Question

Consider a large 5-cm-thick brass plate \((k=\) \(111 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) ) in which heat is generated uniformly at a rate of \(2 \times 10^{5} \mathrm{~W} / \mathrm{m}^{3}\). One side of the plate is insulated while the other side is exposed to an environment at \(25^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(44 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Explain where in the plate the highest and the lowest temperatures will occur, and determine their values.

Short answer

Answer: The highest temperature in the brass plate is 350°C, which occurs at the insulated side (x = 0). The lowest temperature is 625°C, which occurs at the exposed side (x = \(x_{i}\)).

Step-by-step solution

Finding the heat generation in the plate

The heat generation rate in the plate is given as \(q_g = 2 \times 10^5 \text{ W/m}^3\), which remains constant throughout the plate. To find the total heat generated in the plate, we need to multiply \(q_g\) by the volume of the plate. Since we're considering a large brass plate, we can assume that it behaves like an infinite plate in the horizontal directions. Therefore, the heat generated in the plate would be per unit area as follows: \(q_G = q_g \times t\), where \(t\) is the thickness of the plate (0.05 m).

Finding the heat transfer from the exposed side

The heat transfer from the exposed side to the environment is given by Newton's Law of Cooling as: \(q'=hA_{s}(T_{s}-T_{\infty})\), where \(q'\) is the heat transfer per unit area, A\(_{s}\) is the surface area, \(T_{s}\) is the surface temperature of the exposed side, \(T_{\infty}\) is the environment temperature (\(25^{\circ} \mathrm{C}\)), and \(h\) is the heat transfer coefficient (44 W/m²·K). Since the insulated side doesn't allow any heat flow, the heat generated in the plate is equal to the heat transferred to the environment. So, we can write: \(q_G = q'\).

Finding the temperature distribution in the plate

Using Fourier's law of heat conduction, we can write the temperature distribution as: \(q' = -kA_s \frac{dT}{dx}\). Now, combine the equations from Step 2 and Step 3, \(q_G = -k \frac{dT}{dx}\). To find the temperature distribution, rearrange this equation and then integrate w.r.t x: \(\int_{T_{s}}^{T(x)} dT = -\int_{0}^{x} \frac{q_G}{k} dx\). Integrate and solve for the temperature distribution, T(x): \(T(x)=T_{s} + \frac{q_{G}}{k}(x - x_{i})\).

Identifying the highest and the lowest temperatures

The highest temperature in the plate will occur at the insulated side (x = 0), while the lowest temperature will occur at the exposed side (x = \(x_{i}\)). We know that the heat generated equals the heat transferred to the environment, so we can substitute the q\(_G\) value into the equation for Newton's Law of Cooling to find T\(_s\): \(q_{G} = h(T_{s} - T_{\infty})\). Now, we can find T(0): \(T(0)=T_{s} + \frac{q_{G}}{k}(0 - x_{i})=T_{s} - \frac{q_{G}}{k}x_{i}\). And, T(\(x_{i}\)): \(T(x_{i})=T_{s}\).

Determining the values of the highest and the lowest temperatures

First, find T\(_s\) by solving the equation from Step 4: \(T_{s} = T_{\infty}+\frac{q_G}{h}=\left(25+\frac{2\times 10^{5}\times 0.05}{44}\right)^\circ \mathrm{C}=625^\circ \mathrm{C}\). Now, find the highest temperature at the insulated side (T(0)): \(T(0)=625 - \frac{2\times 10^{5}\times 0.05}{111 \mathrm{~W} / \mathrm{m}\cdot \mathrm{K}}=350^\circ \mathrm{C}\). The highest temperature in the plate occurs at the insulated side, and it is \(350^\circ\mathrm{C}\). The lowest temperature occurs at the exposed side and is equal to T(\(x_{i}\)), which is \(625^\circ\mathrm{C}\).

Key concepts

Fourier's Law of Heat Conduction

When we explore the phenomenon of heat transfer in materials like a brass plate, Fourier's Law of Heat Conduction is a fundamental concept to understand. This law states that the rate of heat transfer through a material is proportional to the negative of the gradient of the temperature and to the area, A, through which the heat is flowing perpendicular to that gradient. The mathematical expression is given by the formula:
\( q' = -kA \frac{dT}{dx} \),
where \( q' \) is the rate of heat transfer per unit area, \( k \) is the thermal conductivity of the material, \( A \) is the cross-sectional area, and \( \frac{dT}{dx} \) represents the temperature gradient in the direction of heat flow. This law implies that heat flows from regions of higher temperature to regions of lower temperature, across the temperature gradient, and that the rate of this flow is dependent on the material's thermal conductivity.
In the context of the brass plate exercise, Fourier's Law helps us deduce that heat generated inside the plate will flow toward the side with lower temperature, which in this case is the exposed side, until a steady state is reached. Understanding this concept is essential for analyzing thermal systems and predicting how temperatures will distribute themselves within a material over time.

Newton's Law of Cooling

Newton's Law of Cooling is another pivotal theory that helps us analyze how a hot object exchanges heat with its cooler surroundings. Formulated by Sir Isaac Newton, this principle asserts that the rate of heat loss of a body is proportional to the difference in temperatures between the body and its environment. The law can be expressed as:
\( q' = hA_s(T_s - T_{\text{env}}) \),
where \( q' \) represents the heat transfer rate per unit area, \( h \) denotes the heat transfer coefficient, \( A_s \) is the surface area of the object, \( T_s \) is the temperature of the object's surface, and \( T_{\text{env}} \) is the temperature of the environment. Newton's Law of Cooling is particularly useful when attempting to predict the cooling pattern of a warm object in cooler surroundings. This law is applied in the brass plate exercise to determine how heat from the exposed side of the plate is transferred to the surrounding environment at a known temperature. By knowing the surface temperature and heat transfer coefficient, the cooling rate can be calculated, aiding in the quest to find the temperature distribution across the plate.

Steady State Heat Generation

The concept of steady state heat generation is essential when analyzing thermodynamic systems where heat production is constant and uniform, such as in our brass plate scenario. Steady state refers to a condition where the temperature distribution within an object does not change over time, meaning that there is a balance between the heat being generated and the heat being dissipated or transferred.
In our brass plate example, we assume that heat is generated uniformly throughout the plate at a constant rate. The term 'steady state' implies that after a certain period, the temperature gradient within the plate will remain constant, leading to a linear temperature distribution from one side of the plate to the other in the absence of phase changes or other complications. This steady state concept is critical for designing and analyzing heat transfer systems in engineering where maintaining a constant temperature or managing heat generation is vital for the system's function and integrity.