A spherical vessel used as a reactor for producing...

A spherical vessel used as a reactor for producing pharmaceuticals has a 10 -mm-thick stainless steel wall $(k=17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ and an inner diameter of $1 \mathrm{~m}$. The exterior surface of the vessel is exposed to ambient air $\left(T_{\infty}=25^{\circ} \mathrm{C}\right)$ for which a convection coefficient of $6 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ may be assumed. (a) During steady-state operation, an inner surface temperature of $50^{\circ} \mathrm{C}$ is maintained by energy generated within the reactor. What is the heat loss from the vessel? (b) If a 20 -mm-thick layer of fiberglass insulation $(k=0.040 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ is applied to the exterior of the vessel and the rate of thermal energy generation is unchanged, what is the inner surface temperature of the vessel?

A spherical vessel used as a reactor for producing pharmaceuticals has a 10 -mm-thick stainless steel wall $(k=17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ and an inner diameter of $1 \mathrm{~m}$. The exterior surface of the vessel is exposed to ambient air $\left(T_{\infty}=25^{\circ} \mathrm{C}\right)$ for which a convection coefficient of $6 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ may be assumed.
(a) During steady-state operation, an inner surface temperature of $50^{\circ} \mathrm{C}$ is maintained by energy generated within the reactor. What is the heat loss from the vessel?
(b) If a 20 -mm-thick layer of fiberglass insulation $(k=0.040 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ is applied to the exterior of the vessel and the rate of thermal energy generation is unchanged, what is the inner surface temperature of the vessel?

Key Concepts

Thermal Resistance Network

A thermal resistance network is an approach that models different layers or paths of heat transfer (such as conduction and convection) as resistances in series or parallel. This method simplifies the calculation of total heat transfer by adding individual resistances, which is crucial when dealing with composite systems like a reactor with a metal wall and additional insulation.

Convection Heat Transfer

Convection is the mechanism by which heat is transferred from a solid surface to a fluid (or vice versa) due to the movement of the fluid. In this scenario, the external surface of the reactor is exposed to ambient air, and the convective heat transfer coefficient quantifies how efficiently heat is removed by the air from the surface.

Spherical Geometry in Heat Transfer

When dealing with a spherical vessel, the geometry affects the calculation of thermal resistances, especially in conduction. The heat transfer rate and temperature distribution depend on the inner and outer radii of the sphere, which must be considered in the formulation of the conduction resistance for spherical shells.

Steady-State Heat Transfer

This concept refers to a condition where temperatures at all points in a system remain constant with time, meaning the rate of energy generated or entering a system is balanced by the rate of energy leaving it. In analyzing the spherical vessel, steady-state assumptions simplify the energy balance and allow the use of constant thermal resistances to calculate heat flows.

Conduction Through Solids

Conduction is the process of heat transfer through a material without any bulk motion of the medium. In the context of the spherical reactor, conduction describes how thermal energy passes through the stainless steel wall and any added insulation, governed by the material's thermal conductivity and the geometry of the system.

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