Hot water at an average temperature of 90^∘ C is flowing through a 15-m section of a cast iron p

step 1 So here in this question the rate of heat loss from the hot water can be calculated using the eq

Hot water at an average temperature of 90^∘ C is flowing...

Hot water at an average temperature of $90^{\circ} \mathrm{C}$ is flowing through a $15-\mathrm{m}$ section of a cast iron pipe $(k=52 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ whose inner and outer diameters are $4 \mathrm{~cm}$ and $4.6 \mathrm{~cm}$, respectively. The outer surface of the pipe, whose emissivity is $0.7$, is exposed to the cold air at $10^{\circ} \mathrm{C}$ in the basement, with a heat transfer coefficient of $15 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The heat transfer coefficient at the inner surface of the pipe is $120 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Taking the walls of the basement to be at $10^{\circ} \mathrm{C}$ also, determine the rate of heat loss from the hot water. Also, determine the average velocity of the water in the pipe if the temperature of the water drops by $3^{\circ} \mathrm{C}$ as it passes through the basement.

Key Concepts

Thermal Resistance Network

This concept involves modeling the entire heat transfer process as a series of resistances—each corresponding to a different mode of heat transfer (convection, conduction, radiation). By summing these resistances, one can determine the overall thermal resistance between two regions. This approach simplifies the analysis of complex thermal systems by breaking them down into manageable parts.

Conduction Heat Transfer

Conduction is the transfer of heat through a material due to a temperature gradient, as described by Fourier’s law. It is particularly important in analyzing the heat transfer through the pipe wall, where the material’s thermal conductivity and geometry determine the rate at which heat is conducted.

Convection Heat Transfer

Convection refers to the transfer of heat between a surface and a moving fluid. It is characterized by the convective heat transfer coefficient, which quantifies the effectiveness of heat exchange at the fluid-solid interface. In problems involving pipes, convection occurs on both the inner surface where the fluid contacts the wall and the outer surface exposed to the ambient environment.

Radiative Heat Transfer

Radiative heat transfer involves the emission and absorption of thermal radiation between surfaces. It is influenced by the emissivity of the surface and the temperature difference between the surface and its surroundings. This mode of heat transfer can be significant when there is exposure to environments like the cold basement air, where the surface radiates energy.

Energy Balance in Fluid Flow

An energy balance in fluid flow connects the heat lost by the fluid (due to a temperature drop) to its mass flow rate and specific heat capacity. By equating the energy loss to the heat transfer rate, one can determine operational parameters such as the fluid's velocity within the pipe, making it an essential concept in thermal-fluid analysis.

Fluid Flow and Velocity Calculation

This concept deals with the relationship between the volumetric flow rate, the cross-sectional area of the pipe, and the average flow velocity. Understanding this relationship is crucial when assessing how the energy loss (or temperature drop) in a fluid translates into a specific flow rate or velocity, facilitating the analysis of dynamic thermal systems.

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