The temperature gradient at the Earth's surface is 0.03 K/m. Assuming a homogeneous Earth with uniform distribution of heat-emitting radioactive substances, the thermal conductivity coefficient is 1 W/mK. The temperature profile is described by the equation \(-\lambda \nabla^2 T = q\), leading to \(T(r) = -(q / 6 \lambda) r^2 + const\). The constant is determined by the condition that the heat radiated into space equals the total heat generated by the radioactive sources. This results in \(T(r) = -\frac{1}{2 R} \frac{dT}{dr} |_{r=R} ( R^2 - r^2 ) + ( -\frac{\lambda}{\sigma} \frac{dT}{dr} |_{r=R} )^{1/4}\), where \(R \approx 6400\) km. The temperature at the center of the Earth exceeds the surface temperature by \(9.6 \times 10^4\) K, and the surface temperature is underestimated to be 27 K. Light shines on one side of a one-meter-thick layer of ice surrounded by air at -10°C. The incident energy flux is 100 W/m², and the absorption coefficient for light in ice is 2 m⁻¹. The thermal conductivity coefficient of ice is 2.2 W/mK. The temperature profile is given by the diffusion equation \(d^2 T / dz^2 = -q(z) / \lambda\). The absorbed light is represented as an internal heat source \(q = \mu j\), leading to \(T(z) = \frac{j_0}{\lambda \mu} \{ 1 - \exp(-\mu z) - \frac{z}{h} [ 1 - \exp(-\mu h) ] \} + T_0\). The highest temperature within the layer is at a distance of 0.419 m from the irradiated surface and is equal to -5.36°C. Water flows through a 10 m-long tube with an inner diameter of 2 cm and a wall thickness of 1 cm. The inlet temperature of the water is 100°C, and the ambient temperature is 0°C. The flow rate is 2 m³/s, and the thermal conductivity coefficient of the wall is 6 W/mK. Assuming a small decrease in water temperature and turbulent flow, the rate of heat flow through the tube is given by \(\dot{Q} = -2 \pi \lambda h (T_2 - T_1) / \ln(r_2 / r_1)\). The temperature at the outlet is calculated using the boundary conditions and the heat flow rate, considering the cooling of the water due to heat transfer to the environment.The temperature gradient at the Earth's surface is 0.03 K/m. Assuming a homogeneous Earth with uniform distribution of heat-emitting radioactive substances, the thermal conductivity coefficient is 1 W/mK. The temperature profile is described by the equation \(-\lambda \nabla^2 T = q\), leading to \(T(r) = -(q / 6 \lambda) r^2 + const\). The constant is determined by the condition that the heat radiated into space equals the total heat generated by the radioactive sources. This results in \(T(r) = -\frac{1}{2 R} \frac{dT}{dr} |_{r=R} ( R^2 - r^2 ) + ( -\frac{\lambda}{\sigma} \frac{dT}{dr} |_{r=R} )^{1/4}\), where \(R \approx 6400\) km. The temperature at the center of the Earth exceeds the surface temperature by \(9.6 \times 10^4\) K, and the surface temperature is underestimated to be 27 K. Light shines on one side of a one-meter-thick layer of ice surrounded by air at -10°C. The incident energy flux is 100 W/m², and the absorption coefficient for light in ice is 2 m⁻¹. The thermal conductivity coefficient of ice is 2.2 W/mK. The temperature profile is given by the diffusion equation \(d^2 T / dz^2 = -q(z) / \lambda\). The absorbed light is represented as an internal heat source \(q = \mu j\), leading to \(T(z) = \frac{j_0}{\lambda \mu} \{ 1 - \exp(-\mu z) - \frac{z}{h} [ 1 - \exp(-\mu h) ] \} + T_0\). The highest temperature within the layer is at a distance of 0.419 m from the irradiated surface and is equal to -5.36°C. Water flows through a 10 m-long tube with an inner diameter of 2 cm and a wall thickness of 1 cm. The inlet temperature of the water is 100°C, and the ambient temperature is 0°C. The flow rate is 2 m³/s, and the thermal conductivity coefficient of the wall is 6 W/mK. Assuming a small decrease in water temperature and turbulent flow, the rate of heat flow through the tube is given by \(\dot{Q} = -2 \pi \lambda h (T_2 - T_1) / \ln(r_2 / r_1)\). The temperature at the outlet is calculated using the boundary conditions and the heat flow rate, considering the cooling of the water due to heat transfer to the environment.