Article pubs.acs.org/EF
Noninvasive Ultrasound Measurements of Temperature Distribution and Heat Fluxes in Solids Yunlu Jia and Mikhail Skliar* Department of Chemical Engineering, University of Utah, Salt Lake City, Utah 84112, United States ABSTRACT: A method for noninvasive ultrasound measurements of temperature distribution in solids is described and experimentally demonstrated in the estimation of an axial distribution of the temperature and the heat flux along a cementitious sample. It is further shown that by supplementing the ultrasound measurement with the surface temperature measurements, the entire volumetric distribution of the internal temperature inside of the solid sample and the corresponding heat fluxes can be reconstructed noninvasively. The unique capability for noninvasive characterization of the temperature and heat flux distributions in solids makes the developed approach particularly appealing for thermal characterization of solid components, structures, and containments of energy conversion processes, subsurface and nuclear applications, and other extreme environments, in which conventional sensors degrade quickly or their insertion is undesirable, difficult, or impossible.
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INTRODUCTION The velocity of ultrasound (US) propagation in solids changes with temperature. This dependence is the basis of several temperature measurement techniques, such as those reported in refs 1−6. When the relationship c = f (T )
Unlike the isothermal case (eq 3), there are arbitrarily many temperature distributions T(z) such that, when used in eq 4 to predict tof, the result will be identical to the measured time-offlight. Consequently, the deconvolution of the measurement in eq 4 is an ill-posed problem. The method for UltraSound Measurements of Segmental Temperature Distribution (US-MSTD) in solids6,8,9 resolves the lack of unique dependence of the measured tof on the temperature distribution by (a) using a structured propagation path with multiple echogenic features in order to create a train of n ultrasound echoes, the time-of-flight of which encodes the temperature distribution in different segments of the waveguide and (b) parametrizing “admissible” temperature distributions within each segment by prescribing a functional form that depends on one or more unknown parameters, which are then found from ultrasound and, perhaps, other measurements. The initial demonstration of the method8 reconstructed the segmental distribution of axial temperature as a piecewiseconstant function with discontinuities at the boundaries of adjacent segments. Here, we demonstrate that the US-MSTD method can be used to noninvasively reconstruct the axial temperature distribution in a way that avoids infeasible discontinuities. We showed that, by parametrizing the unknown temperature distribution by a thermal conductivity model, the volumetric temperature distribution inside solid samples can be reconstructed. The effect of different temperature parametrizations on the estimated distributions is examined in experiments conducted with a cementitious sample maintained at a nonuniform temperature. It is furthermore shown that the obtained temperature distribution can then be used to noninvasively estimate the heat fluxes inside and on the surface of a solid sample.
(1)
between the propagation velocity, c, and the uniform temperature, T, of a solid sample is known, finding an unknown T becomes a simple matter of measuring c (colloquially referred to as the speed of sound, SOS) and inverting the correlation in eq 1. In the pulse-echo mode, we can find c by applying an US excitation pulse, generated by a pulser-receiver coupled to a proximal surface of the sample, and measuring the time-of-flight (TOF), tof, of the returned echo, reflected from the distal end of the sample: c=
2L tof
(2)
where L is the length of the sample. With the measured tof, the sample’s uniform temperature is found as
⎛ 2L ⎞ T = f −1 ⎜⎜ ⎟⎟ ⎝ tof ⎠
(3)
where we assumed that the inversion of eq 1 is unique. Note that, in addition to the time-of-flight, other ultrasound characteristics, such as a phase change of echoes produced by a tone burst excitation,7 may be used to characterize temperature-dependent variations in the speed of sound. The described approach becomes problematic when the temperature inside the sample is nonuniform. In this case, the measured tof depends on the spatial distribution of temperature, T(z), along the entire US propagation path as tof = 2
∫0
L
1 dz f (T (z))
Received: January 11, 2016 Revised: March 31, 2016 Published: April 18, 2016
(4) © 2016 American Chemical Society
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Figure 1. (a) An excitation pulse created by the US transducer propagates through the structured containment and encounters n echogenic features along the way, which produce a train of echoes acquired by the receiver. The TOF difference between consecutive echoes is used to estimate the temperature distribution in the corresponding segment of the containment. By sequentially estimating the temperature of each segment, the temperature distribution along the entire path of ultrasound propagation is obtained. (b) Experimental setup used to measure the temperature distribution across the cementitious sample, which was structured into four layers, L1, ..., L4. During the calibration experiments, layers L2−L4 were maintained at a uniform temperature controlled by the side heater. Layer L1 was used as a delay line. A nonuniform temperature distribution was created by using the base heater only.
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METHOD
tofzi = 2
Figure 1(a) gives the graphical summary of the US-MSTD method for the case when the temperature across the containment of an aggressive process is measured. The essential components of the method are described in ref 6 and are briefly summarized below. Structured US Propagation Path. The US-MSTD method uses a structured US propagation path that incorporates echogenic features at known locations. These features may be purposefully engineered into a waveguide, may occur naturally as stratifications, inclusions, or geometric changes, or may have been introduced for reasons unrelated to the implementation of the US-MSTD method. For example, a rifling inside a gun barrel produces two echoes (from the interior and the rifled surfaces) when the ultrasound excitation pulse is applied by a transducer attached to the exterior of the barrel.5 A combination of features purposefully introduced, naturally occurring, or included in the design of the waveguide for unrelated reasons may also be used, provided their spatial locations are precisely characterized. The echogenic features redirect a portion of the US energy of the propagating excitation pulse back to the transducer, where the train of echoes is recorded. The difference in the TOF of these echoes encodes the temperature distribution specific to individual segments of the waveguide, bound by the locations of echogenic features. For cementitious samples used in this study, several methods for producing internal echogenic features were previously investigated.8 It was found that inclusions, stratifications, and variations in the waveguide geometry are the adequate means in creating the structured US propagation path. For example, it was found that by casting multiple cementitious layers of identical composition and allowing time for a partial curing before consecutive castings, enough variation is introduced to create partial US reflections at the interface of the layers. With such implementation, thermal, chemical, and mechanical properties remain essentially constant throughout the structured material. Such an approach is particularly appealing when additive fabrication techniques are used to manufacture components in which we wish to monitor the temperature distribution. In another example,6,10 echogenic features were obtained by drilling small holes along the length of a ceramic (alumina) waveguide. Signal Acquisition. The echo waveforms of returned echoes, acquired by the US transducer, are the primary data used to measure the segmental time-of-flight, and to estimate the temperature distribution in each segment of the ultrasound propagation path. Similarly to eq 4, the arrival delay of the i-th echo, produced by an echogenic feature located at zi, has the following dependence on the temperature distribution:
∫0
zi
1 dz f (T (z))
(5)
where z = z0 = 0 is the location of the transducer/receiver. The segmental time-of-flight tof i
tof = tofzi − tofzi − 1 i
(6)
is equal to twice the time the ultrasound pulse travels through the i-th segment, bound by the echogenic features of the waveguide located at zi and zi−1. This segmental TOF is related to the segmental temperature distribution T(z), zi−1 ≤ z ≤ zi as
tof = 2 i
∫z
zi i−1
1 dz f (T (z))
(7)
where zi − zi−1 is the length of the segment. The arrival delay of the nth echo is equal to the sum of segmental TOFs: n
tofzn = tof =
∑ tofi i
(8)
where tof is given by eq 4 and the distal end of the waveguide has the coordinate z = zn = L. Calibration. The segmental time of fight measurements allow us to calculate the velocity of the ultrasound propagation in different segments of the waveguide. The dependence of the SOS on temperature, eq 1, is material specific and, in this study, was obtained experimentally by measuring the speed of sound through the segment(s) of the waveguide maintained at known uniform temperatures in the range of interest. The length of the US propagation also changes with temperature due to thermal expansion or contractions. For most materials, the length of the ultrasound propagation increases with temperature, while the SOS goes down. Therefore, both factors typically lengthen the echoes’ arrival delay as the temperature rises. If the coefficient of thermal expansion is known, it is possible to separate the contribution of the two factors to the measured time-of-flight, though, for relatively small temperature changes, the contribution of thermal expansion to the change in the segmental time-of-flight with temperature is small for most materials. Furthermore, as long as the calibration in eq 1 describes the combined effect of both factorsas is done in this studythe effect of the thermal expansion is (at least, partially) compensated,6 which makes it unnecessary to adjust the length of the propagation path in eqs 4, 5, and 7 for the dimensional changes with temperature. 4364
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Energy & Fuels Estimation of Temperature Distribution. The temperature distribution within the i-th segment is estimated in measured segmental time-of-flight tof i by parametrizing the segmental temperature distribution in the measurement model (eq 4). All parametrization options used in this study are summarized below. Piecewise-Constant Parametrization. This distribution, first utilized in ref 8, is obtained by assuming a constant speed of sound within each segment. Under this assumption, eq 5 gives the constant SOS in the i-th segment equal to ci =
2(zi − zi − 1) tof
ρCp
q = h(Te − T )
The corresponding constant temperature is obtained by inverting eq 1. After repeating the process for all segments, the entire temperature distribution along the waveguide is approximated by a piecewiseconstant function. An undesirable feature of this result is the presence of infeasible temperature discontinuities occurring at the locations of the echogenic features. Nevertheless, the accuracy of such an approximation is significantly better than the assumption of a constant temperature along the entire path of ultrasound propagation, and can be further improved by finer segmentation of the waveguide using a larger number of echogenic features. Piecewise Linear Parametrization. This parametrization allows us to enforce the continuity of the estimated temperature across the entire US propagation path. In this case, the temperature distribution in the i-th segment is assumed to have the following functional form:
T (z) = miz + ni ,
zi − 1 ≤ z ≤ zi
1
z1
∫0
1 dz f (m1z + n1)
(10)
q(z) = − k
tof = 2 2
∫z
1
1 dz f (m2z + n2)
n2 = (m1 − m2)z1 + n1
∂T ∂z
(16)
One form of an approximate differentiation of a piecewise-constant temperature profile gives the following estimation of the axial heat flux through the i-th segment of the sample:
(11)
Similarly, for the second segment the unknown slope m2 and intercept n2 are obtained from the solution of the following two equations: z2
(15)
where h is the overall heat transfer coefficient and Te is the ambient temperature. If multiple echoes are produced by the pulse propagating through the structured waveguide, then the measurements of their arrival times may provide sufficient data to estimate all required boundary conditions without the need for independent measurements. If such independent measurements are nevertheless available, they can be incorporated into the US-MSTD method to improve the accuracy and the robustness of the estimated temperature distribution. Estimation of Heat Fluxes. Conductive heat fluxes in solids are typically measured by surface flux sensors that are only sensitive to fluxes in the immediate proximity of the surface. The US-MSTD method, on the other hand, can profile the temperature distribution across the entire sample and thus provide the information needed to noninvasively estimate conductive heat fluxes at a considerable distance from the surface where the US transducer is located. Specifically, by differentiating (exactly or approximately) the estimated temperature distribution T(z), the axial component of the conductive heat flux, q, is found as
where mi and ni are unknown parameters. If the temperature of the sample’s proximal end z = 0 is measured independently (e.g., by using a surface thermocouple), the intercept value for the first segment is fixed, n1 = T(0), and the unknown slope, m1, is determined by the following equation: tof = 2
(14)
where r is the radial position and ρ, k, and Cp are the density, the thermal conductivity, and the heat capacity of the waveguide material, respectively. Three boundary conditionsat the proximal, distal, and cylindrical surfaces of the waveguideare required to completely define the model. If the temperatures of the distal and proximal ends of the waveguide are independently measured, then the measured tof can be used to specify the third needed boundary condition, such as the heat flux through the cylindrical boundary of the waveguide:
(9)
i
⎛ 1 ∂ ∂T ⎞ ∂ 2T ∂T r ⎟+ = k⎜ ⎝ r ∂r ∂r ⎠ ∂t ∂z 2
qi(z) ≈ − k
Ti − Ti − 1 , zi − zi − 1
zi − 1 ≤ z ≤ zi
(17)
When a piecewise linear temperature distribution is known, the exact differentiation gives the changes in the axial heat fluxes across the i-th segment of the waveguide as
(12) (13)
q(z)i = − kmi
where tof 2 is the difference in the time-of-flight between the second and first echoes. Here, the second equation enforces the continuity of the temperature at the interface between the first and second segments located at z = z1. The process continues for all remaining segments until the piecewise linear temperature distribution across the entire sample is obtained. Model-Based Parametrization. The temperature parametrization by a one-dimensional heat conduction model was used before.4,5 In both cases, the arrival delay of a single US echo was used to estimate the boundary condition at the distal end of the propagation path. The second required boundary condition was set to be the temperature at z = 0, which was measured independently. The distal boundary condition was adjusted until the TOF predicted by the measurement model (eq 4) matched the measured TOF of an echo produced by a reflection of the excitation pulse from the distal end of the sample. The estimated temperature distribution was given as the solution of the conduction model with the obtained boundary conditions. When a two- or three-dimensional model is needed to provide an adequately accurate description of the temperature distribution in the sample, additional measurements will be required to reconstruct the temperature distribution. For example, consider the case of axial propagation of a US pulse along a cylindrical waveguide. Assuming radial symmetry, the temperature distribution inside the sample must satisfy the following 2D heat transport model in cylindrical coordinates:
(18)
where k generally changes with temperature and, thus, position. Note that, though an improvement, this approximation predicts discontinuous fluxes at the interfaces of the adjacent segment. In the absence of internal sources and sinks of heat, the continuity in fluxes is required to satisfy the energy balance. Such continuity may be maintained if the heat flux is calculated based on the temperature profile that was parametrized to satisfy the heat conduction model. Furthermore, by using 2- or 3D models to estimate the temperature distribution, not just the axial component of the heat flux, but the overall flux vector can be estimated as q⃗ = −k∇T, where the gradient is calculated exactly or approximately. Segmental Characterization of Material Properties. The proposed method essentially depends on the measurement of the segmental speed of sound (or its change) in a solid sample. Properties other than temperature influence the speed of sound in different segments and thus can be estimated under isothermal conditions. These include the density of the waveguide material and its elastic properties. For example, the velocity of longitudinal waves (p-waves) in the i-th segment of a “thin” waveguide maintained at isothermal conditions is equal to ci = 4365
Ei ρi
(19) DOI: 10.1021/acs.energyfuels.6b00054 Energy Fuels 2016, 30, 4363−4371
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Figure 2. (A) Typical waveforms of ultrasound echoes reflected from the interfaces between different layers of the cementitious sample (inset) and the sample−air interface at the distal end of the waveguide. The measurements were acquired at the reference temperature of 20 °C. (B) Envelopes of echo waveforms collected at different temperatures. where E and ρ are, respectively, the Young’s modulus and the density of the waveguide material. If the temperature and one of these properties (e.g., density) remain constant, the segmental changes of the other property (Young’s modulus) can be estimated from segmental changes in the SOS. Spatial variations in the material property may be estimated under different assumptions, such as piecewise-constant or piecewise-linear changes in different segments, or other suitable parametrizations.
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Table 1. Location of Echogenic Features, zi (mm) z0 (proximal end) z1 z2 z3 z4 (distal end)
EXPERIMENTAL DEMONSTRATION
0 26.54 52.74 80.84 101.154
SOS as a Function of Temperature. The correlation between the speed of sound and the temperature, eq 1, for the fabricated sample is the “calibration” curve for the US-MSTD method. For the fabricated sample, it was obtained experimentally. The waveguide was placed inside the heating fixture depicted in Figure 1(b). The fixture consists of a thermally insulated steel container and an internal heating blanket (silicon rubber blanket by BriskHeat, Columbus, OH) that tightly surrounded the sample. The temperature of the heating blanket was measured by a thermocouple and controlled by a PID controller. Independent temperature measurements were provided by four thermocouples (Type T, model 5TC-GG-T-30-36; OMEGA Engineering, Inc., Stamford, CT; the manufacturer-reported accuracy of these thermocouples is ±0.5 °C), which we attached to the cylindrical surface of the waveguide with high-temperature adhesive tape, ensuring that the thermocouple tips were located in the middle of each layer. Two additional thermocouples (not shown in Figure 1(b)) were used to measure the temperature of the top and bottom surfaces of the sample. During the calibration experiments, the side heating blanket was used to maintain a uniform temperature of layers L2, L3, and L4. This temperature was changed in 10 °C increments, from 20 to 100 °C. After each temperature change, sufficient time was allowed for thermal equilibration to occur before acquiring the waveforms of the US echoes. The ultrasound tests utilized a Panametrics pulser/receiver (model 5072PR; Olympus IMS, Waltham, MA) and a Panametrics immersion transducer with a central frequency of 1 MHz (model V302). The data
Structured Waveguide. The experiments were conducted with a layered cementitious sample approximately 4″ long (pictured as the inset in Figure 2). It was fabricated by sequentially casting four ∼1″thick layers (depicted schematically as L1, ..., L4 in Figure 1(b)) of identical mixture of water and Portland Type I/II cement into a vertically oriented 2″ ID cylindrical mold and allowing for their partial curing prior to casting the next layer. The mold was vibrated by an external vibrator after pouring each new layer to ensure a uniform settling and to remove air bubbles. The solidified sample was taken out of the mold and allowed to completely cure and age at ambient temperature. Despite the identical composition of all layers, the described process introduced variations in the material properties sufficient to create partial US reflections from the interface between the layers.8 After the aging, the waveforms of the four US echoes, produced by the three internal interfaces and the distal end of the sample, became stable. These stable echoes obtained at the reference temperature Tref = 20 °C are shown in Figure 2(A). The overall length of the layered waveguide L = z4 was measured by a micrometer. The SOS at the reference temperature was calculated from eq 4, where tof = tof4 is the time-of-flight of the last echo in Figure 2(A). The location of each interface between consecutively cast layers, z1...z3, was determined from the obtained SOS and the time-of-flight of the preceding echoes, tof1···3. The result is shown in Table 1, where z0 = 0 is the transducer location. 4366
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When the temperature in the i-th segment is unknown and nonuniform, the values of tof i(Tref) and tof i−1(Tref), obtained using the envelope cross-correlations method, give the following variant of eq 7:
were acquired using an oscilloscope (Tektronix Inc., Beaverton, OR; model MSO 2024) interfaced to a computer. The ultrasound transducer was coupled to the top surface of layer L1, which extended above the heating fixture to prevent the thermal damage to the transducer. In this arrangement, Layer 1 is effectively used as a delay line. The experiments at different temperatures were repeated at least 6 times in random order. The sequence of the experiments was determined by randomizing the entries in the list containing all test temperatures and their repeats. Random staging was found to be important in avoiding measurement bias, which is more likely to occur when the test temperature changed monotonically from one experiment to the next. Signal Processing. Figure 2(A) shows typical waveforms of echoes reflected from the interfaces between consecutive layers plus the reflection from the distal end of the sample. Note the waveform broadening and distortion for the echoes that travel longer distances through the waveguide prior to reaching the receiver. This outcome is typical for dissipative waveguides, which attenuate higher frequency ultrasound more strongly. The observed waveform distortion complicates the signal analysis used to time the arrival of echoes and makes the time-of-flight measurements susceptible to significant errors.10 We timed the arrival of ultrasound echoes by cross-correlating their envelopes. The envelope, AT(t), of the acquired waveform, s(t), is the absolute value of the corresponding analytic signal, sa(t): A T (t ) = |sa(t )| =
s 2(t ) + s 2̂ (t )
tof (T ) = (tof (Tref ) − tof (Tref )) + (Δtofzi(T ) − Δtofzi − 1(T )) i
=2
ci = =
∫0
1 ⎤⎥ dz f (T (z)) ⎥⎦
ATref (t )A T (t + τ )dt
i−1
For all but the first segment, the time-of-flight through the ith segment at the reference conditions, tof i(Tref) − tof i−1(Tref), was calculated as a value that maximizes the envelope crosscorrelation of the echoes produced by echogenic features that bound the segment. This approach cannot be used for the first segment because the shape of the “initial bang” (occurs close to t = 0 in Figure 2A) is dissimilar from the waveform of the first echo. Instead, tof1(Tref) was found as 2z1 , where z1 is the c2
measured thickness of the first layer and c2 is the speed of sound in the second layer at the reference conditions. Using eq 26, the SOS in different segments of the waveguide was calculated using the values of tof i(T), which were found following the described signal analysis procedure. Figure 3, first reported in ref 8, summarizes the result of the SOS measurements for all four layers. The data points in this figure show the average SOS obtained in 6 repeated calibration experiments as a function of the temperature measured by a thermocouple attached to the middle of the corresponding layer. The linear regression of all data points has the following form:
(22)
c = f (T ) = −1.06T + 3240.3,
20°C ≤ T ≤ 100°C (27)
and is also shown in Figure 3. The shaded area in the figure indicates the 95% confidence interval for this regression. The interval widens as the temperature moves further away from the reference value. By selecting the reference temperature in the middle of the range of the expected temperatures and/or by splitting the overall range into subranges, each with its own Tref, such behavior may be avoided. Estimated Temperature Distribution. The developed US-MSTD method was applied to the measurement of nonuniform temperature distribution established when the waveguide was heated by only the base heater shown in Figure 1(b). After the surface temperatures measured by thermocouples stabilized at constant values, an ultrasound excitation pulse
(23)
+∞
∫−∞
2(zi − zi − 1) (Tref )) + (Δtofzi(T ) − Δtofzi−1(T )) (26)
In this work, the temperature-induced change in the TOF of ultrasound echoes was found as the shift τ needed to maximize the correlation between the envelopes of the given and the reference echoes, and calculated as the solution of the following optimization problem: Δtofzi(T ) = argmax τ
(tof (Tref ) − tof i
(20)
This change is then used to calculate the corresponding change in the speed of sound relative to its value at Tref. When the temperature of the sample is nonuniform, the TOF difference relative to the reference condition is related to the temperature distribution T(z) as zi
2(zi − zi − 1) tof (T ) i
(21)
⎡ z = 2⎢ i − ⎢⎣ c(Tref )
(25)
RESULTS Calibration. The calibration curve (eq 1) was obtained by finding the speed of sound in the i-th segment maintained at a known uniform temperature T. Using the measurements of the change in the time-of-flight, Δtzofi−1(T) and Δtzofi (T), of the echoes produced by two consecutive echogenic features, the segmental speed of sound was calculated as
Here, j = −1 and ŝ(t) is the Hilbert transform of s(t). The envelopes of the echo waveforms in Figure 2(A), which were acquired at the reference temperature Tref = 20 °C, are shown as the top trace in Figure 2(B). The remaining charts in Figure 2(B) give the envelopes of the waveforms collected when the sample was maintained at the elevated uniform temperatures equal to 50, 70, and 100 °C, respectively. The visual comparison reveals a district trend toward an increased time-of-flight of echoes as the temperature rises, which corresponds to the reduction in the speed of sound at the elevated temperatures. When the sample is maintained at a uniform temperature T, we quantify the change in the time-of-flight of the i-th echo relative to its TOF at the reference temperature:
Δtofzi(T )
i−1
■
2
⎡ z zi ⎤ ⎥ Δtofzi(T ) = tofzi(Tref ) − tofzi(T ) = 2⎢ i − c(T ) ⎥⎦ ⎢⎣ c(Tref )
∫z
i−1
1 dz f (T (z))
from which an unknown temperature distribution T(z) within the segment must be found by using the discussed options for the temperature parametrization.
where sa(t) is defined as
sa(t ) = s(t ) + js (̂ t )
i
zi
(24)
The found value is related to the speed of sound by either eq 22 or eq 23, depending on whether the sample is maintained at a uniform or nonuniform temperature, respectively. 4367
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1.89(zi − zi − 1) , tof i
z ∈ [zi , zi − 1] (28)
Applied to all segments, this parametrization (previously used in ref 8) results in a piecewise-constant approximation of the temperature distribution shown in Figure 4(A). For comparison, the thermocouple measurements of the surface temperature in the middle of each segment are shown as black dots. Though this simple parametrization leads to a discontinuous change in the estimated temperature at the locations of echogenic features, the trend in the temperature distribution along the waveguide is, nevertheless, correctly captured. Piecewise-Linear Distribution. When used in eq 7 for each segment of the waveguide, the linear parametrization (eq 10) leads to the following equation in unknown parameters mi and ni: tof = 2
Figure 3. Calibration results for the speed of sound as a function of temperature. The SOS in different segments (layers of the waveguide) was calculated from the measurements of the change in the time-offlight of the four echoes produced by the structured waveguide, relative to their TOF at the reference temperature Tref = 20 °C. The shown linear fit is based on the data for all layers. The shaded area gives the 95% confidence interval for the regression model.
i
∫z
zi i−1
1 dz a(miz + ni) + b
(29)
where a = −1.06 and b = 3240.3 define the linear relationship between the speed of sound and the temperature (eq 27). After performing the integration and by requiring that the temperature remains continuous at the boundary of two adjacent segments, we obtain the following two equations, from which the unknown slope mi and the intercept ni can be found:
was applied to the sample and four return echoes were acquired. For each echo, the value of tof i(T) was calculated by the described envelope cross-correlation method and the result used on the left-hand-side of eq 25. The unknown temperature distribution T(z) in that equation was then obtained under different parametrizations. In each case, the parameters defining the distribution were selected to satisfy eq 25 exactly or with the smallest error in the least-squares sense. Piecewise-Constant Distribution. The assumption of constant temperature in each segment, coupled with the linear calibration (eq 27), leads to the following expression for the temperature of the i-th segment of the waveguide:
tof = i
2 ln[a(miz + ni) + b]zzii−1 ami
mizi + ni = mi − 1zi + ni − 1
(30) (31)
For the first segment, n1 = T(0), where T(0) is the measured temperature at the transducer location provided by a thermocouple. By combining eqs 30 and 31 for all segments of the waveguide, a sufficient number of conditions is obtained to find all unknown parameters. The result for all segments is summarized in Table 2, and the corresponding segmental temperatures and the overall temperature distribution across the entire waveguide is plotted in Figure 4(A).
Figure 4. (A) Estimated temperature distributions based on piecewise-constant and piecewise-linear parametrizations and their comparison with the measurements of the surface temperature provided by thermocouples attached in the middle of each segment. (B) Temperature distribution along the centerline and the surface of the waveguide, estimated to satisfy the two-dimensional thermal conductivity model, in comparison with the thermocouple measurements. 4368
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k = 26.91105 − 0.2477056T + 8.606168 × 10−4T 2
Table 2. Coefficients Defining the Piecewise-Linear Distribution of the Segmental Temperature Waveguide segment
m, °C/m
n, °C
1 2 3 4
443.20 486.36 698.08 1058.88
40 38.86 27.69 −1.48
− 1.00482 × 10−6T 3
(COMSOL database)
(32)
The temperature distribution at the proximal and distal ends of the waveguide and the heat flux through its cylindrical surface were used as the three boundary conditions required to completely define the model. The temperature distributions on the proximal and distal surfaces of the waveguide (z0 = 0 and z4 = 101.154 mm) were assumed to be radially symmetric and decrease linearly from the centerline (r = 0) to the cylindrical edge (r = 25.4 mm). These centerline and edge temperatures on the proximal (T(0,0) and T(0,25.4)) and distal (T(101.6,0) and T(101.6,25.4)) ends of the waveguides were measured using the following procedure. The thermal images of the proximal end of the waveguide (z = 0) were acquired with an infrared camera (model T300, FLIR Systems, Inc., Wilsonville, OR). The average of three such images (one of which is shown in Figure 5A), obtained in repeat experiments, gave T(0,0) = 40 °C and T(0,25.4) = 38.73 °C. These values were then used to find the radially symmetric temperature distribution, that decreased linearly from the center to the edge of the proximal surface. The temperature distribution of the hot distal end was characterized by quickly removing the waveguide from the test fixture and thermally imaging the distal surface. The averaging of three thermal images (one of which is shown in Figure 5(B)), acquired in repeat experiments, gave the centerline and edge temperatures, T(101.6,0) = 115 °C and T(101.6,25.4) = 109.92 °C, from which the temperature distribution on the distal surface was found. The obtained centerline temperature on the distal surface was verified by thermocouple measurements obtained without removing the waveguide from the heating fixture and was found to be in good agreement with the imaging results.
Though the estimated piecewise-linear temperature distribution avoids infeasible discontinuities and better agrees with the independent measurements, a substantial difference (∼10 °C at z = 40 mm) with the thermocouple data is still apparent. This difference may be, at least partially, explained by lower surface temperatures measured by the thermocouples compared to higher internal temperatures, which are noninvasively probed by the ultrasound measurements. In order to confirm this hypothesis and quantify the difference between surface and internal temperatures, the US-MSTD method must use the temperature parametrization that accounts for the heat loss through a cylindrical surface of the waveguide. One such parametrization, given by a two-dimensional heat transfer model, is considered next. Parameterization by a Two-Dimensional Thermal Conductivity Model. The two-dimensional heat conduction model (eq 14) accounts for the radial heat losses through the wall of the test fixture. It was implemented in COMSOL Multiphysics (COMSOL, Inc., Burlington, MA) modeling software. The following model parameters for the cementitious waveguide were used: ρ = 1,200 kg/m3 (measured value); Cp = 1.55 kJ/ (kg K); and
Figure 5. Thermal images of the proximal (A) and distal (B) ends of the waveguide. The estimated volumetric temperature distribution inside the waveguide, obtained with the parametrization provided by the thermal conductivity model, is shown in (C). 4369
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Energy & Fuels The heat transfer coefficient h was found through an interactive line search that minimized the normalized sum of two error types: (a) The errors between the measured and the predicted time-of-flight for each segment, where the predicted TOF values were obtained by finding the temperature distribution T(z, r) by numerically solving the heat transport model (eq 14) in COMSOL with the boundary condition defined by the current value of h, and using the centerline temperature T(z,0) in the measurement model (eq 5) to predict the segmental time of flights. (b) The sum of squared errors between the surface temperature measured by the four thermocouples and the corresponding model predictions of the pointwise temperatures T(13.3,25.4), T(39.6,25.4), T(66.8,25.4), and T(90.0,25.4) in the corresponding locations on the cylindrical surface of the waveguide. This search produced the value of h equal to 14 W/(m2·K). The volumetric temperature distribution T(z, r), shown in Figure 5(C), was reconstructed by numerically solving the model in eq 14 with the three boundary conditions obtained from thermal imaging and the ultrasound measurements. All segmental times of flight of ultrasound echoes, predicted by using the centerline temperature of this distribution in the measurement model (eq 5), differed from the measured values by less than 1%. The agreement of the reconstructed T(z, r) with the independent temperature measurements was assessed by comparing the thermocouple data with the pointwise values of the temperature distribution at the coordinates describing the location of the thermocouples. The maximum difference between the measured and the reconstructed values was found to be less than 2 °C, which is a close agreement given that the accuracy of the thermocouple measurements is ±0.5 °C. The centerline (T(z,0)) and surface (T(z,25.4)) temperatures of the reconstructed volumetric distribution are plotted in Figure 4(B), which graphically illustrates the close agreement with the thermocouple measurements. The result also indicates that the centerline temperature is indeed higher than the surface temperature by as much as 7 °C. Estimation of Heat Fluxes. By differentiating the obtained temperature distributions and using the result in eq 16, the estimation of the axial and volumetric heat fluxes was obtained. The estimated axial heat fluxes along the centerline of the waveguide are shown in Figure 6. For the piecewise-constant parametrization, eq 17 gives only three distinct values of q(z), which, in Figure 6, are placed at the interfaces between the waveguide layers. The depicted values were obtained by setting the temperature-dependent thermal conductivity k, appearing in eq 17, to the value given by eq 32 for the average of two adjacent temperatures. When the same approximation was used with the thermocouple measurements, three distinct values of q were again obtained, as shown in the same figure. Both heat flux estimatesbased on the thermocouple measurements and the piecewise-constant temperature reconstruction of the axial temperatureare consistent despite the fact that the surface and centerline axial fluxes are the same only when the waveguide is perfectly insulated, which is not our case. A higher disagreement is observed toward the hot end of the waveguide, where the thermal losses to the environment, and thus the difference between the surface and the centerline temperatures, are higher.
Figure 6. Axial heat fluxes along the centerline of the waveguide estimated using temperature distribution measured under three different parametrizations: piecewise constant, piecewise linear, and parametrization by the two-dimensional heat conductivity model. For comparison, the flux along the surface of the waveguide was estimated based on the thermocouple measurements of surface temperatures.
The spatial derivative of the piecewise-linear temperature profile can be calculated exactly for every spatial location along the axis of the waveguide. This leads to the axial heat flux distribution that is given by eq 18. If a constant value of the heat capacity is assumed for each segment of the waveguide, eq 18 estimates q(z) as the piecewise-constant distribution plotted in Figure 6, where the segmental k was calculated from eq 17 for the average temperature in each segment. A nearly constant flux in segments L1 and L2 is consistent with an approximately equal temperature slope in these two segments. When k in eq 18 is calculated for the spatially varying temperature, the heat flux distribution will no longer be piecewise constant; however, the flux discontinuities at the locations of the echogenic features will still be present. The flux discontinuity incorrectly implies that heat sources/ sinks are present inside the waveguide. These infeasible flux discontinuities can be avoided by explicitly requiring that the temperature distribution, used in the heat flux calculations, satisfies the energy balance without internal heat sources, such as the heat conduction model in eq 14. The continuous axial flux along the centerline and the cylindrical surface of the waveguide, shown in Figure 6, are such estimates. They were obtained by differentiating the cubic spline interpolation of the axial and surface temperature distributions shown in Figure 4(B), which were obtained to satisfy eq 14, and by using the result to calculate the heat flux, where k in eq 16 was again obtained from eq 32 for every centerline and surface location of the waveguide. The radial component of the heat flux vector can be similarly obtained, while the azimuthal contribution is zero due to the axial symmetry of the reconstructed temperature distribution.
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DISCUSSION We described the concept, the development, and the experimental demonstration of the method for noninvasive ultrasound measurements of the temperature and the heat flux distributions in solids. Two essential components of the method are the structured ultrasound propagation path containing echogenic features and the techniques needed to 4370
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distribution in solids using the US-MSTD method and at the pilot-scale conditions.
deconvolute the measured arrival delays of the acquired ultrasound echoes, produced by these features, into the temperature distribution. If the propagation path is not structured, the method can still be applied as a limiting case of reconstructing the temperature distribution in a single segment. The finer segmentation, however, will generally improve the accuracy of the reconstruction. The echogenic features that segment the waveguide could occur naturally in the material or may be present in the structure for reasons unrelated to the US-MSTD method. Multiple options exist for their purposeful introduction along the length of the ultrasound propagation, including placement of embedded US scatterers,8 lamination and layering (used in the current work and in ref 8), and the geometric changes.10 Additive manufacturing techniques, such as powder sintering, sputtering, lamination, and 3D printing, can be readily used to fabricate components incorporating the required echogenic features into their structure. In the case of walls, valve bodies, vessels, and other containments and components, waveguide inserts may be cemented or embedded into the structure, thermally fused, or otherwise incorporated into the components. In a more conventional implementation, the engineered waveguides may be inserted to provide the structured ultrasound propagation path. The developed method allows for noninvasive estimation of the temperature distribution by coupling the transducer to an easily accessible surface of the waveguide that was engineered to be an integral part of the structure. This capability is particularly appealing when the access to deploy traditional insertion probes is limited, and when the measurements are needed to characterize extreme environments, in which conventional sensors fail quickly. With an appropriate selection of the waveguide material, the temperature distribution can be measured over large ranges and in real time.6 Once the temperature distribution is obtained, the result can be used to estimate the heat fluxes across a solid sample. Currently, there is no other alternative for measuring heat fluxes deep inside solid materials. When used at isothermal conditions, the proposed method may be adapted to characterize segmental changes in such material properties as density or elasticity. The results of the US-MSTD method depend on the selected parametrization of the temperature distribution. Three different options were compared in this paper. The simplest piecewiseconstant parametrization produces the least accurate estimation of the temperature distribution but still correctly captures the trend. Piecewise-linear parametrization improves the accuracy of the estimation and avoids discontinuities in the estimated temperature profile. However, the corresponding distribution of the heat flux remains discontinuous under this parametrization. The highest achievable accuracy and the ability to avoid discontinuous artifacts in the estimated temperature and heat flux distributions may be obtained when the results of the US-MSTD method are parametrized to satisfy the thermal transport models. This higher accuracy was demonstrated experimentally by reconstructing the temperature distribution and the corresponding heat fluxes that satisfy a two-dimensional heat conduction model and the boundary conditions, which were selected to provide the simultaneous agreement with the measurements of the segmental times-of-flight of ultrasound echoes and the pointwise surface temperatures measured by the thermocouples. This is the first demonstration of noninvasive measurements of volumetric temperature
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Author Contributions
M.S. conceived the method, Y.J. conducted the experiments, Y.J. and M.S. analyzed the results. Both authors wrote and edited the manuscript. Notes
The authors declare the following competing financial interest(s): M.S. co-founded a company, Clovis Point Innovations LLC, to commercialize the described technology.
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ACKNOWLEDGMENTS The authors acknowledge financial support from the National Energy Technology Laboratory of the U.S. Department of Energy under award number DEFG2611FE0006947.
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REFERENCES
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