**4.2 Temperature-dependent thermal conductivity**

*<sup>θ</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup> *λ*<sup>2</sup> � *λ*<sup>1</sup>

*Inverse Heat Conduction and Heat Exchangers*

p

physical properties of the material are: *ρ* = 7900 kg m�<sup>3</sup>

solution of the linear transient heat conduction problem [30] is

X∞ *n*¼1

and solved with the following boundary and initial conditions.

*<sup>∂</sup>θ*ð Þ 0, *<sup>τ</sup>*

*<sup>∂</sup>θ*ð Þ 1, *<sup>τ</sup>*

where *λ*<sup>1</sup> ¼ �2 � *Bi* þ

*K* (average) = 35 Wm�<sup>1</sup> K�<sup>1</sup>

**4.1 Average thermal conductivity**

*θ*ð Þ¼ *X*, *τ* 1 � 2

equation.

and

**168**

determine the unknown parameter.

ð Þ *λ*<sup>2</sup> þ 2*Bi e*

**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 thermo-

gas temperature *Tg* = 2946.2 K are used in the solution of the heat conduction

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

> *Bi Bi*<sup>2</sup> <sup>þ</sup> *<sup>λ</sup>*<sup>1</sup> *<sup>n</sup>* <sup>þ</sup> *Bi* � �

For estimating unknown boundary condition, the heat conduction equation is

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.

*λ*2*e*

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

and *λ*<sup>2</sup> ¼ �2 � *Bi* �

*<sup>λ</sup>*1*<sup>τ</sup>* � *<sup>λ</sup>*1*<sup>e</sup>*

*<sup>θ</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> *λ*<sup>2</sup> � *λ*<sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *Bi*<sup>2</sup> <sup>þ</sup> <sup>4</sup> *<sup>λ</sup>*1*<sup>τ</sup>* � ð Þ *<sup>λ</sup>*<sup>1</sup> <sup>þ</sup> <sup>2</sup>*Bi <sup>e</sup>*

. Initial temperature *Ti* = 300 K and combustion

cos½ � *λn*ð Þ 1 � *X* cos *λ<sup>n</sup>*

*<sup>∂</sup><sup>X</sup>* <sup>¼</sup> *Bi*½ � *<sup>θ</sup>*ð Þ� 0, *<sup>τ</sup>* <sup>1</sup> , *<sup>τ</sup>*><sup>0</sup> (15)

*<sup>∂</sup><sup>X</sup>* <sup>¼</sup> 0, *<sup>τ</sup>*><sup>0</sup> (16)

*θ*ð Þ¼ *X*, 0 0, for all *X* (17)

*λ* tan *λ* ¼ *Bi* (14)

*e* �*λ*<sup>2</sup>

*<sup>n</sup><sup>τ</sup>* (13)

*<sup>λ</sup>*2*<sup>τ</sup>* � � (11)

p

*<sup>λ</sup>*2*<sup>τ</sup>* � � (12)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *Bi*<sup>2</sup> <sup>þ</sup> <sup>4</sup>

.

, *Cp* = 545 J kg�<sup>1</sup> K�<sup>1</sup>

,

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 Wm<sup>1</sup> K<sup>1</sup> and 2.718 Wm<sup>1</sup> K<sup>2</sup> , respectively. The advantage of using the exact solution is found directly at specified location and time as compared to the numerical method which needs the computation from the initial state.
