMAKS

admon
Télécharger la présentation

CHAPTER 13 Transient Temperature Measurement

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. CHAPTER 13Transient Temperature Measurement

  2. The True Meaning of a Term Is to Be Found by Observing What a Man Does With It, Not What He Says About It.

  3. 13. 1 GENERAL REMAKS Because of inertia, no instrument (or anything else for that matter) responds instantly or with perfect fidelity to a change in its environment. In mechanical systems, mass is the familiar measure of inertia, whereas in electric and thermal systems inertia is characterized by capacitance. We are concerned here with the response of a temperature sensor to a change in its environmental temperature. The simplified, hence manageable, temperature changes considered here are: (1) The ramp change, in which the environment temperature shifts linearly with time from T1 to T2 ;

  4. (2) The step change, in which the temperature of the sensor environment shifts instantaneously from T1 to T2 ; and (3) The periodic change, in which the environment temperature alternates sinusoidally with time between +T2to -T2 (figure 13.1). We seek answers to the following questions: what is the speed or measure of the sensor response? What is the fidelity or faithfulness of the sensor response? Thermal response belongs, fundamentally, in the realm of transient heat transfer. The rate of response of a temperature sensor clearly depends on the physical properties of the sensor, the physical properties of its environment, as well as the dynamical properties of its environment.

  5. Amplifying on this, we note that because physical properties normally change with temperature, it follows that the response time of a sensor will vary with the temperature level. Because heat transfer coefficients are strongly dependent on the Reynolds number, it follows that sensor response will vary with the mass velocity of its environment. It is common practice to characterize the response of a temperature sensor to a nonisothermal change of state of its environment by a thermal time constant. Although a single time constant can exactly describe the response behavior of only the simplest of systems, it is nonetheless common practice to consider first-order response only.

  6. In Section 13.2, we confine our attention to first-order systems in which the sensor exhibits a rate of change in temperature that is exactly proportional to the temperature difference between the sensor and its environment. In Section 13. 3, second-order systems are considered. We defer until Section 13. 6 the complications that arise from the additional considerations of conduction, radiation, temperature level, turbulence, and distributed thermal capacities.

  7. 13. 2 MATHEMATICAL DEVELOPMENT OF FIRST -ORDER RESPONSE A simplified one-dimensional heat balance can be written for a temperature sensor subjected to a time-varying environmental temperature. We assume that all the heat transferred to the sensor is by convection , and that all this heat is retained by the sensor, that is , the thermal resistance of the system is lumped in the convective heat transfer film around the sensor, and the thermal capacity of the system is lumped in the sensor itself ( Figure 13.2 ) .

  8. Thus the heat transfer rate through the film to the sensor exactly equals the rate of heat storage in the sensor. Expressed in terms of Newton’s law of cooling and Black’s heat capacity equation , we have (13.1)

  9. where h is the convective heat transfer coefficient of the fluid film surrounding the sensor A is the surface area of the sensor through which heat is transferred , TC is the environment temperature at time t , T is the sensor temperature at time t , M is the mass of the sensing portion ( it can also be expressed as , that is , density times volume ) , and c is the specific heat capacity of the sensing portion . Separating the variables in equation (1 3. 1) yields

  10. (13.2) where the quantity in parentheses is taken to be a lumped constant to have the dimensions of time , and called the time constant . Thus, (13.3)

  11. Since thermal conductance is the reciprocal of thermal resistance, equation(13.3) also indicates that ,which is exactly analogous to the time constant of an electric circuit . The first -order, first - degree linear differential equation expressed by equation (13. 2) has the general solution [1]-[3] where C is a constant of integration, which is determined by inserting the proper boundary conditions . (13.4)

  12. 13.2. 1 Ramp Change Under this condition at t=0,T1=T=C,although in general Tc=T2+Rt where R represents the rate of change of the environment temperature Insertion of these boundary values in equation ( 1 3 . 4 ) yields (13.15) Evaluating equation (1 3. 5), we have

  13. which, expressed in terms of the temperature difference ,becomes (13.7)

  14. Where the terms involving approach zero, and equation (13.4 )-(13.7) reduce to According to equation(13.8 ) , the time constant for a ramp change can be defined as follows : If an element is immersed for a long time in an environment whose temperature is rising at a constant rate (i.e., a ramp change), τis the interval between the time when the environment reaches a given temperature and the time when the element indicates this temperature; that is, is the number of seconds the element lags its environment (Figure 13. 1). (13.8)

  15. 13.2.2 Step Change Under this condition at ,although in general Insertion of these boundary values in equation (13.4) yields which, expressed in terms of the temperature difference, becomes According to equation (13.10), the time constant for a step change can be defined as follows: (13.9) (13.10)

  16. If an element is plunged into a constant-temperature environment (i.e., a step change), τis the time required for the temperature difference between the environment and the element to be reduced to l/e of the initial difference, that is, τis the number of seconds for the element to reach 63.2% of the initial temperature difference (Figure 13.1).

  17. 13.2.3 Periodic Change Under this condition , although in general Whereωrepresents the frequency of the forcing oscillations of the environment in radians per unit of time. Insertion of these boundary values in equation (13.4) yields (13.11)

  18. When the last term above approaches zero and the sensor response will lag the environment by the phase angle which in time units corresponds to a lag of (13.12) (13.13) Whereas the ratio of the sensor amplitude to that of the environment is given by (13.14)

  19. Although no general definition of the time constant for a periodic change is forthcoming, a restricted definition can be given as follows: If an element is immersed for a long time in an environment whose temperature is varying sinusoidally(i.e., a periodic change), and if the frequency is much less than 1/τthen, to a close approximation, τis the number of seconds the element lags its environment (Figure 13. 1).

  20. In equation (13.4) and in all equations thereafter, the time constant consistently has represented the lumped constant • Thus it follows that , whenever all conditions stated tender the three separate sections on the forcing functions are met.

  21. 13.3 SECOND-ORDER RESPONSE When several thermal resistances are combined along with several capacities, as in the case of a temperature sensor inserted in a thermometer well, an equivalent electric circuit more complex than that shown in Figure 13.2 must be considered. Also, solutions more complex than those given for first-order systems, namely, equations (13.8) and (13.10) must be used.

  22. 13.3.1 Step Change A second-order thermal system, as a thermometer well-sensor system, and its equivalent circuit are given in Figure 13.3. An appropriate solution describing the temperature-time response of the thermal system of Figure 13.3 to a step change in temperature can be given [4]-[7] as where T is the sensor temperature at any time t, is the step change in temperature, with the initial temperature normalized at zero, and where (13.16)

  23. and represent the roots of the second-order quadratic, with (13.17) (13.18) The thermal resistances and capacities, as shown in Figure 13.3, can be defined mathematically as follows: The sensor capacity is

  24. The well capacity is The sum of the well-side resistances is where and represents the heat transfer film resistance outside the well, and

  25. represents the well-to-capacity resistance. The sum of the sensor-side resistances is where represents the well-to-sensor resistance, and represents the air film between well and sensor, and represents the sensor-to-capacity resistance.

  26. The geometric definitions are as follows: = outer diameter of well = inner diameter of well = mean diameter of well =outer diameter of sensor =inner diameter of sensor = mean diameter of sensor Equation (13.15) is the second-order counterpart of the first-order solution given by equation (13. 10). The roots and of equations (13.15) and (13.16) are reciprocals of the time constants, that is,

  27. When one time constant dominates the other, that is, when equation (13.15) can he reduced to the form of a first-order solution, namely, In similar form, equation (13.10), the first-order solution can be written The conclusion expressed by equation (13.29) is in agreement with Looney[5] and Coon [6] to the affect that in most cases a single, representing the 63.2% definition of the time constant, is adequate to represent even the more complex temperature-sensing systems.

  28. 13.3.2 Ramp Orange The same equations for r1 and r2 hold for the ramp change as for the step change, that is, equations (13.16) to (13.18) and (l3.28). Hence the same time constants apply as well. Just as stated after equations (13.7), after a time lapse of about five dominant time constants, the temperature-sensing systems will follow a temperature-time ramp of the same slope as its environment (Figure 13.1), and the lag for the second-order systems will be as compared to the first-order equation (13.8).

  29. 13.3.3 Periodic Change When the environment temperature is varying sinusoidally, at a frequency below the temperature-sensing system behaves essentially as a single time constant system with an effective equaling the lag of equation (13.30).

  30. 13.4 EXPERIMENTAL DETERNUNATION of TIME CONSTANT The ramp change definition of the time constant provides one method for determining τ. The sensor, initially at some uniform temperature, is inserted into an environment whose rate of change of temperature with time is fixed and known. However, several problems are encountered. The environment temperature must be known as a function of time, and this requires a sensor of known τ or a sensor having an insignificantly small τ. In addition and this is a problem common to all methods of determining τ, the film coefficient of the environment-sensor interface must be known [4]. [8], [9], By the Nunsselt equation for heat transfer by forced convection [see equation (12.29)]

  31. where Nu = Nusselt number= Re = Reynolds number= Pr = Prandtl number= It follows that, along with the physical properties of the film and sensor, a knowledge of the mass velocity G of the environment relative to the sensor is requited if the film coefficient is to be determined within a reasonable uncertainty.

  32. The step change definition of the time constant provides the usual method of determining . The sensor is plunged from an initially different temperature into a constant-temperature bath. It does not matter whether the sensor is heated or cooled by the step change, but we should not start timing the response at the instant of the plunge.

  33. At least four should elapse first to allow stabilization of the fluid film on the sensor, since during this initial time the response of the sensor is not well approximated by the first-order equation. As in the ramp change, the film coefficient must be known. In a stagnant bath, reliable repeatable values for the film coefficient cannot be obtained because of the variableness of the natural convection currents set up in the bath by the temperature gradients.

  34. Murdock, Foltz, and Gregory [10] have discussed a practical method for determining response times of thermometers in stirred-liquid baths. • Their detailed method is somewhat as follows: Noting that the fluid properties K,μ and Cp can be taken as constants for a given liquid bath, equation (13.32) reduce to

  35. Evaluation of the effective mass velocity around the sensor is complicated by the fact that the liquid swirls in a three-dimensional flow pattern. It varies not only with the rate of agitation but also with the physical location of the sensor in the bath. By fixing the sensor location in the bath, we can express the effective value of G in terms of the stirrer speed as Hence equation (13.33) also can be given as

  36. Murdock et al. [10] summarize their experimental data for cylindrical sensors in their stirred-liquid salt with variable agitation by the empirical equation where the film coefficient h is in ,the sensor diameter D is in feet, and the stirrer speed N is in revolutions per minute. Equation (13.36) indicates that the exponents m and c of equation (13.35) are equal.

  37. It is important to note that equation (13.36) is only one particularization of equation (13.35). It cannot apply exactly in the general case. For example, in another stirred-salt bath there has resulted • It is clear that either equation (13.36) or equation (13.36') • yields close estimates to the film coefficient in a stirred-salt • bath. • If the stirrer speed also is fixed, equation (13.35) reduces • further to

  38. For a stirred-oil bath at about 4000F, Murdock's data can be expressed for constant stirrer speed as Again, this is one particularization of equation (13.35). In another bath, at a different but unrecorded stirrer speed, there has resulted It is clear from a comparison of equations (13.38) and (13.38') that both the stirrer speed and perhaps the bath geometry are of importance in determining the film coefficient by this method. Far example, at D = 1 in, equation (13.38) yields h = 17.5 while equation (13.38') yields h=32

  39. By expressing equation (13.3) for cylindrical sensors in the form of equation (13.37), the unknown function of D can be obtained. That is, for a hallow cylinder and neglecting end effects, the volume of the sensor is

  40. Furthermore, for radial heat transfer the surface area of the sensor is Hence According to the method under discussion, the physical properties of the sensor in equation(13.41) vary almost linearly with temperature and are to be evaluated at the average temperature corresponding to 36.8% of the difference between the initial and final temperature. Thus this method provides an experimental determination of the effective film coefficient of a stirred-liquid bath according to equation (13.36) and (13.41).

  41. To determinate τ, one starts with a liquid bath of known physical properties, stirred at known speed, held at a known temperature. A sensor of known physical properties is plunged into a fixed location in the bath and after waiting an appropriate time, one starts timing its response. The timing is stopped at a predetermined percentage of the temperature difference from start to final temperature, where the 63.2% mark yields the first-order time constant directly. As indicated in Figure 13.1, a first order response plots as an exponential curve on linear coordinates and as a straight line on a semilog grid. Since it is usually desirable to come closer to the final temperature than 63.2%, equation (13.10) can be rewritten as a=(TC-T)/(TC -Ti)=e-t/τ

  42. where a indicates how close the sensor temperature is to the final temperature. Equation (13.42) yields, for a few points, the following table:

  43. 13.5 APPLYING THE TIME CONSTANT The ideal model for the time constant of equation (13.3), that is, has four assumptions built into it.

  44. These are: 1. All thermal resistance to heat transfer is lumped in the fluid surrounding the sensor-well system. 2. All thermal capacitance of the system is lumped in the sensor-well system. 3. All the heat received through convection is stored in the sensor-well system. 4. The heat transfer is one-dimensional. In the following three examples, the idealized lumped relation of equation (13.43) is used as the basis of solution.

  45. Example 1. Predict the idealized first-order time constant for a thermocouple embedded to the center of a hollow stainless-steel cylinder of 1-in outside diameter and 0. 26 in inside diameter when plunged into a salt bath that is stirred at 800 r/min. Assume for the properties of steel those given in Table 13.1

  46. Solution By equation (13.36), though this is an effective h, as noted after equation (13.41), because it includes the cylinder thermal resistance as well as the film resistance, it is the h normally used to characterize the bath-sensor film. By equation (13.41)

  47. More precisely, for the hollow cylinder, following equation (13. 39)

More Related