Question

A plane wall of thickness $L$ is subjected to convection at both surfaces with ambient temperature $T_{\mathrm{\infty }1}$ and heat transfer coefficient $h_1$ at inner surface, and corresponding $T_{\mathrm{\infty }2}$ and $h_2$ values at the outer surface. Taking the positive direction of $x$ to be from the inner surface to the outer surface, the correct expression for the convection boundary condition is (a) $k\frac{dT(0)}{dx}=h_1[T(0)-T_{\mathrm{o}1})]$ (b) $k\frac{dT(L)}{dx}=h_2[T(L)-T_{\mathrm{\infty }2})]$ (c) $-k\frac{dT(0)}{dx}=h_1[T_{\mathrm{\infty }1}-T_{\mathrm{\infty }2})]$ (d) $-k\frac{dT(L)}{dx}=h_2[T_{\mathrm{\infty }1}-T_{\mathrm{\infty }22})]$ (e) None of them

Short answer

The correct expressions for the convection boundary conditions at the inner and outer surfaces are: (a) $k\frac{dT(0)}{dx}=h_1[T(0)-T_{\mathrm{\infty }1}]]$ for the inner surface (x=0) (b) $k\frac{dT(L)}{dx}=h_2[T(L)-T_{\mathrm{\infty }2}]]$ for the outer surface (x=L)

Step-by-step solution

Understand the convection boundary conditions

The convection boundary conditions are expressions relating the heat transfer through a surface to the temperature difference between the surface and the ambient air. At the inner surface (at $x=0$), we have a heat transfer coefficient $h_1$ and ambient temperature $T_{\mathrm{\infty }1}$. Similarly, at the outer surface (at $x=L$), we have a heat transfer coefficient $h_2$ and ambient temperature $T_{\mathrm{\infty }2}$.

Analyze the heat transfer at the inner and outer surfaces

We will use Fourier's law to describe heat transfer through the wall. It states that the heat flux (rate of heat transfer per unit area) is proportional to the temperature gradient along the wall: $q_x=-k\frac{dT}{dx}$ where $q_x$ is the heat flux in the x-direction, $k$ is the thermal conductivity, and $\frac{dT}{dx}$ is the temperature gradient along the wall.

Apply the heat transfer coefficients to the inner and outer surfaces

The heat transfer coefficients $h_1$ and $h_2$ represent the proportionality of the heat flux on the inner and outer surfaces of the wall, respectively. We will apply these coefficients at the inner surface (x=0) and the outer surface (x=L): 1. At the inner surface (x=0), the heat transfer coefficient $h_1$ represents the proportionality between the heat flux and the temperature difference as follows: $q_x=h_1[T(0)-T_{\mathrm{\infty }1}]$ 2. At the outer surface (x=L), the heat transfer coefficient $h_2$ represents the proportionality between the heat flux and the temperature difference as follows: $q_x=h_2[T(L)-T_{\mathrm{\infty }2}]$

Compare the expressions for the convection boundary conditions

Now, let's examine the given expressions to find the correct convection boundary condition expression: (a) $k\frac{dT(0)}{dx}=h_1[T(0)-T_{\mathrm{\infty }1}]]$ This expression represents the correct convection boundary condition at the inner surface (x=0) since it describes how the heat flux is related to the heat transfer coefficient $h_1$ and the temperature difference between the inner surface and its ambient temperature. (b) $k\frac{dT(L)}{dx}=h_2[T(L)-T_{\mathrm{\infty }2}]]$ This expression represents the correct convection boundary condition at the outer surface (x=L) since it describes how the heat flux is related to the heat transfer coefficient $h_2$ and the temperature difference between the outer surface and its ambient temperature. (c) $-k\frac{dT(0)}{dx}=h_1[T_{\mathrm{\infty }1}-T_{\mathrm{\infty }2}]]$ This expression is incorrect because it does not represent the temperature difference between the inner surface and its ambient temperature. (d) $-k\frac{dT(L)}{dx}=h_2[T_{\mathrm{\infty }1}-T_{\mathrm{\infty }2}]]$ This expression is incorrect because it does not represent the temperature difference between the outer surface and its ambient temperature. (e) None of them Since we have already found two expressions (a and b) that represent the correct convection boundary conditions at the inner and outer surfaces, this answer is incorrect. The correct answer is (a) and (b).

Key concepts

Convection Boundary Conditions

Convection boundary conditions play a vital role in the analysis of heat transfer across surfaces that are exposed to a moving fluid, like air. These conditions establish a relationship between the heat exchange happening at a surface and the temperature variance between the surface itself and the surrounding fluid.

When a body comes into contact with a fluid, heat transfer doesn’t only depend on conduction inside the material itself but also on how effectively the fluid can carry away or supply heat. This is where the heat transfer coefficient, usually represented by the symbol $h$, steps in. To define the boundary conditions, we distinguish the different temperatures involved:

• The temperature at the surface, let's call it $T_s$.
• The ambient fluid temperature $T_{\mathrm{\infty }}$.

At any point where the surface and fluid meet, the convection heat transfer can be expressed mathematically using the formula:
$$q=h(T_s-T_{\mathrm{\infty }})$$
Here, $q$ stands for the heat flux, $h$ is the heat transfer coefficient, and $T_s$ and $T_{\mathrm{\infty }}$ are the temperatures at the surface and in the ambient air, respectively. This formula is applicable at any point where heat is exchanging between a solid boundary and a fluid abode.

Fourier's Law

Fourier’s Law is a crucial foundational principle in the study of heat transfer. It provides a mathematical description of how heat flows through a material due to the temperature gradient. Simply put, it tells us that the rate of heat transfer is directly proportional to the negative gradient of temperatures defining the direction of heat flow, and the material’s properties expressed as thermal conductivity.

In a one-dimensional scenario, this can be demonstrated as:

• Heat flux, $q_x$, is directly proportional to the temperature gradient $\frac{dT}{dx}$.

The respective mathematical formulation is:
$$q_x=-k\frac{dT}{dx}$$
In this equation, $k$ represents thermal conductivity, $q_x$ depicts the heat flux in the $x$-direction, and $\frac{dT}{dx}$ is the temperature gradient. The negative sign indicates that heat flows in the direction of decreasing temperature, which is intuitive as heat naturally moves from hotter to cooler regions.

Fourier’s Law allows engineers and physicists to predict how heat will move through a material, which is essential for designing better insulating materials and managing thermal processes efficiently.

Thermal Conductivity

Thermal conductivity is an essential property of materials, symbolized by $k$, which indicates how well a material can conduct heat. It describes how heat is transferred through a material by direct contact of molecules within the material itself. High thermal conductivity means that the material conducts heat very effectively, while low thermal conductivity means that it acts as an insulator.

For instance, metals typically possess a high thermal conductivity which is why they feel cold to the touch – they rapidly conduct heat away from your warm hand. On the other hand, materials like rubber or wood have low thermal conductivity, which makes them effective insulative materials.

In the context of heat transfer through a medium, thermal conductivity is crucial for understanding and predicting the rate and efficiency of heat energy transfer across that medium. The rate of heat transfer in a material, according to Fourier's Law, depends on both its thermal conductivity $k$ and the temperature gradient $\frac{dT}{dx}$. Hence, understanding thermal conductivity is essential for a wide range of engineering applications, from designing thermal insulators to improving energy efficiency in buildings.

Heat Transfer Coefficient

The heat transfer coefficient, denoted as $h$, is a parameter that characterizes the efficiency of transfer of heat between a solid surface and a fluid flowing over or around it. It plays a pivotal role in the equation of convection heat transfer $q=h(T_s-T_{\mathrm{\infty }})$, where $h$ acts as the constant of proportionality.

The value of the heat transfer coefficient depends on several factors including:

• Type of flow (laminar or turbulent),
• Fluid properties (such as viscosity and thermal conductivity),
• Surface character (smooth or rough),
• Temperature of the surface and fluid.

The heat transfer coefficient is crucial in scenarios where there's a need to measure and improve the efficiency of heat exchangers. It helps engineers design systems where maximum heat transfer between surfaces and the fluid is desired, such as radiators, economizers, and condensers, by balancing the thermal resistance offered by the fluid and the surface. Understanding and manipulating the heat transfer coefficient can significantly impact the performance of thermal equipment and systems.