Question

Consider the base plate of an \(800-W\) household iron with a thickness of \(L=0.6 \mathrm{~cm}\), base area of \(A=160 \mathrm{~cm}^{2}\), and thermal conductivity of \(k=60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The inner surface of the base plate is subjected to uniform heat flux generated by the resistance heaters inside. When steady operating conditions are reached, the outer surface temperature of the plate is measured to be \(112^{\circ} \mathrm{C}\). Disregarding any heat loss through the upper part of the iron, \((a)\) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the plate, \((b)\) obtain a relation for the variation of temperature in the base plate by solving the differential equation, and (c) evaluate the inner surface temperature. Answer: (c) \(117^{\circ} \mathrm{C}\)

Short answer

Answer: The inner surface temperature of the base plate is \(117^{\circ}C\).

Step-by-step solution

Express the differential equation and boundary conditions for steady one-dimensional heat conduction through the plate

Since we are considering steady one-dimensional heat conduction through the plate, we can use Fourier's Law of heat conduction: $$q = -k\frac{dT}{dx}$$ As the system is at steady-state, the temperature gradient within the plate will be constant, and the energy conservation equation for this case becomes: $$\frac{d^2T}{dx^2} = 0$$ Now we need to establish the boundary conditions. We have two boundary conditions, one at the inner surface (x = 0) and one at the outer surface (x = L): 1. At the inner surface (x = 0), the heat generated by the resistance heaters inside: $$q(0) = -k \frac{dT}{dx}(0)$$ 2. At the outer surface (x = L), the measured temperature is \(112^{\circ}C\): $$T(L) = 112^{\circ}C$$

Obtain a relation for the temperature variation within the base plate

To find the temperature profile, we need to solve the differential equation, considering the boundary conditions. Since the second derivative of the temperature with respect to x is equal to zero, the temperature profile will be linear: $$T(x) = Ax + B$$ Now, we apply the boundary conditions to find A and B: 1. At the inner surface (x = 0), $$q = -k \frac{dT}{dx}(0) = -k(A)$$ This equation gives us: $$A = -\frac{q}{k}$$ 2. At the outer surface (x = L), $$T(L) = AL + B = 112^{\circ}C$$ Substituting A from the previous result, we get: $$B = 112^{\circ}C - L\left(-\frac{q}{k}\right)$$ Substituting A and B into the temperature profile equation: $$T(x) = -\frac{q}{k}x + 112^{\circ}C + L\left(-\frac{q}{k}\right)$$

Evaluate the inner surface temperature

To find the inner surface temperature, we need to determine the value of the heat flux, q. From the given information, the power generated by the iron is \(800 W\). If we assume that the entire power is being transferred through the base plate, we can find the heat flux as: $$q = \frac{800 W}{A}$$ Plugging the area \(A=160 cm^2 = 0.016 m^2\): $$q = \frac{800 W}{0.016 m^2} = 50000 \frac{W}{m^2}$$ Now, we can use the temperature profile equation to find the inner surface temperature (T(0)): $$T(0) = -\frac{q}{k}(0) + 112^{\circ}C + L\left(-\frac{q}{k}\right)$$ Plugging in the given values, \(k = 60 W/(m\cdot K)\), and \(L = 0.6 cm = 0.006 m\): $$T(0) = -\frac{50000 W/m^2}{60 W/(m\cdot K)}(0) + 112^{\circ}C + 0.006 m\left(-\frac{50000 W/m^2}{60 W/(m\cdot K)}\right)$$ Solve for T(0): $$T(0) = 112^{\circ}C - 0.006 m\left(-\frac{50000 W/m^2}{60 W/(m\cdot K)}\right) = 117^{\circ}C$$ The inner surface temperature of the base plate is \(117^{\circ}C\).

Key concepts

Fourier's Law of Heat Conduction

Understanding the basics of heat conduction is crucial in thermodynamics, and Fourier's Law is at the core of this concept. It simply states that the rate of heat transfer (heat flux, \(q\)) through a material is proportional to the negative of the temperature gradient (\(\frac{dT}{dx}\)) across the material. The law is expressed mathematically as \(q = -k\frac{dT}{dx}\), where \(k\) represents the material's thermal conductivity, denoting its ability to conduct heat. This law is applicable for steady one-dimensional heat transfer where the temperature gradient is constant over time. The negative sign indicates that heat flows from higher to lower temperatures.

Using Fourier's Law, we're able to link the physical properties of a material with the thermal gradient and heat flux, allowing us to calculate how quickly or slowly heat will move through the material, which is pivotal in designing engineering systems that manage heat effectively.

Thermal Conductivity

Thermal conductivity (\(k\)) is a material-specific property that quantifies the ability of a material to conduct heat. The higher the thermal conductivity, the more efficiently the material can transfer heat. It is usually expressed in units of \(W/(m\text{·}K)\). In our example of the iron, with a given thermal conductivity of \(60 W/(m\text{·}K)\), the iron's base plate can easily conduct the heat generated by the internal resistance heaters to its surface.

The value of thermal conductivity plays an essential role in heat conduction analysis since it directly affects the rate of heat transfer; for instance, materials with low thermal conductivity, such as insulators, are used to reduce heat transfer where it's undesirable.

Differential Equation for Heat Transfer

In heat transfer analysis, we often derive a differential equation that describes how temperature changes within a material over space and sometimes over time. For steady-state situations, where no change over time occurs, we can work with a simplified form of the temperature distribution equation, such as \(\frac{d^2T}{dx^2} = 0\). This indicates that the temperature gradient is constant across the material.

The differential equation provides the framework for calculating the temperature at any given point within the material when combined with appropriate boundary conditions. As seen in the iron's base plate example, the equation simplifies the problem of finding the temperature along the plate by providing a fundamental relationship that must be satisfied at all points within the material.

Temperature Profile

The temperature profile refers to how the temperature varies across a material or system. Solving the heat transfer differential equation gives us this profile, which can be represented as a function of position. For a steady-state, one-dimensional case with constant thermal properties, the profile is typically linear, as was seen in our iron base plate example: \(T(x) = Ax + B\).

The temperature profile is foundational for understanding heat transfer within a solid because it informs us about the distribution of temperatures across the system, which is critical for ensuring materials are used within their proper thermal limits and for verifying that a system operates as expected under thermal loads.

Boundary Conditions

Setting the correct boundary conditions is essential for solving any differential equation accurately, including those in heat transfer problems. Boundary conditions are constraints that provide the necessary information to determine the unique solution of a differential equation. In the context of heat conduction, they often take the form of known temperatures or heat fluxes at specific locations, as presented in the iron's base plate problem.

Typically, we have to specify the temperature or the rate of heat transfer at the boundaries of the domain we're considering. In application, without appropriate boundary conditions, it would not be possible to solve for the constants in the general solution of the temperature profile, thus leaving us without a practical way to describe the temperature distribution within the material.

Heat Flux Calculation

Heat flux calculation is a pivotal step in many engineering problems that involve thermal analysis. It's a measure of the rate at which heat energy passes through a certain area and is typically expressed in \(W/m^2\). Knowing the heat flux can help in estimating the temperature distribution within an object and the total heat transfer over time.

In practice, heat flux can be directly measured or calculated from known quantities, such as the total power transferred and the area through which the transfer occurs, as demonstrated when we found the inner surface temperature of the iron's base plate. Being able to calculate heat flux is important for designing thermal systems, such as in ensuring that components do not exceed temperature limitations and in optimizing thermal efficiency.