Data Reduction K. RENGANATHAN
from Thin-Film Heat Gauges in Fluidized Beds and R. TURTON
West Virginia University, (Received
249 - 254
in revised form July
heat gauges are used to measure heat transfer coefficients, h, in fluidized beds to immersed surfaces. Usually, the amplitude of fluctuation in the sensor temperature is taken as a representation of the amplitude of fluctuation in h or a semi-infinite medium assumption for the substrate is used to transform the surface temperature into h-values. This paper is a numerical study of a typical thin-film heat gauge using a 1-D heat conduction model with finite depth for the substrate and a sinusoidal variation of h with time. Computer simulations of the gauge for different thermophysical properties of the substrate show that the substrate acts as a thermal capacitor and dampens the amplitude of fluctuations in sensor temperature otherwise driven by fluctuations in h. This damping effect increases with increasing thermal conductivity, and heat capacity of the substrate and decreasing thermal diffusivity, and width of the substrate. This may be the reason for some of the discrepancies in reported values of fluctuations in h in the literature. Thin-film
For over 40 years, numerous investigators have studied the phenomenon of heat exchange between fluidized beds and immersed surfaces. Despite a wealth of information, there still exists a number of unanswered questions regarding this phenomenon. One such question pertains to the nature of the bed to immersed surface or wall heat transfer coefficient h. It is not clear whether the hvalues are quasi-steady, time-varying deterministic or stochastic in nature. Despite advances in instrumentation techniques, few 0032-5910/89/$3.50
workers have reported experimental measurements of the time-varying bed to walI hvalues. The majority of the probes used were thin-film heat gauges made up of a thin metallic film, a few microns thick, deposited on a substrate and powered by dc circuitry. Changes in h-values were detected by changes in the dc measurements of voltage and current. However, there are significant differences in the reported values of h under similar fluidizing conditions. For instance, Baskakov et al. [l] report fluctuations in their measured h-values of over lOO%, whereas Syromyatnikov et al.  report h-values under similar fluidizing conditions which fluctuate by only 4%. One explanation for this discrepancy lies in the errors involved in the data reduction techniques used by these workers to translate their measurements into h data. This paper evaluates the mathematical techniques used to convert measurements into h data in fluidized bed-surface heat transfer. Apart from ascertaining the reason for the reported discrepancy in reported data, this study also explores the factors that must be considered in the choice of substrate for thinfilm heat gauges. Generally, thin-film heat gauges are ohmic resistance elements placed on an immersed surface and powered by a dc circuit. Both voltage and current are measured and from these the power dissipated by the sensor P(t) and the temperature of the sensor T(t) are determined. The response time of the sensor depends on its thermal mass. For example, a thin platinum film 1 pm thick has a response time of approximately 1 ms. Mickley et al.  were the first investigators to use a thin platinum foil attached to a vertical tube to measure bed-surface heat transfer coefficients. Subsequent workers [l, 2, 4 - 61 made use of thin-film technology with variations in the circuitry used to power @ Elsevier
in The Netherlands
these devices. Baskakov et al. [l] and Syromyatnikov et al.  used a Wheatstone bridge network to determine the fluctuations in the temperature T(t) and power P(t) dissipated by the thin-film device. Catipovic [ 51 controlled the temperature of the device using feedback control, while Gloski et al.  used a capacitor discharge circuit. In translating the measurements of current and voltage obtained in such experiments into heat transfer coefficients, it is necessary to calculate the fraction of the power dissipated by the sensor which is transferred to the bed, the balance of this power being dissipated to the sensor substrate and associated leads. Previously this estimation of power loss has been done in one of two ways. (i) By calibrating the thin-film sensor at different temperatures in a fluid for which the heat transfer coefficient is known. Thus a functional relationship between the sensor temperature and the heat losses to the substrate is obtained. For the case when the sensor is immersed in a fluidized bed, the temperature of the metallic film will fluctuate but the heat losses are estimated from the sensor temperature and the previously obtained calibration. In this way, Baskakov et al. [l] and Syromyatnikov et al. [ 21 calculated surface-bed heat transfer coefficients. (ii) By the solution of the energy balance written for the sensor. This requires a heat conduction model for the substrate to estimate the heat dissipated into the substrate qL, from T measurements, by modeling the substrate as a semi-infinite heat medium and using Duhamel’s theorem. This type of approach has been mentioned by other workers [5, 61 in their evaluation of bedsurface heat transfer coefficients. In the first method, the calibration is carried-out under static (steady state) conditions. The rationale for using the results of such a calibration to infer the heat losses during a dynamic (time-varying) experiment such as that occurring in a fluidized bed is questionable. In the dynamic experiment, the temperature-time history of the sensor substrate will also play an important role in determining the heat losses from the metallic film. The object of this work is, therefore, to investigate the methods of data reduction for thin-film temperature sensors in dynamic, time-varying environments. In the second
method, the validity of the semi-infinite medium assumption for the substrate should also be investigated. The following section develops the theory to analyze thin-film sensors and is subsequently used to analyze the effects of physical properties of the sensor on the time response of the probe. Specifically, the efficacy of using the fluctuations in the sensor’s temperature to represent the fluctuations in the bed to surface h-values is addressed.
The following theoretical development is based on a probe subjected to a sinusoidally time-varying heat transfer coefficient h at its surface. The model assumes that the substrate has a finite width and that the heat transfer through the substrate is described by a onedimensional conduction equation. Consider a thin-film heat gauge with a substrate of known thermal properties. Let us assume that a constant current source powers the gauge delivering current I. Information regarding bed to wall h-values would be in the form of voltage signals. These signals need to be translated into heat transfer coefficients. This requires a model of the probe system which includes the bed environment, sensing element and its backing substrate. A simple energy balance on the heat sensor with the assumption that the temperature T is invariant with position in the sensor can be written as
(pvC,)$=I’R,(l+PT)-hA(T-T,)-q, (1) where (PVC,) is the thermal mass of the sensor, I the constant current supplied to the sensor, R, the resistance in ohms of the sensor at 0 “C, 0 the temperature coefficient of resistance, A the surface area of the sensor exposed to the bed, TB the bed temperature and qL the heat loss into the substrate. Equation (1) defines the heat transfer coefficient h in terms of the area of the sensor A and the instantaneous temperature difference between the sensor surface and the average bulk bed temperature (T - TB). Equation (1) is valid as long as the end effects, i.e., heat losses to the periphery of the sensor, can be
neglected. By an order of magnitude analysis, it is noted that due to the typically small value of u (lo-i6 m3), the volume of the metal film, the term due to the accumulation in the foil can be neglected and the expression for h now becomes h
I’RoU + PT) - q, A(T--- TB)
In eqn. (2), T is calculated from the known relation between the electrical resistance and temperature. The heat lost into the substrate q,, however, needs to be estimated. The problem here is that qL is dependent on the sensor temperature T and h. A model is necessary to estimate qL from T us. t data. Most other workers either assume the substrate to be a semi-infinite medium and use Duhamel’s theorem for the estimation of q,  or obtain qL by static calibration of the sensor in air. In order to gain better insight into this problem, a detailed parametric investigation on the effect of substrate properties, current density and probe geometry was carried out by computer simulation. The most general assumptions were made about the substrate, leading to the most general boundary conditions for the problem. The energy balance for heat flow in the substrate, assuming constant physical properties, can be written as (3) Let
Ts--T, TB (4)
Then eqn. (3) becomes ae -=a?-
where hair L
B&i, = -
The second boundary condition is derived from writing an energy balance for the heat sensor, namely eqn. (l), with the inclusion of a conduction term into the substrate. For ease of derivation, let us define R,, as the resistance of the heat sensor at the bed temperature. Then 12R,, =--
+ 12R,/3T,8 k,AT, L
The second boundary condition is therefore
A time-varying value of h of the following form was assumed: h = 400[2
This was done to generate the temperature of the sensor T(t) and to study the effect of substrate properties on the transformation of T( t)-values to h-values. A sinusoidal variation with period 2 seconds was chosen to simulate a time-varying condition at the probe surface consistent with the situation occurring in a bubbling fluidized bed. Once T(t) is computed, this can be converted back into hvalues using the different techniques used by other investigators and compared with the h-values that were used to begin with. This procedure allows the evaluation of the different methods used. In particular, the efficacy of using the fluctuations in T-values to represent the fluctuations in h-values was investigated. Equation (5) was discretized and the appropriate finite difference equations written. The temperature profile in the substrate was obtained by numerical solution of the finite difference equations. A matrix inversion technique similar to that proposed by Crank and Nicholson  was used.
8(X, 0) = 0 Initial condition RESULTS
In each of the computer runs after sufficient time-intervals, the sustained solution
T&x, t) was obtained. At this point, the amplitude of the fluctuation of the temperature of the sensor, T(t), was computed and nondimensionalized. This parameter was then compared with the amplitude of the fluctuation of the temperature of the sensor if the substrate were a perfect insulator, i.e., with zero thermal conductivity. This ratio expressed as 9%fluctuation sustained is the ordinate in Fig. 1, where the abscissa is thermal conductivity of the substrate. Different lines in the plot are for different values of thermal diffusivity of the substrate. If the insulation were perfect, then the fluctuation in h would be directly reflected in the fluctuations in T. Thus, at k, = 0, the % AT sustained is 100% of the fluctuation in h. With increasing thermal conductivity k,, the substrate dampens the fluctuation in h. This may be the reason why Syromyatnikov et al.  reported only 4% fluctuations and Baskakov et al. [l] reported 100% fluctuations in h-values. The latter used a polymerized epoxy as the substrate of the sensor, which has a much lower thermal conductivity (h, < 0.1 W/(m K) than mica (Iz, = 0.5 W/(m K), which was used by the former. This means that in Syromyatnikov’s case, the damping effect of the insulator was higher than in Baskakov’s case. From Fig. 1 it can be seen that in Baskakov’s case over 80% of the amplitude of the fluctuations in h-values is sustained by the temperature fluctuations of the sensor, whereas in Syromyatnikov’s case only 20% of the amplitude of fluctuations in h-values is sustained by the sensor. Since both workers took the fluctuations in
Fig. 1. Simulation of thin-film heat gauge. Effect of thermal conductivity and thermal diffusivity on the temperature fluctuations measured by the heat sensor.
T-values to represent the fluctuations in h-values, they reported different fluctuations in h-values. Apart from resolving the discrepancy in the reported literature, another interesting feature falls out of this analysis. When the h-value increases, the T-value decreases as more heat is removed from the sensor. During the times when the surface temperature drops for a finite depth into the substrate, the temperature values are greater than that of the surface. As a result, the substrate acts as a heat source rather than a heat sink. The periodic change in nature of the substrate from a sink to a source causes the damping effect. An examination of Fig. 1 reveals that with increasing h, and decreasing CY,,the damping effect increases. Let us consider a vertical line in the graph - i.e., at constant h, we decrease a,. Why does the damping effect increase? The constant h, means that the thermal mass p$,, changes with (Y,. As CV,decreases, the thermal mass increases. The density of the substrate is relatively fixed for most materials at constant h,. So the parameter that changes as we descend a vertical line in Fig. 1 is the thermal capacity C,,. As oy,decreases at constant K,, the heat capacity increases. Materials with greater heat capacities can act as a capacitor and charge and discharge thermally at appropriate times more effectively thus causing greater damping. The effect of the assumption of a semiinfinite medium in modeling the substrate of the heat sensor was studied by varying the length of the substrate in the computer simulations. The results are presented in Fig. 2. The effect of length on the damping of the amplitude of the fluctuations in the temperature of the sensor is significant for small widths, low thermal diffusivities, and high thermal conductivity of the substrate. It can be seen from Fig. 2 that the damping effect is insignificant when the length of the substrate is very large (> 1 m). This means that any model of the substrate that assumes a semi-infinite medium would predict that the damping effect is insignificant. But the typical thickness of the substrate used by experimental workers is of the order of a few millimetres. As a result, the assumption of a semi-infinite medium to calculate instantaneous h-values from temperature measure-
Fig. 2. Simulation of thin-film heat gauge. Effect of thermal conductivity and substrate depth on the temperature fluctuations measured by the heat sensor.
ments is questionable. This assumption is good only for low &-values, high a,-values and deep substrates. This may be another reason for the discrepancy in the reported values between Baskakov et al. [l] and Syromyatnikov et al. . The thickness of the mica substrate used by Syromyatnikov et al was 0.55 mm. From Fig. 2 it can be deduced that less than 8% of the fluctuations in h-values were retained by the temperature of the sensor in Syromyatnikov’s case, whereas in Baskakov’s case, since the value of h, is low, the damping effect was insignificant.
Thin-film heat gauges are generally used to measure instantaneous, local heat transfer coefficients in fluidized beds to immersed surfaces. Investigators estimate the fraction of power dissipated by the thin-film heat sensor to its substrate by either a static calibration of the gauge in air or by modeling the substrate as a semi-infinite medium. In order to fully understand the effect of the thermophysical properties of the substrate on the data reduction techniques, computer simulations of a typical thin-film heat gauge were carried out. For the purposes of the study, a sinusoidally time-varying heat transfer coefficient was used and finite difference equations for the temperature profiles of the substrate were set up. The substrate properties such as its thermal conductivity, thermal diffusivity, depth and heat capacity were varied, in order to capture the transient
behavior of thin-film gauges in an environment of fluctuating heat transfer coefficients. It was found that the insulating substrate has a dampening effect on the fluctuations of the temperature of the heat sensor driven by the fluctuating heat transfer environment. The substrate acts as a thermal capacitor and charges and discharges at various times. Thus, at certain times, the substrate acts as a heat source rather than a heat sink. This damping effect of the substrate on the heat sensor increases with increasing thermal conductivity, decreasing thermal diffusivity, increasing heat capacity and decreasing widths of the substrate. Neither the static calibration technique nor the semi-infinite medium model technique captures this phenomenon. As a result, the amplitude of fluctuations of the sensor temperature may not directly correspond to the amplitude of fluctuations of the measured heat transfer coefficient. The semiinfinite medium assumption for the substrate is reasonable only if the substrate has a thermal conductivity less than 0.2 W/(m K) and/or is very deep. In light of the above findings, subsequent studies of experimental measurement of timevarying heat transfer coefficients in fluidized beds to immersed surfaces using thin-film heat gauges should clearly account for the dynamic behavior of the substrate by means of an accurate algorithm for data reduction.
The authors would like to acknowledge that this work was carried out under NSF grant No. CDR-8715496.
A B&i, c PS C* h
area of sensor, m2 Biot number (h,i,L/h,), heat capacity of substrate, J/(kg K) heat capacity of sensor, J/(kg K) heat transfer coefficient between bed and surface, W/(m’ K) heat transfer coefficient between substrate and surrounding ambient air, W/(m* K) constant current delivered to sensor, A
L P(t) (IL R3
R TB t T(t) T&v t) TB X
thermal conductivity of substrate, W/(m K) length of substrate, m power dissipated from sensor, W heat loss to substrate, W ohmic resistance of sensor at 273K,CZ ohmic resistance of sensor at temperature of fluidized bed, s1 time, s temperature of sensor, K temperature of substrate, K temperature of fluidized bed, K coordinate of position, m dimensionless-distance (X = x/L) volume of sensor, m3
Greek symbols thermal diffusivity of substrate, % m2/s temperature coefficient of resisP tance of sensor, K-’ bulk density of sensor, kg/m3 P bulk density of substrate, kg/m3 PS 8 nondimensionalized temperature
(0 = (Ts- Td/Td, -
non-dimensionalized time (T = cJlL2), current density (I’R.,/A), W/m2
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