**3. 2D algorithm for heat flux estimation**

Although the TF and improved Duhamel's theorem can accurately estimate the surface heat flux with multilayer mediums, it still failed to solve 2D IHCPs. Moreover, large nonuniformity radial and temporal surface temperature variations during CSC often occurred [14, 32]. Therefore, lateral heat transfer must be considered. The 2D algorithm for 2D multilayer medium is urgent to be developed. In this chapter, the 2D filter solution was proposed and validated based on the three-layer FTC temperature measurement.

#### **3.1 2D filter solution and optimal comparison criterion**

The 2D general model with K heat fluxes, J temperature sensors, and I layers is shown in **Figure 5**. The multiple surface heat fluxes are assumed to be uniform in a specific upper lateral surface area (e.g., 0 ≤ *x* ≤ *x*1). The temperature sensors are placed at the lateral midpoint of each uniform surface heat flux range in the *i*th layer, and the interval between two thermocouples is Δ*x*. All sensors are placed at an identical depth (*y* = *H*T) from the upper surface. The adiabatic boundary condition is considered on the three other sides (*x* = 0, *x* = *W*, and *y* = *H*). The initial temperature of the substrates is *T*0.

The discrete solution of temperature for direct heat conduction problems with K heat fluxes and J sensors can be presented in a matrix form as follows [37].

$$\mathbf{T} = \mathbf{Xq} \tag{16}$$

**X** ¼

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

að Þ¼ *n*

*ϕ* can be presented as follows:

*ajk*ð Þ¼ *n*

*ϕjk*ð Þ¼ *n T x*TC*<sup>j</sup>*

<sup>S</sup> <sup>¼</sup> ð Þ <sup>Y</sup>‐<sup>T</sup> <sup>0</sup>

q^ ¼ X<sup>0</sup>

q^<sup>i</sup> ¼ Xi 0

X þ αtHt

Xi þ αtHt

0

T *\_*

Hs ½ �‐<sup>1</sup>

Notably, **X***<sup>i</sup>* is different from **X** in the solution of single-layer IHCP. **X***<sup>i</sup>* is calculated by using (I � *i* + 1) layers below the *i*th layer. The estimated interface

*∂ T x*TC*<sup>j</sup>*

*, H*T*, nt*<sup>d</sup>

estimated and measured temperatures plus one-order regularization, *S*, is

ð Þþ <sup>Y</sup>‐<sup>T</sup> <sup>α</sup>t½ � Htq <sup>0</sup>

0

<sup>H</sup>*<sup>s</sup>* ½ ��<sup>1</sup>

Ht þ αsHs

Generally, the 2D multilayer IHCPs are solved layer by layer, starting from the *i*th layer with the known experimental measured temperature. The heat flux is estimated at the interface with the (*i*�1)th layer. Thereafter, the interface heat flux between the (*i*�1)th and (*i*�2)th layers can be calculated by using the known interface heat flux as the input. Finally, the surface heat flux is estimated [18]. The interface heat fluxes **q**^*<sup>i</sup>* between the *i*th and (*i*�1)th layers can be described as

Ht þ αsHs

0

0

where

obtained [8]:

temperature **T**^*<sup>i</sup>* yields

**133**

a 1ð Þ 0 0 ⋯ 0 a 2ð Þ a 1ð Þ 0 ⋯ 0 a 3ð Þ a 2ð Þ a 1ð Þ ⋯ ⋮ ⋮ ⋮ ⋮⋱ 0 a Nð Þ a Nð Þ � 1 a Nð Þ � 2 ⋯ a 1ð Þ

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

*a*11ð Þ *n a*12ð Þ *n a*13ð Þ *n* ⋯ *a*1Kð Þ *n a*21ð Þ *n a*22ð Þ *n a*23ð Þ *n* ⋯ *a*2Kð Þ *n a*31ð Þ *n a*32ð Þ *n a*33ð Þ *n* ⋯ *a*3Kð Þ *n* ⋮ ⋮ ⋮⋱⋮ *a*J1ð Þ *n a*J2ð Þ *n a*J3ð Þ *n* ⋯ *a*JKð Þ *n*

*,*WT*, t* � � � *<sup>T</sup>*<sup>0</sup> � �

where N and *n* are total time steps and current time step. Excessive temperature

� � *qk*¼<sup>1</sup>*, <sup>q</sup>*1¼*q*2¼⋯⋯¼*q*K¼<sup>0</sup> �

½ �þ Htq αs½ � Hsq <sup>0</sup>

X0

Xi 0

*<sup>i</sup>* ¼ **X***i*q^*<sup>i</sup>* (24)

where *t*<sup>d</sup> is the time step. The sum of the squares of the errors between the

where **Y** is the measured temperature matrix. *α<sup>t</sup>* and *α<sup>s</sup>* are regularization parameters with respect to temporal and spatial terms. The superscript 'denotes the transpose of a matrix. **Ht** and **Hs** are temporal and spatial regularization matrixes. Minimizing *S*, estimated heat flux matrix **q**^ using the entire domain data can be

*<sup>∂</sup> qk*ð Þ<sup>1</sup> � � <sup>¼</sup> *<sup>φ</sup>jk*ð Þ *<sup>n</sup>* (19)

� � *T*<sup>0</sup> (20)

½ � Hsq (21)

Y ¼ FY (22)

Y ¼ FiY (23)

(17)

(18)

where **T** is the real temperature matrix, **X** is the sensitivity matrix, and **q** denotes multiple heat fluxes. The sensitivity matrix **X** is [19]

**Figure 5.** *Geometry model of a 2D multilayer IHCP [36].*

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

$$\mathbf{X} = \begin{bmatrix} \mathbf{a}(\mathbf{1}) & \mathbf{0} & \mathbf{0} & \cdots & \mathbf{0} \\ \mathbf{a}(\mathbf{2}) & \mathbf{a}(\mathbf{1}) & \mathbf{0} & \cdots & \mathbf{0} \\ \mathbf{a}(\mathbf{3}) & \mathbf{a}(\mathbf{2}) & \mathbf{a}(\mathbf{1}) & \cdots & \vdots \\ \vdots & \vdots & \vdots & \ddots & \mathbf{0} \\ \mathbf{a}(\mathbf{N}) & \mathbf{a}(\mathbf{N}-\mathbf{1}) & \mathbf{a}(\mathbf{N}-\mathbf{2}) & \cdots & \mathbf{a}(\mathbf{1}) \end{bmatrix} \tag{17}$$

where

*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 his-

Although the TF and improved Duhamel's theorem can accurately estimate the surface heat flux with multilayer mediums, it still failed to solve 2D IHCPs. Moreover, large nonuniformity radial and temporal surface temperature variations during CSC often occurred [14, 32]. Therefore, lateral heat transfer must be considered. The 2D algorithm for 2D multilayer medium is urgent to be developed. In this chapter, the 2D filter solution was proposed and validated based on the three-layer

The 2D general model with K heat fluxes, J temperature sensors, and I layers is shown in **Figure 5**. The multiple surface heat fluxes are assumed to be uniform in a specific upper lateral surface area (e.g., 0 ≤ *x* ≤ *x*1). The temperature sensors are placed at the lateral midpoint of each uniform surface heat flux range in the *i*th layer, and the interval between two thermocouples is Δ*x*. All sensors are placed at an identical depth (*y* = *H*T) from the upper surface. The adiabatic boundary condition is considered on the three other sides (*x* = 0, *x* = *W*, and *y* = *H*). The initial

The discrete solution of temperature for direct heat conduction problems with K

T ¼ **X**q (16)

heat fluxes and J sensors can be presented in a matrix form as follows [37].

denotes multiple heat fluxes. The sensitivity matrix **X** is [19]

where **T** is the real temperature matrix, **X** is the sensitivity matrix, and **q**

tory, with a larger *q*max than that predicted by the SFS method.

**3.1 2D filter solution and optimal comparison criterion**

**3. 2D algorithm for heat flux estimation**

*Inverse Heat Conduction and Heat Exchangers*

FTC temperature measurement.

temperature of the substrates is *T*0.

*Geometry model of a 2D multilayer IHCP [36].*

**Figure 5.**

**132**

$$\mathbf{a}(n) = \begin{bmatrix} a\_{11}(n) & a\_{12}(n) & a\_{13}(n) & \cdots & a\_{1\mathbf{K}}(n) \\ a\_{21}(n) & a\_{22}(n) & a\_{23}(n) & \cdots & a\_{2\mathbf{K}}(n) \\ a\_{31}(n) & a\_{32}(n) & a\_{33}(n) & \cdots & a\_{3\mathbf{K}}(n) \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a\_{\mathbf{J}1}(n) & a\_{\mathbf{J}2}(n) & a\_{\mathbf{J}3}(n) & \cdots & a\_{\mathbf{JK}}(n) \end{bmatrix} \tag{18}$$
 
$$a\_{\vec{\mu}}(n) = \frac{\partial \left( T \left( \mathbf{x\_{TC\_j}}, \mathbf{W\_{T\_I}}t \right) - T\_0 \right)}{\partial \left[ q\_k(1) \right]} = \rho\_{\vec{\mu}k}(n) \tag{19}$$

where N and *n* are total time steps and current time step. Excessive temperature *ϕ* can be presented as follows:

$$\phi\_{jk}(n) = T\left(\mathcal{X}\_{\text{TC}\_{j}}, H\_{\text{T}}, nt\_{\text{d}}\right)\big|\_{q\_{\text{h}}=1,\ q\_{\text{l}}=q\_{\text{l}}=\dots=q\_{\text{k}}=0} - T\_{\text{0}}\tag{20}$$

where *t*<sup>d</sup> is the time step. The sum of the squares of the errors between the estimated and measured temperatures plus one-order regularization, *S*, is

$$\mathbf{S} = (\mathbf{Y} \cdot \mathbf{T})'(\mathbf{Y} \cdot \mathbf{T}) + \mathbf{a}\_t [\mathbf{H}\_t \mathbf{q}]'[\mathbf{H}\_t \mathbf{q}] + \mathbf{a}\_s [\mathbf{H}\_s \mathbf{q}]'[\mathbf{H}\_s \mathbf{q}] \tag{21}$$

where **Y** is the measured temperature matrix. *α<sup>t</sup>* and *α<sup>s</sup>* are regularization parameters with respect to temporal and spatial terms. The superscript 'denotes the transpose of a matrix. **Ht** and **Hs** are temporal and spatial regularization matrixes. Minimizing *S*, estimated heat flux matrix **q**^ using the entire domain data can be obtained [8]:

$$\hat{\mathbf{q}} = \left[\mathbf{X}'\mathbf{X} + \mathbf{a}\_t\mathbf{H}\_t'\mathbf{H}\_t + \mathbf{a}\_s\mathbf{H}\_s'\mathbf{H}\_t\right]^{-1}\mathbf{X}'\mathbf{Y} = \mathbf{F}\mathbf{Y} \tag{22}$$

Generally, the 2D multilayer IHCPs are solved layer by layer, starting from the *i*th layer with the known experimental measured temperature. The heat flux is estimated at the interface with the (*i*�1)th layer. Thereafter, the interface heat flux between the (*i*�1)th and (*i*�2)th layers can be calculated by using the known interface heat flux as the input. Finally, the surface heat flux is estimated [18]. The interface heat fluxes **q**^*<sup>i</sup>* between the *i*th and (*i*�1)th layers can be described as

$$\hat{\mathbf{q}}\_{\text{i}} = \left[\mathbf{X}\_{\text{i}}^{\prime}\mathbf{X}\_{\text{i}} + \mathbf{a}\_{\text{t}}\mathbf{H}\_{\text{t}}^{\prime}\mathbf{H}\_{\text{t}} + \mathbf{a}\_{\text{s}}\mathbf{H}\_{\text{s}}^{\prime}\mathbf{H}\_{\text{s}}\right]^{\cdot \cdot \mathbf{1}}\mathbf{X}\_{\text{i}}^{\prime}\mathbf{Y} = \mathbf{F}\_{\text{i}}\mathbf{Y} \tag{23}$$

Notably, **X***<sup>i</sup>* is different from **X** in the solution of single-layer IHCP. **X***<sup>i</sup>* is calculated by using (I � *i* + 1) layers below the *i*th layer. The estimated interface temperature **T**^*<sup>i</sup>* yields

$$
\hat{\mathbf{T}}\_i = \mathbf{X}\_i \hat{\mathbf{q}}\_i \tag{24}
$$

The interface heat fluxes (**q**^*<sup>i</sup>* ) and temperature (**T**^*i*) are used as the known boundary condition to solve the solution for the (*i*�1)th layer, where the heat fluxes are unknown at *<sup>y</sup>* <sup>=</sup> *<sup>H</sup>*<sup>1</sup> <sup>+</sup> *<sup>H</sup>*<sup>2</sup> <sup>+</sup> ��� <sup>+</sup> *Hi*�2. The interface heat fluxes **<sup>q</sup>**^*i*�<sup>1</sup> between the (*i*�1)th and (*i*�2)th layers can be expressed as

$$\hat{\mathbf{q}}\_{i-1} = \left[\mathbf{X}\_{i\text{,d}}\prime \mathbf{X}\_{i\text{,d}} + a\_{\text{l}} \mathbf{H}\_{\text{t}}\prime \mathbf{H}\_{\text{t}} + a\_{\text{s}} \mathbf{H}\_{\text{s}}\prime \mathbf{H}\_{\text{s}}\right]^{-1} \mathbf{X}\_{\text{d}}\prime \hat{\mathbf{T}}\_{i} - \mathbf{X}\_{i\text{,u}} \hat{\mathbf{q}}\_{\text{i}} = \mathbf{F}\_{i\text{,d}} \hat{\mathbf{T}}\_{i} - \mathbf{X}\_{i\text{,u}} \hat{\mathbf{q}}\_{\text{i}} \tag{25}$$

where **X***i*,d and **X***i*,u have a similar form as the sensitivity matrix and **X** is for singlelayer IHCP. However, excessive temperature *ϕ* in **X** is different. For **X***i*,d, *ϕ* yields

$$\phi\_{jk}(n) = T\big(\propto\_{\text{TC}\_{\text{P}}} H\_1 + H\_2 + \dots + H\_{i-1}, nt\_{\text{d}}\big)\big|\_{q\_k=1,\ q\_1=q\_2=\dots=q\_K=0} - T\_0\tag{26}$$

For **X***i*,u, *ϕ* can be described as

$$\phi\_{jk}(n) = T\big(\mathfrak{x}\_{\text{TC}}, H\_1 + H\_2 + \dots + H\_{i-1}, nt\_{\mathbf{d}}\big)|\_{q\_k=1, \ q\_1=q\_2=\dots=q\_K=0} - T\_0 \tag{27}$$

By substituting Eq. (23) into Eq. (25), the estimated interface heat flux is derived as

$$
\hat{\mathbf{q}}\_{i-1} = \mathbf{F}\_{i,\mathbf{d}} (\mathbf{X}\_i - \mathbf{X}\_{i,\mathbf{u}}) \mathbf{F}\_i \mathbf{Y} = \mathbf{F}\_{i-1} \mathbf{Y} \tag{28}
$$

and *Eq*,*rand* in the filter form

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

**3.2 Validation of 2D filter solution**

.

*Geometry model of three-layer IHCP with six FTCs [36].*

*Ytr* **<sup>F</sup>**<sup>T</sup>**<sup>F</sup>** � � <sup>¼</sup> *<sup>σ</sup>*<sup>2</sup>

*qMRE* <sup>¼</sup> <sup>1</sup>

spray diameter because the computational domain is symmetric.

N � mf

*<sup>Y</sup>* <sup>N</sup> � mp <sup>þ</sup> mf <sup>þ</sup> <sup>1</sup> � � � � <sup>X</sup>

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

To examine the accuracy of the estimated heat fluxes, the mean relative error (MRE) between estimated and hypothetical heat fluxes was employed as follows:

> N <sup>X</sup>�mf *n*¼1

The 2D three-layer geometry (**Figure 6**) with six FTC sensors was used to validate the accuracy of surface heat flux estimated by 2D filter solution. For FTC measurement, the multiple surface heat fluxes were assumed to be uniform in a specific upper lateral surface area (e.g., 0 ≤ *x* ≤ *x*1) and different from each other. Temperature was measured from the spray center (*x* = 0 mm, TC1) to the periphery (*x* = 10 mm, TC6), and the lateral distance between two FTCs was Δ*x* = 2 mm. The geometry width with TFTC measurement was *W* = 11 mm, which is half that of the

The optimal comparison criterion developed for 2D IHCPs was employed in this study to optimize the Tikhonov regularization parameters (*α<sup>t</sup>* and *αs*). As shown in **Figure 7**, logarithmic random component *Eq*,*rand* increased as the regularization parameters *α<sup>t</sup>* and *α<sup>s</sup>* decreased. *Eq*,*bias* and *E*(Rq) reached their minimum value at

As shown in **Figure 8**, the estimated heat fluxes agreed well with the hypothetical ones. However, small deviations were observed at the descent stage after the maximum heat flux for *q*<sup>1</sup> to *q*6. The *qMRE* of *q*1–*q*<sup>6</sup> for TFTC measurement was 2.63%, 2.66%, 2.76%, 2.78%, 3.00%, and 5.18%, respectively. Adding the random noise at the measured point, the maximum *qMRE* increased to 3.71%, which indicated that the accuracy and stability of the filter solution are satisfactory.

. Eventually, the optimum regularization parameters were determined

� � � � K

ð Þ mp X þmfþ1

*f* 2

*k,j*‐<sup>1</sup> <sup>∗</sup> <sup>J</sup>þ*<sup>k</sup>* (33)

(34)

*j*¼1

� � � �

*k*¼1

*q n* ^ð Þ� *q n*ð Þ *q n*ð Þ

*Eq, rand* <sup>¼</sup> *<sup>σ</sup>*<sup>2</sup>

*α<sup>t</sup>* = *α<sup>s</sup>* = 10�<sup>9</sup>

**Figure 6.**

**135**

to be *α<sup>t</sup>* = *α<sup>s</sup>* = 10�<sup>9</sup>

Interface heat flux **<sup>q</sup>**^*<sup>i</sup>*�<sup>2</sup>, **<sup>q</sup>**^*<sup>i</sup>*�<sup>3</sup>,���**q**^<sup>2</sup> can be calculated by repeating Eqs. (25)-(28). Thereafter, surface heat flux matrix **q**^ can be estimated. The solutions for 2D singlelayer and multilayer IHCPs are then obtained.

Given that most filter coefficients can be disregarded except for those of the (mp + mf + 1) time step, the solutions for IHCPs (Eqs. (22) and (28)) can be simplified into a general filter solution, thereby allowing the real-time heat flux monitoring and small computational load [8, 19, 38]:

$$q\_k(n) = \sum\_{m=1}^{\binom{m\_\mathbf{p} + m\_\mathbf{f} + 1}{m} \times \binom{n}{k\_\mathbf{p}}} \left( f\_{k\_\mathbf{p} m} Y\_{\left(n - m\_\mathbf{p} - 1\right) \times \mathbf{J} + m} \right) \tag{29}$$

where *fk*,*<sup>m</sup>* denotes the filter coefficient in the *m*th column from one row of **F** associated with the *k*th unknown heat flux. Notably, *α<sup>t</sup>* and *α<sup>s</sup>* significantly affect the accuracy of the estimated heat flux, which should be determined for a specific IHCP [9, 18, 19]. These parameters can be optimized by the optimal comparison criterion developed for solving 2D single-layer and multilayer IHCPs [36]. The sum of the squares of the errors, Rq, between the estimated and real heat fluxes was used to examine the accuracy of the calculation:

$$\mathbf{R}\_{\mathbf{q}} = (\hat{\mathbf{q}} - \mathbf{q})'(\hat{\mathbf{q}} - \mathbf{q}) = (\mathbf{F}\mathbf{Y} - \mathbf{q})'(\mathbf{F}\mathbf{Y} - \mathbf{q})\tag{30}$$

A more efficient means is to minimize the expected value of Rq [19, 38] as follows:

$$E(\mathbf{R\_{q}}) = \left[ (\mathbf{FX} - \mathbf{I})\mathbf{q} \right]' \left[ (\mathbf{FX} - \mathbf{I})\mathbf{q} \right] + \sigma\_Y^2 tr(\mathbf{F'F}) \tag{31}$$

where *E* is the expected value, *tr* denotes the sum of the diagonal elements, and *σ<sup>Y</sup>* (0.01°C [38]) is the uniform measurement errors. *E*(Rq) contains two components: *Eq*,*bias*:

$$E\_{q\_2 \text{ bias}} = [(\mathbf{FX} - \mathbf{I})\mathbf{q}]'[(\mathbf{FX} - \mathbf{I})\mathbf{q}] \tag{32}$$

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

and *Eq*,*rand* in the filter form

The interface heat fluxes (**q**^*<sup>i</sup>*

*Inverse Heat Conduction and Heat Exchangers*

0

*ϕjk*ð Þ¼ *n T x*TC*<sup>j</sup>*

*ϕjk*ð Þ¼ *n T x*TC*<sup>j</sup>*

<sup>q</sup>^*i*�<sup>1</sup> <sup>¼</sup> **<sup>X</sup>***i,* <sup>d</sup>

(*i*�1)th and (*i*�2)th layers can be expressed as

0

� ��<sup>1</sup>

**Ht** þ *αs***Hs**

*, H*<sup>1</sup> þ *H*<sup>2</sup> þ ⋯ þ *Hi*�1*, nt*<sup>d</sup> � �

*, H*<sup>1</sup> þ *H*<sup>2</sup> þ ⋯ þ *Hi*�<sup>1</sup>*, nt*<sup>d</sup> � �

<sup>q</sup>^*<sup>i</sup>*�<sup>1</sup> <sup>¼</sup> **<sup>F</sup>***i,* <sup>d</sup> **<sup>X</sup>***<sup>i</sup>* � **<sup>X</sup>***i,* <sup>u</sup>

ð Þ mp<sup>þ</sup> Xmfþ<sup>1</sup> �<sup>J</sup>

*m*¼1

0 **Hs**

**X***i,* <sup>d</sup> þ *αt***H**<sup>t</sup>

For **X***i*,u, *ϕ* can be described as

layer and multilayer IHCPs are then obtained.

monitoring and small computational load [8, 19, 38]:

*qk*ð Þ¼ *n*

to examine the accuracy of the calculation:

*E* Rq

nents: *Eq*,*bias*:

**134**

Rq ¼ ð Þ q^ � **q** <sup>0</sup>

� � <sup>¼</sup> ½ � ð Þ FX � <sup>I</sup> <sup>q</sup> <sup>0</sup>

*Eq, bias* ¼ ½ � ð Þ FX � I q <sup>0</sup>

) and temperature (**T**^*i*) are used as the known

*<sup>i</sup>* � **X***i,* uq^*<sup>i</sup>* ¼ **F***i,* dT

*qk*¼1*, <sup>q</sup>*1¼*q*2¼⋯⋯¼*q*K¼<sup>0</sup>

*qk*¼<sup>1</sup>*, <sup>q</sup>*1¼*q*2¼⋯⋯¼*q*K¼<sup>0</sup>

� �**F***i***<sup>Y</sup>** <sup>¼</sup> **<sup>F</sup>***<sup>i</sup>*�<sup>1</sup>**<sup>Y</sup>** (28)

*\_*

� � *T*<sup>0</sup> (26)

� � *T*<sup>0</sup> (27)

*<sup>i</sup>* � **X***i,* uq^*<sup>i</sup>* (25)

(29)

ð Þ **FY** � **q** (30)

*Ytr* **F**<sup>0</sup> ð Þ **F** (31)

½ � ð Þ FX � I q (32)

boundary condition to solve the solution for the (*i*�1)th layer, where the heat fluxes are unknown at *<sup>y</sup>* <sup>=</sup> *<sup>H</sup>*<sup>1</sup> <sup>+</sup> *<sup>H</sup>*<sup>2</sup> <sup>+</sup> ��� <sup>+</sup> *Hi*�2. The interface heat fluxes **<sup>q</sup>**^*i*�<sup>1</sup> between the

> **X**<sup>d</sup> 0 T *\_*

where **X***i*,d and **X***i*,u have a similar form as the sensitivity matrix and **X** is for singlelayer IHCP. However, excessive temperature *ϕ* in **X** is different. For **X***i*,d, *ϕ* yields

By substituting Eq. (23) into Eq. (25), the estimated interface heat flux is derived as

Interface heat flux **<sup>q</sup>**^*<sup>i</sup>*�<sup>2</sup>, **<sup>q</sup>**^*<sup>i</sup>*�<sup>3</sup>,���**q**^<sup>2</sup> can be calculated by repeating Eqs. (25)-(28). Thereafter, surface heat flux matrix **q**^ can be estimated. The solutions for 2D single-

Given that most filter coefficients can be disregarded except for those of the (mp + mf + 1) time step, the solutions for IHCPs (Eqs. (22) and (28)) can be simplified into a general filter solution, thereby allowing the real-time heat flux

where *fk*,*<sup>m</sup>* denotes the filter coefficient in the *m*th column from one row of **F** associated with the *k*th unknown heat flux. Notably, *α<sup>t</sup>* and *α<sup>s</sup>* significantly affect the accuracy of the estimated heat flux, which should be determined for a specific IHCP [9, 18, 19]. These parameters can be optimized by the optimal comparison criterion developed for solving 2D single-layer and multilayer IHCPs [36]. The sum of the squares of the errors, Rq, between the estimated and real heat fluxes was used

ð Þ¼ q^ � **q** ð Þ **FY** � **q** <sup>0</sup>

A more efficient means is to minimize the expected value of Rq [19, 38] as follows:

where *E* is the expected value, *tr* denotes the sum of the diagonal elements, and *σ<sup>Y</sup>* (0.01°C [38]) is the uniform measurement errors. *E*(Rq) contains two compo-

½ �þ ð Þ FX � <sup>I</sup> <sup>q</sup> *<sup>σ</sup>*<sup>2</sup>

�

�

*<sup>f</sup> k, <sup>m</sup>Y*ð Þ *<sup>n</sup>*�mp�<sup>1</sup> �Jþ*<sup>m</sup>* � �

$$E\_{q,\text{ rund}} = \sigma\_Y^2 tr\left(\mathbf{F}^T \mathbf{F}\right) = \sigma\_Y^2 \left(\mathbf{N} - \left(\mathbf{m\_p} + \mathbf{m\_f} + \mathbf{1}\right)\right) \sum\_{k=1}^{K} \sum\_{j=1}^{\left(\mathbf{m\_p} + \mathbf{m\_f} + 1\right)} f\_{k, j \cdot 1 \ast \mid +k}^2 \tag{33}$$

To examine the accuracy of the estimated heat fluxes, the mean relative error (MRE) between estimated and hypothetical heat fluxes was employed as follows:

$$q\_{MRE} = \frac{1}{\mathbf{N} - \mathbf{m}\_{\mathbf{f}}} \sum\_{n=1}^{\mathbf{N} - \mathbf{m}\_{\mathbf{f}}} \left| \frac{\dot{q}(n) - q(n)}{q(n)} \right| \tag{34}$$

#### **3.2 Validation of 2D filter solution**

The 2D three-layer geometry (**Figure 6**) with six FTC sensors was used to validate the accuracy of surface heat flux estimated by 2D filter solution. For FTC measurement, the multiple surface heat fluxes were assumed to be uniform in a specific upper lateral surface area (e.g., 0 ≤ *x* ≤ *x*1) and different from each other. Temperature was measured from the spray center (*x* = 0 mm, TC1) to the periphery (*x* = 10 mm, TC6), and the lateral distance between two FTCs was Δ*x* = 2 mm. The geometry width with TFTC measurement was *W* = 11 mm, which is half that of the spray diameter because the computational domain is symmetric.

The optimal comparison criterion developed for 2D IHCPs was employed in this study to optimize the Tikhonov regularization parameters (*α<sup>t</sup>* and *αs*). As shown in **Figure 7**, logarithmic random component *Eq*,*rand* increased as the regularization parameters *α<sup>t</sup>* and *α<sup>s</sup>* decreased. *Eq*,*bias* and *E*(Rq) reached their minimum value at *α<sup>t</sup>* = *α<sup>s</sup>* = 10�<sup>9</sup> . Eventually, the optimum regularization parameters were determined to be *α<sup>t</sup>* = *α<sup>s</sup>* = 10�<sup>9</sup> .

As shown in **Figure 8**, the estimated heat fluxes agreed well with the hypothetical ones. However, small deviations were observed at the descent stage after the maximum heat flux for *q*<sup>1</sup> to *q*6. The *qMRE* of *q*1–*q*<sup>6</sup> for TFTC measurement was 2.63%, 2.66%, 2.76%, 2.78%, 3.00%, and 5.18%, respectively. Adding the random noise at the measured point, the maximum *qMRE* increased to 3.71%, which indicated that the accuracy and stability of the filter solution are satisfactory.

**Figure 6.** *Geometry model of three-layer IHCP with six FTCs [36].*

**Figure 7.** *Geometry model of three-layer IHCP with six FTCs [36].*

### **3.3 Comparison between 2D filter solution and 1D improved Duhamel's theorem**

To investigate the importance considering lateral heat transfer, the estimated surface heat flux (**Figure 9(a)**) and simulated temperature (**Figure 9(b)**) using the estimated surface heat flux as boundary, calculated by 2D filter solution and 1D improved Duhamel's theorem, were compared taking the example of 2D three-layer geometry with FTC measurement. For FTC measurement containing aluminum film with high heat conductivity coefficient (*λ* = 236 W∙m<sup>1</sup> ∙K<sup>1</sup> ), the maximum heat flux calculated by the 1D method is underestimated by 60% than that calculated by 2D filter solution, indicating that the lateral heat transfer cannot be disregarded, especially when the heat conductivity coefficient of the material is large. As shown in **Figure 9(b)**, the simulated temperature using estimated surface heat flux as boundary calculated by 2D filter solution agreed well with measured temperature, indirectly

> indicating the accuracy of 2D filter solution. However, the large simulated temperature deviation was observed using 1D improved Duhamel's theorem, owing to the

*Variations of surface temperature and heat flux with FTC measurement [36]. (a) measured surface*

*Comparative results at spray center calculated by 2D filter solution and 1D improved Duhamel's theorem [36].*

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

*(a) surface heat flux and (b) temperature at sensor location.*

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

*temperature and (b) estimated surface heat flux.*

The dynamic internal temperature measured by six FTCs with Δ*t* = 50 ms and *L* = 30 mm is depicted in **Figure 10(a)**. The temperature histories were similar, but differences existed in the specific values. The minimum temperatures (*T*min) at *r* = 0, 2, 4, 6, 8, and 10 mm were 43.42, 36.13, 29.55, 28.22, 27.32, and 22.45°C, respectively. Additionally, the measured temperature was lower at the spray center (*r* = 0 mm) than periphery. **Figure 10(b)** presents the estimated heat fluxes calculated by the filter solution for 2D three-layer IHCP with FTC measurement. The estimated heat flux profiles at different locations in this figure were also similar. However, a large difference was observed in heat fluxes at different lateral locations. The best cooling capacity was found at the spray center (*r* = 0–2 mm).

The extremely high global warming potential (GWP = 1430) of commercially used cryogen R134a with boiling point of 26.1°C will cause severe environmental

inaccurate estimated surface heat flux disregarding lateral heat transfer.

**Figure 9.**

**Figure 10.**

**137**

**4. Clinical potential of spray cooling by low GWP R1234yf**

#### **Figure 8.**

*Results of the comparison between estimated heat fluxes (scatters) and hypothetical ones (solid lines) for threelayer IHCP with FTCs [36].*

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

#### **Figure 9.**

**3.3 Comparison between 2D filter solution and 1D improved Duhamel's**

with high heat conductivity coefficient (*λ* = 236 W∙m<sup>1</sup>

*Geometry model of three-layer IHCP with six FTCs [36].*

*Inverse Heat Conduction and Heat Exchangers*

To investigate the importance considering lateral heat transfer, the estimated surface heat flux (**Figure 9(a)**) and simulated temperature (**Figure 9(b)**) using the estimated surface heat flux as boundary, calculated by 2D filter solution and 1D improved Duhamel's theorem, were compared taking the example of 2D three-layer geometry with FTC measurement. For FTC measurement containing aluminum film

calculated by the 1D method is underestimated by 60% than that calculated by 2D filter solution, indicating that the lateral heat transfer cannot be disregarded, especially when the heat conductivity coefficient of the material is large. As shown in **Figure 9(b)**, the simulated temperature using estimated surface heat flux as boundary calculated by 2D filter solution agreed well with measured temperature, indirectly

*Results of the comparison between estimated heat fluxes (scatters) and hypothetical ones (solid lines) for three-*

∙K<sup>1</sup>

), the maximum heat flux

**theorem**

**Figure 7.**

**Figure 8.**

**136**

*layer IHCP with FTCs [36].*

*Comparative results at spray center calculated by 2D filter solution and 1D improved Duhamel's theorem [36]. (a) surface heat flux and (b) temperature at sensor location.*

#### **Figure 10.**

*Variations of surface temperature and heat flux with FTC measurement [36]. (a) measured surface temperature and (b) estimated surface heat flux.*

indicating the accuracy of 2D filter solution. However, the large simulated temperature deviation was observed using 1D improved Duhamel's theorem, owing to the inaccurate estimated surface heat flux disregarding lateral heat transfer.

The dynamic internal temperature measured by six FTCs with Δ*t* = 50 ms and *L* = 30 mm is depicted in **Figure 10(a)**. The temperature histories were similar, but differences existed in the specific values. The minimum temperatures (*T*min) at *r* = 0, 2, 4, 6, 8, and 10 mm were 43.42, 36.13, 29.55, 28.22, 27.32, and 22.45°C, respectively. Additionally, the measured temperature was lower at the spray center (*r* = 0 mm) than periphery. **Figure 10(b)** presents the estimated heat fluxes calculated by the filter solution for 2D three-layer IHCP with FTC measurement. The estimated heat flux profiles at different locations in this figure were also similar. However, a large difference was observed in heat fluxes at different lateral locations. The best cooling capacity was found at the spray center (*r* = 0–2 mm).
