Problem 7.1: The temperature gradient at Earth's surface is 0.03 K/m. Assuming Earth is homogeneous and has uniform radioactive heat sources, the temperature at the center is calculated using the heat diffusion equation. The solution gives the center temperature as 9.6×10⁴ K higher than the surface temperature, which is estimated at 27 K. This model severely underestimates the surface temperature. Problem 7.2: A 1-meter-thick ice layer is exposed to light with an energy flux of 100 W/m². The absorption coefficient is 2 m⁻¹, and the thermal conductivity of ice is 2.2 W/mK. The highest temperature within the layer is found to be -5.36°C at a depth of 0.419 m from the surface. Problem 7.3: Water flows through a 10 m-long tube with inner diameter 2 cm and wall thickness 1 cm. The inlet water temperature is 100°C, and the outer temperature is 0°C. The flow rate is 2 l/s. Assuming turbulent flow and constant water temperature across the cross-section, the heat transfer through the wall is calculated. The temperature of the water at the outlet is determined based on the heat loss to the environment. The water temperature at the outlet is found to be lower than the inlet temperature due to heat loss.Problem 7.1: The temperature gradient at Earth's surface is 0.03 K/m. Assuming Earth is homogeneous and has uniform radioactive heat sources, the temperature at the center is calculated using the heat diffusion equation. The solution gives the center temperature as 9.6×10⁴ K higher than the surface temperature, which is estimated at 27 K. This model severely underestimates the surface temperature. Problem 7.2: A 1-meter-thick ice layer is exposed to light with an energy flux of 100 W/m². The absorption coefficient is 2 m⁻¹, and the thermal conductivity of ice is 2.2 W/mK. The highest temperature within the layer is found to be -5.36°C at a depth of 0.419 m from the surface. Problem 7.3: Water flows through a 10 m-long tube with inner diameter 2 cm and wall thickness 1 cm. The inlet water temperature is 100°C, and the outer temperature is 0°C. The flow rate is 2 l/s. Assuming turbulent flow and constant water temperature across the cross-section, the heat transfer through the wall is calculated. The temperature of the water at the outlet is determined based on the heat loss to the environment. The water temperature at the outlet is found to be lower than the inlet temperature due to heat loss.