Question

An AISI 316 stainless steel spherical container is used for storing chemicals undergoing exothermic reaction that provides a uniform heat flux of \(60 \mathrm{~kW} / \mathrm{m}^{2}\) to the container's inner surface. The container has an inner diameter of \(1 \mathrm{~m}\) and a wall thickness of \(5 \mathrm{~cm}\). For safety reason to prevent thermal burn on individuals working around the container, it is necessary to keep the container's outer surface temperature below \(50^{\circ} \mathrm{C}\). If the ambient temperature is \(23^{\circ} \mathrm{C}\), determine the necessary convection heat transfer coefficient to keep the container's outer surface temperature below \(50^{\circ} \mathrm{C}\). Is the necessary convection heat transfer coefficient feasible with free convection of air? If not, discuss other option to prevent the container's outer surface temperature from causing thermal burn.

Short answer

Answer: The necessary convection heat transfer coefficient to keep the container's outer surface temperature below 50°C is approximately 2222 W/(m²K). This value is higher than the typical range for free convection of air (5 to 25 W/(m²K)), so it is not feasible. Alternative options should be considered, such as forced convection or adding insulation material.

Step-by-step solution

Calculate Surface Area and Total Heat Generated

First, we need to find the inner surface area of the container, from which the heat is being generated. The formula for the surface area of a sphere is: \[A = 4 \pi r^{2}\] where A is the surface area and r is the radius. In this case, the inner diameter is given as 1m, so the radius is 0.5m. Therefore, the surface area is: \[A = 4 \pi (0.5)^{2} = \pi \mathrm{m^{2}}\] Next, we need to find the total heat being generated. We are given that the heat flux is 60 kW/m². So, the total heat generated (Q) can be calculated as: \[Q = q_{\text{flux}} \times A\] \[Q = 60 \mathrm{kW/m^{2}} \times \pi \mathrm{m^{2}} = 60\pi \mathrm{kW}\]

Heat Transfer Through the Container Wall

Now, we need to determine the heat transfer through the container wall. A formula for the heat transfer rate through a sphere is given by: \[Q = \frac{kA\Delta T}{r}\] where k is the thermal conductivity of the material (AISI 316 stainless steel has k ≈ 14.9 W/m·K), A is the surface area, ΔT is the temperature difference across the wall, and r is the radius of the sphere. We want to keep the outer surface temperature below 50°C, and the ambient temperature is 23°C. Solving for ΔT, we find the temperature difference across the wall: \[\Delta T = T_{\text{inside}} - T_{\text{ambient}}\] We can solve this equation for the temperature difference across the wall, and then use the heat transfer formula to find the temperature difference between the inside and outside of the container.

Calculate Required Convection Heat Transfer Coefficient

The convective heat transfer formula states that: \[Q = hA\Delta T_{\text{conv}}\] where h is the convection heat transfer coefficient, A is the surface area, and ΔT_conv is the temperature difference between the outer surface of the container and the ambient temperature. We know that ΔT_conv = 50 - 23 = 27°C. We also know the total heat transfer (Q) from step 1. Now, we will solve for h: \[h = \frac{Q}{A\Delta T_{\text{conv}}}\] \[h = \frac{60\pi \mathrm{kW}}{\pi \mathrm{m^{2}} \times 27 \mathrm{K}} = \frac{60\pi \times 10^3 \mathrm{W}}{\pi \mathrm{m^{2}} \times 27 \mathrm{K}} \approx 2222 \mathrm{W/(m^{2}K)}\]

Feasibility and Alternative Options

The typical range of convection heat transfer coefficients for free convection of air is around 5 to 25 W/(m²K). Our calculated value of 2222 W/(m²K) is considerably higher than the typical range, which means free convection of air is not sufficient to prevent the outer surface temperature from exceeding 50°C. Alternative options to prevent thermal burn on the container's outer surface can include: 1. Forced convection (using fans or blowers) which can provide higher convection heat transfer coefficients. 2. Adding insulation material on the container's outer surface to reduce heat transfer and decrease the exposed temperature.

Key concepts

Heat Flux Calculation

Heat flux is a measure of the rate of heat energy transfer through a given surface per unit time. In our case, the given heat flux is a uniform value of 60 kW/m². To understand how much heat is actually being transferred, we need to account for the surface area of the inner surface of the sphere from which the heat is conducted. By multiplying this heat flux by the spherical surface area \( A = 4 \pi r^{2} \) (where \( r \) is the sphere's radius), we calculate the total heat generation. For the stainless steel container, this resulted in \( Q = q_{\text{flux}} \times A = 60 \pi \mathrm{kW} \) of heat energy being generated. It's crucial that students acknowledge that heat flux provides a rate, and by itself does not indicate the total heat transferred without also considering the area.

In practical situations, like in engineering applications, understanding heat flux helps in designing systems for thermal management, which might involve choosing materials or designing for proper ventilation.

Spherical Heat Transfer

Spherical heat transfer differs from planar or cylindrical heat transfer because the surface area through which heat is transferred changes with the radius. This fundamental characteristic impacts how heat transfer calculations are approached. In our textbook example, the spherical nature of the container means that the heat transfer rate through the stainless steel wall can be calculated using the formula \( Q = \frac{kA\Delta T}{r} \), emphasizing the relationship between thermal conductivity (\( k \)), surface area (\( A \)), temperature difference across the wall (\( \Delta T \)), and the sphere's radius (\( r \)).

The spherical shape ensures that heat is distributed uniformly from every point on the surface, which is an important concept when designing spherical containers or vessels in industries that require strict temperature maintenance, such as chemical storage or reactors.

Thermal Conductivity

Thermal conductivity, represented by \( k \), is a material-specific value that measures a material’s ability to conduct heat. High thermal conductivity materials transfer heat efficiently, while low thermal conductivity materials act as insulators. In our textbook problem, AISI 316 stainless steel has a thermal conductivity of approximately 14.9 W/m·K. The value of \( k \) is crucial for calculating the rate of heat transfer through the spherical wall and allows engineers to select appropriate materials for construction based on thermal requirements.

Students should appreciate that while metals typically have high thermal conductivity and are therefore good at transferring heat, materials with lower thermal conductivity are better suited for thermal insulation purposes.

Surface Area Calculation

The calculation of surface area is essential in heat transfer problems because the rate at which heat is exchanged depends on the surface area exposed to the transfer process. For spherical shapes, the surface area formula \( A = 4 \pi r^{2} \) is used, where \( r \) is the sphere’s radius. In our container example, we utilized this formula to determine the surface area involved in heat transfer, which directly impacted the total heat being generated.

It's important for students to be comfortable with surface area calculations, as they frequently appear in problems involving radiation, convection, and conduction. Understanding how to calculate surface area correctly is key in tackling multidisciplinary problems involving heat transfer, such as in environmental engineering or architectural design.

Forced Convection

Forced convection is a mechanism of heat transfer where fluid motion is generated by an external source, like a pump or a fan. It’s contrasted with natural or free convection, where fluid motion is due to buoyancy forces that arise from density differences caused by temperature variations within the fluid. The calculated convection heat transfer coefficient necessary to keep the outer surface temperature below 50°C in our problem indicated that free convection was insufficient.

Forced convection can provide much higher heat transfer coefficients; hence, it's often used in industrial processes to enhance heat dissipation. For students, understanding when to apply forced convection is critical in the design of thermal systems to ensure safety and efficiency, such as in the cooling systems of electronics or the heating systems of buildings.

Thermal Insulation

Thermal insulation refers to the use of materials with low thermal conductivity to reduce heat transfer between objects or environments. In our exercise, insulating the spherical container could be effective to reduce heat transfer to the external environment and maintain the outer surface temperature below the safety threshold, preventing thermal burns.

This concept is applied extensively in building envelopes to improve energy efficiency and in industrial applications to maintain process temperatures. For students, understanding thermal insulation is essential for designing systems aimed at energy conservation, environmental control, and personal safety. Selecting the right insulation material and thickness is a key component of solving real-world engineering challenges.