**4. Description of the numerical method**

Velocity, pressure, and temperature of the steam, as well as the temperature of the tube wall with the complex cross section, will be computed using the ANSYS/CFX v13.0 software [25].

The direct and inverse problems will be solved. In the inverse problem, the steam temperature at the superheater inlet and the heat transfer coefficient on the flue-gas side will be determined using measured steam temperatures at selected locations in the superheater. If the flue-gas side heat transfer coefficient is determined based on the measured steam temperature at the superheater outlet, the secant method is used. In this case, it is assumed that the inlet steam temperature is also measured. The Levenberg–Marquardt method is used to solve the least squares problem when the inlet steam temperature and the flue-gas side heat transfer coeffi‐ cient are determined based on the measured steam temperature at the selected points along the steam flow path. At every iteration step, a direct problem was solved using the ANSYS/CFX v13.0 software. Time-averaged three-dimensional equations of conservation of mass, momentum and energy are solved using an element-based finite volume method. The heat transfer in the flowing steam was modeled using equations of mass, momentum, and energy conservation which can be written as follows [25]:

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{U}) = 0 \tag{19}$$

$$\frac{\partial(\rho \mathbf{U})}{\partial t} + \nabla \cdot (\rho \mathbf{U} \otimes \mathbf{U}) = -\nabla p + \nabla \cdot \boldsymbol{\tau} + \mathbf{S}\_{\mathbf{M}} \tag{20}$$

$$\frac{\partial(\rho h)}{\partial t} - \frac{\partial p}{\partial t} + \nabla \cdot (\rho \,\mathrm{U}h) = \nabla \cdot (k \nabla T) + \mathbf{U} \cdot \nabla p + \tau : \nabla \mathbf{U} + \mathbf{S}\_{\mathrm{E}} \tag{21}$$

where the stress tensor *τ* is related to the strain rate by

$$
\tau = \mu (\nabla \mathbf{U} + (\nabla \mathbf{U})^T - \frac{2}{3} \delta \nabla \cdot \mathbf{U}) \tag{22}
$$

Temperature distribution in double omega tube walls (**Figure 18**) is governed by the heat conduction equation

$$\mathcal{L}\_w \rho\_w \frac{\partial T\_w}{\partial t} = \nabla \cdot \left[ k\_w(T\_w) \nabla T\_w \right] \tag{23}$$

Two equation turbulence model SST (Shear Stress Transport), which was developed based on the Wilcox *k*–*ω* and *k*–*ε* turbulence models [53], was used to model the flow turbulence. The baseline *k*–*ω* model is a two equations turbulence model that offers a good compromise between numerical effectiveness and computational accuracy. This model is a combination of the *k*–*ω* and *k*–*ε* models.

The Wilcox *k*–*ω* model is given by two equations:

**4. Description of the numerical method**

240 Numerical Simulation - From Brain Imaging to Turbulent Flows

energy conservation which can be written as follows [25]:

*t* r

where the stress tensor *τ* is related to the strain rate by

t m

*t t* r

¶ ¶

conduction equation

*t* r

( ) ( ) *<sup>p</sup>*

( ) ( ) () : *h p h kT p*

¶ ¶ - +Ñ× =Ñ× Ñ + ×Ñ + Ñ +

<sup>2</sup> ( () ) <sup>3</sup> *T*

Temperature distribution in double omega tube walls (**Figure 18**) is governed by the heat

[() ] *<sup>w</sup> w w ww w <sup>T</sup> c kT T*

¶ =Ñ× Ñ

*t* r

d

= Ñ + Ñ - Ñ× **UU U**

r

r

¶

Velocity, pressure, and temperature of the steam, as well as the temperature of the tube wall with the complex cross section, will be computed using the ANSYS/CFX v13.0 software [25].

The direct and inverse problems will be solved. In the inverse problem, the steam temperature at the superheater inlet and the heat transfer coefficient on the flue-gas side will be determined using measured steam temperatures at selected locations in the superheater. If the flue-gas side heat transfer coefficient is determined based on the measured steam temperature at the superheater outlet, the secant method is used. In this case, it is assumed that the inlet steam temperature is also measured. The Levenberg–Marquardt method is used to solve the least squares problem when the inlet steam temperature and the flue-gas side heat transfer coeffi‐ cient are determined based on the measured steam temperature at the selected points along the steam flow path. At every iteration step, a direct problem was solved using the ANSYS/CFX v13.0 software. Time-averaged three-dimensional equations of conservation of mass, momentum and energy are solved using an element-based finite volume method. The heat transfer in the flowing steam was modeled using equations of mass, momentum, and

( )0

 t

**<sup>U</sup> U U <sup>S</sup>** (20)

 t

**U U USE** (21)

¶ (23)

**U** (19)

(22)

r

¶ +Ñ× =

¶ + Ñ × Ä = -Ñ + Ñ × + ¶ **<sup>M</sup>**

$$\frac{\partial \left(\rho k\right)}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_j} \left(\rho U\_j k\right) = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left(\mu + \frac{\mu\_t}{\sigma\_{k1}}\right) \frac{\partial k}{\partial \mathbf{x}\_j} \right] + P\_k - \beta^\* \rho k o o \tag{24}$$

$$\frac{\partial \left(\rho oo\right)}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_{j}} \left(\rho U\_{j}\rho o\right) = \frac{\partial}{\partial \mathbf{x}\_{j}} \left[ \left(\mu + \frac{\mu\_{r}}{\sigma\_{a1}}\right) \frac{\partial oo}{\partial \mathbf{x}\_{j}} \right] + \alpha\_{1} \frac{oo}{k} P\_{k} - \beta\_{1} {}^{\*}\rho oo^{2} \tag{25}$$

To develop the baseline *k*–*ω* turbulence model, the transformed *k*–*ε* model is also used:

$$\frac{\partial \left(\rho k\right)}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_{j}} \left(\rho U\_{j} k\right) = \frac{\partial}{\partial \mathbf{x}\_{j}} \left[ \left(\mu + \frac{\mu\_{i}}{\sigma\_{k2}}\right) \frac{\partial k}{\partial \mathbf{x}\_{j}} \right] + P\_{i} - \beta^{\*} \rho k o o \tag{26}$$

$$\frac{\partial(\rho o)}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_j}(\rho U\_j o) = $$

$$= \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_r}{\sigma\_{a2}} \right) \frac{\partial o}{\partial \mathbf{x}\_j} \right] + 2\rho \frac{1}{\sigma\_{a2}} \frac{\partial k}{\partial \mathbf{x}\_j} \frac{\partial o}{\partial \mathbf{x}\_j} + \alpha\_2 \frac{o}{k} P\_k - \beta\_2 \rho o^2 \tag{27}$$

Multiplying Eqs. (24) and (25) of the Wilcox model by weighting function *F*1, the transformed *k*–*ε* Eqs. (26) and (27) by a weighting function (1 – *F*1) and adding the transformed equations gives the baseline *k*–*ω* model:

$$\frac{\partial(\rho k)}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_j} (\rho U\_j k) = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_r}{\sigma\_{\mathbb{k}3}} \right) \frac{\partial k}{\partial \mathbf{x}\_j} \right] + P\_k - \beta^\* \rho k o o + P\_{\mathbb{k}b} \tag{28}$$

$$\begin{aligned} \frac{\partial(\rho o)}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_j} (\rho U\_j o) &= \\ = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_t}{\sigma\_{o3}} \right) \frac{\partial o}{\partial \mathbf{x}\_j} \right] + \left( 1 - F\_i \right) 2 \rho \frac{1}{\sigma\_{o2}} \frac{\partial k}{\partial \mathbf{x}\_j} \frac{\partial o}{\partial \mathbf{x}\_j} + \alpha\_3 \frac{o}{k} P\_k - \beta\_3 \rho o^2 + P\_{ob} \end{aligned} \tag{29}$$

The baseline *k*–*ω* model has the advantages of the Wilcox *k*–*ω*, and the *k*–*ε* but is not able to predict onset and amount of flow separation from smooth surfaces [53]. The main reason for this drawback is neglecting of the transport of the turbulent shear stress. This causes an overprediction of the eddy-viscosity. In order to eliminate this deficiency, the shear stress transport (SST) turbulence model was developed [53], in which the turbulent viscosity *ν<sup>t</sup>* is limited by the following equation:

$$\nu\_i = \frac{\alpha\_i k}{\max(\alpha\_i o, SF\_1)}\tag{30}$$

$$\begin{array}{c} \frac{\partial \ T\_w}{\partial n} = 0 \\\\ \frac{\partial T\_w}{\partial n} = 0 \\\\ \frac{\partial T\_w}{\partial n} \end{array}$$

$$\sigma\_w^{\text{h}\_\mathbf{g}}$$

**Figure 19.** One-fourth of a tube cross section.

Because of the symmetry, only one-fourth of a tube cross section was analyzed (**Figure 19**). Heat is transferred by convection from the combustion chamber to the tube through the surface E–F. The surfaces A–F and B–C–D–E are thermally insulated. The tube inner surface A–B is cooled by the flowing live steam. The following boundary conditions were assumed for the tube wall (**Figure 19**):

Solving Inverse Heat Transfer Problems When Using CFD Modeling http://dx.doi.org/10.5772/63807 243

$$\left.T\right|\_{A-B} = T\_w \left|\_{A-B}, \quad k \right. \frac{\left.\hat{\mathcal{O}}T}{\left.\hat{\mathcal{O}}\,\mathbf{n}}\right|\_{A-B} = -k\_w \left.\frac{\left.\hat{\mathcal{O}}T\_w}{\left.\hat{\mathcal{O}}\,\mathbf{n}}\right|\_{A-B}\right|\right. \tag{31}$$

$$k\_w \frac{\partial T\_w}{\partial n}|\_{B-C-D-E} = 0\tag{32}$$

$$k\_w \frac{\partial T\_w}{\partial n}|\_{E-F} = h\_g \left(T\_g - T\_w|\_{E-F}\right) \tag{33}$$

$$k\_{\boldsymbol{w}} \frac{\partial \boldsymbol{T}\_{\boldsymbol{w}}}{\partial \boldsymbol{n}} \mid\_{\mathcal{A}-\boldsymbol{F}} = \mathbf{0} \tag{34}$$

Also, the fluid temperature and the steam mass flow rate or steam velocity are known at the inlet of the tube. The steam pressure is given at the outlet of the tube. The direct conjugate heat transfer problem described by Eqs. (19) and (29) with appropriate boundary conditions was solved using the ANSYS/CFX software.
