Consider a long concrete dam \((k=0.6 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), (es \(\alpha_{s}=0.7\) ) of triangular cross section whose exposed surface is subjected to solar heat flux of \(\dot{q}_{s}=\) \(800 \mathrm{~W} / \mathrm{m}^{2}\) and to convection and radiation to the environment at \(25^{\circ} \mathrm{C}\) with a combined heat transfer coefficient of \(30 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The \(2-\mathrm{m}\)-high vertical section of the dam is subjected to convection by water at \(15^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and heat transfer through the 2-m-long base is considered to be negligible. Using the finite difference method with a mesh size of \(\Delta x=\Delta y=1 \mathrm{~m}\) and assuming steady two-dimensional heat transfer, determine the temperature of the top, middle, and bottom of the exposed surface of the dam.

Step by step solution

01

Governing Equation for Steady, Two-Dimensional Heat Transfer

For steady, two-dimensional heat transfer in a material of thermal conductivity k, we can use the Laplace's equation for temperature distribution T(x,y): \( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0 \)

02

Apply Boundary Conditions

Let's apply the given boundary conditions: 1. Exposed surface subjected to solar heat flux, convective, and radiative heat transfer with a combined heat transfer coefficient h (30 W/m²K): \( -k \frac{dT}{dx} + h(T - T_\text{env}) = \dot{q}_s \) 2. Vertical section subjected to convection by water at temperature T_water (15°C) with heat transfer coefficient h_water (150 W/m²K): \( -k \frac{dT}{dy} + h_\text{water}(T - T_\text{water}) = 0 \) 3. Heat transfer through the 2-meter base is negligible, so we can consider it as adiabatic (no heat transfer): \( \frac{dT}{dy} = 0 \)

03

Set up Finite Difference Grid

We set up a grid with mesh size Δx = Δy = 1m. We only need a grid of 3 points in while x lay in [1,2] and only one point at y = 1m Let the temperature at these points be T1, T2, and T3.

04

Solve Finite Difference Equation System

Using the central difference approximation to the second derivatives in the Laplace's equation, we can write the finite difference equations for T1, T2, and T3 considering the boundary conditions: Boundary Condition 1: Exposed surface For x = 1m, only T2 exists, so: \( -k(T_\text{top} - T_\text{middle}) + h(T_\text{top} - T_\text{env}) = \dot{q}_s \) For x = 2m, only T3 exists, so: \( -k(T_\text{bottom} - T_\text{middle}) + h(T_\text{bottom} - T_\text{env}) = \dot{q}_s \) Boundary Condition 2: Vertical section For T1, on the vertical section (y = 1m, x = 2m): \( -k(T_\text{middle} - T_\text{bottom}) + h_\text{water}(T_\text{middle} - T_\text{water}) = 0 \) Boundary Condition 3: Base (adiabatic) It is adiabatic, so no contribution in the equations. Solve this system of 3 equations to get the temperatures at the top, middle, and bottom of the exposed surface.

05

Find Temperature at Top, Middle, and Bottom

Solving the above three equations simultaneously, we can find the values for T_top, T_middle, and T_bottom. These values are the temperature at the top, middle, and bottom of the exposed surface of the dam, which is the final answer.

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