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Chapter 2: Problem 1

How does transient heat transfer differ from steady heat transfer? How does one-dimensional heat transfer differ from two-dimensional heat transfer?

Short Answer

Expert verified
Answer: The main differences between transient and steady heat transfer are related to the constancy of the temperature distribution within the system: transient heat transfer involves temperature distribution changes over time, while steady heat transfer maintains constant temperature distribution. The primary differences between one-dimensional and two-dimensional heat transfers involve spatial considerations: one-dimensional heat transfer assumes heat transfer occurs in one direction, while two-dimensional heat transfer accounts for variations in temperature and heat flow in two spatial directions.

Step by step solution

01

Definition of Transient Heat Transfer

Transient heat transfer refers to the process where the temperature distribution within a system changes over time. This occurs when the heat transfer rate within the system is time dependent, which means that the rate at which heat is transferred is not constant.
02

Definition of Steady Heat Transfer

Steady heat transfer, on the other hand, is the process where the temperature distribution within a system remains constant over time. In this case, the heat transfer rate is time independent, meaning that the rate at which heat is being transferred remains constant.
03

Differences between Transient and Steady Heat Transfer

The main difference between transient and steady heat transfer is the constancy of the temperature distribution within the system. In transient heat transfer, the temperature distribution changes over time, whereas in steady heat transfer, the temperature distribution remains constant. In other words, a system experiencing transient heat transfer will eventually reach a steady state, where the temperature distribution stops changing, and steady heat transfer takes place.
04

Definition of One-dimensional Heat Transfer

One-dimensional heat transfer is a simplification of the heat transfer process in which heat transfer is assumed to take place only in one direction within the system. This type of heat transfer is usually applied to cases with large differences in conductivity or thickness in one direction compared to others, such as a long rod or a wall.
05

Definition of Two-dimensional Heat Transfer

Two-dimensional heat transfer accounts for heat transfer in two spatial directions within a system. This is a more realistic representation of the heat transfer process and is typically used when the temperature distribution varies in two directions. Examples of two-dimensional heat transfer include heat transfer in a rectangular plate or across the corner of a wall.
06

Differences between One-dimensional and Two-dimensional Heat Transfer

The primary difference between one-dimensional and two-dimensional heat transfer lies in the spatial considerations. One-dimensional heat transfer is a simplification that assumes heat transfer occurs only in one direction, while two-dimensional heat transfer accounts for variations in temperature and heat flow in two spatial directions. Understanding the correct dimensionality of a given problem is essential for accurate analysis and modeling of the heat transfer process.

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Key Concepts

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

Transient Heat Transfer
Transient heat transfer occurs when temperature changes over time. This means the heat transfer rate varies as the system evolves.
Think of it like a boiling pot of water, where the temperature heats up until it reaches boiling point. During transient heat transfer, the system is not in equilibrium. Everything is in motion鈥攅nergy is being exchanged.
Factors that influence transient heat transfer include:
  • Material properties: Different materials conduct heat at different rates.
  • Initial conditions: The starting temperature distribution of the system.
  • Boundary conditions: How heat enters or leaves the system.
Understanding transient heat transfer is crucial for designing systems that need to reach desired temperatures over time.
Steady Heat Transfer
Steady heat transfer is characterized by a constant temperature distribution over time. Once equilibrium is achieved, the temperature stops changing.
This is like steady sunshine warming a brick wall to a certain temperature, which then remains stable. The system is in balance鈥攖he rate of heat entering is the same as the rate of heat leaving.
Key aspects of steady heat transfer include:
  • Constant thermal conditions: No change in temperature as time progresses.
  • Fixed heat transfer rate: Energy flows consistently over time.
This concept is widely used in engineering, as it simplifies calculations and designs for thermal systems in equilibrium.
One-dimensional Heat Transfer
In one-dimensional heat transfer, heat moves in just one direction. Solutions to these problems assume minimal variation across other dimensions.
Picture a long, heated metal rod. Heat travels along its length, but not much elsewhere. This simplification is useful when differences in geometry or material properties are significant along one axis.
Essential points of one-dimensional heat transfer:
  • Suitable for slender configurations: Long rods, thin slabs.
  • Assumes negligible lateral heat conduction.
This approach is often the first step in analyzing complex heat transfer scenarios by reducing the problem to its simplest form.
Two-dimensional Heat Transfer
Two-dimensional heat transfer accounts for heat flow in two perpendicular directions. This offers a more realistic depiction of how heat moves through surfaces, like in a flat plate.
Imagine heat spreading across a wall corner. It doesn鈥檛 just flow straight down; it also spreads sideways. This type of analysis is vital for structures with significant temperature gradients in two dimensions.
Important considerations for two-dimensional heat transfer include:
  • Complex geometry: Often applied to plates, walls, or regions with uneven heat application.
  • More intricate calculations: Requires more advanced mathematical methods.
Understanding this concept helps engineers design more efficient cooling and heating systems across various fields.
Temperature Distribution
Temperature distribution refers to how temperature varies within a material or system. It can be a simple gradient or a complex pattern depending on conditions and materials.
Imagine a pie cooling down on a window sill. The temperature is higher in the center and cooler at the edges, forming a gradient.
Factors influencing temperature distribution:
  • Material properties: Conductivity, density, and specific heat.
  • Heat source and sink locations: Where heat is added or removed.
  • System geometry: Shape and size affect distribution patterns.
Understanding temperature distribution is key for predicting system behavior and optimizing designs for efficiency and safety.

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Most popular questions from this chapter

Starting with an energy balance on a ring-shaped volume element, derive the two-dimensional steady heat conduction equation in cylindrical coordinates for \(T(r, z)\) for the case of constant thermal conductivity and no heat generation.

In order to size the compressor of a new refrigerator, it is desired to determine the rate of heat transfer from the kitchen air into the refrigerated space through the walls, door, and the top and bottom section of the refrigerator. In your analysis, would you treat this as a transient or steady- state heat transfer problem? Also, would you consider the heat transfer to be one-dimensional or multidimensional? Explain.

A 6-m-long 3-kW electrical resistance wire is made of \(0.2\)-cm-diameter stainless steel \((k=15.1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). The resistance wire operates in an environment at \(20^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(175 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) at the outer surface. Determine the surface temperature of the wire \((a)\) by using the applicable relation and \((b)\) by setting up the proper differential equation and solving it. Answers: (a) \(475^{\circ} \mathrm{C}\), (b) \(475^{\circ} \mathrm{C}\)

Can a differential equation involve more than one independent variable? Can it involve more than one dependent variable? Give examples.

In subsea oil and natural gas production, hydrocarbon fluids may leave the reservoir with a temperature of \(70^{\circ} \mathrm{C}\) and flow in subsea surrounding of \(5^{\circ} \mathrm{C}\). As a result of the temperature difference between the reservoir and the subsea surrounding, the knowledge of heat transfer is critical to prevent gas hydrate and wax deposition blockages. Consider a subsea pipeline with inner diameter of \(0.5 \mathrm{~m}\) and wall thickness of \(8 \mathrm{~mm}\) is used for transporting liquid hydrocarbon at an average temperature of \(70^{\circ} \mathrm{C}\), and the average convection heat transfer coefficient on the inner pipeline surface is estimated to be \(250 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The subsea surrounding has a temperature of \(5^{\circ} \mathrm{C}\) and the average convection heat transfer coefficient on the outer pipeline surface is estimated to be \(150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If the pipeline is made of material with thermal conductivity of \(60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), by using the heat conduction equation (a) obtain the temperature variation in the pipeline wall, \((b)\) determine the inner surface temperature of the pipeline, \((c)\) obtain the mathematical expression for the rate of heat loss from the liquid hydrocarbon in the pipeline, and \((d)\) determine the heat flux through the outer pipeline surface.
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