Consider a long solid bar \((k=28 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\alpha=\) \(12 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\) ) of square cross section that is initially at a uniform temperature of \(32^{\circ} \mathrm{C}\). The cross section of the bar is \(20 \mathrm{~cm} \times 20 \mathrm{~cm}\) in size, and heat is generated in it uniformly at a rate of \(\dot{e}=8 \times 10^{5} \mathrm{~W} / \mathrm{m}^{3}\). All four sides of the bar are subjected to convection to the ambient air at \(T_{\infty}=30^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(h=45 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Using the explicit finite difference method with a mesh size of \(\Delta x=\Delta y=10 \mathrm{~cm}\), determine the centerline temperature of the bar \((a)\) after \(20 \mathrm{~min}\) and \((b)\) after steady conditions are established.

Step by step solution

01

Understand the Explicit Finite Difference Method

The explicit finite difference method (EFDM) is a numerical method used to approximate the solution of partial differential equations (PDEs) such as the heat conduction equation. In this method, we will discretize the time and space into intervals and solve the temperature for each node using the neighboring nodes' values. We will be using the heat conduction equation for this purpose.

02

Discretize the heat conduction equation using the explicit finite difference method

The heat conduction equation for a 2-D Cartesian coordinate system is given by: \(\frac{\partial T}{\partial t} = \alpha(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}) + \frac{\dot{e}}{k}\) The explicit finite difference equation for this PDE is: \( T_{i,j}^{n+1} = T_{i,j}^n + \Delta t \left[ \alpha(\frac{T_{i-1,j}^n - 2 T_{i,j}^n + T_{i+1,j}^n}{\Delta x^2} + \frac{T_{i,j-1}^n - 2 T_{i,j}^n + T_{i,j+1}^n}{\Delta y^2}) + \frac{\dot{e}}{k} \right] \) We will use this equation to find the centerline temperature of the bar.

03

Set up the initial conditions and mesh

Given that the bar has an initial uniform temperature of 32°C, we can assign this as the initial temperature at each node in our mesh. The given mesh size is \(\Delta x = \Delta y = 10 \mathrm{~cm} = 0.1 \mathrm{~m}\).

04

Calculate the temperature after 20 minutes

For part (a), we need to find the temperature after 20 minutes (i.e., 1200 seconds). We first need to calculate Δt from the mesh size and given thermal diffusivity α: \( \Delta t = \frac{\Delta x^{2}}{4\alpha} \) \( \Delta t = \frac{(0.1)^2}{4 \times 12 \times 10^{-6}} \) \( \Delta t = 2.08 \) seconds Now, we will use the EFDM equation from Step 2 to find the temperature at each node after 20 minutes (1200 seconds). We will update the temperature values at each iteration until 1200 seconds.

05

Compute steady-state temperature

For part (b), we need to find the steady-state temperature. For steady-state conditions, we require the change in temperature at each node to be minimal: \(\frac{\partial T}{\partial t} = 0\) Using the equation from Step 2 for these conditions, we calculate the centerline temperature after the steady-state is achieved.

06

Calculate the centerline temperature

At the end of Steps 4 and 5, we get temperature values for each node in the mesh. To calculate the centerline temperature of the bar, we take the average of the temperatures at the intersection of the two centerlines after 20 minutes and steady-state conditions. Following the steps mentioned above and using the given parameters, you can find the centerline temperature of the bar using the explicit finite difference method for both cases: (a) after 20 minutes and (b) after steady conditions are established.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!