/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 130 Steel-reinforced concrete pillar... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 130

Steel-reinforced concrete pillars are used in the construction of large buildings. Structural failure can occur at high temperatures due to a fire because of softening of the metal core. Consider a 200 -mm-thick composite pillar consisting of a central steel core \((50 \mathrm{~mm}\) thick) sandwiched between two 75 -mm-thick concrete walls. The pillar is at a uniform initial temperature of \(T_{i}=\) \(27^{\circ} \mathrm{C}\) and is suddenly exposed to combustion products at \(T_{\infty}=900^{\circ} \mathrm{C}, h=40 \mathrm{~W} / \mathrm{m}^{2}+\mathrm{K}\) on both exposed surfaces. The surroundings temperature is also \(900^{\circ} \mathrm{C}\). (a) Using an implicit finite difference method with \(\Delta x=10 \mathrm{~mm}\) and \(\Delta t=100 \mathrm{~s}\), determine the temperature of the exposed concrete surface and the center of the steel plate at \(t=10,000 \mathrm{~s}\). Steel properties are: \(k_{s}=55 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho_{s}=7850 \mathrm{~kg} / \mathrm{m}^{3}\), and \(c_{s}=450 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). Concrete properties are: \(k_{c}=1.4 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho_{c}=2300 \mathrm{~kg} / \mathrm{m}^{3}, c_{c}=880 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) and \(\varepsilon=0.90\). Plot the maximum and minimum concrete temperatures along with the maximum and minimum steel temperatures over the duration \(0 \leq t \leq 10,000 \mathrm{~s} .\) (b) Repeat part (a) but account for a thermal contact resistance of \(R_{t, c}^{\prime}=0.20 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\) at the concretesteel interface. (c) At \(t=10,000 \mathrm{~s}\), the fire is extinguished, and the surroundings and ambient temperatures return to \(T_{\infty}=T_{\text {sur }}=27^{\circ} \mathrm{C}\). Using the same convection heat transfer coefficient and emissivity as in parts (a) and (b), determine the maximum steel temperature and the critical time at which the maximum steel temperature occurs for cases with and without the contact resistance. Plot the concrete surface temperature, the concrete temperature adjacent to the steel, and the steel temperatures over the duration \(10,000 \leq t \leq 20,000 \mathrm{~s} .\)

Short Answer

Expert verified
In summary, to solve the given problem, we use the implicit finite difference method to determine the temperature of the exposed concrete surface and the center of the steel plate at \(t = 10,000 s\). We discretize the domain, initialize the parameters, calculate the boundary conditions, formulate the implicit finite difference method, perform a time loop to update temperatures, obtain results for the maximum and minimum concrete and steel temperatures, and plot the temperature profiles. By considering thermal contact resistance, we can analyze the effect on the overall temperature distribution in the composite pillar after the fire is extinguished.

Step by step solution

01

Discretization and Initialization

Divide the domain into segments with a size of \(\Delta x = 10 \ mm\). Define the parameters for each segment, including temperature, thermal conductivity, and density. Initialize the temperatures (interior nodes) as \(T_i = 27^{\circ} C\). Initialize the time step \(\Delta t = 100 s\), and the final time \(t_{final} = 10,000 s\). ##Step 2: Boundary Conditions##
02

Boundary Conditions

Calculate the boundary conditions at each time step. The heat transfer from exposed surfaces to the surroundings is given by: \(q = h(T_{\infty} - T)\). Set the temperatures for the exposed concrete surfaces considering the convection heat transfer coefficient \(h = 40 \mathrm{~W} / \mathrm{m}^{2}+\mathrm{K}\) and the surrounding temperature \(T_{\infty}=900^{\circ} \mathrm{C}\). ##Step 3: Formulation of Implicit Finite Difference Method##
03

Formulation of Implicit Finite Difference Method

Formulate the implicit finite difference method using the given parameters (thermal conductivity, density, and heat capacity) for each material (concrete and steel). Use the matrix equation: \([I - \Delta tA]T^{n+1} = T^{n}\) to update the temperatures at each time step. ##Step 4: Time Loop and Temperature Update##
04

Time Loop and Temperature Update

Perform the time loop by repeating steps 2 and 3 until reaching the final time (\(t_{final} = 10,000 s\)). Update the temperatures for the nodes at every time step. ##Step 5: Obtain Results##
05

Obtain Results

1. Determine the maximum and minimum concrete temperatures, and the maximum and minimum steel temperatures over the duration \(0 \leq t \leq 10,000 s\). 2. Repeat part 1, considering the thermal contact resistance between the concretesteel interface. 3. Determine the maximum steel temperature and the critical time at which the maximum steel temperature occurs for cases with and without the contact resistance after the fire is extinguished. ##Step 6: Plot##
06

Plot

Plot the concrete surface temperature, the concrete temperature adjacent to the steel, and the steel temperatures over the duration \(10,000 \leq t \leq 20,000 \mathrm{~s}\). The steps outlined above describe the procedure to solve the given problem. The implementation of the implicit finite difference method can be done using programming languages such as MATLAB, Python, or C++.

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Key Concepts

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

Thermal Conductivity
Thermal conductivity is a material's ability to conduct heat. It measures how quickly heat energy is transferred through a material. Materials with high thermal conductivity, like metals, allow heat to flow through them rapidly. On the other hand, materials with low thermal conductivity, such as concrete, slow down this heat transfer.

In the context of the given exercise, the steel core of the pillar has a thermal conductivity (\(k_s = 55\ \mathrm{W}/\mathrm{m}\cdot \mathrm{K}\)) that is significantly higher than that of the surrounding concrete walls (\(k_c = 1.4\ \mathrm{W}/\mathrm{m}\cdot \mathrm{K}\)). This discrepancy is essential for understanding the temperature distribution in the pillar when subjected to high temperatures.
  • High thermal conductivity in steel allows the heat to reach the center of the steel faster.
  • Low thermal conductivity in concrete slows down the heat flow from the outside to the inner steel core.
Understanding these differences is crucial for engineers when designing structures to be resilient in high-temperature scenarios.
Heat Transfer
Heat transfer involves three main mechanisms: conduction, convection, and radiation. In this exercise, both conduction and convection play fundamental roles. Conduction refers to the transfer of heat between substances that are in physical contact, while convection refers to the transfer of heat between a surface and a fluid in motion (such as air or water).

For the construction pillar:
  • Conduction occurs between the concrete and the steel in the pillar as it is subjected to high temperatures.
  • Convection happens at the surface of the concrete exposed to the high-temperature surrounding, transferring heat to the cooler concrete body.
The heat transfer between the concrete surfaces and the external environment is governed by Newton's law of cooling: \[ q = h(T_\infty - T) \]where \(h\) is the heat transfer coefficient (\(40\ \mathrm{W}/\mathrm{m}^2\cdot\mathrm{K}\)) in this scenario, \(T_\infty\) the surrounding temperature (\(900^{\circ} \mathrm{C}\)), and \(T\) is the surface temperature of the concrete. This calculation allows thermal engineers to predict how fast structures may heat up during a fire.
Thermal Contact Resistance
Thermal contact resistance is the resistance to heat flow across the interface of two materials. Even with good overall thermal conductivities, the contact surface can act as a bottleneck to heat transfer. This feature can be critical in composite materials, such as the steel-reinforced concrete pillar.

In the exercise, the inclusion of a thermal contact resistance (denoted as \(R_{t, c}^{\prime}=0.20\ \mathrm{m}^2\cdot \mathrm{K}/\mathrm{W}\)) between the concrete and steel affects the overall heat transfer process. This resistance needs to be considered in calculations for accurate results.
  • A larger thermal contact resistance results in decreased heat flow between the materials.
  • This resistance must be incorporated when using methods such as the implicit finite difference method to solve heat transfer problems involving such interfaces.
Addressing thermal contact resistance in simulations helps ensure realistic predictions of temperature profiles, which is vital in evaluating a structure's performance under thermal stress scenarios.

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Most popular questions from this chapter

A very thick plate with thermal diffusivity \(5.6 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\) and thermal conductivity \(20 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) is initially at a uniform temperature of \(325^{\circ} \mathrm{C}\). Suddenly, the surface is exposed to a coolant at \(15^{\circ} \mathrm{C}\) for which the convection heat transfer coefficient is \(100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Using the finite- difference method with a space increment of \(\Delta x=15 \mathrm{~mm}\) and a time increment of \(18 \mathrm{~s}\), determine temperatures at the surface and at a depth of \(45 \mathrm{~mm}\) after \(3 \mathrm{~min}\) have elapsed.

A one-dimensional slab of thickness \(2 L\) is initially at a uniform temperature \(T_{i}\). Suddenly, electric current is passed through the slab causing uniform volumetric heating \(\dot{q}\left(\mathrm{~W} / \mathrm{m}^{3}\right)\). At the same time, both outer surfaces \((x=\pm L)\) are subjected to a convection process at \(T_{\infty}\) with a heat transfer coefficient \(h\). Write the finite-difference equation expressing conservation of energy for node 0 located on the outer surface at \(x=-L\). Rearrange your equation and identify any important dimensionless coefficients.

Spheres of \(40-\mathrm{mm}\) diameter heated to a uniform temperature of \(400^{\circ} \mathrm{C}\) are suddenly removed from the oven and placed in a forced-air bath operating at \(25^{\circ} \mathrm{C}\) with a convection coefficient of \(300 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the sphere surfaces. The thermophysical properties of the sphere material are \(\rho=3000 \mathrm{~kg} / \mathrm{m}^{3}, c=850 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and \(k=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). (a) How long must the spheres remain in the air bath for \(80 \%\) of the thermal energy to be removed? (b) The spheres are then placed in a packing carton that prevents further heat transfer to the environment. What uniform temperature will the spheres eventually reach?

A wall \(0.12 \mathrm{~m}\) thick having a thermal diffusivity of \(1.5 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\) is initially at a uniform temperature of \(85^{\circ} \mathrm{C}\). Suddenly one face is lowered to a temperature of \(20^{\circ} \mathrm{C}\), while the other face is perfectly insulated. (a) Using the explicit finite-difference technique with space and time increments of \(30 \mathrm{~mm}\) and \(300 \mathrm{~s}\), respectively, determine the temperature distribution at \(t=45 \mathrm{~min}\). (b) With \(\Delta x=30 \mathrm{~mm}\) and \(\Delta t=300 \mathrm{~s}\), compute \(T(x, t)\) for \(0 \leq t \leq t_{\mathrm{ss}}\), where \(t_{\mathrm{ss}}\) is the time required for the temperature at each nodal point to reach a value that is within \(1^{\circ} \mathrm{C}\) of the steady-state temperature. Repeat the foregoing calculations for \(\Delta t=75 \mathrm{~s}\). For each value of \(\Delta t\), plot temperature histories for each face and the midplane.

Two plates of the same material and thickness \(L\) are at different initial temperatures \(T_{i, 1}\) and \(T_{i, 2}\), where \(T_{i, 2}>T_{i, 1}\). Their faces are suddenly brought into contact. The external surfaces of the two plates are insulated. (a) Let a dimensionless temperature be defined as \(T *(F o) \equiv\left(T-T_{i, 1}\right) /\left(T_{i, 2}-T_{i, 1}\right)\). Neglecting the thermal contact resistance at the interface between the plates, what are the steady-state dimensionless temperatures of each of the two plates, \(T_{s s, 1}^{*}\) and \(T_{s s, 2}^{*}\) ? What is the dimensionless interface temperature \(T_{\text {in }}^{*}\) at any time? (b) An effective overall heat transfer coefficient between the two plates can be defined based on the instantaneous, spatially averaged dimensionless plate temperatures, \(U_{\mathrm{eff}}^{*} \equiv q^{*} /\left(\bar{T}_{2}^{*}-\bar{T}_{1}^{*}\right)\). Noting that a dimensionless heat transfer rate to or from either of the two plates may be expressed as \(q^{*}=d\left(Q / Q_{o}\right) / d F o\), determine an expression for \(U_{\text {eif }}^{*}\) for \(F o>0.2\).
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