**1. Introduction**

Heat flux is an important parameter to characterize heat transfer performance in many industrial applications, such as thermal protection of space shuttles [1], thermal management of electronic devices [2], metal heat treatment [3], maintenance of boilers [4] and nuclear reactors [5], spray cooling [6], geophysics [7], etc. Heat flux is often estimated by surface or internal temperature, which is also termed as inverse heat conduction problem (IHCP).

IHCPs are mathematically ill-posed, and a small error in temperature may significantly affect the accuracy of heat flux estimation [8, 9]. Several analytical and numerical methods have been proposed for the solution of IHCPs, such as

sequential function specification (SFS) method, Tikhonov regularization (TR) method, transfer function (TF) method, Duhamel's theory, etc. The SFS method is commonly used to solve IHCPs by minimizing the effect of random errors using temperature data at future time steps based on the least square method [10]. The TR method estimates all of the heat flux simultaneously for all time steps and is usually presented as whole time domain form, which often causes heavy computational load [11, 12]. The TF method analogizes the heat conduction problems to dynamic systems, in which heat flux is treated as the input of the system and the temperature profile as the response [13]. This method is simple in concept and one of the most accurate ways of estimating surface heat flux. However, it is relatively difficult to determine the analytic solution of the transfer function for the complex geometry problem. Duhamel's theorem is based on the principle of superposition and assumes that the substrate thermal response at *t* equals the total sum of what the substrate experienced in small steps prior to *t* [14]. This method is simple and widely used in the surface heat flux estimation with known surface temperature. However, its assumption that the internal temperature equals to the surface one for indirect temperature measurement often causes significant calculation errors. Thus, Duhamel's theory needs to be improved.

consists of three layers, namely, aluminum, thermal paste, and epoxy resin. Moreover, the radial and temporal surface temperature variations during CSC result in significant nonuniformity of the surface heat flux [14, 32]. Therefore, lateral heat transfer must

In this chapter, the SFS, TF, and Duhamel's theorem methods for TFTC and FTC

The SFS method is widely used to solve IHCPs by minimizing the error between the measured temperature *Yk* and estimated temperature *Tk* for the current time

*Yk*þ*r*�<sup>1</sup> � *Ti, <sup>k</sup>*þ*r*�<sup>1</sup>

Several functional forms of *q*(*t*) from *tk* to *tk* <sup>+</sup> *<sup>r</sup>*�<sup>1</sup> have been proposed. The

Then, the temperature distribution is represented as a function of surface heat flux *qk*, and the temperature field is expanded in a Taylor series about a known

*Zi, <sup>k</sup>* � *<sup>∂</sup>Ti, <sup>k</sup> qk*

The solution for the estimated surface heat flux at time *tk* can be obtained by

P*<sup>R</sup> <sup>i</sup>*¼<sup>1</sup> *<sup>X</sup><sup>k</sup> i,r*�<sup>1</sup>

The values of *r* are selected based on the residual principle [33], which are Eqs. (3) and (4) for the direct and indirect temperature measurement methods,

� � <sup>þ</sup> *Zi, <sup>k</sup> qk* � *<sup>q</sup>*^*<sup>k</sup>*

� �

*∂qk*

*<sup>i</sup>*¼<sup>1</sup> *Yk*þ*r*�<sup>1</sup> � *Tk*þ*r*�<sup>1</sup> *<sup>q</sup>*^*<sup>k</sup>* � � � � *Z<sup>k</sup>*

� �<sup>2</sup> (1)

*qk* <sup>¼</sup> *qk*þ<sup>1</sup> <sup>¼</sup> <sup>⋯</sup> <sup>¼</sup> *qk*þ*<sup>r</sup>* (2)

� � <sup>þ</sup> <sup>⋯</sup> (3)

*i,r*�<sup>1</sup>

� �<sup>2</sup> (5)

(4)

be considered. For generality, 2D multilayer IHCPs need to be developed.

*The Fundamental and Application of Surface Heat Flux Estimation by Inverse Method…*

**2. 1D algorithms for heat flux estimation**

*DOI: http://dx.doi.org/10.5772/intechopen.89095*

**2.1 Sequential function specification method**

simplest one is that *q*(*t*) is a constant

value of surface heat flux *q*^*<sup>k</sup>* as

minimizing Eq. (1) with respect to *qk*:

**127**

*Ti, <sup>k</sup> qk*

*qk* ¼ *q*^*<sup>k</sup>* þ

where *Zi,k* is the sensitivity coefficient defined by

and *r* future time steps based on the least square method [10]

*SM* <sup>¼</sup> <sup>X</sup> *R*

*i*¼1

� � <sup>¼</sup> *Ti, <sup>k</sup> <sup>q</sup>*^*<sup>k</sup>*

P*<sup>R</sup>*

measurements were compared based on one hypothetical heat flux. Duhamel's theorem was improved to increase the accuracy of surface heat flux when using the indirect three-layer FTC measurement. Afterwards, the 2D filter solution was proposed to calculate the surface heat flux for 2D multilayer mediums. Six hypothetical triangular pulse heat fluxes were used to examine the accuracy and sensitivity of the algorithm. Finally, the 2D filter solution was employed to calculate the surface heat flux to investigate the cooling performance. Using estimated heat flux as evaluation index, the possibility of substituting the commercial cryogen R134a with high GWP (1430) by environment friendly cryogen with low GWP (<1) was discussed.

Recently, a filter solution based on TR has attracted the interest of many researchers [9, 15–19], which minimizes the sum of the squares of the errors between estimated and measured temperatures and stabilized by Tikhonov regularization. This solution is expressed in a digital filter form, allowing for a real-time heat flux estimation, and has been used for heat flux estimation of directional flame thermometer [20]. This method demonstrates superiority when solving IHCPs with a complex geometry. However, it can only be used to solve 1D single-layer IHCPs. The multidimensional, multilayer IHCP has yet to be solved.

Although IHCPs have been extensively investigated with regard to various other applications, little work has been conducted related to surface heat flux estimation during cryogen spray cooling (CSC) in laser dermatology. Spray cooling is widely applied in metallurgy, electronics, power plant, and laser dermatology for vascular skin lesions [21–27], because of the superiorities of high power density, ultrafast cooling rate, uniformity of heat removal, and low fluid inventory. In the laser treatment of vascular skin lesions (e.g., port wine stain, PWS), CSC can be implemented to prevent unwanted thermal damage of the epidermis induced by high laser absorption of melanin. Different with traditional steady spray cooling, CSC is a highly transient process with several tens of milliseconds to avoid cold injury. The transient surface heat flux is crucial for cooling performance evaluation, which needs accurate heat flux estimation method and rapid temperature measurement with fast response and small lag, as well as damping of algorithm [28].

In transient CSC, two typical temperature measurements of fine thermocouple (FTC) and thin-film thermocouple (TFTC) are widely used to measure internal and surface temperature. Aguilar et al. [29] used the SFS method to estimate surface heat flux by internal temperature measured by a type-T FTC placed underneath a thinlayer aluminum foil, positioned on the top of epoxy resin surface to provide rapid heat transfer and mechanical support. Zhou et al. [28, 30, 31] measured time-dependent surface temperature by a type-T TFTC with thickness of 2 μm directly deposited onto the epoxy resin surface; this measurement accurately captured the temperature variation owing to its ultrafast thermal response (1.2 μs). Then, the surface heat flux was estimated by Duhamel's theorem. However, TFTC cannot be used to measure the metal material temperature due to the electrical conductivity. Moreover, TFTC corrodes and oxidizes easily in high-temperature environments. Therefore, FTC measurement is widely used in many industries owing to its reliability and stability. Different with TFTC measurement with single-layer geometry, FTC measurement

*The Fundamental and Application of Surface Heat Flux Estimation by Inverse Method… DOI: http://dx.doi.org/10.5772/intechopen.89095*

consists of three layers, namely, aluminum, thermal paste, and epoxy resin. Moreover, the radial and temporal surface temperature variations during CSC result in significant nonuniformity of the surface heat flux [14, 32]. Therefore, lateral heat transfer must be considered. For generality, 2D multilayer IHCPs need to be developed.

In this chapter, the SFS, TF, and Duhamel's theorem methods for TFTC and FTC measurements were compared based on one hypothetical heat flux. Duhamel's theorem was improved to increase the accuracy of surface heat flux when using the indirect three-layer FTC measurement. Afterwards, the 2D filter solution was proposed to calculate the surface heat flux for 2D multilayer mediums. Six hypothetical triangular pulse heat fluxes were used to examine the accuracy and sensitivity of the algorithm. Finally, the 2D filter solution was employed to calculate the surface heat flux to investigate the cooling performance. Using estimated heat flux as evaluation index, the possibility of substituting the commercial cryogen R134a with high GWP (1430) by environment friendly cryogen with low GWP (<1) was discussed.

## **2. 1D algorithms for heat flux estimation**

#### **2.1 Sequential function specification method**

The SFS method is widely used to solve IHCPs by minimizing the error between the measured temperature *Yk* and estimated temperature *Tk* for the current time and *r* future time steps based on the least square method [10]

$$\mathbf{S}\_{\mathbf{M}} = \sum\_{i=1}^{R} \left( Y\_{k+r-1} - T\_{i,k+r-1} \right)^{2} \tag{1}$$

Several functional forms of *q*(*t*) from *tk* to *tk* <sup>+</sup> *<sup>r</sup>*�<sup>1</sup> have been proposed. The simplest one is that *q*(*t*) is a constant

$$q\_k = q\_{k+1} = \dots = q\_{k+r} \tag{2}$$

Then, the temperature distribution is represented as a function of surface heat flux *qk*, and the temperature field is expanded in a Taylor series about a known value of surface heat flux *q*^*<sup>k</sup>* as

$$T\_{i\_2,k}(q\_k) = T\_{i\_2,k}(\dot{q}\_k) + Z\_{i,k}(q\_k - \dot{q}\_k) + \cdots \tag{3}$$

where *Zi,k* is the sensitivity coefficient defined by

$$Z\_{i,k} \equiv \frac{\partial T\_{i,k}(q\_k)}{\partial q\_k} \tag{4}$$

The solution for the estimated surface heat flux at time *tk* can be obtained by minimizing Eq. (1) with respect to *qk*:

$$q\_k = \hat{q}\_k + \frac{\sum\_{i=1}^{\mathbb{R}} \left[ Y\_{k+r-1} - T\_{k+r-1}(\hat{q}\_k) \right] Z\_{i,r-1}^k}{\sum\_{i=1}^{\mathbb{R}} \left( X\_{i,r-1}^k \right)^2} \tag{5}$$

The values of *r* are selected based on the residual principle [33], which are Eqs. (3) and (4) for the direct and indirect temperature measurement methods,

sequential function specification (SFS) method, Tikhonov regularization (TR) method, transfer function (TF) method, Duhamel's theory, etc. The SFS method is commonly used to solve IHCPs by minimizing the effect of random errors using temperature data at future time steps based on the least square method [10]. The TR method estimates all of the heat flux simultaneously for all time steps and is usually presented as whole time domain form, which often causes heavy computational load [11, 12]. The TF method analogizes the heat conduction problems to dynamic systems, in which heat flux is treated as the input of the system and the temperature profile as the response [13]. This method is simple in concept and one of the most accurate ways of estimating surface heat flux. However, it is relatively difficult to determine the analytic solution of the transfer function for the complex geometry problem. Duhamel's theorem is based on the principle of superposition and assumes that the substrate thermal response at *t* equals the total sum of what the substrate experienced in small steps prior to *t* [14]. This method is simple and widely used in the surface heat flux estimation with known surface temperature. However, its assumption that the internal temperature equals to the surface one for indirect temperature measurement often causes significant calculation errors. Thus,

Recently, a filter solution based on TR has attracted the interest of many researchers [9, 15–19], which minimizes the sum of the squares of the errors between estimated and measured temperatures and stabilized by Tikhonov regularization. This solution is expressed in a digital filter form, allowing for a real-time heat flux estimation, and has been used for heat flux estimation of directional flame thermometer [20]. This method demonstrates superiority when solving IHCPs with a complex geometry. However, it can only be used to solve 1D single-layer IHCPs.

Although IHCPs have been extensively investigated with regard to various other applications, little work has been conducted related to surface heat flux estimation during cryogen spray cooling (CSC) in laser dermatology. Spray cooling is widely applied in metallurgy, electronics, power plant, and laser dermatology for vascular skin lesions [21–27], because of the superiorities of high power density, ultrafast cooling rate, uniformity of heat removal, and low fluid inventory. In the laser treatment of vascular skin lesions (e.g., port wine stain, PWS), CSC can be implemented to prevent unwanted thermal damage of the epidermis induced by high laser absorption of melanin. Different with traditional steady spray cooling, CSC is a highly transient process with several tens of milliseconds to avoid cold injury. The transient surface heat flux is crucial for cooling performance evaluation, which needs accurate heat flux estimation method and rapid temperature measurement with fast response and small lag, as well as damping of algorithm [28]. In transient CSC, two typical temperature measurements of fine thermocouple (FTC) and thin-film thermocouple (TFTC) are widely used to measure internal and surface temperature. Aguilar et al. [29] used the SFS method to estimate surface heat flux by internal temperature measured by a type-T FTC placed underneath a thinlayer aluminum foil, positioned on the top of epoxy resin surface to provide rapid heat transfer and mechanical support. Zhou et al. [28, 30, 31] measured time-dependent surface temperature by a type-T TFTC with thickness of 2 μm directly deposited onto the epoxy resin surface; this measurement accurately captured the temperature variation owing to its ultrafast thermal response (1.2 μs). Then, the surface heat flux was estimated by Duhamel's theorem. However, TFTC cannot be used to measure the metal material temperature due to the electrical conductivity. Moreover, TFTC corrodes and oxidizes easily in high-temperature environments. Therefore, FTC measurement is widely used in many industries owing to its reliability and stability. Different with TFTC measurement with single-layer geometry, FTC measurement

The multidimensional, multilayer IHCP has yet to be solved.

Duhamel's theory needs to be improved.

*Inverse Heat Conduction and Heat Exchangers*

**126**

respectively. The detailed information about the SFS method can be found in previous publications [10, 34].

#### **2.2 Transfer function method**

The transfer functions establish the relationship between the input and output in a dynamic system, which can also be used to solve the linear heat conduction problems, where the heat flux is treated as the input of the system and the temperature is treated as the response [13].

The estimated surface heat flux using Laplace transform can be described as

$$q(t) = L^{-1} \left[ \frac{1}{H\_c(s)} \right] \* \theta\_c(t) \tag{6}$$

where *φ*(*x*,*t*) is the temperature distribution response unit step function of the substrate and *f*(0) and *T*(*τ*) are the initial and time-varying surface temperature.

> 2 ffiffiffiffi *<sup>α</sup><sup>t</sup>* <sup>p</sup>

> > *dT*

*f* ¼ *f i*ð Þ � Δ*t , i* ¼ 1*,* 2*,* …*, n,* (13)

*f*ð Þ 0 � Δ*t f*ð Þ 1 � Δ*t* ⋮ *f n* ð Þ ð Þ� ‐<sup>1</sup> <sup>Δ</sup>*<sup>t</sup>*

� � (11)

*<sup>d</sup><sup>τ</sup> <sup>d</sup><sup>τ</sup>* (12)

(14)

(15)

*Tc*ð Þ 0 � Δ*t Tc*ð Þ 1 � Δ*t* ⋮ *Tc*ð Þ *n* � Δ*t*

*<sup>φ</sup>*ð Þ¼ *x, t* <sup>1</sup> � *erf <sup>x</sup>*

*The Fundamental and Application of Surface Heat Flux Estimation by Inverse Method…*

Solving Eq. (10), the temperature gradient at the surface can be obtained:

0

The simplification that the internal temperature equals the surface ones ignoring the heat dissipating capacity of the materials above the thermocouple, thereby often causing significant heat flux errors. Therefore, Duhamel's theorem needs to be improved to deal with this problem, rather than directly estimate surface heat flux from the measured temperature data. Firstly, the real surface temperature is needed to be calculated from the internal measured temperature at *x* = *c* location, which can

where *f* is the surface temperature at the time of *i*�Δ*t*. The temperature at *x* = *c*

where Δ*φ*(*i*Δ*t*) represents *φ*(*x*, *i*�Δ*t*)�*φ* (*x*, (*i*�1)�Δ*t*). The surface temperature can be obtained by multiplying the inverse temperature transformation matrix on both sides of the equation from the measured internal temperature data. After-

Two different methods were employed to measure the surface temperature during CSC. **Figure 1** shows a schematic of the thin-film thermocouple (TFTC) and the fine thermocouple (FTC) measurements. The type-T TFTC with a thickness of 2 μm was directly deposited onto the epoxy resin surface using the magnetron technique. It has perfect contact with the underlying substrate and a fast response time (�1 μs). The FTC measurement with a response time of 3.33 ms [36] consists of a fine type-T thermocouple (�10 μm bead diameter) underneath a thin layer of aluminum foil (�10 μm), which is positioned on the surface of the epoxy resin

*Tc*ð Þ¼ *n* � Δ*t f*ð Þ 0 *φ*ð Þþ *x, n* � Δ*t* ½ � *f*ð Þ� 1 � Δ*t f*ð Þ 0 *φ*ð Þ *x, n*ð Þ� � 1 Δ*t* þ ½ � *f*ð Þ� 2 � Δ*t f*ð Þ 1 � Δ*t φ*ð Þ *x, n*ð Þ� � 2 Δ*t* ⋯ þ ½ � *f n*ð Þ� � Δ*t f n* ð Þ ð Þ� � 1 Δ*t φ*ð Þ *x,* 0

1 ffiffiffiffiffiffiffiffiffiffi *<sup>t</sup>* � *<sup>τ</sup>* <sup>p</sup>

ffiffiffiffiffiffiffi *λρc π* r ð*<sup>t</sup>*

The unit step function for a semi-infinite planar solid is [35].

*DOI: http://dx.doi.org/10.5772/intechopen.89095*

*q t*ðÞ¼

be expressed by a piecewise constant function of time as

location can be solved by Duhamel's theorem as [35].

Eq. (14) can be written in expanded matrix form as

wards, the surface heat flux can be estimated using Eq. (12).

Δ*φ*ð Þ Δ*t* 0 ⋯ 0 Δ*φ*ð Þ 2Δ*t* Δ*φ*ð Þ Δ*t* ⋯ 0 ⋮ ⋮⋱ 0 Δ*φ*ð Þ *n*Δ*t* Δ*φ*ð Þ ð Þ *n* � 1 Δ*t* ⋯ Δ*φ*ð Þ Δ*t*

**2.4 Validation of 1D filter solution**

**129**

where *θc*(*t*) is the temperature allowance (*θc*(*t*) = *Tc*(*t*) � *T*0), the subscript *c* denotes the measurement position, the superscript\* is the convolution integral operator, and *T*<sup>0</sup> is the initial temperature. *L*�<sup>1</sup> [1/*H*(*s*)] is the Laplace inverse transform of the transfer function, which can be written as the function of time *t*, as follows:

$$L^{-1}\left[\frac{\mathbf{1}}{H(s)}\right] = f(t) \tag{7}$$

Substituting Eq. (7) into Eq. (6), *q*(*t*) becomes

$$q(t) = \int\_0^t \theta\_\epsilon(t-\tau) f(\tau) d\tau \tag{8}$$

After dispersing Eq. (8), it takes the following form:

$$
\begin{bmatrix} q\_1 \\ q\_2 \\ \vdots \\ q\_n \end{bmatrix} = \begin{bmatrix} \theta\_{\ell,1} & 0 & \cdots & 0 \\ \theta\_{\ell,2} & \theta\_{\ell,1} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ \theta\_{\ell,n} & \theta\_{\ell,n-1} & \cdots & \theta\_{\ell,1} \end{bmatrix} \begin{bmatrix} f\_0 \\ f\_1 \\ \vdots \\ f\_n \end{bmatrix} \tag{9}
$$

An assist question can be established to solve the temperature allowance *θ* assuming the boundary condition *q*(*t*) = 1. Substituting the numerical solution into Eq. (8), the surface heat flux *q*(*t*) can be obtained.

#### **2.3 Duhamel's theorem method**

Duhamel's theorem directly treated the measured internal temperature as the surface temperature of the substrate when using indirect FTC temperature measurement. A one-dimensional, direct heat conduction problem for the two surface temperature measurements can be solved to calculate the temperature distribution in the substrate [14, 35]:

$$T(\mathbf{x},t) = f(\mathbf{0})\rho(\mathbf{x},t) + \int\_0^t \rho(\mathbf{x}, t-\tau) \frac{dT(\tau)}{d\tau} d\tau \tag{10}$$

*The Fundamental and Application of Surface Heat Flux Estimation by Inverse Method… DOI: http://dx.doi.org/10.5772/intechopen.89095*

where *φ*(*x*,*t*) is the temperature distribution response unit step function of the substrate and *f*(0) and *T*(*τ*) are the initial and time-varying surface temperature. The unit step function for a semi-infinite planar solid is [35].

$$\rho(\mathbf{x},t) = \mathbf{1} - \epsilon \eta f\left(\frac{\mathbf{x}}{2\sqrt{at}}\right) \tag{11}$$

Solving Eq. (10), the temperature gradient at the surface can be obtained:

$$q(t) = \sqrt{\frac{\lambda \rho c}{\pi}} \int\_0^t \frac{1}{\sqrt{t - \pi}} \frac{dT}{d\tau} d\tau \tag{12}$$

The simplification that the internal temperature equals the surface ones ignoring the heat dissipating capacity of the materials above the thermocouple, thereby often causing significant heat flux errors. Therefore, Duhamel's theorem needs to be improved to deal with this problem, rather than directly estimate surface heat flux from the measured temperature data. Firstly, the real surface temperature is needed to be calculated from the internal measured temperature at *x* = *c* location, which can be expressed by a piecewise constant function of time as

$$f = f(i \cdot \Delta t), \quad i = 1, 2, \dots, n,\tag{13}$$

where *f* is the surface temperature at the time of *i*�Δ*t*. The temperature at *x* = *c* location can be solved by Duhamel's theorem as [35].

$$\begin{split} T\_{\varepsilon}(\boldsymbol{n}\cdot\Delta t) &= f(\boldsymbol{0})\rho(\mathbf{x},\boldsymbol{n}\cdot\Delta t) + [f(\mathbf{1}\cdot\Delta t) - f(\mathbf{0})]\rho(\mathbf{x},(\boldsymbol{n}-\mathbf{1})\cdot\Delta t) \\ &\quad + [f(\mathbf{2}\cdot\Delta t) - f(\mathbf{1}\cdot\Delta t)]\rho(\mathbf{x},(\boldsymbol{n}-\mathbf{2})\cdot\Delta t) \cdots \\ &\quad + [f(\mathbf{n}\cdot\Delta t) - f((\mathbf{n}-\mathbf{1})\cdot\Delta t)]\rho(\mathbf{x},\mathbf{0}) \end{split} \tag{14}$$

Eq. (14) can be written in expanded matrix form as

$$
\begin{bmatrix}
\Delta\boldsymbol{\rho}(\Delta t) & \mathbf{0} & \cdots & \mathbf{0} \\
\Delta\boldsymbol{\rho}(2\Delta t) & \Delta\boldsymbol{\rho}(\Delta t) & \cdots & \mathbf{0} \\
\vdots & \vdots & \ddots & \mathbf{0} \\
\Delta\boldsymbol{\rho}(n\Delta t) & \Delta\boldsymbol{\rho}((n-1)\Delta t) & \cdots & \Delta\boldsymbol{\rho}(\Delta t)
\end{bmatrix}
\begin{bmatrix}
f(\mathbf{0}\cdot\Delta t) \\
f(\mathbf{1}\cdot\Delta t) \\
\vdots \\
f((n\cdot\mathbf{1})\cdot\Delta t)
\end{bmatrix} = \begin{bmatrix}
T\_{\varepsilon}(\mathbf{0}\cdot\Delta t) \\
T\_{\varepsilon}(\mathbf{1}\cdot\Delta t) \\
\vdots \\
T\_{\varepsilon}(n\cdot\Delta t)
\end{bmatrix} \tag{15}
$$

where Δ*φ*(*i*Δ*t*) represents *φ*(*x*, *i*�Δ*t*)�*φ* (*x*, (*i*�1)�Δ*t*). The surface temperature can be obtained by multiplying the inverse temperature transformation matrix on both sides of the equation from the measured internal temperature data. Afterwards, the surface heat flux can be estimated using Eq. (12).

#### **2.4 Validation of 1D filter solution**

Two different methods were employed to measure the surface temperature during CSC. **Figure 1** shows a schematic of the thin-film thermocouple (TFTC) and the fine thermocouple (FTC) measurements. The type-T TFTC with a thickness of 2 μm was directly deposited onto the epoxy resin surface using the magnetron technique. It has perfect contact with the underlying substrate and a fast response time (�1 μs). The FTC measurement with a response time of 3.33 ms [36] consists of a fine type-T thermocouple (�10 μm bead diameter) underneath a thin layer of aluminum foil (�10 μm), which is positioned on the surface of the epoxy resin

respectively. The detailed information about the SFS method can be found in

a dynamic system, which can also be used to solve the linear heat conduction problems, where the heat flux is treated as the input of the system and the temper-

*q t*ðÞ¼ *<sup>L</sup>*�<sup>1</sup> <sup>1</sup>

The transfer functions establish the relationship between the input and output in

The estimated surface heat flux using Laplace transform can be described as

*Hc*ð Þ*s* � �

where *θc*(*t*) is the temperature allowance (*θc*(*t*) = *Tc*(*t*) � *T*0), the subscript *c* denotes the measurement position, the superscript\* is the convolution integral

form of the transfer function, which can be written as the function of time *t*, as

*<sup>L</sup>*�<sup>1</sup> <sup>1</sup> *H s*ð Þ � �

> ð*t* 0

*θc,* <sup>1</sup> 0 ⋯ 0 *θc,* <sup>2</sup> *θc,* <sup>1</sup> ⋯ 0 ⋮ ⋮ ⋱⋮ *θc, <sup>n</sup> θc, <sup>n</sup>*�<sup>1</sup> ⋯ *θc,* <sup>1</sup>

An assist question can be established to solve the temperature allowance *θ* assuming the boundary condition *q*(*t*) = 1. Substituting the numerical solution into

Duhamel's theorem directly treated the measured internal temperature as the surface temperature of the substrate when using indirect FTC temperature measurement. A one-dimensional, direct heat conduction problem for the two surface temperature measurements can be solved to calculate the temperature distribution

> ð*t* 0

*<sup>φ</sup>*ð Þ *x, t* � *<sup>τ</sup> dT*ð Þ*<sup>τ</sup>*

*<sup>d</sup><sup>τ</sup> <sup>d</sup><sup>τ</sup>* (10)

*q t*ðÞ¼

After dispersing Eq. (8), it takes the following form:

*T x, t* ð Þ¼ *f*ð Þ 0 *φ*ð Þþ *x, t*

∗ *θc*ð Þ*t* (6)

[1/*H*(*s*)] is the Laplace inverse trans-

¼ *f t*ð Þ (7)

*θc*ð Þ *t* � *τ f*ð Þ*τ* d*τ* (8)

*f* 0 *f* 1 ⋮ *f n*

(9)

previous publications [10, 34].

*Inverse Heat Conduction and Heat Exchangers*

**2.2 Transfer function method**

ature is treated as the response [13].

follows:

operator, and *T*<sup>0</sup> is the initial temperature. *L*�<sup>1</sup>

Substituting Eq. (7) into Eq. (6), *q*(*t*) becomes

*q*1 *q*2 ⋮ *qn*

Eq. (8), the surface heat flux *q*(*t*) can be obtained.

**2.3 Duhamel's theorem method**

in the substrate [14, 35]:

**128**

**2.5 1D heat flux estimation**

*DOI: http://dx.doi.org/10.5772/intechopen.89095*

(350 kW/m<sup>2</sup>

**Figure 3.**

**Figure 4.**

**131**

*measurement.*

**Figure 3** shows the variation in surface temperature as a function of time, measured by the direct measurement method with the TFTC and the indirect measurement method with FTC. The temperature first decreased rapidly when the R404A droplets impinge on the substrate surface, following a relatively slow change as the thin liquid film forms on the surface. It began to resume to the ambient temperature after the liquid film completely evaporated. It is notable that the surface temperature measured by the TFTC measurement decreased faster than that by the indirect FTC measurement method, which showed a definite delay. **Figure 4(a)** depicts the time-varied surface heat flux predicted by Duhamel's theorem, SFS method, and transfer function method with TFTC measurement. It can be seen that all the three algorithms predicted very similar results. They first increased rapidly to their maximum values, then dropped quickly to a certain value

*The Fundamental and Application of Surface Heat Flux Estimation by Inverse Method…*

), and finally gradually decreased to zero. Note that the estimated

result by the SFS method increased faster and had a slightly greater maximum heat

**Figure 4(b)** represents the heat flux predicted by different algorithms under the FTC measurement. As mentioned above, Duhamel's theorem was inappropriate for predicting surface heat flux directly from internal measured temperature. Thus, the

flux than that obtained by the other two algorithms.

*Variations of surface temperature measured by TFTC and FTC measurements [28].*

*Variations of surface heat flux predicted by different algorithms [28]. (a) TFTC measurement and (b) FTC*

#### **Figure 1.**

*Schematic of the TFTC and FTC measurements with single temperature sensor [28]: (a) TFTC measurement and (b) FTC measurement.*

#### **Figure 2.**

*Results of the comparison between estimated heat fluxes (scatters) and hypothetical ones (solid lines) [28]. (a) single-layer IHCP with single TFTC and (b) three-layer IHCP with single FTC.*

substrate (5 mm), as shown in **Figure 1(b)**. The thermal paste with thickness of 100 μm is placed between the aluminum foil and the epoxy resin, ensuring good thermal contact and providing mechanical support.

A hypothetical triangular pulse heat flux is commonly used to examine the accuracy and sensitivity to random noises of the algorithms [10, 13]. This given heat flux is first set as the surface boundary condition, and then the temperature adding random noise (*ε* = 1°C) at the measured point corresponding to the two different measurement methods is obtained by solving the direct heat conduction problem. New surface heat flux can be predicted based on the calculated temperature through different algorithms.

**Figure 2** shows the new estimated heat fluxes using different algorithms with random noise using TFTC and FTC measurements. The heat fluxes calculated by Duhamel's theorem, SFS, and the transfer function method all agreed well with the hypothetical ones using the TFTC measurement. When using FTC measurement (**Figure 2(b)**), the estimated heat flux obtained from Duhamel's theorem and SFS methods changed nonlinearly over time. Furthermore, they can hardly match the exact heat flux well. Thus, they were unsuitable for estimating the surface heat flux in the indirect surface temperature measurement with FTC. In comparison, both the transfer function and improved Duhamel's theorem can provide linear and accurate results that match the given heat flux exactly.

*The Fundamental and Application of Surface Heat Flux Estimation by Inverse Method… DOI: http://dx.doi.org/10.5772/intechopen.89095*

### **2.5 1D heat flux estimation**

**Figure 3** shows the variation in surface temperature as a function of time, measured by the direct measurement method with the TFTC and the indirect measurement method with FTC. The temperature first decreased rapidly when the R404A droplets impinge on the substrate surface, following a relatively slow change as the thin liquid film forms on the surface. It began to resume to the ambient temperature after the liquid film completely evaporated. It is notable that the surface temperature measured by the TFTC measurement decreased faster than that by the indirect FTC measurement method, which showed a definite delay.

**Figure 4(a)** depicts the time-varied surface heat flux predicted by Duhamel's theorem, SFS method, and transfer function method with TFTC measurement. It can be seen that all the three algorithms predicted very similar results. They first increased rapidly to their maximum values, then dropped quickly to a certain value (350 kW/m<sup>2</sup> ), and finally gradually decreased to zero. Note that the estimated result by the SFS method increased faster and had a slightly greater maximum heat flux than that obtained by the other two algorithms.

**Figure 4(b)** represents the heat flux predicted by different algorithms under the FTC measurement. As mentioned above, Duhamel's theorem was inappropriate for predicting surface heat flux directly from internal measured temperature. Thus, the

**Figure 4.**

*Variations of surface heat flux predicted by different algorithms [28]. (a) TFTC measurement and (b) FTC measurement.*

substrate (5 mm), as shown in **Figure 1(b)**. The thermal paste with thickness of 100 μm is placed between the aluminum foil and the epoxy resin, ensuring good

*Results of the comparison between estimated heat fluxes (scatters) and hypothetical ones (solid lines) [28].*

*Schematic of the TFTC and FTC measurements with single temperature sensor [28]: (a) TFTC measurement*

A hypothetical triangular pulse heat flux is commonly used to examine the accuracy and sensitivity to random noises of the algorithms [10, 13]. This given heat flux is first set as the surface boundary condition, and then the temperature adding random noise (*ε* = 1°C) at the measured point corresponding to the two different measurement methods is obtained by solving the direct heat conduction problem. New surface heat flux can be predicted based on the calculated temperature

**Figure 2** shows the new estimated heat fluxes using different algorithms with random noise using TFTC and FTC measurements. The heat fluxes calculated by Duhamel's theorem, SFS, and the transfer function method all agreed well with the hypothetical ones using the TFTC measurement. When using FTC measurement (**Figure 2(b)**), the estimated heat flux obtained from Duhamel's theorem and SFS methods changed nonlinearly over time. Furthermore, they can hardly match the exact heat flux well. Thus, they were unsuitable for estimating the surface heat flux in the indirect surface temperature measurement with FTC. In comparison, both the transfer function and improved Duhamel's theorem can provide linear and

thermal contact and providing mechanical support.

*(a) single-layer IHCP with single TFTC and (b) three-layer IHCP with single FTC.*

accurate results that match the given heat flux exactly.

through different algorithms.

**Figure 1.**

**Figure 2.**

**130**

*and (b) FTC measurement.*

*Inverse Heat Conduction and Heat Exchangers*

*q*max and the magnitude of its increase rate by Duhamel's theorem were both far lower than the case with other algorithms. It seems that the heat flux was significantly delayed and reduced. Other surface heat fluxes predicted by the transfer function and newly improved Duhamel's theory had almost the same varying history, with a larger *q*max than that predicted by the SFS method.
