A 30 -cm-diameter, 4-m-high cylindrical column of a house made of concrete \(\left(k=0.79 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \alpha=5.94 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\right.\), \(\rho=1600 \mathrm{~kg} / \mathrm{m}^{3}\), and \(\left.c_{p}=0.84 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right)\) cooled to \(14^{\circ} \mathrm{C}\) during a cold night is heated again during the day by being exposed to ambient air at an average temperature of \(28^{\circ} \mathrm{C}\) with an average heat transfer coefficient of \(14 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Using analytical one-term approximation method (not the Heisler charts), determine \((a)\) how long it will take for the column surface temperature to rise to \(27^{\circ} \mathrm{C},(b)\) the amount of heat transfer until the center temperature reaches to \(28^{\circ} \mathrm{C}\), and (c) the amount of heat transfer until the surface temperature reaches to \(27^{\circ} \mathrm{C}\).

Step by step solution

01

Calculate the Biot number

The Biot number (Bi) is calculated by the following formula: \(B i=\frac{h L_{c}}{k}\) Where: \(h\) is the heat transfer coefficient (\(14 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\)) \(L_c\) is the characteristic dimension of the column, which for a cylinder is \(L_c = \dfrac{V}{A_s} = \dfrac{\dfrac{\pi d^2}{4}h}{\pi dh} = \dfrac{d}{4}\); with \(d\) being the diameter (\(0.3\,\text{m}\)) and \(h\) being the height (\(4\,\text{m}\)) of the column \(k\) is the thermal conductivity (\(0.79 \,\mathrm{W} / \mathrm{m} \cdot \mathrm{K}\)) Calculating the Biot number: \(B i = \dfrac{14\frac{0.3}{4}}{0.79}\)

02

Calculate the Fourier number

The Fourier number (Fo) is calculated by the following formula: \(F o=\frac{\alpha t}{L_{c}^{2}}\) Where: \(\alpha\) is the thermal diffusivity (\(5.94 \times 10^{-7}\,\mathrm{m}^{2} / \mathrm{s}\)) \(t\) is the time we need to determine \(L_c\) is the characteristic dimension obtained in Step 1 We will use the Fourier number in the following steps to find the temperature at different points of the column after a certain time.

03

Calculate the surface and center temperatures using one-term approximation method

Using the analytical one-term approximation method, the temperature distribution in the cylinder will follow this equation: \(\dfrac{T-T_{\infty}}{T_{i}-T_{\infty}}=\Phi_{2, n}(\text { Bi, Fo, r } / L_{c})\) Where: \(\Phi_{2, n}\) is the analytical function that calculates the transient solution for a cylinder \(T\) is the temperature at a desired radian or center position \(T_{\infty}\) is the ambient temperature (\(28^{\circ} \mathrm{C}\)) \(T_i\) is the initial temperature of the column (\(14^{\circ} \mathrm{C}\)) \(r\) is the radial position in the cylinder \(L_c\) is the characteristic dimension obtained in Step 1 With this equation, we can find the surface temperature (\(r=L_c\)) and center temperature (\(r=0\)).

04

Solve for the time when the surface temperature reaches \(27^{\circ} \mathrm{C}\)

Set \(T = 27^{\circ} \mathrm{C}\) and \(r=L_c\), then solve the equation from Step 3 for time \(t\). We use the known values of Bi and Fo, and perform an iterative solution to obtain the time required for the surface temperature to reach \(27^{\circ} \mathrm{C}\).

05

Calculate the amount of heat transfer until the center temperature reaches \(28^{\circ} \mathrm{C}\)

Once we have found the time when the center temperature of the column reaches \(28^{\circ} \mathrm{C}\), we can calculate the heat transfer using the following formula: \(Q = \rho V c_p(T - T_i)\) Where: \(Q\) is the heat transfer \(\rho\) is the density of the concrete (\(1600 \,\mathrm{kg} / \mathrm{m}^{3}\)) \(V\) is the volume of the column (\(V=\dfrac{\pi d^2}{4} h\)) \(c_p\) is the specific heat capacity of the concrete (\(0.84 \,\mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}\)) \(T\) is the center temperature of the column (\(28^{\circ} \mathrm{C}\)) \(T_i\) is the initial temperature of the column (\(14^{\circ} \mathrm{C}\))

06

Calculate the amount of heat transfer until the surface temperature reaches \(27^{\circ} \mathrm{C}\)

Similar to Step 5, now set \(T=27 ^{\circ} \mathrm{C}\) when calculating the heat transfer using the same formula: \(Q = \rho V c_p(T - T_i)\)

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Key Concepts

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

Biot Number

The Biot number (Bi) is a dimensionless quantity in heat transfer calculations that helps determine the relation between conductive heat resistance within a body and convective heat transfer across its surface. It is generally given by the equation Bi = \frac{hL_c}{k}; where \(h\) is the heat transfer coefficient, \(L_c\) is the characteristic length, and \(k\) is the thermal conductivity of the material.

For a cylindrical object like the concrete column in our exercise, the characteristic length is computed based on the volume to surface area ratio. A smaller Biot number indicates that the heat conduction within the object is fast compared to the convective heat transfer on its surface, which justifies using a lumped capacitance model for simplicity. Conversely, a larger Biot number suggests that the internal resistance to heat conduction cannot be neglected.

Fourier Number

The Fourier number (Fo) is another dimensionless number, crucial for analyzing transient heat conduction. It is represented by the formula Fo = \frac{αt}{L_c^2} where \(α\) is the thermal diffusivity, \(t\) stands for the time, and \(L_c\) is the characteristic length as previously defined.

The Fourier number essentially compares the rate of heat conduction within the material to the rate of thermal energy storage. A higher Fourier number indicates that more time has elapsed, or in simpler terms, the heat has had more time to diffuse through the material. When dealing with heat transfer questions, particularly for non-steady states, the Fourier number helps in predicting temperature distribution over time.

Heat Transfer Coefficient

The heat transfer coefficient (symbolized as \(h\)) is an important factor in the convective heat transfer process. It quantifies the convective heat transfer rate per unit area per unit temperature difference between a solid surface and the fluid around it, measured in \(W/m^2K\).

In our exercise, the heat transfer coefficient determines how effectively the ambient air transfers heat to the concrete column. The value of \(h\) can significantly affect how quickly the column's surface temperature rises to the desired level. It’s essential to accurately determine \(h\) in practice as it can be influenced by various factors including fluid velocity, fluid properties, and the nature of the surface.

Thermal Conductivity

Thermal conductivity (\(k\)) is a material property that indicates how well a material can conduct heat. It is given in units of \(W/mK\).

The thermal conductivity of concrete, used in the provided cylinder example, directly affects how heat is conducted through the material from its heated surface to the cooler interior. In our exercise, the value of \(k\) for concrete influences the calculation of the Biot number, which determines whether simple models can be applied to predict the temperature distribution within the column.

Thermal Diffusivity

Thermal diffusivity (\(α\)) combines the effects of thermal conductivity, density (\(ρ\)), and specific heat capacity (\(c_p\)) into a single term. It is defined by \(α = \frac{k}{ρc_p}\) and measures how quickly a material can respond to changes in temperature, given in \(m^2/s\).

High thermal diffusivity means the material adjusts its temperature rapidly in response to a temperature change at its boundaries, making it significant in transient heat conduction analysis. In the context of our concrete column, the thermal diffusivity affects the rate at which the temperature increases towards the target of \(27°C\) when exposed to the ambient air.

Specific Heat Capacity

Specific heat capacity (\(c_p\)) is a measure of the amount of heat energy required to raise the temperature of a unit mass of a substance by one degree Celsius (or Kelvin), expressed in \(kJ/kgK\).

For the concrete cylinder in our exercise, the specific heat capacity of concrete helps determine how much heat needs to be absorbed to achieve a certain temperature change. This property plays a key role in the calculation of the total heat transfer required to raise the column’s temperature as part of the solution process.

Transient Heat Conduction

Transient heat conduction refers to the situation where temperature varies with both position and time within a material. Unlike steady-state conduction, transient analysis must account for the time-dependent nature of heat flow.

In the exercise, we examine how a concrete column responds over time as it is exposed to a different ambient temperature. The analytical one-term approximation method leverages the Biot and Fourier numbers to approximate the transient behavior and predict how long it takes for the column to reach certain temperatures. It highlights the dynamic process of heat absorption and distribution throughout the column until a steady state is eventually reached. The application of these principles is crucial in fields such as building design, where material choice and dimensions impact thermal comfort and energy efficiency.