Question

Consider a long rectangular bar of length \(a\) in the \(x-\) direction and width \(b\) in the \(y\)-direction that is initially at a uniform temperature of \(T_{i}\). The surfaces of the bar at \(x=0\) and \(y=0\) are insulated, while heat is lost from the other two surfaces by convection to the surrounding medium at temperature \(T_{\infty}\) with a heat transfer coefficient of \(h\). Assuming constant thermal conductivity and transient two-dimensional heat transfer with no heat generation, express the mathematical formulation (the differential equation and the boundary and initial conditions) of this heat conduction problem. Do not solve.

Short answer

The mathematical formulation consists of the governing equation, boundary conditions, and initial conditions: 1. Governing equation: $$ \frac{\partial T}{\partial t} = \alpha \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} \right) $$ 2. Boundary conditions: $$ \frac{\partial T}{\partial x} \Big|_{x=0} = 0 \quad \text{and} \quad \frac{\partial T}{\partial y} \Big|_{y=0} = 0 $$ $$ -k \frac{\partial T}{\partial x} \Big|_{x=a} = h(T(a,y,t)-T_{\infty}) \quad \text{and} \quad -k \frac{\partial T}{\partial y} \Big|_{y=b} = h(T(x,b,t)-T_{\infty}) $$ 3. Initial condition: $$ T(x,y,0) = T_{i} $$

Step-by-step solution

Write the governing equation for heat conduction

The two-dimensional heat conduction equation is given by: $$ \frac{\partial T}{\partial t} = \alpha \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} \right) $$ where \(T\) is the temperature, \(t\) is the time, \(\alpha\) is the thermal diffusivity, and \(x,y\) are spatial coordinates.

Specify the boundary conditions

We have the following boundary conditions: 1. Insulated surfaces at \(x=0\) and \(y=0\): $$ \frac{\partial T}{\partial x} \Big|_{x=0} = 0 \quad \text{and} \quad \frac{\partial T}{\partial y} \Big|_{y=0} = 0 $$ 2. Convection at surfaces \(x=a\) and \(y=b\): $$ -k \frac{\partial T}{\partial x} \Big|_{x=a} = h(T(a,y,t)-T_{\infty}) \quad \text{and} \quad -k \frac{\partial T}{\partial y} \Big|_{y=b} = h(T(x,b,t)-T_{\infty}) $$ where \(k\) is the thermal conductivity of the bar, and \(T_{\infty}\) is the surrounding temperature.

Specify the initial conditions

The initial condition is given by the uniform temperature \(T_{i}\): $$ T(x,y,0) = T_{i} $$ In summary, the mathematical formulation of the given heat conduction problem consists of the governing equation, boundary conditions, and initial conditions: 1. Governing equation: $$ \frac{\partial T}{\partial t} = \alpha \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} \right) $$ 2. Boundary conditions: $$ \frac{\partial T}{\partial x} \Big|_{x=0} = 0 \quad \text{and} \quad \frac{\partial T}{\partial y} \Big|_{y=0} = 0 $$ $$ -k \frac{\partial T}{\partial x} \Big|_{x=a} = h(T(a,y,t)-T_{\infty}) \quad \text{and} \quad -k \frac{\partial T}{\partial y} \Big|_{y=b} = h(T(x,b,t)-T_{\infty}) $$ 3. Initial condition: $$ T(x,y,0) = T_{i} $$

Key concepts

Transient Heat Transfer

Transient heat transfer, also known as unsteady-state heat transfer, refers to the situation where the temperature within an object changes over time. Unlike steady-state heat transfer, where temperatures remain constant over time, transient heat transfer involves temperature gradients that vary with time. Understanding transient heat transfer is crucial in many practical applications, from industrial processes to thermal management in electronics.

When dealing with problems involving transient heat transfer, we must consider how heat diffuses through materials over time. This often requires solving the heat conduction equation, which relates changes in temperature to time and space. In educational examples, simplified models, such as the one-dimensional or two-dimensional bar, are used to illustrate these concepts and allow for easier mathematical handling.

Thermal Conductivity

Thermal conductivity, denoted by the symbol \(k\), is a material property that measures a substance’s ability to conduct heat. High thermal conductivity indicates that the material is a good conductor of heat and can quickly transfer thermal energy. Conversely, a low thermal conductivity means the material is a thermal insulator.

In the context of the exercise, the bar has a constant thermal conductivity, suggesting that its ability to conduct heat is uniform throughout the material. This property, along with thermal diffusivity (\(\alpha\)), plays a critical role in heat transfer calculations. Thermal diffusivity combines thermal conductivity, material density, and specific heat capacity to describe how fast heat propagates through the material.

Boundary Conditions

Boundary conditions are essential for solving heat transfer problems as they provide additional information required to find a unique solution to the governing differential equation. There are three common types of boundary conditions in heat transfer: Dirichlet, which specifies temperature values; Neumann, which involves heat flux; and Robin (also called convective), which involves convection at the surface.

In our exercise, there are two types of boundary conditions. First, we have the insulated surfaces at \(x=0\) and \(y=0\) where the heat flux is zero — this represents a Neumann boundary condition. Second, for the surfaces at \(x=a\) and \(y=b\), we have a convective boundary condition where heat is lost through convection, described by the heat transfer coefficient \(h\) and the surrounding temperature \(T_{\infty}\). These boundary conditions significantly influence the temperature distribution within the bar over time.

Mathematical Formulation of Heat Transfer

The mathematical formulation of heat transfer problems involves creating a set of equations that describe how heat moves within a body. For the given exercise, the cornerstone of the formulation is the partial differential equation known as the heat conduction equation, which relates the rate of change of temperature \(T\) to its spatial distribution within the bar.

In the heat conduction equation \[\frac{\partial T}{\partial t} = \alpha \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} \right)\], \(\alpha\) is thermal diffusivity, integrating thermal conductivity \(k\), density, and specific heat capacity. This equation, together with the specified initial and boundary conditions, forms a complete mathematical description that, theoretically, can be solved to predict the transient temperature distribution in the bar. However, such problems typically require the application of numerical methods, as analytic solutions are often not feasible for complex geometries or conditions.