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Sections of the trans-Alaska pipeline run above the ground and ate supported by vertical steel shafts \((k=25\) W/m \(-\mathrm{K})\) that are \(1 \mathrm{~m}\) long and have a cross-sectional area of \(0.005 \mathrm{~m}^{2}\). Under normal cperating conditions, the temperature variation along the length of a shaft is known to be governed by an expression of the form $$ T=100-150 x+10 x^{2} $$ where \(T\) and \(x\) have units of \({ }^{\circ} \mathrm{C}\) and meters, respectively. Temperature variations are small over the shaft cross section. Evaluate the temperature and conduction heat rate at the shaft-pipeline joint \((x=0)\) and at the shaft-ground interface \((x=1 \mathrm{~m})\). Explain the difference in the heat rates.

Step by step solution

01

Temperature at Shaft-Pipeline Joint

To find the temperature at the shaft-pipeline joint, substitute \(x = 0\) into the given temperature expression:\[ T = 100 - 150(0) + 10(0)^2 = 100^\circ \text{C} \] The temperature at the shaft-pipeline joint is 100°C.

02

Temperature at Shaft-Ground Interface

To find the temperature at the shaft-ground interface, substitute \(x = 1\) into the given expression:\[ T = 100 - 150(1) + 10(1)^2 = 100 - 150 + 10 = -40^\circ \text{C} \] The temperature at the shaft-ground interface is -40°C.

03

Calculate Temperature Gradient

To evaluate the conduction heat rate, calculate the temperature gradient \(\frac{dT}{dx}\). Differentiate the given temperature expression:\[ \frac{dT}{dx} = -150 + 2 \cdot 10x = -150 + 20x \]

04

Temperature Gradient at Shaft-Pipeline Joint

Substitute \(x = 0\) into the gradient obtained in Step 3:\[ \left(\frac{dT}{dx}\right)_{x=0} = -150 + 20(0) = -150 \text{ °C/m} \]

05

Temperature Gradient at Shaft-Ground Interface

Substitute \(x = 1\) into the gradient obtained in Step 3:\[ \left(\frac{dT}{dx}\right)_{x=1} = -150 + 20(1) = -130 \text{ °C/m} \]

06

Calculate Heat Rate at Shaft-Pipeline Joint

Use Fourier's Law, \(q = -kA \frac{dT}{dx}\), where \(k = 25\text{ W/m-K}\), and \(A = 0.005\text{ m}^2\) to find the heat rate:\[ q_{x=0} = -25 \cdot 0.005 \cdot (-150) = 18.75 \text{ W} \]

07

Calculate Heat Rate at Shaft-Ground Interface

Use Fourier's Law again with the gradient at \(x = 1\):\[ q_{x=1} = -25 \cdot 0.005 \cdot (-130) = 16.25 \text{ W} \]

08

Explanation of Heat Rate Difference

The difference in the heat rates is due to the change in temperature gradient along the length of the shaft. The gradient is more negative at the pipeline joint \((x=0)\), resulting in a higher heat flow rate compared to the ground interface \((x=1)\).

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Key Concepts

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

Fourier's Law

Fourier's Law is a fundamental principle of heat conduction. It states that the rate of heat transfer through a material is proportional to the negative of the temperature gradient and the cross-sectional area through which the heat flows. The formula for Fourier's Law is:

• \( q = -k A \frac{dT}{dx} \)

Here, \( q \) is the heat conduction rate (W), \( k \) is the thermal conductivity of the material (W/m-K), \( A \) is the cross-sectional area (m²), and \( \frac{dT}{dx} \) is the temperature gradient (°C/m).
This law is critical in understanding how heat moves through materials and designing systems like cooling devices and heat exchangers.
In our exercise, Fourier's Law helps calculate the heat transferred from the pipeline to the ground via the steel shaft.

Temperature Gradient

The temperature gradient is the rate of change of temperature with respect to distance. In mathematical terms, it is the derivative of temperature \( T \) with respect to position \( x \), \( \frac{dT}{dx} \).
In the context of heat conduction, it tells us how quickly the temperature changes along the material. A higher gradient indicates a steeper change in temperature, leading to more significant heat transfer.
In our problem, you need to calculate the temperature gradient at two points on the shaft to understand the heat conduction rate. At \( x = 0 \), the gradient is steeper compared to \( x = 1 \), indicating more heat flow near the pipeline joint.

Conduction Heat Rate

Conduction heat rate is the amount of thermal energy transferred per unit time from one point to another within a material due to a temperature difference.
Using Fourier's Law, we calculate this rate:

• For \( x = 0 \), the heat rate is found to be 18.75 W.
• For \( x = 1 \), the heat rate reduces to 16.25 W.

This reduction demonstrates the dependency of the conduction heat rate on the temperature gradient.
It indicates how efficiently heat is being transferred through the steel shaft, providing insights for thermal insulation or heating system improvements in real-world applications.

Trans-Alaska Pipeline

The Trans-Alaska Pipeline is a major oil pipeline that runs through the state of Alaska. Due to extreme environmental conditions, parts of the pipeline are supported by steel shafts to prevent the oil from cooling and freezing.
The thermal conductivity of these steel shafts is crucial in regulating the temperature of the oil they transport. Engineers must carefully consider the heat conduction properties to maintain optimal temperatures.
The temperature changes along the shaft, impacting the heat flow rate, as observed in the given exercise.
This understanding ensures that the pipeline operates efficiently and safely, preventing blockages or potential hazards.

Differentiation in Thermodynamics

Differentiation is a mathematical tool used extensively in thermodynamics to analyze how a quantity changes with another, such as temperature with distance. It allows us to derive relationships and optimize systems for desired thermal performance.
In our exercise, differentiation helps determine the temperature gradient. By differentiating the temperature expression \( T = 100 - 150x + 10x^2 \), we find \( \frac{dT}{dx} = -150 + 20x \).
This equation lets us evaluate the gradient at different points along the shaft, essential for calculating the heat conduction rate using Fourier's Law.
This approach is central in heat transfer analysis, ensuring better system design and energy efficiency.