Applications of Digital Holographic Interferometry in Heat Transfer Measurements from Heated Industrial Objects

Abstract

Digital holographic interferometry (DHI) is used worldwide for many scientific and industrial applications. In DHI, two digital holograms; one in the reference/ambient state of the object and another in changed state of object are recorded by electronic imaging sensors (such as CCD/CMOS) as reference holograms and object holograms, respectively. Phase of object wavefronts in different states of the object is numerically reconstructed from digital holograms. The interference phase is reconstructed by subtracting the phase of reference hologram from the phase of object hologram, without performing any phase-shifting interferometry. Thus, no extra effort is needed in DHI for calculating the interference phase. Apart from direct reconstruction of interference phase from two digital holograms, the recent development, availability of recording devices at video rate, and high-performance computers make the measurements faster, reliable, robust, and even real-time. In this chapter, DHI is presented for the investigation of temperature distribution and heat transfer parameters such as natural convective heat transfer coefficient and local heat flux around the surface of industrial heated objects such as cylindrical wires and heat sinks.

1. Introduction

The study of heat flow/heat transfer from industrial heated objects such as wires, cylinders, conductors, and heats sinks is involved in many scientific and industrial applications. The knowledge of surface temperature and temperature profile normal to the surface of heated objects is sought for the measurements of heat transfer parameters such as convective heat transfer coefficients and heat flux [1, 2, 3]. The temperature measurements by intrusive methods such as thermocouple, thermistor, hot wire anemometer, and resistance temperature detector (RTD) are not reliable because these contact type methods disturb the heat flow and also lack in providing full field temperature information [3]. In the past, many optical interferometric methods have been widely used for the measurements of temperature, temperature profile, and heat transfer studies. These methods being non-invasive and non-contact type in nature provide reliable measurement of temperature. Other advantages of optical methods are (1) these methods provide full field temperature measurements in contrast to contact-type methods which provide point-by-point measurements and (2) measurements are inertia free so rapidly time-varying phenomena can also be investigated [3, 4]. Careful use of optical methods leads to more accurate and precise temperature measurements.

In the past, most of the temperature and heat transfer studies such as natural convection, forced convections, heat flux in liquids, and gases from heated objects such as cylinders, vertical plates, horizontal plates, finned plates, compact channels, microprocessor chips, grooved channels, downward facing heater surfaces, rib-roughened vertical surfaces, and horizontal cylinders were performed using classical interferometry (Mach–Zehnder Interferometry [MZI] and Michelson Interferometry [MI]), and holographic interferometry (HI) [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. Goldstein used MZI to study the aerodynamics and heat transfer [20] and Haridas et al. used both MI and MZI to measure the forced convection in compact channels [21]. In these studies it was concluded that MI provides a larger number of fringes (data points) in the temperature fields in comparison to MZI, thus MI is a better method of temperature field measurement in compact channels [21]. MI is more sensitive to temperature change than MZI but MI is not good for studying dynamic/rapidly varying phenomena as the light has to travel the test section/object field two times.

In the temperature and heat transfer studies using classical interferometry and HI mentioned in the above paragraph, the full field quantitative temperature field was not measured but the qualitative information on temperature was visualized. The temperature was calculated at each fringe position (i.e., isothermal line) using fringe count and fringe shift methods. The temperature profile was plotted by fitting the temperature data calculated at the location of each fringe. HI has several advantages in comparison to classical interferometry. HI does not require high-quality optical components in comparison to classical interferometry, because in HI, only the relative phase change of object wavefronts is evaluated. This leads to reduced costs of holographic interferometric equipment. HI provides the major advantage of greater experimental and technical simplicity. Thus, the time required for adjustment of equipment and taking measurement is also reduced considerably [26].

Furthermore, several optical interferometric methods such as classical interferometry [5, 6, 35, 36], holographic interferometry [22, 23, 24, 37], moiré deflectometry [38], speckle shearing interferometry [39], shearing interferometry [40, 41], Talbot interferometry (TI) [42, 43], Lau phase interferometry (LPI) [44], digital speckle pattern interferometry (DSPI) [45], holo-shear lens based lateral shear interferometry [46, 47, 48], and DHI [49, 50, 51, 52, 53, 54, 55, 56, 57, 58] have been investigated to measure the temperature field inside the axi-symmetric gaseous flames and transparent hot objects. TI, Moiré deflectometry, lateral shear interferometry, and holo-shear lens-based interferometry are common path interferometric methods and hence these methods are relatively less sensitive to vibrations or external perturbations as compared to speckle shearing interferometry, HI, and DHI. In shearing interferometry and speckle shearing interferometry, prior information regarding temperature profile direction is required. Prior information regarding temperature profile direction is also needed, if TI and LPI are performed with linear grating. However, TI and LPI using circular grating do not require any prior information regarding temperature field direction [39]. Also, in TI and LPI with linear gratings, the alignment between the gratings is difficult to achieve. LPI uses white light source, so the advantage of using LPI over TI is that it reduces coherent speckle noise in the interferogram, and thus provides high signal-to-noise ratio [39, 59]. TI, LPI, and holo-shear lens-based interferometry require high-quality optical components (such as gratings, lenses, and holo-shear lens) as compared to HI. The interference fringes in speckle shearing interferometry, and DSPI are not of good contrast because these fringes are formed due to speckles. Speckle filtering techniques are used to minimize the speckle noise to get usable information [60, 61].

In digital holography (DH), holograms are recorded in digitized form by using CCD or CMOS sensors and complex amplitude of object wave-front from recorded digital hologram is reconstructed by numerical methods using computers [62, 63, 64]. Thus precise quantitative reconstruction/evaluation of amplitude and phase of an object wave-front from single digital hologram is possible in DH using numerical reconstruction methods. In double exposure DHI, two digital holograms, one in the ambient/reference state of object and other in changed state of objects are recorded. The interference phase is directly calculated from two digital holograms by subtracting the numerically reconstructed phase of reference hologram from the phase of object hologram without performing any phase-shifting interferometry. Thus no extra efforts are required in DHI for the reconstruction of interference phase. Also in DH, the data acquisition rate is also high because by using CCD/CMOS sensor, signals are available at video frequencies. The data acquisition rate in DH is only limited by the frame rate of recording CCD/CMOS sensor. The reconstruction rate is also high (∼1 Hz) and can be controlled by using fast computers and efficient reconstruction algorithms [65]. Hence, the processes of measurement become faster, reliable, and even almost real-time. The sensitivity of photographic plate is ∼10⁵ photons/μm², whereas the sensitivity of CCD can be ∼1 photon/μm² [66]. This leads to a large reduction in exposure time as well as simpler requirements on experimental set-up stability against external vibration and perturbations/disturbances. These parameters make the system simple, reliable, robust, more precise, accurate, and more suitable for industrial environment. Whereas, in conventional HI the time-consuming and tedious development process of recording materials makes the use of HI difficult in industrial environment.

In the past, DHI has been extensively used for many applications such as measurement of temperature [49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 67, 68, 69], heat transfer from surface of heated objects [52, 54], diffusion coefficient [70, 71], flow field analysis [72], observation of decomposition of crude oil under dispersant fluid [73], refractive index of liquids [74], heat conduction in glass [75], solution concentration variation in protein crystallization process [76], visualization of thermal gradients in bulb [77], and human skin temperature [78].

In this chapter, DHI in lenless Fourier transform (LLFT) configuration is presented for the investigation of temperature distribution and heat transfer parameters such as natural convective heat transfer coefficient and local heat flux around the surface of industrial heated objects such as cylindrical wires and heat sinks.

2. Theory/methodology

In DH, numerical methods are applied for hologram reconstruction in which diffraction of digital reference wave at the micro interference fringes of digital hologram is simulated using scalar diffraction theory. Several numerical reconstruction methods of hologram such as Fresnel reconstruction [62, 63, 64], convolution [79], and angular reconstruction method [80] have been developed. But, these reconstruction methods suffer from the following shortcomings (1) lateral resolution of the numerically reconstructed images is limited by the inherent coherent speckle noise. For getting high lateral resolution in reconstructed image, the speckle size should be minimized. (2) Due to low spatial resolution (∼125 lines pair per mm for CCD with pixel size 4 μm) of CCD/CMOS sensor, the object needs to be placed at a minimum distance from the sensor to ensure that the angle between the object beam and reference beam does not exceed a certain value at all points of the surface of sensor so that sampling theorem is not violated which requires that fringe spacing of interference pattern in hologram is larger than double the pixel size [64]. (3) Most of holographic set-ups use plane reference waves for hologram reconstruction, and hence the angle between object and the reference beam varies over entire sensor chip and so does the spatial frequency. Thus, the full spatial bandwidth of sensor is not utilized in these set-ups. As the lateral resolution in the reconstruction relies on the complete evaluation of information captured by sensor, so full spatial bandwidth of the sensor should be utilized. Also, in Fresnel and convolution methods, several Fourier transforms and complex multiplications have to be performed. Therefore, to enhance the speed of digital image processing, computation of image reconstruction algorithm should be simpler and fast. All these problems as mentioned above are tackled by lensless Fourier transform digital holography (LLFTDH). Improved lateral resolution and use of the full spatial bandwidth of the CCD sensor can be achieved through LLFTDH. In LLFTDH, set-up, a spherical reference wave is used instead of plane reference wave and the point source of spherical wave is kept in the plane of object. This ensures that the angle between the interfering reference and object beams remains approximately same over the entire sensor chip. Thus, one can ensure that the sampling theorem is not violated over the full area of recording sensor and complete spatial-frequency spectrum of sensor is used at all point of sensor surface area. The entire spatial bandwidth of sensor can be utilized for complete evaluation of all the information one get from the sensor, if the ratio of object size and the distance between the object and the CCD target is optimized [3, 64]. Fresnel reconstruction method is used to derive the reconstruction algorithm of LLFTDH. The digitized form of complex amplitude of object wavefront is expressed as [63].

$$\begin{array}{c}O\left(m\mathrm{\Delta }{X}_I,n\mathrm{\Delta }{Y}_I\right)=\frac{i}{\lambda d}exp\left(-i\frac{2\pi }{\lambda }d\right)exp\left[-i\pi \lambda d\left(\frac{m^2}{M^2\mathrm{\Delta }{X}^2}+\frac{n^2}{N^2\mathrm{\Delta }{Y}^2}\right)\right]\\ \times \mathit{\text{IFFT}}\left\{E_R\left(p,q\right)H\left(p,q\right)exp\left[-i\frac{\pi }{\lambda d}\left(p^2\mathrm{\Delta }{X}^2+q^2\mathrm{\Delta }{Y}^2\right)\right]\right\}.\end{array}$$E1

In Eq. (1), M × N is the total number of pixels on the CCD/CMOS sensor, $m,p=0,1,2,3,..\dots \dots \dots M-1$ and $n,q=0,1,2,3,..\dots \dots \dots N-1$, and ∆X, ∆Y are pixel size on the sensor chip in X and Y directions, respectively. “λ” is the wavelength of laser light, “d” is the distance between object and recording plane (imaging sensor), $E_R\left(p,q\right)$ is the digitized reference wave, and $H\left(p,q\right)$ is the digital hologram. $\mathrm{\Delta }{X}_I$ and $\mathrm{\Delta }{Y}_I$ are the pixel sizes in the reconstructed image. Pixel sizes in image plane are connected with the pixel sizes of the recording/hologram plane as [81, 82].

$$\mathrm{\Delta }{X}_I=\frac{\lambda d}{M\mathrm{\Delta }X};\mathrm{\Delta }{Y}_I=\frac{\lambda d}{N\mathrm{\Delta }Y}.$$E2

In LLFT configuration of DH, the object and point source of spherical wave are kept in same plane. Thus in this configuration, the spherical phase factor associated with the Fresnel diffraction of the transmitted wave through hologram is eliminated by using of a spherical reference wave $E_R\left(p,q\right)$ with the same average curvature that was used during the recording [63].

$$E_R\left(p,q\right)=\left(\text{const}.\right)exp\left[i\frac{\pi }{\mathit{\lambda d}}\left(p^2\mathrm{\Delta }{X}^2+q^2\mathrm{\Delta }{Y}^2\right)\right].$$E3

By substituting $E_R\left(p,q\right)$ in Eq. (1), a simpler algorithm of complex amplitude of object wavefront reconstruction in LLFTDH is obtained and expressed as [63].

$$O\left(m\mathrm{\Delta }{X}_I,n\mathrm{\Delta }{Y}_I\right)=\frac{i}{\lambda d}exp\left(-i\frac{2\pi }{\lambda }d\right)exp\left[-i\pi \lambda d\left(\frac{m^2}{M^2\mathrm{\Delta }{X}^2}+\frac{n^2}{N^2\mathrm{\Delta }{Y}^2}\right)\right]\times \mathit{\text{IFFT}}\left\{H\left(p,q\right)\right\}.$$E4

In Eq. (4), inverse fast Fourier transform (IFFT) of only single term, that is, recorded digital hologram is evaluated apart from some multiplicative constant. Also an autofocus image is obtained without tuning the distance “d”. Thus, this method is simpler and faster as compared to other reconstruction methods such as Fresnel reconstruction and convolution method, in which combination of several Fourier transform and complex multiplications need to be evaluated. The simpler and faster reconstruction algorithm of LLFTDH enhances the possibility of real-time applications [3]. The spatial phase distribution from reconstructed object wavefront is evaluated as [3].

$$\phi \left(m\mathrm{\Delta }{X}_I,n\mathrm{\Delta }{Y}_I\right)=arctan\frac{Im\left[O\left(m\mathrm{\Delta }{X}_I,n\mathrm{\Delta }{Y}_I\right)\right]}{Re\left[O\left(m\mathrm{\Delta }{X}_I,n\mathrm{\Delta }{Y}_I\right)\right]}.$$E5

In the above equation, the operators “Re” and “Im” provide the real and imaginary part of complex functions, respectively. In double exposure DHI, two digital holograms, one in Ref. or ambient state of object, and other in changed state of object are recorded. Phases of two different states of the object wavefronts are calculated using Eq. (5). Let ${\phi }_1\left(m,n\right)$ be the phase of object wavefront in ambient state and ${\phi }_2\left(m,n\right)$ be the phase of changed state of object. The interference phase, which is the phase difference between ambient and changed state of object, is calculated by modulo 2π subtraction as [83].

$$\mathrm{\Delta }\phi \left(m\mathrm{\Delta }{X}_I,n\mathrm{\Delta }{Y}_I\right)=\left\{\begin{array}{l}{\phi }_1\left(m\mathrm{\Delta }{X}_I,n\mathrm{\Delta }{Y}_I\right)-{\phi }_2\left(m\mathrm{\Delta }{X}_I,n\mathrm{\Delta }{Y}_I\right)\mathit{\text{if}}{\phi }_1\left(m\mathrm{\Delta }{X}_I,n\mathrm{\Delta }{Y}_I\right)\ge {\phi }_2\left(m\mathrm{\Delta }{X}_I,n\mathrm{\Delta }{Y}_I\right),\\ {\phi }_1\left(m\mathrm{\Delta }X{}_I,n\mathrm{\Delta }{Y}_I\right)-{\phi }_2\left(m\mathrm{\Delta }{X}_I,n\mathrm{\Delta }{Y}_I\right)+2\pi \mathit{\text{if}}{\phi }_1\left(m\mathrm{\Delta }{X}_I,n\mathrm{\Delta }{Y}_I\right)<{\phi }_2\left(m\mathrm{\Delta }{X}_I,n\mathrm{\Delta }{Y}_I\right).\end{array}\right.$$E6

From this equation, one can directly evaluate the interference phase from the digital holograms. This phase is wrapped between 0 and 2π. A continuous phase difference is obtained by applying appropriate phase unwrapping algorithm [84]. The physical parameters of interest like deformation, refractive index, temperature, etc. are calculated from the unwrapped phase difference.

3. Applications of LLFTDHI in heat transfer measurements

3.1 Measurement of heat transfer parameters along the surface of heated wire

Measurement of natural convective heat transfer from heated wires/cylinders is used in several scientific and industrial applications such as quality control of electrical wires, to find ideal insulation properties of current carrying conductors, vertical tubes of HVAC (Heating, Ventilation and Air Conditioning), in resistive heating of electronic components, space shuttle launch pads, waste nuclear rods stored in repositories, refrigerating coils, and hot radiators [3, 57, 85]. Surface temperature and temperature gradient normal to the surface of the heated object is needed for the calculation of heat transfer parameters such as local convective heat transfer coefficient “h_c” and heat flux “Q(y)”. DHI is very suitable for the measurement of the heat transfer parameters. Phase difference of the order of λ/30 (0.209 radians) could be measured using DHI in our lab. Hence, the measurement accuracy of local “h_c” and local Q(y), which essentially depend on the phase difference measurement from DHI system is expected to be high [3].

In this section, an application of LLFTDHI is presented for the measurement of the natural local (h_c) and Q(y) along the surface of electrically heated tungsten wire of different diameters and different heating conditions. The schematic of the LLFTDHI experimental set-up used for the measurement of heat transfer parameters is shown in Figure 1(a) and photograph of spring loaded mount is shown in Figure 1(b). Photograph of experimental set-up while conducting the experiment is shown in Figure 1(c). In the experiment, a 30 mW He-Ne laser (Make - Spectra Physics, λ = 632.8 nm) is used as a light source. Beam splitter divides laser beam into object and reference beams. Object beam illuminates the air around the wire placed in vertical position as can be seen in Figure 1(c). The interference pattern formed by the superposition of spherical reference beam and object beam was captured by CCD camera (Lumenera’s Infinity3-1 M). First, a digital hologram of ambient air is recorded without heating the wire as a reference hologram. A known amount of current and voltage is applied across the wire using a variable dc power supply and second, digital hologram is recorded in steady state convective flow of air around the wire.

Figure 1.

(a) Schematic of experimental set-up for the measurement of temperature and heat transfer parameters, (b) photograph of spring-loaded mount used to clamp the wire in vertical position, and (c) photograph of experimental set-up [3, 58].

Phase difference maps of heated air and ambient air around the wire were calculated from the two digital holograms by using Eq. (6), and unwrapped phase difference map was calculated using the Goldstein phase unwrapping method [84]. The phase change $\mathrm{\Delta }\phi \left(X\right)$ along a line at distance X from axis of wire is given as [52].

$$\mathrm{\Delta }\phi \left(X\right)={\phi }_2\left(X\right)-{\phi }_1\left(X\right)=\frac{2\pi }{\lambda }\underset{0}{\overset{L}{\int }}\left[n_r\left(r\right)-n_0\left(r\right)\right]\mathit{dZ}$$E7

Here r is the radial distance, z is the direction of propagation of light, λ is the wavelength of light used in the experiment, L is the distance traveled by light in heated air, and n_r(r) and n₀(r) arerefractive indices of heated air and ambient air, respectively (see Figure 2).

Figure 2.

Top view of the cross-section of an axisymmetric refractive index distribution [3].

Experiment was conducted in a closed room with stable air, refractive index of heated air around wire can be considered cylindrically symmetrical. For the axisymmetric objects, the evaluated phase difference map can be transformed into a radial distribution of refractive index difference distribution by using inverse Abel transforms as [52].

$$\mathrm{\Delta }n\left(r\right)=\frac{-\lambda }{2{\pi }^2}{\int }_r^R\frac{d\left(\mathrm{\Delta }\phi \right)/\mathit{dX}}{{\left(X^2-r^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\mathit{dX}$$E8

where R is the radial distance up to which hot air is present, outside the R, refractive index is “n_o” and Y is along the axis of symmetry (see Figure 2). After calculating refractive index difference data, temperature distribution can be obtained by using Lorentz–Lorenz formula [39, 52, 86].

$$T=T_0{\left[\left(\frac{n-n_0}{n_0}\right)\left(\frac{3\mathit{PA}+2{\mathit{RT}}_0}{3\mathit{PA}}\right)+1\right]}^{-1},$$E9

where T_o is the room temperature and n₀ is the refractive index of ambient air in the laboratory. P is the atmospheric pressure and A is the molar refractivity of air. R is the universal gas constant. The local natural “h_c” and local Q(y) can be calculated as [52, 54].

$$h_c=-\frac{K_s}{\left(T_s-T_0\right)}{\left(\frac{\partial T}{\partial r}\right)}_s;Q\left(y\right)=-K_s{\left(\frac{\partial T}{\partial r}\right)}_s$$E10

where $T_s$, and $T_0$ are surface temperature of the heated object (wire/heat sink), and ambient temperature, respectively. $K_s$ is the thermal conductivity of air at the surface temperature of heated wire, and ${\left(\frac{\partial T}{\partial r}\right)}_s$ is the derivative of temperature normal to surface of wire.

Initially, an experiment was conducted on three different heating conditions of tungsten wire of diameter 0.4 mm, when applied power across the wire were (I) P₁ = 70 Watt, (II) P₂ = 46.8 Watt, and (III) P₃ = 37 Watt. Different reconstruction steps involved in the temperature profile calculation from the wrapped phase difference map in case when P = 46.8 W, are shown in Figure 3. From Figure 3(d), experimentally calculated surface temperature of wire is found to be 477 K. The surface temperature of wire was also verified with the temperature measured by thermocouple. The thermocouple readings were also in the range 475–479 K (Avg. Temperature 477 K ± 2 K). Temperature profile along the lines AB, CD, EF and GH shown in Figure 3(d) are used to calculate the natural local “h_c”.

Figure 3.

(a) Wrapped phase difference map of hot air and ambient air when applied power P = 46.8 W; (b) 3D unwrapped phase difference map corresponding to Figure 3(a); (c) refractive index difference distribution along the lines AB, CD, EF, and GH; and (d) temperature profile of hot air along the AB, CD, EF, and GH.

Experimental results corresponding to power P = 70 W and P = 37 W are shown in Figure 4. Experimentally calculated values of “h_c” along the surface of wire at four different heights for three different heating conditions are compared in Table 1. The results indicate that “h_c” decreases as height increases from bottom to top of wire. The reason for the increasing value of “h_c” is the broadening of isothermal line around the wire. Experimentally measured values are within range as given in [14]. Results indicate that measured values of h_c increases as the power applied across of wire of equal length and diameter increases.

Figure 4.

Wrapped phase difference map of heated air and ambient air when applied power across the wire is (a₁) P = 37 W, (a₂) P = 70 W; (b₁–b₂) temperature profiles corresponding to (a₁) and (a₂) at four different heights of wire.

Height	Applied input power = 70 W, Tw = 517 K, wire diameter d1 = 0.4 mm	Applied input power = 46.8 W, Tw = 477 K, wire diameter d2 = 0.4 mm	Applied input power = 37 W, Tw = 457 K, wire diameter d3 = 0.4 mm
	hc (W/m^2 K)	hc(W/m^2 K)	hc (W/m^2 K)
Along AB	13.94	11.55	10.87
Along CD	13.30	10.50	9.56
Along EF	12.20	8.93	8.41
Along GH	10.84	8.60	6.92

Table 1.

Experimentally calculated values of local natural convective heat transfer coefficient for three heating conditions of wire.

A comparison of temperature measured by DHI and thermocouple is shown in Figure 5, and a good agreement is observed between the two measurements. Deviations in the temperature measured by DHI and thermocouple are found to be within 0.82–5.2%. The value of “h_c” measured with thermocouple is 13.9 W/(m² K), while with DHI is 13.30 W/(m² K). The difference between the two values is 4.31%.

Figure 5.

Comparison between the temperature profile measured by DHI and thermocouple [3].

Further, an experiment was conducted on wires of different diameters (d₁ = 0.4 mm, d₂ = 0.6 mm, d₃ = 0.78 mm) of equal length with a constant power (70 W) applied across the wire. Results are shown in Figure 6. A comparison of experimentally calculated values of local “h_c” and Q(y) using DHI for wires of three different diameters are shown in Table 2. Experimentally measured values are within range as given in literature [14]. Results indicate that measured values of local natural “h_c” and Q(y) decreases as the diameter of wire increases. The reason for this decreasing trend of “h_c” and Q(y) is that when same electrical power is applied across the wire of different diameters of same length, surface temperature of wire decreases as diameter of wire increases. This happens because of increasing surface area of wire. The thermal conductivity “K_s” of air depends on temperature and it decreases as surface temperature of wire (T_s) decreases. The temperature gradient also decreases because of its dependence on the temperature difference (T_s-T_o). Both “h_c” as well as “Q(y)” depend on the value of temperature gradient normal to surface of wire, and K_s. Hence, local natural “h_c”, and Q(y) decreases. Table 2 also shows a decreasing trend in the values of local “h_c”, and Q(y) with the increase in height from the bottom of wire.

Figure 6.

(a₁–a₃) Wrapped phase difference map of heated air and ambient air around the wire for different diameters (d₁ = 0.4 mm, d₂ = 0.6 mm, d₃ = 0.78 mm) of wire with equal power applied across the wire of equal length; (b₁–b₃) temperature profiles corresponding to (a₁–a₃) at four different heights of wire.

Height	Applied input power = 70 W, Tw = 517 K, wire diameter d1 = 0.4 mm		Applied input power = 70 W, Tw = 491 K, wire diameter d2 = 0.6 mm		Applied input power = 70 W, Tw = 465 K, wire diameter d3 = 0.78 mm
	hc (W/m^2 K)	Q (W/m^2)	hc (W/m^2 K)	Q (W/m^2)	hc (W/m^2 K)	Q (W/m^2)
Along AB	13.94	3136.5	12.38	2463.62	10.28	1778.44
Along CD	13.3	2992.5	10.78	2145	8.80	1522
Along EF	12.2	2745	10.12	2013	8.37	1448
Along GH	10.84	2439	9.37	1864	7.72	1335.56

Table 2.

Calculated values of convective heat transfer coefficient and local heat flux using digital holographic interferometry (DHI) for wires of three different diameters.

3.2 Heat Flow performances of heat sinks

Electronics circuits, components/devices are highly demanded in every aspect of modern world ranging from toys, appliances to high performance computers. In recent, with increasing demand in processing, manufacturer of electronic circuits/equipment are forced to increase the performance and functionality of electronics components/ICs chips in addition to reducing their sizes. In high performance electronic components /ICs/computers, rate of heat generation is comparatively high. This elevated heat generation from high performance electronic components/devices produces high temperature level in the electronics components/devices, which may lead to their failure due to overheating. Operating temperature of high performance components/devices/computers plays a vital role in maintaining trouble free functioning. So, there is requirement to keep these electronic devices in reasonable temperature range suitable for their functioning by enhancing the transfer of generated heat. The heat sink as a source of heat transfer plays an important role in effective thermal management strategies, which makes the optimization of heat sink design a useful area of investigation. Heat sink are used in effective thermal management for increased reliability and proper functioning of such devices, while micro-channel heat sinks are used in miniaturized heat transport systems like compact heat exchangers, fuel cell powered mechanisms, components of advanced propulsion systems, micro-electromechanical systems (MEMS) applications, and cooling system for fusion reactor, rocket nozzles, avionics, hybrid vehicle power electronics and systems, etc. [3, 54, 56, 85]. The temperature distribution surrounding the heat sink must be measured for investigating the heat flow performance of heat sinks.

This section presents the investigations about heat flow performance from plate fin heat sink using LLFTDHI. Visual inspection of reconstructed phase difference maps of heated air and ambient air around the heat sink provide qualitative information about the variation of temperature and heat dissipation process. Quantitative information of temperature distribution is obtained from the relationship between the digitally reconstructed phase difference map of ambient air and heated air. The effect of fin spacing on the heat flow performance of heat sink is experimentally studied in case of natural heat convection. From experimental data, heat transfer parameters such as local heat flux and convective heat transfer coefficients are also calculated. Same LLFTDHI set-up is used in this application. Figure 7 shows a photograph of DHI set-up used to study the heat flow performance of plate fin heat sink. In experiment, aluminum heat sink is attached with a load resistor in its bottom to heat the heat sink. A known value of voltage and current (5 V and 2.4 A) is applied across the load resistor to heat the heat sink.

Figure 7.

Photograph of experimental setup, and heat sink with its dimensional representations.

During the heating process, several digital holograms are recorded at different time interval. Reconstructed wrapped phase difference maps of hot and ambient air surrounding the heat sink at different times are shown in Figure 8(a)–8(h). Wrapped phase difference maps at time 75 and 90 min. are same. It indicates that that after 75 minute temperature surrounding the heat sink has reached in steady state. It is assumed that the refractive index of air along the direction of propagation of light ray is uniform, then the relationship between the unwrapped phase difference data $\mathrm{\Delta }\phi \left(x,y\right)$ and refractive index change $\mathrm{\Delta }n\left(x,y\right)$ can be expressed as [54].

Figure 8.

(a)–(h) Wrapped phase difference maps of air surrounding the heat sink at different times during heating process, (i) temperature profile along line AB at different times in vertical direction corresponding to Figure 8(a)–(g).

$$\mathrm{\Delta }\phi \left(x,y\right)=\frac{2\pi }{\lambda }L\mathrm{\Delta }n\left(x,y\right)=\frac{2\pi }{\lambda }L\left[n\left(x,y\right)-n_o\right]$$E11

where “L” is the geometrical path length that the laser light travels in temperature field, “λ” is the wavelength of laser light, and n₀ is the refractive index of ambient air. Refractive index of hot air $n\left(x,y\right)$ can be related to temperature distribution T(x,y) using Gladstone-Dale relation and ideal gas equation through a relation $T\left(x,y\right)=\frac{\mathit{uPK}}{R\left[n\left(x,y\right)-1\right]}$ [54]. Here “u” is the molar mass of air, “P” is atmospheric pressure, K is Gladstone–Dale constant, and R is universal gas constant. Substituting this relation in Eq. (11), temperature distribution can be obtained from phase difference map as [54]

$$T\left(x,y\right)={\left[\frac{\mathrm{\Delta }\phi \left(x,y\right)\mathit{\lambda R}}{2\mathit{\text{πLuPK}}}+\frac{1}{T_0}\right]}^{-1}$$E12

In Eq. (12), T₀ is the ambient temperature. Thus, one can calculate the temperature distribution by knowing T_o and phase difference distribution $\mathrm{\Delta }\phi \left(x,y\right)$. In the experiment, ambient temperature T₀ was measured using K type (Chromel–Alumel) thermocouple with multi-logger. Ambient temperature was 305.5 K. Temperature distribution was reconstructed from the phase difference map using Eq. (12). Figure 8(i) shows the reconstructed temperature profiles along the line AB at different times. Same trend was observed in the temperature profiles drawn along the central line AB at different times. Temperature gradient was observed to be maximum in steady state. It reveals that heat dissipation capability of heat sink is maximum in the steady state.

Further an experiment was conducted to study the effect of channel width/fin spacing (S) on heat flow performance of the plate fin heat sink. For this purpose, heat sinks with different fin spacing (Number of channels = 6, S₁ = 3.25 mm; Number of channels = 5, S₂ = 5 mm; Number of channels = 3, S₃ = 9 mm) were chosen and all other dimensional parameters of heat sink were kept same. An equal power of P = 16.8 W was applied across the load resistor for all the three heat sinks. In steady state of hot air, experimentally reconstructed phase difference maps of hot air and ambient air for the heat sinks of three different fin spacing are shown in Figure 9(a1)–9(a3). The reconstructed 2D temperature distribution around the heat sink of three channel widths corresponding to Figure 9(a1)–9(a3) are shown in Figure 9(b1)–9(b3). The color variations shown in Figure 9(b1)–9(b3) indicate that the temperature distribution/variation between the adjacent fins. Temperature is observed to be almost constant (i.e., almost negligible temperature gradient) in the case of heat sink of smaller fin spacing (S₁ = 3.25 mm), and temperature gradient is the highest in case of the heat sink of largest channel width (S₃ = 9 mm). It indicates that heat is accumulated between the adjacent fins of a heat sink with smaller fin spacing and it is more difficult to diffuse the heat than that of heat sinks with larger fin spacing in the case of natural convection.

Figure 9.

(a₁–a₃) Reconstructed wrapped phase difference map of air field when S₁ = 3.25 mm, S₂ = 5 mm, and S₃ = 9 mm, respectively; (b₁–b₃) reconstructed temperature distribution corresponding to (a₁–a₃); (c) temperature profiles along the line AB corresponding to (a₁–a₃), respectively.

Temperature profiles along the line AB for the heat sinks of different fin spacing are shown in Figure 9(c). The temperature gradient along AB is observed to be highest for heat sink of wider fin spacing/channel width S₃ = 9 mm than that of the heat sinks of lower channel widths (i.e., S₁ = 3.25 mm, and S₂ = 5 mm). This is the reason why natural local convective “Q” and “h_c” increases with increase in fin spacing of heat sink.

Table 3 shows a comparison of experimentally calculated values of local convective “Q” and “h_c” for the heat sinks of three different channel widths. From the Table 3, it can be observed that heat sink with wider channel width S₃ = 9 mm has higher value of local convective “Q” and “h_c” than the heat sinks of lower channel widths (i.e., S₂ = 5 mm and S₁ = 3.25 mm). Therefore, for better heat dissipation, there should be an optimization between the no. of channels and fin spacing so that heat does not accumulate inside the fins.

Fin spacing or channel width S (mm)	Dimensions of heat sink; Length L = 50 mm, Width W = 32 mm, Height H = 15 mm, base thickness h = 3.5 mm, fin thickness b = 1.5 mm; Room temperature = 32.5°C, applied power across load resistor = 16.8 W
	Calculated base temperature Tb (K)	Thermocouple value of Tb (K)	Local convective heat flux (w/m^2) $Q\left(x\right)=-K_s{\left(\frac{\mathit{dT}}{\mathit{dy}}\right)}_s$	Experimentally calculated hc W/(m^2 K)
S1 = 3.25	402	404	182.9	1.89
S2 = 5	397.2	400	243.6	2.64
S3 = 9	396.3	399	305	3.34

Table 3.

Experimentally calculated values of natural local heat flux and local convective heat transfer coefficient for heat sinks with different channel widths [54].

4. Conclusions

In conclusion, this chapter demonstrates LLFTDHI technique for the investigation of temperature distribution, and measurement of heat transfer parameters around the heated industrial objects such as wires/conductors and heat sinks. The LLFT configuration of DHI offers following advantages (1) due to a simpler hologram reconstruction algorithm, it provides faster reconstruction than other counterpart methods such as Fresnel and Convolution approach, (2) it can be implement almost in real time, and (3) LLFTDH provides improved lateral resolution in the reconstructed image because entire spatial bandwidth can be utilized for complete evaluation of information from sensor by optimizing the ratio of object size and distance between object and sensor. The LLFTDHI system is investigated for the following scientific and industrial applications (1) Measurement of temperature profile, natural local convective heat transfer coefficient and heat flux along the surface of electrically heated wire. These investigations are helpful to study the heat transfer characteristics of wires/conductors, and finding ideal insulation properties of wires/conductors. (2) The heat dissipation performances of heat sinks. Heat dissipation capabilities of heat sinks are determined by qualitative and quantitative estimations of temperature distribution and heat transfer parameters. These investigations can be helpful in making the proper choice of the heat sink in the electronic circuit, and are also required in making electronic circuits of high reliability for critical industrial as well as for space application. Based on investigated applications, it can be concluded that LLFTDHI provides a feasible and effective method for temperature measurements and heat transfer studies.