91Ó°ÊÓ

A plane wall \(\left(\rho=4000 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=500 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.\), \(k=10 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) of thickness \(L=20 \mathrm{~mm}\) initially has a linear, steady-state temperature distribution with boundaries maintained at \(T_{1}=0^{\circ} \mathrm{C}\) and \(T_{2}=100^{\circ} \mathrm{C}\). Suddenly, an electric current is passed through the wall, causing uniform energy generation at a rate \(\dot{q}=2 \times 10^{7} \mathrm{~W} / \mathrm{m}^{3}\). The boundary conditions \(T_{1}\) and \(T_{2}\) remain fixed. (a) On \(T-x\) coordinates, sketch temperature distributions for the following cases: (i) initial condition \((t \leq 0)\); (ii) steady-state conditions \((t \rightarrow \infty\) ), assuming that the maximum temperature in the wall exceeds \(T_{2}\); and (iii) for two intermediate times. Label all important features of the distributions. (b) For the system of three nodal points shown schematically (1, \(m, 2)\), define an appropriate control volume for node \(m\) and, identifying all relevant processes, derive the corresponding finitedifference equation using either the explicit or implicit method. (c) With a time increment of \(\Delta t=5 \mathrm{~s}\), use the finitedifference method to obtain values of \(T_{m}\) for the first \(45 \mathrm{~s}\) of elapsed time. Determine the corresponding heat fluxes at the boundaries, that is, \(q_{x}^{\prime \prime}\) \((0,45 \mathrm{~s})\) and \(q_{x}^{\prime \prime}(20 \mathrm{~mm}, 45 \mathrm{~s})\). (d) To determine the effect of mesh size, repeat your analysis using grids of 5 and 11 nodal points ( \(\Delta x=5.0\) and \(2.0 \mathrm{~mm}\), respectively).

Step by step solution

01

(a) Sketch the temperature distributions

(i) Initial condition (\(t \leq 0\)): The initial temperature distribution is linear, with the boundaries maintained at \(T_1 = 0^{\circ} C\) and \(T_2 = 100^{\circ} C\). (ii) Steady-state conditions (\(t \rightarrow \infty\)): In this case, calculate the maximum temperature in the wall when an electric current is passed through it indicating that the maximum temperature exceeds \(T_2\). (iii) Two intermediate times: Sketch the temperature distribution at two arbitrary intermediate times.

02

(b) Derive a finite difference equation

For this step, a control volume around the node m is defined. Applying the energy conservation law and identifying relevant processes (heat conduction and energy generation), derive the finite difference equation for the temperature at node m using either the explicit or implicit method.

03

(c) Calculate \(T_m\) and the heat fluxes at the boundaries

For calculating \(T_m\) for the specified time period, implement the derived finite difference equation from part (b) with a time increment of \(\Delta t = 5 s\) and a nodal point configuration of 3 nodes (1, m, 2). After finding \(T_m\), the heat fluxes at the boundaries can be calculated using Fourier's law of heat conduction: \[ q_x^{\prime\prime}(x, t) = -k \frac{\partial T}{\partial x} \] Calculate the heat fluxes at both boundaries, i.e., \(q_x^{\prime\prime}(0, 45~s)\) and \(q_x^{\prime\prime}(20~mm, 45~s)\).

04

(d) Analyze the effect of mesh size

Repeat the analysis from part (c) but use grids of 5 and 11 nodes (\(\Delta x = 5.0\) and \(2.0~mm\), respectively). Compare the results obtained for different mesh sizes and analyze the effect of mesh size on the calculated temperature values and heat fluxes.

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Key Concepts

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

Finite Difference Method

The Finite Difference Method is a numerical approach used to approximate solutions to differential equations that describe various physical processes, such as heat transfer. It is particularly useful for solving problems with boundary and initial conditions, which can be complex to manage analytically.

In the context of heat transfer, this method discretizes the temperature distribution in a medium by dividing the medium into small sections or nodes. Each node represents a specific point within the material where temperature and other properties are calculated.

Key steps include:

• Defining a grid of nodes along the medium.
• Applying the conservation of energy principle to each control volume.
• Deriving equations that relate temperatures at each node to its neighbors.

This approach enables us to predict how temperature evolves over time at different points in the system without solving complex differential equations analytically. In practice, an explicit or implicit finite difference equation can be used depending on the specific requirements of stability and time-stepping.

Thermal Conductivity

Thermal conductivity is a material property that measures a material’s ability to conduct heat. It is a critical concept in heat transfer analysis, as it directly affects the rate at which heat can pass through a material.

The symbol for thermal conductivity is usually denoted as 'k' and it has units of Watts per meter-Kelvin (W/m·K). In the given problem, the wall has a thermal conductivity of 10 W/m·K, which indicates how well heat travels through this specific material.

Higher thermal conductivity means better heat conduction through the material.

• Good conductors, like metals, have high thermal conductivity.
• Poor conductors, like wood or rubber, have low thermal conductivity.

This property is crucial for determining the temperature distribution in a material subjected to heat. For instance, using Fourier’s Law of Heat Conduction: \[ q'' = -k \frac{dT}{dx} \]
where \( q'' \) is the heat flux and \( \frac{dT}{dx} \) is the temperature gradient, thermal conductivity links the rate of heat flow to the temperature change across a distance.

Steady-State Conditions

In heat transfer, steady-state conditions refer to a state where the temperature of the system does not change with time. This implies that, although heat continues to flow through the system, the amount of heat entering and exiting each control volume remains constant over time.

For the given problem, initially, the wall is in steady-state, with a linear temperature distribution from 0°C to 100°C maintained between boundaries. When the electric current starts, energy generation within the wall continues until a new steady-state is achieved.

Key characteristics of steady-states include:

• Constant temperature profile over time.
• The sum of heat gains and losses in the control volume equals zero.

Steady-state analysis simplifies the calculations since it reduces time-dependent terms in the heat transfer equations. While reaching the steady-state after energy generation starts, the temperature at all points evolves but eventually stabilizes when equilibrium is achieved.

Temperature Distribution

Temperature distribution in a material is the variation of temperature within the material as a function of position. In our problem, it plays a crucial role in understanding how the wall responds to internal and external thermal influences.

Initially, the wall, being in steady-state, exhibits a linear temperature gradient from 0°C to 100°C, due to fixed boundary conditions. As energy generation begins, this distribution changes, raising the temperatures throughout the wall.

Key points in understanding temperature distribution:

• Temperature gradients drive heat flow, based on Fourier’s Law.
• Uneven energy generation (like electric heating) alters the gradient.
• The distribution at any time combines initial, boundary, and internal conditions.

Graphically sketching these distributions for different times (initial, intermediate, and steady-state) provides insight into the dynamic thermal behavior of the wall. It enables prediction of maximum temperatures and mapping of thermal problems.