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10. Heat is conducted through a material with a temperature gradient of -9000 °C/m. The conductivity of the material is 25 W/m.K. If this heat is convected to surrounding at 30 °C with a convection coefficient of 345 W/m²K. determine the surface temperature. ​

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Answer:

The surface temperature \( T_s \) can be calculated using the heat fluxes from conduction and convection:

Given:

[tex] \text{Thermal conductivity } k = 25 \ \text{W/m·K}[/tex]

[tex]text \: {Temperature \: gradient } \frac{dT}{dx} = -9000 \ \text{°C/m}[/tex]

[tex]text{Convection coefficient } h = 345 \ \text{W/m}^2\text{K}[/tex]

[tex]text \: {Surrounding \: temperature } \: T_\infty = 30 \ \text{°C}[/tex]

1. Heat flux by conduction:

[tex]q = -k \frac{dT}{dx} = -25 \times (-9000) = 225000 \ \text{W/m}^2[/tex]

2. Surface temperature by convection:

[tex]T_s = \frac{q}{h} + T_\infty = \frac{225000}{345} + 30 \approx 652.17 + 30 = 682.17 \ \text{°C}[/tex]

Thus, the surface temperature ( T_s ) is approximately

[tex]\( 682.17 \ \text{°C} \).[/tex]

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