91Ó°ÊÓ

The structural components of modern aircraft are commonly fabricated of high- performance composite materials. These materials are fabricated by impregnating mats of extremely strong fibers that are held within a form with an epoxy or thermoplastic liquid. After the liquid cures or cools, the resulting component is of extremely high strength and low weight. Periodically, these components must be inspected to ensure that the fiber mats and bonding material do not become delaminated and, in turn, the component loses its airworthiness. One inspection method involves application of a uniform, constant radiation heat flux to the surface being inspected. The thermal response of the surface is measured with an infrared imaging system, which captures the emission from the surface and converts it to a colorcoded map of the surface temperature distribution. Consider the case where a uniform flux of \(5 \mathrm{~kW} / \mathrm{m}^{2}\) is applied to the top skin of an airplane wing initially at \(20^{\circ} \mathrm{C}\). The opposite side of the 15 -mm-thick skin is adjacent to stagnant air and can be treated as well insulated. The density and specific heat of the skin material are \(1200 \mathrm{~kg} / \mathrm{m}^{3}\) and \(1200 \mathrm{~J} / \mathrm{kg}+\mathrm{K}\), respectively. The effective thermal conductivity of the intact skin material is \(k_{1}=1.6 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). Contact resistances develop internal to the structure as a result of delamination between the fiber mats and the bonding material, leading to a reduced effective thermal conductivity of \(k_{2}=1.1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). Determine the surface temperature of the component after 10 and \(100 \mathrm{~s}\) of irradiation for (i) an area where the material is structurally intact and (ii) an adjacent area where delamination has occurred within the wing.

Step by step solution

01

Define known variables

The initial temperature: \(T_i = 20^{\circ}\mathrm{C}\) The applied heat flux at the surface: \(q'' = 5\,\mathrm{kW/m^2}\) Density of skin material: \(\rho = 1200\,\mathrm{kg/m^3}\) Specific heat capacity of skin material: \(c_p = 1200\,\mathrm{J/(kg\cdot K)}\) Thermal conductivity of intact skin material: \(k_1 = 1.6\,\mathrm{W/(m\cdot K)}\) Thermal conductivity of delaminated skin material: \(k_2 = 1.1\,\mathrm{W/(m\cdot K)}\) Time of inspection: \(t = 10\,\mathrm{s}\) and \(t = 100\,\mathrm{s}\).

02

Compute the thermal diffusivity for both cases

- For intact skin material: \[\alpha_{1} = \frac{k_1}{\rho c_p} = \frac{1.6}{1200 \cdot 1200}\,\mathrm{m^2/s}\] - For delaminated skin material: \[\alpha_{2} = \frac{k_2}{\rho c_p} = \frac{1.1}{1200 \cdot 1200}\,\mathrm{m^2/s}\]

03

Calculate the surface temperature after 10 and 100 seconds for the intact skin material

Recall the temperature distribution formula for a semi-infinite solid: \[ T(x,t) = T_i + q''\sqrt{\frac{4\alpha t}{\pi}}\exp(-\frac{x^2}{4\alpha t}) - q''x\,\mathrm{erf}(\frac{x}{2\sqrt{\alpha t}}) \] At the surface, \(x=0\) and we get: \[T_s(t) = T_i + q''\sqrt{\frac{4\alpha t}{\pi}}\] For the intact skin material (\(\alpha_1\)): - At \(t=1\)s (converting \(q''\) to W): \[T_s(t = 10) = 20 + 5000\sqrt{\frac{4\alpha_1 \cdot 10}{\pi}}\] - At \(t=100\)s: \[T_s(t = 100) = 20 + 5000\sqrt{\frac{4\alpha_1 \cdot 100}{\pi}}\] Calculate the surface temperatures for the intact skin material using the obtained expressions.

04

Calculate the surface temperature after 10 and 100 seconds for the delaminated skin material

For the delaminated skin material (\(\alpha_2\)): - At \(t=10\)s (converting \(q''\) to W): \[T_s(t = 10) = 20 + 5000\sqrt{\frac{4\alpha_2 \cdot 10}{\pi}}\] - At \(t=100\)s: \[T_s(t = 100) = 20 + 5000\sqrt{\frac{4\alpha_2 \cdot 100}{\pi}}\] Calculate the surface temperatures for the delaminated skin material using the obtained expressions.

05

Present the results

Finally, present the calculated surface temperatures after 10 and 100 seconds for both the intact and delaminated skin material cases.

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.

Thermal Conductivity

Thermal conductivity is a crucial property in understanding how materials transfer heat. It tells us how easily heat can flow through a material over a certain distance. The higher the thermal conductivity, the better the material is at conducting heat. This is essential in applications where uniform temperature distribution is key, such as in heat exchangers or building insulation.
For our exercise, the intact aircraft wing skin has a thermal conductivity of 1.6 W/(m·K), meaning it efficiently transfers heat. When delamination occurs, this value drops to 1.1 W/(m·K). This reduction indicates a poorer heat flow capability due to potential gaps or voids between the layers. These internal resistances disrupt the heat flow, potentially affecting the wing's structural integrity.
Understanding thermal conductivity helps in predicting how materials will behave under different heating conditions and is essential in engineering reliable components, especially in safety-critical applications.

Composite Materials

Composite materials are engineered by combining two or more distinct materials to create a new material with enhanced properties. Typically, they consist of a matrix and a reinforcement. The matrix, often a thermoset resin or thermoplastic, supports the reinforcement, which might be fibers like carbon or glass, in maintaining the composite material's overall structure.
One big advantage of composites is their high strength-to-weight ratio, which makes them ideal for aerospace applications. In our exercise, the aircraft's skin is a composite, offering lightweight yet robust protection.
However, composites can face challenges such as delamination, where layers separate due to impacts or stress over time. This not only weakens the material but also alters its thermal properties, as seen with the reduced thermal conductivity in our delaminated wing scenario. Proper inspection and maintenance of composite materials are therefore crucial to ensure their longevity and performance.

Infrared Thermography

Infrared thermography is a non-destructive inspection technique used to measure and visualize temperature distribution across surfaces. This tool is handy for identifying issues like the delamination in composites. By detecting heat emitted in the infrared spectrum, thermography cameras provide thermal images or thermograms without the need for physical contact.
The principle behind this method is quite simple: all objects emit infrared radiation as a function of their temperature. Infrared thermography captures this radiation and converts it into temperature data. During inspections, engineers apply a known heat flux to the surface and capture the thermal response.
In the exercise, infrared thermography helps detect irregularities due to changes in heat transfer properties. For instance, areas with delamination would show a different thermal profile compared to intact areas. This technique is non-invasive, making it perfect for ongoing monitoring of high-value components like airplane wings, ensuring safety without dismantling the structure.