**4.4 Nonlinear boundary condition**

Numerical analysis of nonlinear heat conduction with a radiation boundary condition [36] is carried out to estimate wall heat flux using temperature history on the back wall of the rocket nozzle. The high temperature variation alters thermophysical properties of the material of mild steel. **Table 4** shows comparison


#### **Table 1.**

*Solution of inverse heat conduction problem.*


#### **Table 2.**

*Comparison between iterative and Beck methods.*


#### **Table 3.**

*Wall heat flux at various grid arrangements.*

between the estimated convective heat transfer coefficients with the Bartz solution [33]. Effects of nonlinear IHCP with radiation boundary condition are investigated and results are presented in **Table 4**.

on the estimated values of heat transfer coefficient. **Table 5** shows the effect of

The calculated convective heat transfer coefficients and inner wall temperature are used to determine the wall heat flux and the combustion temperature using Eq. (8). The iterative scheme is based on relation between wall heat flux and convective heat transfer coefficient [35]. **Table 6** shows the predicted values of wall heat flux and convective heat transfer coefficient. The IHCP is extended to determine wall heat flux in conjunction with convective heat transfer coefficient. A similar IHCP but referring to the 122 mm medium-range missile during correction

geometrical parameters on the predicted heat transfer coefficient.

engine operation has been considered by Zmywaczyk et al. [43].

**4.6 Estimation of heat flux and heat transfer coefficient**

*<sup>t</sup>***, s** *To***, K at** *<sup>X</sup>* **= 0** *Tm* **K at** *<sup>X</sup>* **= 1** *qc* **106 W/m<sup>2</sup>** *<sup>h</sup>* **W/m<sup>2</sup>**

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

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

*<sup>t</sup>***, s** *To* **K at** *<sup>X</sup>* **= 0** *Tm* **K at** *<sup>X</sup>* **= 1** *qc* **106 W/m<sup>2</sup>** *<sup>h</sup>***, W/m<sup>2</sup>**

**Table 4.**

**Table 5.**

**171**

*Inverse problem in a hollow cylinder.*

*Solution with nonlinear boundary condition.*

6 659.8 326 2.3547 950.0 2254.2 3137 2946.2 7 801.0 342 2.3899 1019.6 2254.2 3122 2946.2 8 900.7 356 2.2211 992.4 2254.2 3115 2946.2 9 996.3 380 2.6489 1237.1 2254.2 3113 2946.2 10 1050.5 402 2.3670 1135.5 2254.2 3108 2946.2 11 1066.4 425 1.7100 827.3 2254.2 3104 2946.2 12 1201.8 440 2.8144 1459.2 2254.2 3099 2946.2 13 1320.0 460 2.6559 1467.0 2254.2 3098 2946.2 14 1354.8 479 1.7595 991.7 2254.2 3095 2946.2 15 1383.4 507 1.3810 791.4 2254.2 3094 2946.2 16 1414.9 528 1.1684 681.8 2254.2 3094 2946.2

6 1260.2 326 3.6805 1789.6 2254.2 3316 2946 7 1175.9 342 3.3995 1628.0 2254.2 3264 2946 8 1160.7 356 2.4745 1181.4 2254.2 3255 2946 9 1165.8 380 2.5385 1194.7 2254.2 3290 2946 10 1196.0 402 2.5348 1261.1 2254.2 3206 2946 11 1192.3 425 2.3385 1166.4 2254.2 3197 2946 12 1205.8 440 2.2094 1114.8 2254.2 3187 2946 13 1211.0 460 2.1333 1229.5 2254.2 2946 2946 14 1222.1 479 2.0441 1187.5 2254.2 3943 2946 15 1237.1 507 2.0626 1206.7 2254.2 2946 2946 16 1249.1 528 2.0027 1180.9 2254.2 2945 2946

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

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

**K** *Tgc* **K** *Tg* **K**

**K θg, K θgc, K**

#### **4.5 Heat conduction in a hollow cylinder**

A grid point shift strategy [42] is adapted to solve inverse conduction problem in a radial coordinate of rocket nozzle with inner and outer radius of rocket nozzle. The inner and outer radius of the nozzle is 0.0839 m and 0.0105 m, respectively. The purpose of the present example to investigate the influence of radial coordinate

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


#### **Table 4.**

*Solution with nonlinear boundary condition.*


#### **Table 5.** *Inverse problem in a hollow cylinder.*

on the estimated values of heat transfer coefficient. **Table 5** shows the effect of geometrical parameters on the predicted heat transfer coefficient.

#### **4.6 Estimation of heat flux and heat transfer coefficient**

The calculated convective heat transfer coefficients and inner wall temperature are used to determine the wall heat flux and the combustion temperature using Eq. (8). The iterative scheme is based on relation between wall heat flux and convective heat transfer coefficient [35]. **Table 6** shows the predicted values of wall heat flux and convective heat transfer coefficient. The IHCP is extended to determine wall heat flux in conjunction with convective heat transfer coefficient. A similar IHCP but referring to the 122 mm medium-range missile during correction engine operation has been considered by Zmywaczyk et al. [43].

between the estimated convective heat transfer coefficients with the Bartz solution [33]. Effects of nonlinear IHCP with radiation boundary condition are investigated

*t***, s** *θ***(***0***,** *τ***)** *h***, W/m2**

**Iterative method Beck method** *θc***(1, τ) θ***m***(1, τ) Iterative method Beck method**

**Uniform grid Non-uniform grid Moving grid**

*h***c, W/m<sup>2</sup> K** *qw* **106 , W/m<sup>2</sup>**

*h***c, W/m<sup>2</sup> K**

*qw* **<sup>10</sup><sup>6</sup> , W/m<sup>2</sup>**

6 326 3.715 1964.5 3.846 2044.9 4.517 2412.1 7 342 2.700 1408.8 2.848 1449.6 2.818 1485.9 8 356 2.698 1436.9 2.840 1531.6 2.820 1512.8 9 380 2.704 1463.0 2.589 1569.8 2.842 1552.8 10 402 2.705 1491.4 2.858 1603.3 2.846 1586.7 11 425 2.704 1518.9 2.852 1632.6 2.845 1618.5 12 440 2.691 1539.7 2.805 1636.2 2.812 1630.8 13 460 2.683 1564.6 2.776 1649.7 2.791 1650.6 14 479 2.673 1588.1 2.738 1657.1 2.764 1665.9 15 507 2.094 1226.4 2.015 1190.6 2.091 1235.4 16 528 2.086 1231.8 1.981 1178.5 2.067 1231.6

6 0.0883 0.0838 0.0099 0.0098 536.6 581.7 7 0.1067 0.1075 0.0158 0.0159 600.6 587.0 8 0.1144 0.1116 0.0220 0.0212 592.6 598.4 9 0.1367 0.1367 0.0302 0.0302 674.2 685.3 10 0.1522 0.1545 0.0386 0.0385 712.9 693.2 11 0.1654 0.1690 0.0472 0.0472 737.4 730.0 12 0.1686 0.1639 0.0529 0.0529 718.2 721.9 13 0.1773 0.1777 0.0605 0.0605 723.6 725.8 14 0.1844 0.1813 0.0677 0.0676 723.0 725.1 15 0.1944 0.2040 0.0781 0.0782 753.6 765.0 16 0.2083 0.2174 0.0862 0.0862 758.3 770.0

**K**

A grid point shift strategy [42] is adapted to solve inverse conduction problem in a radial coordinate of rocket nozzle with inner and outer radius of rocket nozzle. The inner and outer radius of the nozzle is 0.0839 m and 0.0105 m, respectively. The purpose of the present example to investigate the influence of radial coordinate

and results are presented in **Table 4**.

*Wall heat flux at various grid arrangements.*

*Comparison between iterative and Beck methods.*

*Inverse Heat Conduction and Heat Exchangers*

*qw* **106 , W/m<sup>2</sup>**

*h***c, W/m<sup>2</sup> K**

**Table 2.**

**Table 3.**

**170**

*t***, s** *Tm* **K at** *X* **= 1**

**4.5 Heat conduction in a hollow cylinder**


**Table 6.**

*Wall heat flux and convective heat transfer coefficient.*

## **5. Estimation of heat flux with two-nodes in a sounding rocket**

A two-node exact solution is used to calculate the back-wall temperature as described in Section 3.4. The iterative method described above has been used for estimating aerodynamic heating for a sounding rocket in free flight test. Here, the wall heat flux is estimated using the measured temperature history in conjunction with the iterative technique [30]. The aerodynamic heating rate is estimated for a typical sounding rocket as depicted in **Figure 2**. The location of thermocouple is marked in the diagram. The thermophysical properties of Inconel and wall thickness are *k* = 18 Wm<sup>1</sup> K<sup>1</sup> , *<sup>α</sup>* = 4.47 <sup>10</sup><sup>6</sup> <sup>m</sup><sup>2</sup> /s, *<sup>L</sup>* = 0.7874 <sup>10</sup><sup>3</sup> m. **Figure 3** depicts the measured temperature time history at different locations measured from the tip of the cone in the free flight of a sounding rocket as delineated in **Figure 2**. It can be observed from temperature history that the initial time delay in thermal response is 6 s. The unknown *qw* are estimated using an iterative technique which starts with an initial value of wall heat flux and is repeated until |*F*(*qw*)| ≤ 10<sup>4</sup> .

A two-node exact solution is used to calculate the wall temperature distribution. The unknown *qw* are estimated using an iterative technique which starts with an initial value of wall heat flux and is repeated until |*F*(*qw*)| ≤ 10<sup>4</sup> . **Figure 4** displays the estimated variation of the wall heat flux as a function of flight time of the sounding rocket.

The wall heat flux variation depends on the sounding rocket speed. The increase and decrease of the aerodynamic heating are a function of flight Mach

The estimated wall heat flux is compared with Van Driest's results [44]. **Table 7** depicts the estimated values of wall heat flux as a function of flight time at thermocouple location 29 as shown in **Figure 2**. It can be observed from the table that highest aerodynamic heating occurs during 7–8 s, another significant peak wall heat

number.

**173**

**Figure 4.**

**Figure 3.**

*Measured temperature history in free flight of the sounding rocket.*

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

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

flux was found at 22 s.

*Variations of wall heat flux vs. flight time.*

#### **Figure 2.**

*Schematic sketch of sounding rocket showing location of thermocouple.*

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

**Figure 3.** *Measured temperature history in free flight of the sounding rocket.*

**Figure 4.** *Variations of wall heat flux vs. flight time.*

The wall heat flux variation depends on the sounding rocket speed. The increase and decrease of the aerodynamic heating are a function of flight Mach number.

The estimated wall heat flux is compared with Van Driest's results [44]. **Table 7** depicts the estimated values of wall heat flux as a function of flight time at thermocouple location 29 as shown in **Figure 2**. It can be observed from the table that highest aerodynamic heating occurs during 7–8 s, another significant peak wall heat flux was found at 22 s.

**5. Estimation of heat flux with two-nodes in a sounding rocket**

thickness are *k* = 18 Wm<sup>1</sup> K<sup>1</sup>

*<sup>t</sup>***, s** *T0* **K at** *<sup>X</sup>* **= 0** *Tm* **K at** *<sup>X</sup>* **= 1** *qc* **106**

*Inverse Heat Conduction and Heat Exchangers*

*Wall heat flux and convective heat transfer coefficient.*

.


**Table 6.**

sounding rocket.

**Figure 2.**

**172**

A two-node exact solution is used to calculate the back-wall temperature as described in Section 3.4. The iterative method described above has been used for estimating aerodynamic heating for a sounding rocket in free flight test. Here, the wall heat flux is estimated using the measured temperature history in conjunction with the iterative technique [30]. The aerodynamic heating rate is estimated for a typical sounding rocket as depicted in **Figure 2**. The location of thermocouple is marked in the diagram. The thermophysical properties of Inconel and wall

**, W/m<sup>2</sup>** *h***, W/m<sup>2</sup>**

6 1355.6 326 3.2502 2631.2 2254.2 3351 2946 7 1287.8 342 3.2950 1805.3 2254.2 3113 2946 8 1315.6 356 3.2974 1861.5 2254.2 3087 2946 9 1368.9 380 3.2967 1885.9 2254.2 3117 2946 10 1414.4 402 3.2837 1962.1 2254.2 3088 2946 11 1463.6 425 3.2718 2049.5 2254.2 3060 2946 12 1370.8 440 2.3825 1476.0 2254.2 2985 2946 13 1360.9 460 2.4140 1502.1 2254.2 2968 2946 14 1370.3 479 2.3625 1520.6 2254.2 2924 2946 15 1382.5 507 2.3675 1517.2 2254.2 2943 2946 16 1399.3 528 2.3645 1540.7 2254.2 2934 2946

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

**K** *Tg***, K** *Tgc***, K**

, *<sup>α</sup>* = 4.47 <sup>10</sup><sup>6</sup> <sup>m</sup><sup>2</sup>

A two-node exact solution is used to calculate the wall temperature distribution. The unknown *qw* are estimated using an iterative technique which starts with an

**Figure 3** depicts the measured temperature time history at different locations measured from the tip of the cone in the free flight of a sounding rocket as delineated in **Figure 2**. It can be observed from temperature history that the initial time delay in thermal response is 6 s. The unknown *qw* are estimated using an iterative technique which starts with an initial value of wall heat flux and is repeated until

the estimated variation of the wall heat flux as a function of flight time of the

initial value of wall heat flux and is repeated until |*F*(*qw*)| ≤ 10<sup>4</sup>

*Schematic sketch of sounding rocket showing location of thermocouple.*

/s, *<sup>L</sup>* = 0.7874 <sup>10</sup><sup>3</sup> m.

. **Figure 4** displays


*k0* reference thermal conductivity at *Ti*

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

*x* distance from the inner surface *X* dimensionless coordinate *α* thermal diffusivity

*τ* nondimensional time, *αt*/*L2*

*g* combustion gas temperature

*β* constant thermal conductivity coefficient

Department of Aeronautical Engineering, Noorul Islam Centre for Higher

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: drrakhab.mehta@gmail.com

provided the original work is properly cited.

*p* nondimensional parameter, *α*Δ*T*/(Δ*x*)

*θ* nondimensional temperature, = (*T Tg*)/(*Tg Ti*)

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

2

*L* slab thickness

*qw* wall heat flux *T* temperature

*t* time

*ρ* density

*B* Bartz *c* computed

**Author details**

Rakhab C. Mehta

Education, India

**175**

*i* initial value *m* measured *o* outer wall *w* wall

**Subscripts**

**Table 7.** *Comparison between calculated and Van Driest's heat flux at location 29.*
