Question

A large plane wall, with a thickness \(L\) and a thermal conductivity \(k\), has its left surface \((x=0)\) exposed to a uniform heat flux \(\dot{q}_{0}\). On the right surface \((x=L)\), convection and radiation heat transfer occur in a surrounding temperature of \(T_{\infty}\). The emissivity and the convection heat transfer coefficient on the right surface are \(\bar{\varepsilon}\) and \(h\), respectively. Express the houndary conditions and the differential equation of this heat conduction problem during steady operation.

Short answer

#Answer# The boundary conditions are: 1) At x=0 (left surface): q_0 = -k (dT/dx) |_(x=0) 2) At x=L (right surface): q_R = -k (dT/dx) |_(x=L) The main differential equation is: d^2T/dx^2 = 0

Step-by-step solution

Fourier's Law of heat conduction

(1) Fourier's Law states that the heat conduction in a direction x (perpendicular to wall surface) is proportional to the temperature gradient in that direction: $$ q_x = -k \frac{dT}{dx} $$ where \(q_x\) represents the heat conduction, k is the thermal conductivity, and \(\frac{dT}{dx}\) is the temperature gradient.

Derive the equation for the temperature distribution T(x)

(2) To find the temperature distribution T(x), we apply a steady-state energy balance over a differential volume of the plane wall, assuming that heat conduction is the only mode of heat transfer within the wall. For steady-state operation, the heat conduction in the x direction should be constant: $$ \frac{d^2T}{dx^2} = 0 $$ Integrating this equation twice, we get: $$ T(x) = Ax + B $$ where A and B are constants of integration.

Calculate heat transfer on the right surface by convection and radiation

(3) On the right surface (x=L), we can calculate heat transfer by convection as: $$ q_{conv} = h (T(L) - T_\infty) $$ By using the Stefan-Boltzmann Law, we can calculate heat transfer by radiation as: $$ q_{rad} = \bar{\varepsilon} \sigma (T(L)^4 - T_\infty^4) $$ Total heat transfer on the right surface, qR, is the sum of convection and radiation heat transfer: $$ q_R = q_{conv} + q_{rad} $$

Express the boundary conditions and the differential equation

(4) Now, it's time to express the boundary conditions and derive the main differential equation for this heat conduction problem: Boundary conditions: 1) At x=0 (left surface): $$ q_0 = -k \frac{dT}{dx} |_{x=0} $$ 2) At x=L (right surface): $$ q_R = -k \frac{dT}{dx} |_{x=L} $$ The differential equation was obtained already in Step 2: $$ \frac{d^2T}{dx^2} = 0 $$

Key concepts

Fourier's Law of heat conduction

Fourier's Law of heat conduction is a fundamental principle used to analyze the thermal behavior of materials. It describes how heat moves through materials at a rate that is proportional to the negative gradient of the temperature. In simpler terms, it means heat flows from hotter areas to cooler ones, and the speed of this flow depends on how quickly the temperature changes in space—and this change is what we call the temperature gradient.

Mathematically, it's expressed as: $$ q_x = -k \frac{dT}{dx} $$Here, \(q_x\) represents the heat flux per unit area in the x-direction, \(k\) is the thermal conductivity, a property that measures a material's ability to conduct heat, and \(\frac{dT}{dx}\) is the rate of change of temperature with respect to distance in the x-direction. Fourier's Law thus serves as the cornerstone for solving heat conduction problems in steady-state conditions.

Temperature distribution

Understanding temperature distribution within a material is crucial when examining heat conduction. It indicates how the temperature varies from one point to another inside the material. In the provided problem, we assume steady-state conditions, meaning the temperature at any given point does not change with time.

Finding the temperature distribution, T(x), involves solving a differential equation derived from an energy balance. In the solution, after integrating the second-order differential equation, $$ \frac{d^2T}{dx^2} = 0 $$, we arrive at a linear equation $$ T(x) = Ax + B $$ where A and B are constants determined by the specific boundary conditions of the problem. This linear relationship simplifies analyzing the heat conduction process by making it easier to visualize and calculate the temperature at any point along the wall's thickness.

Boundary conditions

Boundary conditions are the constraints that define how heat transfer interacts with the boundaries of a material. These conditions are necessary to solve the differential equations governing steady-state heat conduction.

In our scenario, there are two key conditions: 1. At the left surface of the wall (x=0), the heat flux is given and is equal to \( \dot{q}_0 \) (imposed heat flux condition). 2. At the right surface (x=L), convection and radiation occur, leading to a combined heat loss through these processes (convective and radiative conditions).

These conditions enable the calculation of the constants A and B in the temperature distribution equation, which in turn enables the prediction of temperature at any location within the wall.

Thermal conductivity

Thermal conductivity (\(k\)) is an intrinsic property of materials that indicates their ability to conduct heat. It determines the rate at which heat is transferred through the material due to a temperature gradient. In the context of our problem, thermal conductivity is a crucial factor as it directly affects the rate of heat conduction through the wall's thickness.

The value of \(k\) can vary greatly depending on the material; for example, metals typically have high thermal conductivity values, while insulating materials have low values. Consequently, materials with high thermal conductivity will facilitate a more significant heat flow for the same temperature gradient, compared to materials with low thermal conductivity.

Convective heat transfer

Convective heat transfer describes the exchange of heat between a surface and a fluid moving past it. It's one of the ways heat can be lost from the wall into the surrounding environment. The heat transfer coefficient (\(h\)) plays an essential role in quantifying the rate of this heat transfer.

In the given problem, the convective heat transfer at the right surface of the wall (x=L) is calculated by the equation $$ q_{conv} = h (T(L) - T_\infty) $$ where \(T(L)\) is the temperature of the wall at x=L, and \(T_\infty\) is the surrounding temperature. The higher the value of \(h\), the more efficient the convection process is at removing heat from the wall.

Radiative heat transfer

Radiative heat transfer is another method of heat exchange that occurs between surfaces and their surroundings. Unlike conduction and convection, radiation does not require a medium and can occur through a vacuum. It involves the emission of electromagnetic waves and is described by the Stefan-Boltzmann Law.

In our example, the radiative heat transfer at the wall's right surface is given by $$ q_{rad} = \bar{\varepsilon} \sigma (T(L)^4 - T_\infty^4) $$ where \(\bar{\varepsilon}\) is the emissivity of the wall surface, \(\sigma\) is the Stefan-Boltzmann constant, \(T(L)\) is the wall's surface temperature at x=L, and \(T_\infty\) is the surrounding temperature. The equation reflects that radiative heat transfer is influenced by the fourth power of temperature, making it a highly temperature-sensitive process.

Differential equation

Differential equations in the context of heat transfer are used to mathematically express how temperature varies with space and sometimes time. In steady-state heat conduction, the temperature within the wall does not change with time, so the differential equation only involves spatial variables.

The exercise provided leads us to a second-order differential equation where the second derivative of temperature with respect to distance is zero (\(\frac{d^2T}{dx^2} = 0\)). Solving this yields a simple equation that relates temperature to position within the wall, facilitating the calculation of the temperature distribution based on known boundary conditions.

Energy balance

An energy balance is a calculation that ensures that the energy going into a system equals the energy coming out plus any accumulation within the system. In the context of steady-state heat transfer, accumulation is zero, meaning all the energy entering one side of the wall must exit from the other side.

In step 2 of the solution, an energy balance is performed on a differential element of the wall to deduce the temperature distribution. Further, the boundary conditions incorporate this balance considering both convection and radiation—integrating the effects of these forms of heat transfer into the overall analysis to ensure energy is conserved. This approach is pivotal for accurately predicting temperatures and heat flows in heat transfer problems.