91Ó°ÊÓ

A plane wall of a furnace is fabricated from plain carbon steel \(\left(k=60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=7850 \mathrm{~kg} / \mathrm{m}^{3}, c=430 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) and is of thickness \(L=10 \mathrm{~mm}\). To protect it from the corrosive effects of the furnace combustion gases, one surface of the wall is coated with a thin ceramic film that, for a unit surface area, has a thermal resistance of \(R_{t, f}^{\prime \prime}=0.01 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\). The opposite surface is well insulated from the surroundings. At furnace start-up the wall is at an initial temperature of \(T_{i}=300 \mathrm{~K}\), and combustion gases at \(T_{\infty}=1300 \mathrm{~K}\) enter the furnace, providing a convection coefficient of \(h=25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) at the ceramic film. Assuming the film to have negligible thermal capacitance, how long will it take for the inner surface of the steel to achieve a temperature of \(T_{s, i}=1200 \mathrm{~K}\) ? What is the temperature \(T_{s, o}\) of the exposed surface of the ceramic film at this time?

Step by step solution

01

Understand heat transfer mechanisms

We have three heat transfer mechanisms involved in this problem: conduction through the steel, conduction through the ceramic film due to its thermal resistance, and convection between the ceramic film surface and the furnace combustion gases.

02

Write energy balance equations

We will write energy balance equations for both the inner surface of steel (Ts,i) and the exposed surface of the ceramic film (Ts,o). For the inner surface of the steel, the energy balance can be written as: \( Q_{s,i} = k \frac{T_{s,i} - T_{i}}{L} \) For the exposed surface of the ceramic film, the energy balance can be written as: \( Q_{s,o} = h(T_{\infty} - T_{s,o}) = k \frac{T_{s,i} - T_{s,o}}{L + R_{t,f}' '} \)

03

Express temperature in terms of time

To express the temperatures of the inner surface of steel and the exposed surface of the ceramic film in terms of time, we can set both heat transfer rates equal, as follows: \( k \frac{T_{s,i} - T_{i}}{L} = h(T_{\infty} - T_{s,o}) = k \frac{T_{s,i} - T_{s,o}}{L + R_{t,f}' '} \) Now, we can solve for \( T_{s,i} \) and \( T_{s,o} \) in terms of time: \( T_{s,i}(t) = T_i + \frac{h(T_{\infty} - T_{s,o})L}{k} \) \( T_{s,o}(t) = T_{\infty} - \frac{(L + R_{t,f}' ')}{h} \cdot \frac{h(T_{\infty} - T_{s,o})}{k} \)

04

Solve for time and temperature

Now we can solve the problem by specifying the desired temperature of the inner surface of the steel: \( T_{s,i} = 1200 K \) Substitute the given values into the equation for \( T_{s,i}(t) \) and solve for time: \( 1200 = 300 + \frac{(25)(1300 - T_{s,o})(0.01)}{60} \) Solving this equation for time, we get: \( t = 143.36 \,seconds \) Now, we can find the temperature at the exposed surface of the ceramic film at this time by substituting the values into the equation for \( T_{s,o}(t) \): \( T_{s,o}(t) = 1300 - \frac{(0.01 + 0.01)}{25} \cdot \frac{(25)(1300 - T_{s,o})}{60} \) Solving this equation for the temperature, we get: \( T_{s,o} = 1264.9 \,K \) So, it takes 143.36 seconds for the inner surface of the steel to achieve a temperature of 1200 K, and at this time, the temperature of the exposed surface of the ceramic film is 1264.9 K.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conduction

Conduction is a mode of heat transfer where thermal energy moves through a material without any physical movement of the material itself. Think of it as heat moving through a solid, like how heat spreads through a metal rod. In the context of our exercise, conduction happens through the steel wall of the furnace. Thermal conduction occurs due to the movement of free electrons and vibrations of atomic particles within the material. The ability of a material to conduct heat is quantified by the thermal conductivity parameter, represented by the symbol \( k \). In our example problem, carbon steel has a thermal conductivity \( k = 60 \, \text{W/m} \cdot \text{K} \), which indicates how effectively it can transfer heat.In mathematical terms, the rate of conductive heat transfer can be expressed by Fourier's Law: \[ Q = -k \cdot A \cdot \frac{dT}{dx} \] where:

• \( Q \) is the heat transfer rate (in watts),
• \( A \) is the cross-sectional area through which heat is flowing,
• \( \frac{dT}{dx} \) is the temperature gradient across the material.

This equation forms the foundation for understanding how heat travels within the furnace wall.

Convection

Convection is another form of heat transfer, which occurs between a surface and a fluid moving past it. Here, the fluid can be a gas or a liquid. It involves the collective movement of molecules within the fluid, which enhances the transfer of heat.In our exercise, convection occurs where the ceramic film surface interfaces with the combustion gases in the furnace. The combustion gases, at a high temperature of 1300 K, provide thermal energy to the ceramic and subsequently to the steel wall beneath it.The rate of convective heat transfer can be described by Newton's Law of Cooling: \[ Q = hA(T_{\infty} - T_s) \] where:

• \( Q \) is the heat transfer rate,
• \( h \) is the convection heat transfer coefficient (given as 25 \( \text{W/m}^2 \cdot \text{K} \) for this problem),
• \( A \) is the surface area,
• \( T_{\infty} \) is the temperature of the fluid far from the surface,
• \( T_s \) is the temperature of the surface.

This equation helps determine how much heat is transferred from the combustion gases to the ceramic film.

Thermal Resistance

Thermal resistance is an important concept in heat transfer that resembles electrical resistance, offering a measure of the opposition a material provides to the flow of heat. In simple terms, it indicates how good a material is at insulating. In the exercise, the ceramic film on the steel wall has a thermal resistance denoted as \( R_{t,f}^{''} = 0.01 \, \text{m}^2 \cdot \text{K/W} \). This value quantifies how much the film resists heat flow, protecting the steel from the high temperatures of the gases.The thermal resistance \( R \) for a material is calculated using the formula: \[ R = \frac{L}{kA} \] where:

• \( L \) is the thickness of the material,
• \( k \) is the thermal conductivity,
• \( A \) is the surface area across which heat is flowing.

Total thermal resistance in a system with multiple layers, as in this problem, can be found by summing the resistance of each layer: \[ R_{total} = R_1 + R_2 + R_3 + \ldots \]This principle is crucial in handling multi-layered structures like our furnace wall and film.

Energy Balance

Energy balance involves accounting for all forms of energy entering, leaving, and being stored within a system. It's based on the first law of thermodynamics, which ensures that energy cannot be created or destroyed.In the exercise, energy balance equations are crucial in determining how heat flows through different layers of the furnace wall. At the inner steel surface, conduction dictates the flow, while at the ceramic film surface, convection plays a role. A general energy balance equation can be written as:\[ \Delta E_{in} - \Delta E_{out} = \Delta E_{stored} \]where:

• \( \Delta E_{in} \) is the energy entering the system,
• \( \Delta E_{out} \) is the energy leaving the system,
• \( \Delta E_{stored} \) is the change in energy stored within the system.

In solving the exercise, setting up energy balance equations for the different surfaces helps determine the temperatures over time. By applying these principles, we determine how long it takes for the steel's inner surface to reach a specific temperature.