Two 3-m-long and 0.4-cm-thick cast iron \((k=\) \(52 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \varepsilon=0.8)\) steam pipes of outer diameter \(10 \mathrm{~cm}\) are connected to each other through two \(1-\mathrm{cm}\)-thick flanges of outer diameter \(20 \mathrm{~cm}\). The steam flows inside the pipe at an average temperature of \(200^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(180 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The outer surface of the pipe is exposed to convection with ambient air at \(8^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) as well as radiation with the surrounding surfaces at an average temperature of \(T_{\text {surr }}=290 \mathrm{~K}\). Assuming steady one-dimensional heat conduction along the flanges and taking the nodal spacing to be \(1 \mathrm{~cm}\) along the flange \((a)\) obtain the finite difference formulation for all nodes, \((b)\) determine the temperature at the tip of the flange by solving those equations, and \((c)\) determine the rate of heat transfer from the exposed surfaces of the flange.

Step by step solution

01

Finite Difference Formulation

To find the finite difference formulation for all nodes, we can use the following equation: $$\frac{T_{i-1}-2 T_{i}+T_{i+1}}{\Delta x^{2}}=\frac{-q_{i}}{k}$$ where \(T_i\) is the temperature at the ith node, \(\Delta x\) is the nodal spacing, \(q_i\) is the heat flux at the ith node, and k is the thermal conductivity of cast iron. Given that the nodal spacing is 1 cm, we can calculate the heat flux at each node as: $$q_i = h(T_{i,\text {air}}-T_i) + \varepsilon\sigma(T_i^4 - T_{\text{surr}}^4)$$ where \(h\) is the heat transfer coefficient due to convection, \(\varepsilon\) is the emissivity of cast iron, \(\sigma\) is the Stefan-Boltzmann constant, \(T_{i,\text {air}}\) is the ambient air temperature, and \(T_{\text{surr}}\) is the surrounding temperature in Kelvin.

02

Determine the Temperature at the Tip of the Flange

The equation we derived above will be our system of equations with 10 nodes, starting from the beginning of the flange up to the tip of the flange. To solve these equations, we can initialize the temperature at each node and then iteratively update the temperatures using the finite difference formulation until the solutions converge. We can do this using a numerical method like the Gauss-Seidel method. Once the temperature values have converged, the temperature at the tip of the flange will be \(T_{10}\).

03

Determine the Rate of Heat Transfer from the Exposed Surfaces of the Flange

Now that we have the temperature at each node, we can determine the rate of heat transfer from the exposed surfaces of the flange. The rate of heat transfer can be calculated by multiplying the heat transfer coefficient from step 1 by the convective heat transfer coefficient \(h\) and the difference between the surface temperature of the flange and the ambient temperature: $$q_{i,\text {total}}=h(T_{i,\text {air}}-T_i) + \varepsilon\sigma(T_i^4 - T_{\text{surr}}^4)$$ We need to sum up the rates of heat transfer from each node to find the total rate of heat transfer from the exposed surfaces of the flange.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Convection Heat Transfer

In the context of heat transfer, convection occurs when heat is carried away by the movement of fluids, which can be liquids or gases. This process is driven by the difference in temperature between the body's surface and the fluid. In our example with steam pipes, the ambient air at a temperature of 8oC acts as the cooling fluid, flowing over the pipe's external surface. The heat transfer coefficient h = 25 W/m2·K quantifies how effectively heat is being convected away from the pipe.

Factors affecting h include fluid velocity, viscosity, and the surface characteristics of the heat-transferring body. To improve understanding of convection heat transfer in exercises, visual aids such as flow patterns around the pipe and comparisons with familiar scenarios, like a cool breeze blowing over a hot surface, can be helpful to students.

Thermal Conductivity

Thermal conductivity, represented by k, is a material-specific value that indicates how well the material can conduct heat. In the case of the cast iron steam pipes, k = 52 W/m·K demonstrates that cast iron has decent thermal conductivity, which allows heat to transfer along the flange effectively. Understanding thermal conductivity is crucial for grasping the finite difference method in heat transfer, as it factors directly into the finite difference formulation used to find temperature distribution across the material.

Enhancing comprehension of thermal conductivity can be achieved by comparing substances with different k values – illustrating that materials like copper conduct heat more efficiently than materials like wood or plastic, for example.

Radiation Heat Transfer

Radiation heat transfer is the emission of thermal energy in the form of electromagnetic waves, which can occur even in a vacuum. No medium is needed unlike convection or conduction. It's governed by the Stefan-Boltzmann law, which posits that the power radiated per unit area is proportional to the fourth power of the emissive body's absolute temperature. The exercise features radiation heat transfer with an emissivity factor ε = 0.8 for the cast iron. Emissivity is a measure of how well a surface emits thermal radiation compared to a perfect black body.

Illustrating radiation concepts can be simplified by highlighting everyday experiences, such as feeling the warmth of the sun or the heat emitted from a hot stovetop, even from a distance. For students, being able to calculate and compare heat transfer rates by convection and radiation offers a more holistic understanding of thermal processes.