**2. Measurements of heat flux absorbed by water walls in combustion chambers**

Measurements of steady-state heat flux and heat transfer coefficient are subject of many current studies [3–10]. Heat flux measurements absorbed by the water walls of the com‐ bustion chamber are used in the design of the steam boilers or in the control systems of the soot blowers. A method for online measurement of the heat flux to the refractory lin‐

ing in steam generators of municipal solid waste incinerators is presented in [9, 10]. The heat flux measurement can be used for the online monitoring of the refractory lining as well as for determining the build up of deposits on the membrane walls [9, 10]. Various methods are used for solving inverse heat conduction problem (IHCP) to estimate surface heat flux or heat transfer coefficient. A proper understanding of combustion and heat transfer in furnaces and heat exchange on the water-steam side in tubes needs an accurate measurement of heat flux which is taken over by membrane furnace walls [1, 8–23]. Iden‐ tification of boundary conditions has also great practical meaning in boiler superheaters [2]. There are three wide categories of heat flux measurements of the boiler water-walls: (1) portable heat flux devices inserted in inspection ports [11–15], (2) Gardon-type heat flux meters welded to the sections of the boiler tubes [11–14], (3) tubular-type instruments located between two neighboring boiler tubes [1, 2, 7, 8, 16–21]. Tubular-type and Gardon meters placed on the furnace tube wall in intense slagging areas can be important boiler diagnostic device for monitoring of slag deposition [2, 22–24]. If a heat flux instrument is to measure the absorbed heat flux accurately, it must be similar to the boiler tube as closely as possible. It is crucial for radiant heat exchange between the flame and measur‐ ing instruments. Two main factors in this respect are the emissivity and the temperature of the absorbing surface. Because the instrument will almost always be coated with ash, mainly the properties of the ash and not the instrument dominate the situation. Due to the significant variation of the thermal conductivity, accurate measurements can be per‐ formed only if the deposit on the meter is corresponding to that on the surrounding tubes. The tubular-type instruments known also as flux tubes satisfy this requirement. In these devices, the measured heat flux tube temperatures are used for indirect measure‐ ment of heat flux.

ured steam temperature at the inlet and outlet of the three pass steam superheater. In the second inverse problem, the inlet steam temperature and the heat transfer coefficient on the flue-gas side were estimated using measured steam temperatures at selected locations of the superheater. The Levenberg–Marquardt method was also used to solve the second inverse problem. At every iteration step, a direct conjugate heat transfer problem was solved using the ANSYS/CFX software. The CFX program was called and controlled by an external program written in Python language. The Levenberg–

**Keywords:** CFD modeling, inverse heat transfer problem, heat flux measurement, pulverized coal-fired boiler, furnace, water wall, circulating fluidized bed boiler, plat‐

The aim of direct conjugate heat transfer problems was determining the distribution of pressure, velocity, and temperature of the fluid and solid in the analyzed area when boundary conditions are known. In inverse problems, based on temperature measurements at chosen points located within the analyzed domain, boundary conditions are determined. Nonlinear inverse problems are solved iteratively. At every iteration, the direct problem is solved. When analyzed domain has a complex shape, then it is necessary to apply CFD software to solve the direct problem. In particular, when conjugate boundary problem is solved, that is, when the temperature fields in the liquid and solid are determined simultaneously, use of CFD software becomes indispensable. Two inverse problems are solved in this chapter. Identification of boundary conditions in the tube of membrane water wall was carried out in the first example. The subject of the second inverse problem is to identify the temperature of the steam entering the superheater and the identification of the heat transfer coefficient on the outer surface of the superheater. The Levenberg–Marquardt method was used to solve both inverse problems. At every iteration step, a direct conjugate heat transfer problem was solved using the ANSYS/ CFX software. The CFX program was called and controlled by an external program written in Python language. The Levenberg–Marquardt method that was used for solving nonlinear least squares problem was also included in the Python program. This chapter is based mainly on

**2. Measurements of heat flux absorbed by water walls in combustion**

Measurements of steady-state heat flux and heat transfer coefficient are subject of many current studies [3–10]. Heat flux measurements absorbed by the water walls of the com‐ bustion chamber are used in the design of the steam boilers or in the control systems of the soot blowers. A method for online measurement of the heat flux to the refractory lin‐

Marquardt algorithm was also included in the Python program.

en superheater

210 Numerical Simulation - From Brain Imaging to Turbulent Flows

two papers published earlier [1, 2].

**chambers**

**1. Introduction**

The measuring tube is equipped with two thermocouples in holes of known radial loca‐ tions *r*1 and *r*2. The thermocouples are led out to the junction box where they are connect‐ ed differentially to give a flux-related electromotive force.

The use of the one-dimensional heat conduction equation for determining temperature distribution in the tube wall gives the simple relationship:

$$q\_w = \frac{k(f\_1 - f\_2)}{r\_o \ln(r\_1/r\_2)}\tag{1}$$

where *f*1 and *f*2 are measured wall temperatures at the locations *r*1 and *r*2, respectively, and *ro* is the outer radius of the tube. The symbol *k* denotes the thermal conductivity of the tube material. The accuracy of this equation is low because of the circumferential heat conduction in the tube wall.

**Figure 1.** Flux tubes made of bare tubes; (a) flux tube with three temperature sensors, (b) flux tube with five tempera‐ ture sensors.

Therefore, the measurement of the heat flux absorbed by water-walls with satisfactory accuracy is a challenging task. Considerable work has been done in recent years in this field [1, 2, 9–21, 24]. Previous studies to precisely measure the local heat flux to membrane water walls in steam boilers failed due to unknown water-side heat transfer coefficients. The heat flux can be only determined accurately if the water-side heat transfer coefficient will be identified experimentally [7, 8, 15–18]. In this section, a numerical method for determining the heat flux in boiler furnaces, based on measured interior flux-tube temperatures, is presented. The tubular type device has been designed (**Figures 1** and **2**) to provide a highly accurate measurement of absorbed heat flux *qm*, water-side heat transfer coefficient *hf* , and water-steam temperature *Tf* . The number of thermocouples is greater than or equal to three. Unlike existing devices, the developed flux tubes are manufactured from bare or longitudinally finned tubes which are not welded to adjacent water-wall tubes. Temperature distribution in the flux tube is symmetric and not distorted by different temperature fields in neighboring tubes. Significant differences in water temperature can occur in the neighboring water wall tubes of sub- and supercritical once-through boilers due to the nonuniform heat flux distribution across the width of the combustion chamber wall. For this reason, the flux tubes presented in the chapter are especially useful for measurements in the once-through boilers. The thermal conductivity of the flux-tube material depends on temperature. The meter is made from a short piece of the eccentric tube. It contains two or four thermocouples on the fireside below the inner and outer surfaces of the tube. The third (**Figure 1a**) or the fifth thermocouple (**Figures 1b**, **2a**, and **2b**) is located at the rear of the tube (on the casing side). The boundary conditions on the outer and inner surfaces of the water flux tube must then be determined from temperature measurements at the interior locations. Two (**Figure 1a**) or four (**Figures 1b** and **2**) K-type sheathed thermo‐ couples, 1 mm in diameter, are inserted into holes, which are parallel to the tube axis, that is, parallel to the direction of flow of the water–steam mixture. The end of the sheathed thermo‐ couple, where the hot junction is situated, was bonded to the bottom of the hole using ceramic adhesive to reduce the contact resistance between the thermocouple jacket and tube flux material. The heat transfer by conduction along the thermocouple is minimized because the thermocouples pass through isothermal holes with a depth of 60 mm. The thermocouples are brought to the rear of the tube in the slot machined in the tube wall. An austenitic guard strip with a thickness of 3 mm—welded to the tube—is used to protect the thermocouples from the incident flame radiation. A K-type sheathed thermocouple with a pad is used to measure the temperature at the rear of the flux tube. This temperature is very close to the water-steam temperature. Because the heat flux on the rear surface of the measuring tube is zero (membrane water-walls are perfectly insulated) or very small (bare tube water-walls), the rear part of the tube is almost unheated. For this reason, there is very little difference in temperature between the wall and the fluid in the rear part of the flux tube. This was proved both, by experiments and computer calculations.

**Figure 1.** Flux tubes made of bare tubes; (a) flux tube with three temperature sensors, (b) flux tube with five tempera‐

Therefore, the measurement of the heat flux absorbed by water-walls with satisfactory accuracy is a challenging task. Considerable work has been done in recent years in this field [1, 2, 9–21, 24]. Previous studies to precisely measure the local heat flux to membrane water walls in steam boilers failed due to unknown water-side heat transfer coefficients. The heat flux can be only determined accurately if the water-side heat transfer coefficient will be identified experimentally [7, 8, 15–18]. In this section, a numerical method for determining the heat flux in boiler furnaces, based on measured interior flux-tube temperatures, is presented. The tubular type device has been designed (**Figures 1** and **2**) to provide a highly accurate

devices, the developed flux tubes are manufactured from bare or longitudinally finned tubes which are not welded to adjacent water-wall tubes. Temperature distribution in the flux tube is symmetric and not distorted by different temperature fields in neighboring tubes. Significant differences in water temperature can occur in the neighboring water wall tubes of sub- and supercritical once-through boilers due to the nonuniform heat flux distribution across the width of the combustion chamber wall. For this reason, the flux tubes presented in the chapter are especially useful for measurements in the once-through boilers. The thermal conductivity of the flux-tube material depends on temperature. The meter is made from a short piece of the eccentric tube. It contains two or four thermocouples on the fireside below the inner and outer surfaces of the tube. The third (**Figure 1a**) or the fifth thermocouple (**Figures 1b**, **2a**, and **2b**) is

. The number of thermocouples is greater than or equal to three. Unlike existing

, and water-steam

measurement of absorbed heat flux *qm*, water-side heat transfer coefficient *hf*

ture sensors.

212 Numerical Simulation - From Brain Imaging to Turbulent Flows

temperature *Tf*

**Figure 2.** Flux tubes for membrane water walls; (a) fins welded to the flux tube, (b) fins welded to adjacent water wall tubes.

An IHCP was solved using the least squares method. Three unknown parameters were determined using the Levenberg–Marquardt method. At every iteration step, the temperature distribution in the cross section of the heat flux tube was computed using the ANSYS/CFX software [25]. Test calculations were conducted to estimate the accuracy of the presented method. The uncertainty in determined parameters was calculated using the Gauss uncer‐ tainty propagation rule. The technique presented in this chapter is appropriate for water walls made of bare tubes (**Figure 1a** and **b**) and membrane water-walls (**Figure 2a** and **b**). The new technique has advantages in view of the simplicity and accuracy.

The method can be used to determine the resistance of scale or iron oxide on the inner surface of the water wall tubes. The thermal resistance *R* of deposits on the inner surface of the measuring device can be determined experimentally using the following relationship

$$R = \frac{1}{h\_f} - \frac{1}{h\_c},\tag{2}$$

where the symbol *hf* stands for the equivalent heat transfer coefficient for the tube inner surface with a scale layer.

The thermal resistance *Rs* = *δs*/*ks* of the scale layer can be determined analytically, but the thickness *δs* and thermal conductivity *ks* of the scale layer are difficult to measure in practice.

The heat transfer coefficient *hc* can be identified after the beginning of the boiler operation when the tube inner surface is clean. Also, the value of *hc* can be estimated after chemical cleaning of the tube internal surfaces. An equivalent heat transfer coefficient *hf* is monitored online during the boiler operation. Another measure of scale presence on the tube inner surface is the maximum temperature of the outer surface of the heat flux tube, which increases with the growth of scale thickness on the inner surface. If the thermal resistance of the scale and the maximum temperature of the heat flux tube is greater than the limit values, the boiler should be cleaned chemically.

#### **2.1. Theory**

The water walls can be made of plain tubes or tubes which are welded together with steel flat bars (longitudinal fins) to form membrane wall panels. The water walls are insulated on the rear side and exposed on the front side to radiation from a combustion chamber. The thermo‐ metric inserts were manufactured in the laboratory and then securely welded to the water wall tubes at various levels in the combustion chamber of the steam boiler.

The following assumptions in a heat conduction model of the flux tube are made:


The temperature distribution *T* is governed by the nonlinear partial differential equation

Solving Inverse Heat Transfer Problems When Using CFD Modeling http://dx.doi.org/10.5772/63807 215

$$\nabla \cdot \left[ k(T) \nabla T \right] = 0 \tag{3}$$

The unknown boundary conditions may be expressed as

distribution in the cross section of the heat flux tube was computed using the ANSYS/CFX software [25]. Test calculations were conducted to estimate the accuracy of the presented method. The uncertainty in determined parameters was calculated using the Gauss uncer‐ tainty propagation rule. The technique presented in this chapter is appropriate for water walls made of bare tubes (**Figure 1a** and **b**) and membrane water-walls (**Figure 2a** and **b**). The new

The method can be used to determine the resistance of scale or iron oxide on the inner surface of the water wall tubes. The thermal resistance *R* of deposits on the inner surface of the

measuring device can be determined experimentally using the following relationship

*R*

1 1 , *f c*

The thermal resistance *Rs* = *δs*/*ks* of the scale layer can be determined analytically, but the thickness *δs* and thermal conductivity *ks* of the scale layer are difficult to measure in practice.

The heat transfer coefficient *hc* can be identified after the beginning of the boiler operation when the tube inner surface is clean. Also, the value of *hc* can be estimated after chemical

online during the boiler operation. Another measure of scale presence on the tube inner surface is the maximum temperature of the outer surface of the heat flux tube, which increases with the growth of scale thickness on the inner surface. If the thermal resistance of the scale and the maximum temperature of the heat flux tube is greater than the limit values, the boiler should

The water walls can be made of plain tubes or tubes which are welded together with steel flat bars (longitudinal fins) to form membrane wall panels. The water walls are insulated on the rear side and exposed on the front side to radiation from a combustion chamber. The thermo‐ metric inserts were manufactured in the laboratory and then securely welded to the water wall

**•** the thermal conductivity of the flux tube and membrane wall may depend on temperature,

**•** the heat transfer coefficient *hc* and the scale thickness *δs* are uniform on the inner tube

The temperature distribution *T* is governed by the nonlinear partial differential equation

cleaning of the tube internal surfaces. An equivalent heat transfer coefficient *hf*

tubes at various levels in the combustion chamber of the steam boiler.

**•** the temperature distribution is two dimensional and steady state,

The following assumptions in a heat conduction model of the flux tube are made:

*h h* = - (2)

is monitored

stands for the equivalent heat transfer coefficient for the tube inner surface

technique has advantages in view of the simplicity and accuracy.

214 Numerical Simulation - From Brain Imaging to Turbulent Flows

where the symbol *hf*

be cleaned chemically.

circumference.

**2.1. Theory**

with a scale layer.

$$\left\| \left[ k(T) \frac{\partial T}{\partial n} \right] \right\|\_{s} = q(s) \tag{4}$$

where *q*(*s*) is the radiation heat flux absorbed by the exposed flux tube and membrane wall surface. The local heat flux *q*(*s*) is a function of the view factor *ψ*(*s*)

$$q(\mathbf{s}) = q\_n \boldsymbol{\wp}(\mathbf{s}) \tag{5}$$

where *qm* is measured heat flux (thermal loading of the heating surface), and *s* denotes the coordinate along the boundary (**Figure 2**). The view factor *ψ*(*s*) from the infinite flame surface to the differential element on the membrane wall surface *ds* can be determined graphically [17], or numerically [25].

In this chapter, *ψ*(*s*) was evaluated analytically for bare tubes and numerically for membrane water-walls using the finite element program ANSYS/CFX [25] as a function of extended coordinate *s*.

The boundary condition on the inner surface of the tube *Sin* is defined by Newton's law of cooling

$$-\left[k(T)\frac{\partial T}{\partial n}\right]\vert\_{s\_u} = h\_f(T\mid\_{s\_u} - T\_f) \tag{6}$$

where *Tf* designates the temperature of the water–steam mixture.

The rear side of the membrane water wall is perfectly insulated. Instead of the boundary condition on the outer surface of the water wall tube, measured values *fi* of metal temperature at internal locations of the flux tubes are known

$$T\_e(\mathbf{r}\_i) = f\_i, \quad i = 1, \ldots, m \tag{7}$$

where *m* = 3 or *m* = 5 designates the number of thermocouples (**Figures 1** and **2**). The unknown parameters: *x*1 = *qm*, *x*2 = *hf* , and *x*3 = *Tf* were estimated using the least-squares method. The symbol *rin* = *df* /2 = *dt* /2 stands for the inner tube radius. The symbols *df* and *dt* designate the inner diameter of the flux and water wall tube, respectively. Unknown parameters **x** = (*x*1, …, *xn*) tr for *n* = 3 are selected so that computed temperatures *T*(**x**, **r***<sup>i</sup>* ) agree within certain limits with the experimentally measured temperatures *fi* . The location of the thermocouple is defined by the position vector **r***<sup>i</sup>* . This may be expressed as

$$T(\mathbf{x}, \mathbf{r}\_i) - f\_i \equiv \mathbf{0}, \quad i = 1, \ldots, m. \tag{8}$$

The least-squares method is used to determine parameters **x**. The sum of squares

$$S(\mathbf{x}) = \sum\_{i=1}^{m} [f\_i - T(\mathbf{x}, \mathbf{r}\_i)]^2 \tag{9}$$

can be minimized by a general unconstrained method.

However, the properties of (9) make it worthwhile to use methods designed specifically for the nonlinear least-squares problem. In this work, the Levenberg–Marquardt method [26, 27] is used to determine the parameters *x*1, *x*2 and *x*3. The Levenberg–Marquardt method performs the *k*-th iteration as

$$\mathbf{x}^{(k+1)} = \mathbf{x}^{(k)} + \boldsymbol{\delta}^{(k)} \tag{10}$$

where

$$\mathbf{J}^{(k)} = [(\mathbf{J}^{(k)})^\nu \mathbf{J}^{(k)} + \mu \mathbf{I}\_\mathbf{n}]^{-1} (\mathbf{J}^{(k)})^\nu [\mathbf{f} - \mathbf{T}(\mathbf{x}^{(k)})], \quad k = 0, 1, \ldots \tag{11}$$

where *μ* is the multiplier, and *I*n is the identity matrix. The Levenberg–Marquardt method is a combination of the Gauss–Newton method (*μ*(*k*) → 0) and the steepest-descent method (*μ*(*k*) → ∞). The latter method is used far from the minimum, changing continuously to the former as the minimum is approached. Initially, a small positive value of *μ* was selected, for example, *μ*(1) = 0.01. If, at the *k*th iteration, the step **δ**(*k*) of (10) reduces S(**x**), then *μ*(k) is decreased by a factor of 10 and trial solution is updated, that is, **x**(*<sup>k</sup>*+1) = **x**(*k*) + **δ**(*k*) . In this way, the algorithm is pushed closer to Gauss–Newton method. If within the *k*th iteration, the step **δ**(*k*) does not reduce S(**x**), *μ*(*k*) is progressively increased by a factor of 10, for ex‐ ample, *μ*(*k*) = 10*μ*(*k*) , each time recomputing **δ**(*k*) until a reduction in S(**x**) is achieved. The *m* × *n* Jacobian matrix **J** of *T*(**x**(*k*) , **r***<sup>i</sup>* ) is defined as

#### Solving Inverse Heat Transfer Problems When Using CFD Modeling http://dx.doi.org/10.5772/63807 217

$$\mathbf{J}^{(k)} = \frac{\partial \mathbf{T}(\mathbf{x})}{\partial \mathbf{x}^{\nu}} \Big|\_{\mathbf{X} - \mathbf{X}^{(k)}} = \begin{bmatrix} \frac{\partial T\_1}{\partial \mathbf{x}\_1} & \cdots & \frac{\partial T\_1}{\partial \mathbf{x}\_n} \\\\ \cdots & \cdots & \cdots \\\\ \cdots & \cdots & \cdots \\\\ \cdots & \cdots & \cdots \\\\ \frac{\partial T\_m}{\partial \mathbf{x}\_1} & \cdots & \frac{\partial T\_m}{\partial \mathbf{x}\_n} \end{bmatrix}\_{\mathbf{X} = \mathbf{X}^{(k)}},\tag{12}$$

where **T**(**x**(*<sup>k</sup>* ) ) = (*T*<sup>1</sup> (*<sup>k</sup>* ), ..., *Tm*(*<sup>k</sup>* ))tr. A finite-difference method was used to estimate the Jacobian matrix at the approximate solution. The sensitivity coefficient defined in Eq. (12) is important indicators of the ability to estimate the unknown parameters. Very small sensitivi‐ ties indicate the parameters difficult to estimate. The inverses of the sensitivities appear in the variance propagation rule given by Eq. (16). Although the unknowns *x1*, *x*2, and *x3*, are not of the same order of magnitude, the normalized and not normalized vector **x** gave the same solutions. Therefore, the vector **x** was not normalized prior to the solution of the problem. The iterative procedure is continued until the variations in *xi* (*k* ) , *i* = 1, …, *n* are less than a small preset tolerance *ε*. At every *k*-th iteration step, the temperature distribution **T**(**x**(*k*) , **r***<sup>i</sup>* ) is calculated using the element-based FVM method.

The estimates for the initial values of the parameters x1 (1), x2 (1), and x3 (1) are

the experimentally measured temperatures *fi*

216 Numerical Simulation - From Brain Imaging to Turbulent Flows

. This may be expressed as

The least-squares method is used to determine parameters **x**. The sum of squares

1 ( ) [ ( , )] *m*

*i S fT* =

can be minimized by a general unconstrained method.

the position vector **r***<sup>i</sup>*

the *k*-th iteration as

d

where

(*μ*(*k*)

the step **δ**(*k*)

= 10*μ*(*k*)

× *n* Jacobian matrix **J** of *T*(**x**(*k*)

ample, *μ*(*k*)

. The location of the thermocouple is defined by

( , ) 0, 1, , . *T fim* **x r***i i* - @ =¼ (8)

**<sup>x</sup>** = - å **x r** (9)

<sup>+</sup> **x x** = + (10)

→ 0) and the steepest-descent method

is progressively increased by a factor of 10, for ex‐

until a reduction in S(**x**) is achieved. The *m*

of (10) reduces S(**x**), then *μ*(k) is

+ **δ**(*k*)

. In this

2

*i i*

However, the properties of (9) make it worthwhile to use methods designed specifically for the nonlinear least-squares problem. In this work, the Levenberg–Marquardt method [26, 27] is used to determine the parameters *x*1, *x*2 and *x*3. The Levenberg–Marquardt method performs

> ( 1) ( ) ( ) *k kk* d

() () () 1 () ( ) [( ) ] ( ) [ ( )], 0,1, *k k tr k k tr <sup>k</sup>*

decreased by a factor of 10 and trial solution is updated, that is, **x**(*<sup>k</sup>*+1) = **x**(*k*)

) is defined as

*<sup>k</sup>* - = + - =¼ **J J I J f Tx <sup>n</sup>**

where *μ* is the multiplier, and *I*n is the identity matrix. The Levenberg–Marquardt method

way, the algorithm is pushed closer to Gauss–Newton method. If within the *k*th iteration,

 → ∞). The latter method is used far from the minimum, changing continuously to the former as the minimum is approached. Initially, a small positive value of *μ* was selected,

(11)

m

is a combination of the Gauss–Newton method (*μ*(*k*)

does not reduce S(**x**), *μ*(*k*)

for example, *μ*(1) = 0.01. If, at the *k*th iteration, the step **δ**(*k*)

, each time recomputing **δ**(*k*)

, **r***<sup>i</sup>*

$$\chi\_1^{(l)} = \frac{k^{(l)}}{r\_o} \frac{0.5(f\_1 + f\_2) - 0.5(f\_3 + f\_4)}{\ln(r\_1 / r\_2)} \tag{13}$$

$$\mathbf{x}\_{2}^{(l)} = \frac{\mathbf{x}\_{1}^{(l)}r\_{o}}{r\_{in}} \frac{1}{0.5(f\_{3} + f\_{4}) - (\mathbf{x}\_{1}^{(l)}r\_{o} \;/\; k^{(l)}) \ln\{r\_{2} \;/\left[r\_{2} - 0.8(r\_{2} - r\_{in})\right]\} - \mathbf{x}\_{3}^{(l)}} \tag{14}$$

$$\mathbf{x}\_3^{(l)} = f\_3 \tag{15}$$

where the radii are: *ro* = 0.035 m, *rin* = 0.020 m, *r*1 = 0.033 m, *r*2 = 0.026 m. The thermal conductivity *k*(1) is evaluated at the mean temperature:

$$f\_w = (f\_1 + f\_2 + f\_3 + f\_4) / \, 4,\\ i.g.k^{(l)} = k(f\_m).$$

**Figure 3.** Flux tube with constant thickness used to determine initial values of unknown sought parameters.

The relationships (13)–(15) were derived assuming one-dimensional temperature distribution in the heat flux tube with constant thickness (**Figure 3**). In the formula (14), the tube wall temperature at the distance: 0.2 (*r*2 – *rin*) from the inner surface was taken as the temperature of the tube inner surface, which results in a lower initial value of the heat transfer coefficient x2 (1).

The Levenberg–Marquardt method allows determining the desired parameters even for an inaccurate estimation of their starting values since the steepest-descent method is used at the beginning of the iteration process. The use of Eqs. (13–15) to select the initial values of parameters makes it possible to find quickly a convergent solution with a small number of iterations. By selecting other starting values, the correct solution is also obtained but with a larger number of iterations.

The Levenberg–Marquardt method works very well in practice and has become the standard of nonlinear least-squares procedures [26, 27]. The boundary value problem that is given by Eq. (3) and boundary conditions (4) and (6) was solved at each iteration step by the element based finite volume method using the ANSYS/CFX software. The CFX program was called and controlled by an external program written in Python programming language [28]. The Levenberg–Marquardt algorithm was also incorporated into the Python program.

#### **2.2. Uncertainty analysis**

The uncertainties of the determined parameters **x**\* will be estimated using the error propaga‐ tion rule of Gauss [29–33]. The propagation of uncertainty in the independent variables: measured wall temperatures *fj* , *j* = 1, …, *m*, thermal conductivity *k*, radial *rj* and angular ϕ*<sup>j</sup>* , *j* = 1, …, *m*, coordinates of the thermocouples is estimated from the following expression

$$2\sigma\_{\mathbf{x}\_{i}} = \left[\sum\_{j=1}^{m} \left[\left(\frac{\partial \mathbf{x}\_{i}}{\partial f\_{j}} 2\sigma\_{f\_{j}}\right)^{2} + \left(\frac{\partial \mathbf{x}\_{i}}{\partial r\_{j}} 2\sigma\_{r\_{j}}\right)^{2} + \left(\frac{\partial \mathbf{x}\_{i}}{\partial \varphi\_{j}} 2\sigma\_{\varphi\_{j}}\right)^{2}\right] + \left(\frac{\partial \mathbf{x}\_{i}}{\partial k} 2\sigma\_{k}\right)^{2}\right]^{\frac{1}{2}},\tag{16}$$

$$i = 1,2,3.$$

The symbols,*<sup>σ</sup> <sup>f</sup> <sup>j</sup>* <sup>2</sup> , *<sup>σ</sup>rj* 2 , *σφ<sup>j</sup>* 2, and *σ<sup>k</sup>* 2 designate variances of measured temperatures, radial, and angular thermocouple locations and thermal conductivity.

The 95% uncertainty in the estimated parameters can be expressed in the form *xi* = *xi* \* ± 2*σxi* , where *xi* = *xi* \* , *i* = 1, 2, 3 represent the value of the parameters obtained using the least squares method.

The sensitivity coefficients ∂*xi* /∂*fj* , ∂*xi* /∂*rj* , ∂*xi* /∂*ϕ<sup>j</sup>* , and ∂*xi* /∂*k* in Eq. (16) were approximated by the central difference quotients

$$\frac{\partial \mathbf{x}\_i}{\partial f\_j} = \frac{\mathbf{x}\_i(f\_1, \dots, f\_j + \delta, \dots, f\_n) - \mathbf{x}\_i(f\_1, \dots, f\_j - \delta, \dots, f\_n)}{2\delta},\tag{17}$$

$$i = 1, \dots, n, \qquad j = 1, \dots, m$$

where *δ* is a small positive number.

The error propagation rule given by Eq. (16), which is also known as the principle of the variance propagation, is used extensively to assess the uncertainty of indirect steady-state measurements.

#### **2.3. Test computations**

(1)

1234 ( ) / 4, . . ( ). *m m f f f f f igk k f* = +++ =

218 Numerical Simulation - From Brain Imaging to Turbulent Flows

**Figure 3.** Flux tube with constant thickness used to determine initial values of unknown sought parameters.

x2 (1).

larger number of iterations.

**2.2. Uncertainty analysis**

The uncertainties of the determined parameters **x**\*

The relationships (13)–(15) were derived assuming one-dimensional temperature distribution in the heat flux tube with constant thickness (**Figure 3**). In the formula (14), the tube wall temperature at the distance: 0.2 (*r*2 – *rin*) from the inner surface was taken as the temperature of the tube inner surface, which results in a lower initial value of the heat transfer coefficient

The Levenberg–Marquardt method allows determining the desired parameters even for an inaccurate estimation of their starting values since the steepest-descent method is used at the beginning of the iteration process. The use of Eqs. (13–15) to select the initial values of parameters makes it possible to find quickly a convergent solution with a small number of iterations. By selecting other starting values, the correct solution is also obtained but with a

The Levenberg–Marquardt method works very well in practice and has become the standard of nonlinear least-squares procedures [26, 27]. The boundary value problem that is given by Eq. (3) and boundary conditions (4) and (6) was solved at each iteration step by the element based finite volume method using the ANSYS/CFX software. The CFX program was called and controlled by an external program written in Python programming language [28]. The

tion rule of Gauss [29–33]. The propagation of uncertainty in the independent variables:

will be estimated using the error propaga‐

Levenberg–Marquardt algorithm was also incorporated into the Python program.

Four thermometric inserts will be investigated. In the first flux tube (**Figure 1a**) made from a plain eccentric tube, the wall temperature is measured at two locations at different radii on the fire side. The third thermocouple is placed on the outer tube surface on the insulated side of the water wall. In the second flux tube (**Figure 1b**), made also from the bare tube, four thermocouples are placed at the forward tube part. The fifth temperature sensor is situated at the same position as in the first heat flux. The third flux tube (**Figure 2a**) has two longitudinal fins, which are not welded to the adjacent water wall tubes, so the temperature distribution in the measuring device is not disturbed by the water wall tubes. The fourth device (**Figure 2b**) is without fins, which are welded to the neighboring water wall tubes. There are four ther‐ mocouples located at different radii and angles in the flux-tube wall on the fire side, and one thermocouple is attached to the outer tube surface on the back side of the water wall.

#### *2.3.1. Flux tube made from bare tube*

First, the temperature distribution in the cross section of the flux tube will be determined. The following data were adopted for the calculation: absorbed heat flux, *qm,e* = 250 000 W/m2 , heat transfer coefficient, *hf,e* = 30 000 W/(m2 K), and the temperature of the watersteam mixture, *Tf,e* = 318°C. The view factor *ψ*(*ϕ*) was evaluated analytically and numeri‐ cally using the ANSYS software. The consistency of analytical and numerical results is quite good (**Figure 4**).


**Table 1.** Thermal conductivity k(T) of steel 20G, as a function of temperature.

The view factor at φ = 180° is about 0.08 since a part of the incident radiation heat flux from the combustion chamber passes through spaces between bare tubes and is reflected by the adiabatic boiler refractory. The material of the heat flux tube is 20G steel. The composition of the 20G mild steel is as follows: 0.17–0.24% C, 0.7–1.0% Mn, 0.15–0.40% Si, 60.04% P, 60.04% S. The thermal conductivity of the heat flux tube is assumed to be temperature dependent (**Table 1**).

**Figure 4.** Comparison of view factor calculated analytically and numerically using ANSYS/CFX for the flux tubes shown in **Figure 1**. The origin of the cylindrical coordinate system is at the center of the outer flux-tube surface.

The 95% uncertainty interval for thermal conductivity was assumed ±0.5 W/(m K) [34]. The thermal conductivity of the flux-tube material (mild steel 20G) was approximated using a simple linear function

$$k(T) = \\$3.26 - 0.0238T \tag{18}$$

First, measured temperatures were generated artificially to test the method developed in the chapter. The division of the flux-tube model into finite elements is shown in **Figure 5**. Three different finite element meshes were used to compute the temperature at the locations 1–5. The temperature distribution in the flux tube was calculated using ANSYS/CFX v13.0.

The coordinates of the temperature measuring points 1–5 are as follows:


is without fins, which are welded to the neighboring water wall tubes. There are four ther‐ mocouples located at different radii and angles in the flux-tube wall on the fire side, and one

First, the temperature distribution in the cross section of the flux tube will be determined. The following data were adopted for the calculation: absorbed heat flux, *qm,e* = 250 000

Temperature, °C 100 200 300 400 Thermal conductivity, W/(m K) 50.69 48.60 46.09 42.30

The view factor at φ = 180° is about 0.08 since a part of the incident radiation heat flux from the combustion chamber passes through spaces between bare tubes and is reflected by the adiabatic boiler refractory. The material of the heat flux tube is 20G steel. The composition of the 20G mild steel is as follows: 0.17–0.24% C, 0.7–1.0% Mn, 0.15–0.40% Si, 60.04% P, 60.04% S. The thermal conductivity of the heat flux tube is assumed to be temperature dependent

**Figure 4.** Comparison of view factor calculated analytically and numerically using ANSYS/CFX for the flux tubes shown in **Figure 1**. The origin of the cylindrical coordinate system is at the center of the outer flux-tube surface.

**Table 1.** Thermal conductivity k(T) of steel 20G, as a function of temperature.

, heat transfer coefficient, *hf,e* = 30 000 W/(m2 K), and the temperature of the watersteam mixture, *Tf,e* = 318°C. The view factor *ψ*(*ϕ*) was evaluated analytically and numeri‐ cally using the ANSYS software. The consistency of analytical and numerical results is

thermocouple is attached to the outer tube surface on the back side of the water wall.

*2.3.1. Flux tube made from bare tube*

220 Numerical Simulation - From Brain Imaging to Turbulent Flows

quite good (**Figure 4**).

W/m2

(**Table 1**).

The temperature at the points 1–5 obtained for various meshes shown in **Figure 5** are sum‐ marized in **Table 2**.

The analysis of the results listed in **Table 2** shows that the temperature calculated using the different grids are very close to each other.


**Table 2.** Flux-tube temperature at the points P1–P5 computed for different finite element meshes shown in **Figure 5**.

**Figure 5.** Division of geometrical model of the heat flux tube into finite elements; (a) hexahedral mesh–1150 elements hex8, 1692 nodes; (b) tetrahedral mesh—1909 elements tet4, 518 nodes; (c) hexahedral mesh—59520 elements hex8, 66759 nodes.

**Figure 6.** Flux-tube temperature at the outer and inner surfaces as a function of angular coordinate *φ*1.

The temperature changes at the inner and outer surface on the flux-tube circumference which were obtained using the mesh shown in **Figure 5a** are depicted in **Figure 6**.

First, the temperatures for the mesh shown in **Figure 5a** were taken as "measured data": *f*1 = 418.31°C, *f*2 = 415.66°C, *f*3 = 374.08°C, *f*4 = 372.22°C, *f*5 = 321.11°C (**Table 2**).

Taking the calculated temperatures as measured temperatures and using the same finite element mesh, the following results are obtained:

**•** For three measuring points: P1, P3, and P5 (**Figure 1a**):

$$q\_m = 2.50\,000\,.19\,\text{W/m}^2,$$

<sup>2</sup> *h* 30 000.38 W/(m K), =

*Tf* 318.00 C = °

**•** For five measuring points: P1, P2, P3, P4, and P5 (**Figure 1b**):

**Figure 5.** Division of geometrical model of the heat flux tube into finite elements; (a) hexahedral mesh–1150 elements hex8, 1692 nodes; (b) tetrahedral mesh—1909 elements tet4, 518 nodes; (c) hexahedral mesh—59520 elements hex8,

66759 nodes.

222 Numerical Simulation - From Brain Imaging to Turbulent Flows

$$q\_m = 249\,999\,94\,\text{W/m}^2,$$

$$h = 30\,0000\,1.6\,\text{W/(m}^2\text{ K)},$$

$$T\_f = 3.18.00 \text{°C}\_\circ$$

When the temperature values obtained for the mesh illustrated in **Figure 5c** were adopted as "measured data" then an inverse calculation using the mesh shown in **Figure 5a** gives little different results:

**•** For three measuring points: P1, P3, and P5 (**Figure 1a**):

```
2 250 892.94 / m , mq =
2 h 30 418.33 / (m K), =
     318.00 C. Tf = °
```
**•** For five measuring points: P1, P2, P3, P4, and P5 (**Figure 1b**):

$$q\_m = 250898.22 \text{ W/m}^2,$$

<sup>2</sup> *h* 30 498.66 W/(m ·K), =

$$T\_f = 3.18.01^\circ \text{C}\_\cdot$$

Analysis of the results demonstrates that for exact "measurement data," the obtained results are in very good agreement with the input values. The impact of measuring point number on the results of the inverse problem solution can be better estimated when the measured temperatures are assumed to be disturbed with pseudorandom errors, which are characterized by the 95% uncertainty intervals: 2*<sup>σ</sup> <sup>f</sup> <sup>j</sup>* = ± 0.2 *K*;2*σrj* = ± 0.05 *mm*;2*σφ<sup>j</sup>* = ± 0.5 , *j* = 1, …, 5; 2*σ<sup>k</sup>* = ± 0.5 *W* / (*m*⋅ *K*). At first, the uncertainties in the estimated parameters: *x1* = *qm*, *x*2 = *h*, *x*3 = *T*<sup>f</sup> will be determined when the flux tube illustrated in **Figure 1a** is used. To estimate uncertainties in determined parameters, Eq. (16) was used. The partial derivatives appearing

in Eq. (16) were calculated using the central difference approximation (17). The results are listed in **Table 3**.

The analysis of the results presented in **Table 3** shows that the largest absolute values have the following coefficients: ∂*qm*/∂*fj* , ∂*hf* /∂*fj* , ∂*qm*/∂*rj* , and ∂*hf* /∂*rj* .

For this reason to obtain small uncertainties in the heat flux *qm* and the heat transfer coefficient *hf* , the random errors in measured temperatures and radial locations of the thermocouples should be small.

Applying Eq. (16) yields: 2σ(x1)= 4073.50 W/m2; 2σ(x2)= 3921.12W/(m2·K); 2σ(x3)= 0.22 K. The 95% uncertainties in the estimated parameters are as follows:

2 250 000.2 4073.5 W/m , *mq* = ±

When the temperature values obtained for the mesh illustrated in **Figure 5c** were adopted as "measured data" then an inverse calculation using the mesh shown in **Figure 5a** gives little

2 250 892.94 / m , *mq* =

<sup>2</sup> *h* 30 418.33 / (m K), =

318.00 C. *Tf* = °

2 250 898.22 W/m , *mq* =

<sup>2</sup> *h* 30 498.66 W/(m ·K), =

318.01 C. *Tf* = °

Analysis of the results demonstrates that for exact "measurement data," the obtained results are in very good agreement with the input values. The impact of measuring point number on the results of the inverse problem solution can be better estimated when the measured temperatures are assumed to be disturbed with pseudorandom errors, which are characterized

= ± 0.2 *K*;2*σrj*

 will be determined when the flux tube illustrated in **Figure 1a** is used. To estimate uncertainties in determined parameters, Eq. (16) was used. The partial derivatives appearing in Eq. (16) were calculated using the central difference approximation (17). The results are

, and ∂*hf*

/∂*rj* .

2*σ<sup>k</sup>* = ± 0.5 *W* / (*m*⋅ *K*). At first, the uncertainties in the estimated parameters: *x1* = *qm*, *x*2 = *h*,

The analysis of the results presented in **Table 3** shows that the largest absolute values have

For this reason to obtain small uncertainties in the heat flux *qm* and the heat transfer coefficient

Applying Eq. (16) yields: 2σ(x1)= 4073.50 W/m2; 2σ(x2)= 3921.12W/(m2·K); 2σ(x3)= 0.22 K. The 95%

, the random errors in measured temperatures and radial locations of the thermocouples

, ∂*qm*/∂*rj*

, ∂*hf* /∂*fj*

uncertainties in the estimated parameters are as follows:

= ± 0.05 *mm*;2*σφ<sup>j</sup>*

= ± 0.5 , *j* = 1, …, 5;

different results:

**•** For three measuring points: P1, P3, and P5 (**Figure 1a**):

224 Numerical Simulation - From Brain Imaging to Turbulent Flows

**•** For five measuring points: P1, P2, P3, P4, and P5 (**Figure 1b**):

by the 95% uncertainty intervals: 2*<sup>σ</sup> <sup>f</sup> <sup>j</sup>*

*x*3 = *T*<sup>f</sup>

*hf*

listed in **Table 3**.

should be small.

the following coefficients: ∂*qm*/∂*fj*

2 30 000.4 3921.1 W/(m K), *<sup>f</sup> h* ± ×=

$$T\_f = \textbf{318.0} \pm \textbf{0.222^\circ C}.$$


**Table 3.** Partial derivatives appearing in Eq. (16) for the evaluation of measurement uncertainty.

If the device with five temperature measurement points is used (**Figure 1b**) then the uncer‐ tainties become smaller: 2*σx*<sup>1</sup> = 3557.73 W/m<sup>2</sup> ;2*σx*<sup>2</sup> = 2379.85W/(m<sup>2</sup> K);2*σx*<sup>3</sup> = 0.22 K.

The limits of the 95% uncertainty interval are as follows:

2 249 999.9 3557.7 W/m , *mq* = ± 2 30 000.16 2379.9 W/(m K), *<sup>f</sup> h* = ±

318.0 0.22 C. *Tf* = ±°

Inspection of the results demonstrates the good accuracy in both cases. If the number of temperature measurement points is equal five then the uncertainties in the estimated heat flux and heat transfer coefficient become smaller, because an imprecise location of one thermo‐ couple has a smaller influence on the estimated parameters. For two temperature sensors placed at the forward part of the heat flux tube, an inaccurately situated temperature sensor influences to a larger extent the results.

If the standard deviations of the variables measured directly are greater also standard deviations of the determined parameters: *qm*, *hf* , and *Tf* will be greater. If the random errors will be doubled, the standard deviation will also be doubled. This conclusion follows from the analysis of Eq. (16). This has no effect on the convergence of the solution. For larger random errors in measured temperatures, a convergent solution is obtained, but with a slightly larger number of iterations.

To demonstrate the influence of the selection of parameter initial values on the solution convergence and on the number of iterations, the computations will be carried out for five measuring temperature points and using two sets of initial values:


Adopting "measured data" from **Table 2** for the mesh shown in **Figure 5a**, the following results were obtained:


Because the start values of parameters are far from the input values (exact solution) the number of iteration is large. However, the solution found is correct in both cases.

If one of the temperature measurements is affected by a significant measurement error then the incorrect temperature measurement must be removed from the analysis. It was

assumed that the measured temperature at the point 4 is equal to the measured tempera‐ ture at the point 5 to assess the influence of the large measurement error. Using the finite element mesh depicted in **Figure 5a** and disturbed "measurement data": *f*1 = 418.31°C, *f*2 = 415.66°C, *f*3 = 374.08°C, *f*4 = 321.11°C, *f*5 = 321.11°C, the following values of the parameters were found: *qm* = 303 248.29 W/m2 , *hf* = 5 705 392.54 W/(m2 K), *Tf* = 306.16°C. One can see that the determined parameters considerably differ from the exact values: *qm* = 250 000 W/m2 , *hf* = 30 000 W/(m2 K), *Tf* = 318°C. In practice, it is easy to notice that one of the measuring points is damaged because during correct measurements the temperatures measured at the points 1 and 2 should be similar, as well as temperatures at the points 3 and 4 should be close to each other.

### *2.3.2. Flux tube for membrane water walls–fins attached to the flux tube*

If the device with five temperature measurement points is used (**Figure 1b**) then the uncer‐

2 249 999.9 3557.7 W/m , *mq* = ±

2 30 000.16 2379.9 W/(m K), *<sup>f</sup> h* = ±

318.0 0.22 C. *Tf* = ±°

Inspection of the results demonstrates the good accuracy in both cases. If the number of temperature measurement points is equal five then the uncertainties in the estimated heat flux and heat transfer coefficient become smaller, because an imprecise location of one thermo‐ couple has a smaller influence on the estimated parameters. For two temperature sensors placed at the forward part of the heat flux tube, an inaccurately situated temperature sensor

If the standard deviations of the variables measured directly are greater also standard

be doubled, the standard deviation will also be doubled. This conclusion follows from the analysis of Eq. (16). This has no effect on the convergence of the solution. For larger random errors in measured temperatures, a convergent solution is obtained, but with a slightly larger

To demonstrate the influence of the selection of parameter initial values on the solution convergence and on the number of iterations, the computations will be carried out for five

Adopting "measured data" from **Table 2** for the mesh shown in **Figure 5a**, the following results

Because the start values of parameters are far from the input values (exact solution) the number

If one of the temperature measurements is affected by a significant measurement error then the incorrect temperature measurement must be removed from the analysis. It was

(1)= 316°C,

(1)= 317°C,

K), *Tf* =318.00°C after 27 iterations,

K), *Tf* =318.00°C after 27 iterations.

measuring temperature points and using two sets of initial values:

, *hf* =29999.78W/ (m2

, *hf* =30000.03W/ (m2

of iteration is large. However, the solution found is correct in both cases.

(1) = 40 000 W / (m2 K), *Tf*

(1) = 10 000 W / (m2 K), *Tf*

, and *Tf*

= 2379.85W/(m<sup>2</sup>

K);2*σx*<sup>3</sup>

= 0.22 K.

will be greater. If the random errors will

;2*σx*<sup>2</sup>

= 3557.73 W/m<sup>2</sup>

The limits of the 95% uncertainty interval are as follows:

226 Numerical Simulation - From Brain Imaging to Turbulent Flows

tainties become smaller: 2*σx*<sup>1</sup>

influences to a larger extent the results.

number of iterations.

(1)= 100 000 W / m<sup>2</sup>

(1)= 400 000 W / m<sup>2</sup>

**•** *qm*

**•** *qm*

were obtained:

**•** *qm* =249999.80W / <sup>m</sup><sup>2</sup>

**•** *qm* =250000.15W / <sup>m</sup><sup>2</sup>

deviations of the determined parameters: *qm*, *hf*

, *hf*

, *hf*

In combustion chambers with membrane water walls, heat flux tube with longitudinal fins may be used (**Figure 2a**). To eliminate the influence of neighboring water wall tubes on the temperature field in the heat flux tube, the longitudinal fins are not welded to the adjacent tubes.

**Figure 7.** View factor distribution on the outer surface of the flux tube and adjacent water-wall tube which are shown in **Figure 2a**.

The distribution of the view factor on the surface of the flux tube, fin, and water wall tube is depicted in **Figure 7**. The origin of the *s* coordinate is shown in **Figure 2**. Because of the symmetry, only the representative water wall section illustrated in **Figure 8** needs to be analyzed.

**Figure 8.** Temperature distribution in the flux-tube cross section for: qm = 150000 W/m2 . Tf = 317°C and hf = 27000 W/(m2 K).

To illustrate that the maximum temperature of the fin tip is lower than the allowable temper‐ ature for the 20G steel, the flux-tube temperature was calculated using ANSYS/CFX package [25]. Variations of the view factor on the flux-tube weld and fin surface were also calculated with ANSYS/CFX. An inspection of the results shown in **Figure 8** shows that the maximum temperature of the fin does not exceed 375°C. Next to illustrate the effectiveness of the presented method test calculations were carried out. The thermal conductivity of the 20G steel was approximated by the function (18).

**Figure 9.** Temperature distribution (a) in the flux tube obtained from the solution of the inverse problem for the "ex‐ act" data: f1 = 419.66°C. f2 = 417.31°C. f3 = 374.90°C. f4 = 373.19°C. f5 = 318.01°C and iteration number (b) for the tem‐ perature T1.

The "measured" temperatures *fi* , *i* = 1, 2, …, 5 were generated artificially by means of ANSYS/ CFX for: *qm* = 250 000 W/m2 , *hf* = 30 000 W/(m2 K) and *Tf* = 318°C. The following values of "measured" temperatures were obtained *f*1 = 419.66°C, *f*2 = 417.31°C, *f*3 = 374.90°C, *f*4 = 373.19°C, *f*<sup>5</sup> = 318.01°C. The temperature distribution in the flux-tube cross section reconstructed on the basis of five measured temperatures is shown in **Figure 9a**. The inverse method presented in the chapter is very accurate because the estimated parameters:

*qm* = 250 000.063 W/m2 , *hf* = 30 000.054 W/(m2 K), and *Tf* = 318.0°C differ slightly from the input values.

**Figure 8.** Temperature distribution in the flux-tube cross section for: qm = 150000 W/m2

To illustrate that the maximum temperature of the fin tip is lower than the allowable temper‐ ature for the 20G steel, the flux-tube temperature was calculated using ANSYS/CFX package [25]. Variations of the view factor on the flux-tube weld and fin surface were also calculated with ANSYS/CFX. An inspection of the results shown in **Figure 8** shows that the maximum temperature of the fin does not exceed 375°C. Next to illustrate the effectiveness of the presented method test calculations were carried out. The thermal conductivity of the 20G steel

hf

= 27000 W/(m2

K).

was approximated by the function (18).

228 Numerical Simulation - From Brain Imaging to Turbulent Flows

. Tf

= 317°C and

**Figure 10.** Temperature distribution (a) in the flux tube obtained from the solution of the inverse problem for the "per‐ turbed" data: f1 = 420.16°C. f2 = 416.81°C. f3 = 375.40°C. f4 = 372.69°C. f5 = 318.01°C and iteration number (b) for the temperature T1.

To show the influence of the measurement errors on the determined parameters, the 95% confidence intervals were estimated. The following uncertainties of the measured values were assumed (at 95% confidence interval): 2*<sup>σ</sup> <sup>f</sup> <sup>j</sup>* <sup>=</sup> <sup>±</sup> 0.4K, 2*σrj* = ± 0.10mm, 2*σφ<sup>j</sup>* = ± 1.0° , *j* =1, …, 5, 2*σ<sup>k</sup>* = ± 1.0W/ (mK). For these test calculations, the 95% uncertainties in the param‐ eters measured directly were taken two times greater than the uncertainties in the previous case analyzed in Section 4.1. The limits of the 95% uncertainty interval are as follows:

2 250 000.06 7102.46 W/m , *mq* = ±

2 30 000.05 4735.71 W/(m K), *<sup>f</sup> h* = ±

318.00 0.41 C. *Tf* = ±°

Despite the doubling of uncertainties, the results are quite good. Larger relative errors in determined the heat transfer coefficient are due to a small difference in temperature between the inner surface of the flux tube and the fluid temperature. For this reason, the impact of the uncertainties in direct measurements on the estimated heat transfer coefficient is greater.

The uncertainties (95% confidence interval) of the coefficients *xi* were determined using the error propagation rule (16). The calculated uncertainties are: ±4.1% for *qm*, ±27.3% for *hf* and ±0.1% for *Tf* . The accuracy of the results obtained is acceptable.

Then, the inverse analysis was carried out for perturbed data: *f*1 = 420.16°C, *f*<sup>2</sup> = 416.81°C, *f*<sup>3</sup> = 375.40°C, *f*4 = 372.69°C, *f*5 = 318.01°C. The reconstructed temperature distribution is illustrated in **Figure 10a**. The obtained results are *qm* = 250 118.613 W/m2 , *hf* = 30 050.041 W/(m2 K) and *Tf* = 317.99°C. The influence of the error in the measured temperatures on the estimated param‐ eters is small.

The number of iterations in the Levenberg–Marquardt procedure is small in both cases (**Figures 9b** and **10b**).
