Question

In a production facility, large plates made of stainless steel \(\left(k=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \alpha=3.91 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)\) of \(40 \mathrm{~cm}\) thickness are taken out of an oven at a uniform temperature of \(750^{\circ} \mathrm{C}\). The plates are placed in a water bath that is kept at a constant temperature of \(20^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(600 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The time it takes for the surface temperature of the plates to drop to \(100^{\circ} \mathrm{C}\) is (a) \(0.28 \mathrm{~h}\) (b) \(0.99 \mathrm{~h}\) (c) \(2.05 \mathrm{~h}\) (d) \(3.55 \mathrm{~h}\) (e) \(5.33 \mathrm{~h}\)

Short answer

Answer: (b) \(0.99 \mathrm{~h}\)

Step-by-step solution

List down the given parameters

We are given: - Thermal conductivity \((k)\): \(15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) - Thermal diffusivity \((\alpha)\): \(3.91 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\) - Plate thickness \((d)\): \(0.40 \mathrm{~m}\) - Initial temperature of plate \((T_\text{initial})\): \(750^{\circ} \mathrm{C}\) - Temperature of water bath \((T_\text{bath})\): \(20^{\circ} \mathrm{C}\) - Heat transfer coefficient \((h)\): \(600 \mathrm{~W} / \mathrm{m}^2 \cdot \mathrm{K}\) - Final surface temperature of plate \((T_\text{final})\): \(100^{\circ} \mathrm{C}\)

Calculate the Biot number

To find the Biot number \((\text{Bi})\), we will use the formula \(\text{Bi} = \frac{h \cdot d}{k}\) Plugging in the given values, we get: \(\text{Bi} = \frac{600 \mathrm{~W} \text{/} \mathrm{m}^2 \cdot \mathrm{K} \cdot 0.40 \mathrm{~m}}{15 \mathrm{~W} \text{/} \mathrm{m} \cdot \mathrm{K}} = 16\)

Use the dimensionless temperature definition

Now, we will define dimensionless temperature (\(\Theta\)) as \(\Theta = \frac{T_\text{final} - T_\text{bath}}{T_\text{initial} - T_\text{bath}}\). Plugging in the given temperatures, we get: \(\Theta = \frac{100^{\circ} \mathrm{C} - 20^{\circ} \mathrm{C}}{750^{\circ} \mathrm{C} - 20^{\circ} \mathrm{C}} = \frac{80}{730} = 0.1096\)

Use the Newton's Law of Cooling formula

Newton's Law of Cooling states that \(\Theta = e^{-\text{Bi} \cdot \text{Fo}}\), where the Fourier number (Fo) is defined as \(\text{Fo} = \frac{\alpha \cdot t}{d^2}\) (with \(t\) being the time). We are trying to find \(t\), so we will rearrange the formula to find \(\text{Fo}\) and then solve for \(t\): 1. \(\text{Fo} = -\frac{1}{\text{Bi}} \cdot \ln{(\Theta)}\) 2. \(t = \frac{\text{Fo} \cdot d^2}{\alpha}\)

Calculate the Fourier number

Using the formula \(\text{Fo} = -\frac{1}{\text{Bi}} \cdot \ln{(\Theta)}\): \(\text{Fo} = -\frac{1}{16} \cdot \ln{(0.1096)} \approx 0.0982\)

Calculate the time

Now, we will use the formula \(t = \frac{\text{Fo} \cdot d^2}{\alpha}\) to calculate the time taken for the surface temperature to drop to \(100^{\circ} \mathrm{C}\). \(t = \frac{0.0982 \cdot (0.40 \mathrm{~m})^2}{3.91 \times 10^{-6} \mathrm{~m}^2 \text{/} \mathrm{s}} \approx 4010 \mathrm{s}\) Lastly, convert the time in seconds to hours: \(t = \frac{4010 \mathrm{s}}{3600 \mathrm{s/h}} \approx 1.11 \mathrm{~h}\) The closest answer is (b) \(0.99 \mathrm{~h}\).

Key concepts

Biot Number

The Biot Number (\(\text{Bi}\)) is a dimensionless quantity that compares the rate of heat transfer across the boundary of a body to the rate of heat conduction within the body. It helps determine whether the temperature within the object can be considered uniform. When dealing with problems involving heat transfer, calculating the Biot Number is crucial to understanding how quickly heat moves through different materials.

To calculate the Biot Number, use the formula:

• \(\text{Bi} = \frac{h \cdot d}{k}\)

Here, \(h\) represents the heat transfer coefficient, \(d\) is the characteristic length (such as the thickness of a plate), and \(k\) stands for thermal conductivity. If the Biot Number is much less than 1, it suggests that surface heat transfer is highly efficient compared to internal conduction. Conversely, if it's much more than 1, the internal conduction may not keep up with the surface heat transfer, indicating the need to analyze the temperature gradient within the material.

Fourier Number

The Fourier Number (\(\text{Fo}\)) is a dimensionless number used to analyze heat conduction problems, especially in transient heat transfer. It is essential for understanding how heat diffuses through a material over time. The Fourier Number helps predict how long it takes for a material to reach a desired temperature.

The Fourier Number is given by:

• \(\text{Fo} = \frac{\alpha \cdot t}{d^2}\)

In this equation, \(\alpha\) represents thermal diffusivity, \(t\) is the time, and \(d\) is the characteristic length.

By connecting these components, the Fourier Number gives insight into how quickly the temperature changes within a material. A higher Fourier Number implies quicker heat diffusion, which is significant in applications like cooling processes where quick temperature changes are desired.

Newton's Law of Cooling

Newton's Law of Cooling describes the rate at which an exposed body changes temperature through radiation and convection. It's fundamental in many heat transfer problems to predict how temperature evolves over time.

The law is expressed mathematically as:

• \(\Theta = e^{-\text{Bi} \cdot \text{Fo}}\)

Here, \(\Theta\) is the dimensionless temperature, \(\text{Bi}\) is the Biot Number, and \(\text{Fo}\) is the Fourier Number.

Newton's Law of Cooling tells us that the rate of temperature change is proportional to the difference in temperature between the body and its environment. This proportionality means larger temperature differences lead to faster cooling rates, crucial in cooling devices or heat exchangers.

Thermal Conductivity

Thermal conductivity (\(k\)) measures a material's ability to conduct heat. It's a fundamental property in the study of heat transfer, dictating how efficiently heat can move through a material. High thermal conductivity implies that heat flows easily, while low thermal conductivity suggests poor heat flow.

In mathematical terms, thermal conductivity appears in various heat transfer equations and is often expressed in units of watts per meter-kelvin (\(W/m \cdot K\)).

In design or engineering calculations, knowing the thermal conductivity of materials helps engineers choose the right materials for insulating systems or heat sinks, ensuring optimal thermal management in various applications.

Thermal Diffusivity

Thermal diffusivity (\(\alpha\)) is an important material property that combines thermal conductivity and specific heat capacity. It measures how quickly a material can respond to changes in temperature, playing a critical role in unsteady or transient heat flow analyses.

It is defined as:

• \(\alpha = \frac{k}{\rho \cdot c}\)

where \(k\) is thermal conductivity, \(\rho\) is density, and \(c\) is specific heat capacity.

A higher thermal diffusivity indicates that a material can quickly reach thermal equilibrium, useful in many industrial processes where rapid temperature adjustments are necessary. Understanding thermal diffusivity helps in designing materials for efficient heating or cooling in various engineering fields.