**3. Heat conduction equation**

#### **3.1 Analytical solution**

The computation of the turbulent convective heat transfer coefficient from combustion gases to the rocket nozzle wall is based on the Bartz's equation [33] incorporating the effects of compressibility, throat curvature and variation of transport properties in the boundary layer. The transient heat conduction in a one-dimensional Cartesian coordinate system having two parallel plane surfaces *Sn* (*n* = 1, 2) of a slab may be written in dimensional form [34] as.

$$\frac{\partial \theta(X,\tau)}{\partial \tau} = \frac{\partial}{\partial X} \left[ K(\theta) \frac{\partial \theta(X,\tau)}{\partial X} \right], \text{in region } R, \tau > 0 \tag{3}$$

with following initial and boundary conditions:

$$\theta(\mathbf{X}, \mathbf{0}) = f\_i(\mathbf{X}), \text{in region } \mathbf{R}, \tag{4}$$

$$-\frac{\partial \theta(\mathbf{S}\_n, \tau)}{\partial \mathbf{X}} = Bi\_n[\theta(\mathbf{S}\_n, \tau)], \text{or } \frac{q\_w L}{(T\_\mathbf{g} - T\_i)} \text{ on boundary } \mathbf{S}\_n, \tau > 0 \tag{5}$$

where *fi* is initial temperature distribution in the region *R* of the slab. Eq. (4) represents both convective heat transfer or heat flux condition as applied to the inner surface.

We now consider the constant thermal property solution and can be written in terms of eigen function *ψ λ*ð Þ *<sup>m</sup>*,*X* as

$$\theta(X,\tau) = \sum\_{m=1}^{\infty} \exp\left[-\lambda\_m^2 \tau\right] \varphi(\lambda\_m, X) \int\_{\mathcal{R}} \varphi\left(\lambda\_m \overline{X}\right) f(\overline{X}) d\overline{X} \tag{6}$$

In the above Eq. (5), *Bi* or *qw* is the unknown parameter to be determined using measured temperature time history at location *xm* as depicted in **Figure 1**. In estimating the unknown condition, one has to minimize the absolute difference between the calculated and measured temperature at specified location and time (*xm*, *τ*) in a prescribed tolerance value using an iteration procedure. The iteration scheme is described in the following sections.

#### **3.2 Inverse algorithm**

The IHCP is solved by comparing calculated and measured temperature using an iterative technique [30]. In estimating *qw*, one minimizes

*Influence of Input Parameters on the Solution of Inverse Heat Conduction Problem DOI: http://dx.doi.org/10.5772/intechopen.91000*

$$F(q\_w) \approx |\theta\_c(X\_m, \tau) - \theta\_m(X\_m, \tau)|\tag{7}$$

where *θ<sup>c</sup>* and *θ<sup>m</sup>* are the calculated and measured temperatures at (*Xm*, *τ*), respectively. The computed temperature is a nonlinear function of unknown parameters such as wall heat flux or convective heat transfer coefficient. Temperature is calculated using Eq. (6) and compared with the measured temperature as expressed in Eq. (7). The inverse problem starts with initial guess value of the unknown parameter. The second step is to correct the previous guessed unknown parameter using the Newton-Raphson method. The sensitivity coefficient can be obtained by differentiating temperature with respect to wall heat flux *qw*. The iteration procedure will continue until <sup>j</sup>*F*(*qw*)<sup>j</sup> <sup>≤</sup> <sup>10</sup>�<sup>4</sup> . This iterative scheme estimates the component of the *qw* at a time and thus may be considered on-line method.

The inverse method for solving a value of *qw*(*0*, *τ*) is as follows. Initiate with an initial guess value of *qw*, satisfy the convergence criterion, and implement the Newton-Raphson to obtain the estimate value.

Now, it is possible to estimate convective heat transfer coefficient and combustion gas temperature in conjunction with measured temperature history [35]. The equation for converting the calculated heat flux to the heat transfer coefficient is

$$Bi = \frac{Lq\_w}{K(\theta) \left(T\_g - T\_i\right)}\tag{8}$$

In the foregoing equation,*T*<sup>g</sup> is an unknown quantity and can be estimated using again the above-mentioned minimization and iteration methods. The convergence criterion for the iterative scheme remains same as mentioned above.

#### **3.3 Numerical methods**

An optimization method based on a direct and systematic search region reduction optimization method [32] can be employed to estimate the unknown convective heat transfer coefficient in a typical rocket nozzle. The most attractive feature of the direct search scheme is the simplicity of computer programming. The pseudo-random algorithm, an effective tool for optimization, does not require computation of derivatives but depends only on function evaluation. It works even when the differentiability requirements cannot be ensured in the feasible domain. For initiating the search only an estimate of the feasible domain is needed. Therefore, another advantage of the method is that the starting condition is not

The computation of the turbulent convective heat transfer coefficient from combustion gases to the rocket nozzle wall is based on the Bartz's equation [33] incorporating the effects of compressibility, throat curvature and variation of transport properties in the boundary layer. The transient heat conduction in a one-dimensional Cartesian coordinate system having two parallel plane surfaces *Sn*

> *<sup>K</sup>*ð Þ*<sup>θ</sup> <sup>∂</sup>θ*ð Þ *<sup>X</sup>*, *<sup>τ</sup> ∂X* � �

> > *Tg* � *Ti*

where *fi* is initial temperature distribution in the region *R* of the slab. Eq. (4) represents both convective heat transfer or heat flux condition as applied to the

We now consider the constant thermal property solution and can be written in

*<sup>m</sup><sup>τ</sup>* � �*ψ λ*ð Þ *<sup>m</sup>*, *<sup>X</sup>*

In the above Eq. (5), *Bi* or *qw* is the unknown parameter to be determined using measured temperature time history at location *xm* as depicted in **Figure 1**. In estimating the unknown condition, one has to minimize the absolute difference between the calculated and measured temperature at specified location and time (*xm*, *τ*) in a prescribed tolerance value using an iteration procedure. The iteration

The IHCP is solved by comparing calculated and measured temperature using an

ð *R*

*ψ λmX* � �*f X*

� �*dX* (6)

, in region *R*, *τ*>0 (3)

ð Þ *X* , in region *R*, (4)

� � on boundary S*n*, *<sup>τ</sup>*>0 (5)

(*n* = 1, 2) of a slab may be written in dimensional form [34] as.

*∂X*

*θ*ð Þ¼ *X*, 0 *fi*

exp �*λ*<sup>2</sup>

*<sup>∂</sup><sup>X</sup>* <sup>¼</sup> *Bin*½ � *<sup>θ</sup>*ð Þ *Sn*, *<sup>τ</sup>* , or *qwL*

with following initial and boundary conditions:

*<sup>∂</sup>θ*ð Þ *<sup>X</sup>*, *<sup>τ</sup> <sup>∂</sup><sup>τ</sup>* <sup>¼</sup> *<sup>∂</sup>*

� *<sup>∂</sup>θ*ð Þ *Sn*, *<sup>τ</sup>*

terms of eigen function *ψ λ*ð Þ *<sup>m</sup>*,*X* as

*<sup>θ</sup>*ð Þ¼ *<sup>X</sup>*, *<sup>τ</sup>* <sup>X</sup><sup>∞</sup>

scheme is described in the following sections.

*m*¼1

iterative technique [30]. In estimating *qw*, one minimizes

inner surface.

**3.2 Inverse algorithm**

**166**

crucial; any reasonable value will do.

*Inverse Heat Conduction and Heat Exchangers*

**3. Heat conduction equation**

**3.1 Analytical solution**

It is not always feasible to obtain analytical solution of temperature-dependent thermal conductivity and radiation boundary condition. The Crank-Nicolson finite difference method with two-time level implicit numerical scheme [36] has been employed to solve the nonlinear conduction problem with the Newton-Raphson method to consider the radiation boundary condition.

Deforming or moving finite elements method [37] is used to solve linear heat conduction equation. The moving finite element [38] is used to consider the time delay in the measurement of back wall temperature.

#### **3.4 Two-nodes system of transient heat conduction equation**

For only two nodes the system of [39] equations reduce to the following pair of equations:

$$\frac{d\theta\_0}{d\tau} = \frac{1}{\left(\Delta X\right)^2} \left[ -\mathcal{Q}(\theta\_1 + Bi\Delta X)\theta\_0 + \mathcal{Q}\theta\_1 \right] \tag{9}$$

$$\frac{d\theta\_1}{d\tau} = \frac{1}{\left(\Delta X\right)^2} \left[2\theta\_0 - 2\theta\_1\right] \tag{10}$$

where 0 and 1 represent node in a slab of finite thickness. These are the exact solutions to the system of two ordinary differential equations which resulted from a two-node finite-difference approximation to the original problem.

$$\theta\_0 = \frac{1}{\lambda\_2 - \lambda\_1} \left[ (\lambda\_2 + 2Bi)e^{\lambda\_1 \tau} - (\lambda\_1 + 2Bi)e^{\lambda\_2 \tau} \right] \tag{11}$$

$$\theta\_1 = \frac{\mathbf{1}}{\lambda\_2 - \lambda\_1} \left[ \lambda\_2 e^{\lambda\_1 \tau} - \lambda\_1 e^{\lambda\_2 \tau} \right] \tag{12}$$

An iterative scheme is used to solve inverse problem [30]. The iteration is carried out till the absolute difference between calculated and measured temperature is

*Influence of Input Parameters on the Solution of Inverse Heat Conduction Problem*

values of the convective heat transfer coefficient based on the exact solution of heat conduction equation with the calculated values of Bartz [33]. Bartz's equation calculates conservative estimates for the convective heat transfer to the wall [40].

An iteration procedure [41] is employed in conjunction with exact solution to predict convective heat transfer coefficient from the measured temperature-time data at the outer wall of the nozzle as shown in **Table 2**. The expression for

temperature-dependent conductivity is *K*(*T*) = *k0 βT*. The value of *k0* and *β* are 57

solution is found directly at specified location and time as compared to the numer-

Deforming or moving finite element is used to consider the time delay in temperature at the outer wall of the slab [37]. Estimated values of wall heat flux and heat transfer coefficient are tabulated in **Table 3**. It can be observed from the table that the estimated wall quantities are having significant influence on the predicted unknown boundary conditions. This example is extended to consider spatial grid changed and temporal dependence on the numerical solution using moving finite

Numerical analysis of nonlinear heat conduction with a radiation boundary condition [36] is carried out to estimate wall heat flux using temperature history on

thermophysical properties of the material of mild steel. **Table 4** shows comparison

6 0.2950 0.0098 0.0096 1821.9 2254.2 7 0.3109 0.0159 0.0158 1810.0 2254.2 8 0.2996 0.0212 0.0211 1610.3 2254.2 9 0.3244 0.0301 0.0302 1690.9 2254.2 10 0.3340 0.0386 0.0385 1669.7 2254.2 11 0.3416 0.0473 0.0472 1641.9 2254.2 12 0.3302 0.0529 0.0529 1497.6 2254.2 13 0.3312 0.0602 0.0604 1443.1 2254.2 14 0.3409 0.0677 0.0676 1387.0 2254.2 15 0.3442 0.0781 0.0782 1413.0 2254.2 16 0.3475 0.0862 0.0861 1383.7 2254.2

the back wall of the rocket nozzle. The high temperature variation alters

*t***, s** *θ***<sup>0</sup> at inner surface** *θ<sup>c</sup>* **at outer surface** *θ<sup>m</sup>* **at outer surface** *h***, W/m<sup>2</sup>**

. **Table 1** exhibits the comparison between the estimated

, respectively. The advantage of using the exact

**K** *hB***, W/m<sup>2</sup>**

**K**

less than or equal to 10<sup>4</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.91000*

Wm<sup>1</sup> K<sup>1</sup> and 2.718 Wm<sup>1</sup> K<sup>2</sup>

element method [38].

**Table 1.**

**169**

*Solution of inverse heat conduction problem.*

**4.4 Nonlinear boundary condition**

**4.2 Temperature-dependent thermal conductivity**

ical method which needs the computation from the initial state.

**4.3 Numerical solution with various computational grids**

where *λ*<sup>1</sup> ¼ �2 � *Bi* þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *Bi*<sup>2</sup> <sup>þ</sup> <sup>4</sup> p and *λ*<sup>2</sup> ¼ �2 � *Bi* � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *Bi*<sup>2</sup> <sup>þ</sup> <sup>4</sup> p .

Solution of the above simultaneous equation calculates the temperature with a given value of *Bi*. The solution is now solving simultaneously Eqs. (11) and (12) to determine the unknown parameter.

#### **4. Inverse problem of heat conduction applied to a rocket nozzle**

The influence of constant (average) thermal conductivity, temperaturedependent thermal conductivity, computational grid in numerical solver, nonlinear boundary condition, cylindrical coordinate and the estimation of the wall heat flux and convective heat transfer is carried out by employing measured temperature history of a rocket nozzle of a solid motor. Solution of linear heat conduction equation is used to estimate the convective heat transfer coefficient with the measured temperature data of outer wall of a rocket nozzle. The running time of rocket motor is 16 s. The nozzle wall thickness *L* = 0.0211 m. The thermophysical properties of the material are: *ρ* = 7900 kg m�<sup>3</sup> , *Cp* = 545 J kg�<sup>1</sup> K�<sup>1</sup> , *K* (average) = 35 Wm�<sup>1</sup> K�<sup>1</sup> . Initial temperature *Ti* = 300 K and combustion gas temperature *Tg* = 2946.2 K are used in the solution of the heat conduction equation.

#### **4.1 Average thermal conductivity**

Prediction of convective heat transfer coefficient is carried out in conjunction with the calculated and measured temperature history at outer surface of nozzle divergent in a solid rocket motor static test. The constant thermal conductivity solution of the linear transient heat conduction problem [30] is

$$\theta(X,\tau) = 1 - 2\sum\_{n=1}^{\infty} \frac{Bi}{\left(Bi^2 + \lambda\_n^1 + Bi\right)} \frac{\cos\left[\lambda\_n(1-X)\right]}{\cos\lambda\_n} e^{-\lambda\_n^2 \tau} \tag{13}$$

$$
\lambda \tan \lambda = Bi \tag{14}
$$

For estimating unknown boundary condition, the heat conduction equation is and solved with the following boundary and initial conditions.

$$\frac{\partial \theta(\mathbf{0}, \tau)}{\partial X} = Bi[\theta(\mathbf{0}, \tau) - \mathbf{1}], \tau > \mathbf{0} \tag{15}$$

$$\frac{\partial \theta(\mathbf{1}, \tau)}{\partial X} = \mathbf{0}, \tau > \mathbf{0} \tag{16}$$

and

$$\theta(X,0) = 0, \text{for all } X \tag{17}$$

Exact analytical solution of transient heat conduction as written in Eq. (13) is used to estimate convective heat transfer on the inner surface of the rocket nozzle. An iterative scheme is used to solve inverse problem [30]. The iteration is carried out till the absolute difference between calculated and measured temperature is less than or equal to 10<sup>4</sup> . **Table 1** exhibits the comparison between the estimated values of the convective heat transfer coefficient based on the exact solution of heat conduction equation with the calculated values of Bartz [33]. Bartz's equation calculates conservative estimates for the convective heat transfer to the wall [40].
