91Ó°ÊÓ

For each of the following cases, determine an appropriate characteristic length \(L_{c}\) and the corresponding Biot number \(B i\) that is associated with the transient thermal response of the solid object. State whether the lumped capacitance approximation is valid. If temperature information is not provided, evaluate properties at \(T=300 \mathrm{~K}\). (a) A toroidal shape of diameter \(D=50 \mathrm{~mm}\) and cross-sectional area \(A_{c}=5 \mathrm{~mm}^{2}\) is of thermal conductivity \(k=2.3 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The surface of the torus is exposed to a coolant corresponding to a convection coefficient of \(h=50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (b) A long, hot AISI 304 stainless steel bar of rectangular cross section has dimensions \(w=3 \mathrm{~mm}\), \(W=5 \mathrm{~mm}\), and \(L=100 \mathrm{~mm}\). The bar is subjected to a coolant that provides a heat transfer coefficient of \(h=15 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) at all exposed surfaces. (c) A long extruded aluminum (Alloy 2024) tube of inner and outer dimensions \(w=20 \mathrm{~mm}\) and \(W=24 \mathrm{~mm}\), respectively, is suddenly submerged in water, resulting in a convection coefficient of \(h=37 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) at the four exterior tube surfaces. The tube is plugged at both ends, trapping stagnant air inside the tube. (d) An \(L=300-m m\)-long solid stainless steel rod of diameter \(D=13 \mathrm{~mm}\) and mass \(M=0.328 \mathrm{~kg}\) is exposed to a convection coefficient of \(h=30 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (e) A solid sphere of diameter \(D=12 \mathrm{~mm}\) and thermal conductivity \(k=120 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) is suspended in a large vacuum oven with internal wall temperatures of \(T_{\text {sur }}=20^{\circ} \mathrm{C}\). The initial sphere temperature is \(T_{i}=100^{\circ} \mathrm{C}\), and its emissivity is \(\varepsilon=0.73\). (f) A long cylindrical rod of diameter \(D=20 \mathrm{~mm}\), density \(\rho=2300 \mathrm{~kg} / \mathrm{m}^{3}\), specific heat \(c_{p}=1750 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and thermal conductivity \(k=16 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) is suddenly exposed to convective conditions with \(T_{\infty}=20^{\circ} \mathrm{C}\). The rod is initially at a uniform temperature of \(T_{i}=200^{\circ} \mathrm{C}\) and reaches a spatially averaged temperature of \(T=100^{\circ} \mathrm{C}\) at \(t=225 \mathrm{~s}\). (g) Repeat part (f) but now consider a rod diameter of \(D=200 \mathrm{~mm}\).

Step by step solution

01

Calculate the characteristic length Lc

Since the torus is cylindrical in nature, calculate the characteristic dimension as the volume/(surface area): \[ L_{c} = \frac{\pi r_{d}^{2} - \pi r_{i}^{2}}{2\pi r_{d} + 2\pi r_{i}} = \frac{A_{c}}{\pi (r_{d} + r_{i})} \] Where \( r_{d} \) and \( r_{i} \) are the outer and inner radii of the torus. Given that \( D = 50 \mathrm{~mm} \) and \( A_{c} = 5 \mathrm{~mm}^{2} \), we can find the inner and outer radii and determine Lc.

02

Determine the Biot number Bi

Using the formula Bi = \( \frac{h L_{c}}{k} \), we can input the known values of h and k and the value of Lc calculated in the previous step to find the Biot number.

03

Assess the validity of the lumped capacitance approximation

If the Biot number is less than or equal to 0.1, the lumped capacitance approximation is valid. (b)

04

Calculate the characteristic length Lc

For an object with a rectangular cross section, we can take the average of the two smallest dimensions (w and W) as the characteristic length: \[ L_{c} = \frac{w + W}{2} \]

05

Determine the Biot number Bi

Using the formula Bi = \( \frac{h L_{c}}{k} \), we can substitute the known values of h, k, and Lc to obtain the Biot number.

06

Assess the validity of the lumped capacitance approximation

If the Biot number is less than or equal to 0.1, the lumped capacitance approximation is valid. (c)

07

Calculate the characteristic length Lc

For a hollow cylinder, we can calculate the characteristic length by taking the volume/(surface area) of the hollow region: \[ L_{c} = \frac{\pi R_{o}^{2} - \pi R_{i}^{2}}{2\pi R_{o} + 2\pi R_{i}} \]

08

Determine the Biot number Bi

Using the formula Bi = \( \frac{h L_{c}}{k} \), we can substitute the known values of h, k, and Lc to obtain the Biot number.

09

Assess the validity of the lumped capacitance approximation

If the Biot number is less than or equal to 0.1, the lumped capacitance approximation is valid. (d)

10

Calculate the characteristic length Lc

Since the rod is cylindrical, we can take half the diameter as the characteristic length: \[ L_{c} = \frac{D}{2} \]

11

Determine the Biot number Bi

Using the formula Bi = \( \frac{h L_{c}}{k} \), we can substitute the known values of h, k, and Lc to obtain the Biot number.

12

Assess the validity of the lumped capacitance approximation

If the Biot number is less than or equal to 0.1, the lumped capacitance approximation is valid. (e)

13

Calculate the characteristic length Lc

For a sphere, the characteristic length is equal to the radius: \[ L_{c} = \frac{D}{2} \]

14

Calculate the Biot number Bi

Since this problem involves radiation instead of convection, we will use the radiation analog of the Biot number, which is \[ Bi_{r} = \frac{2hrL_{c}}{k} \] where hr is the radiation heat transfer coefficient.

15

Assess the validity of the lumped capacitance approximation

If the radiation Biot number is less than or equal to 0.1, the lumped capacitance approximation is valid. (f)

16

Calculate the characteristic length Lc

Since the rod is cylindrical, we can take half the diameter as the characteristic length: \[ L_{c} = \frac{D}{2} \]

17

Determine the Biot number Bi

Using the formula Bi = \( \frac{h L_{c}}{k} \), we can substitute the known values of h, k, and Lc to obtain the Biot number.

18

Assess the validity of the lumped capacitance approximation

If the Biot number is less than or equal to 0.1, the lumped capacitance approximation is valid. (g)

19

Calculate the characteristic length Lc

Since the rod is cylindrical, we can take half the diameter as the characteristic length: \[ L_{c} = \frac{D}{2} \]

20

Determine the Biot number Bi

Using the formula Bi = \( \frac{h L_{c}}{k} \), we can substitute the known values of h, k, and Lc to obtain the Biot number.

21

Assess the validity of the lumped capacitance approximation

If the Biot number is less than or equal to 0.1, the lumped capacitance approximation is valid.

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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 is an important figure in the study of heat transfer, especially when using the lumped capacitance method. It is a dimensionless parameter defined as the ratio of conductive resistance within a body to the convective resistance across the surface of the body. Mathematically, it can be expressed as: \[Bi = \frac{hL_{c}}{k}\] where:

• \(h\) is the heat transfer coefficient (in \(W/m^2 \cdot K\)),
• \(L_{c}\) is the characteristic length (in meters),
• \(k\) is the thermal conductivity of the material (in \(W/m \cdot K\)).

A Biot number less than or equal to 0.1 indicates that the lumped capacitance approximation is valid. This suggests that the temperature within an object can be assumed uniform, simplifying thermal analysis. If the Biot number is greater than 0.1, temperature gradients within the object become significant, meaning more complex models are needed to accurately describe the heat transfer process.

characteristic length

The characteristic length \(L_{c}\) is a measure used to describe the "size" of the object relevant to the heat transfer process. It helps in evaluating the object's thermal behavior during heat transfer:For different shapes, the characteristic length can be calculated differently, as follows:

• **Cylindrical objects** such as rods have a characteristic length calculated as half the diameter: \[ L_{c} = \frac{D}{2} \]
• **Rectangular objects** use the average of the two smallest dimensions: \[ L_{c} = \frac{w + W}{2} \]
• **Hollow cylinders** or toroidal shapes might require a more detailed formula taking both inner and outer surfaces into account: \[ L_{c} = \frac{A_{c}}{2\pi(r_{d} + r_{i})} \]

By calculating this length, it becomes easier to predict how the object will behave under heat transfer conditions. This helps determine whether a simpler model, like the lumped capacitance method, is appropriate.

heat transfer coefficient

The heat transfer coefficient, denoted as \(h\), is crucial in analyzing heat exchange between a solid object and a fluid in contact with it. This coefficient is significant as it reflects how effectively heat is being transferred through the surface of the object.In heat transfer analysis, the value of \(h\) can be influenced by:

• The type of fluid (such as air, water, oil),
• The state of the fluid flow (laminar or turbulent),
• Surface properties of the solid,
• The temperature difference between the solid and the fluid.

It is measured in \(W/m^2 \cdot K\) and can vary widely depending on the condition of the experiment or practical situation. For example, a solid object immersed in water might have a higher \(h\) compared to the same object exposed to air. Understanding the heat transfer coefficient helps in assessing whether simple methods like using the lumped capacitance model are applicable or more complex computational approaches are necessary.