Temperature and Heat Flux Distribution in a Natural Convection Enclosure Flow. Bohn Ren Anderson. Coefficient of thermal expansion e dimensionless temperature, Eq. (3) eb average value of e in the test cell v. Each slot 1as perpendicular to the electrically heated wa 11, and the three slot l ocat i ans were on. GENERATOR FIELD WINDING SHORTED TURN DETECTION TECHNOLOGY Donald R. Albright, David J. Albright and James D. Albright Generatortech, Inc. ABSTRACT The use of air-gap magnetic flux probes has proven effective in the detection of generator rotor. Thermal expansion of a slot’s center. The use of air-gap magnetic flux probes has proven effective in the detection of generator rotor winding shorted turns and has helped to improve the quality of predictive maintenance decisions.

Linear Expansion

Thermal expansion is the tendency of matter to change in volume in response to a change in temperature.

Learning Objectives

Describe volume changes that take place in response to a temperature change

Key Takeaways

Key Points

• Inter-particle potential usually takes an asymmetric form, rather than a symmetric form as a function of particle-particle distance. This is why matters expands and contracts as temperature changes.
• The change in length measurements of an object due to thermal expansion is related to temperature change by a “linear expansion coefficient”, which is given as [latex]alpha_text{L} = frac{1}{text{L}}frac{text{dL}}{text{dT}}[/latex].
• The linear expansion coefficient is as an approximation over a narrow temperature interval only.

Key Terms

• potential: A curve describing the situation where the difference in the potential energies of an object in two different positions depends only on those positions.

Thermal expansion is the tendency of matter to change in volume in response to a change in temperature. (An example of this is the buckling of railroad track, as seen in. ) Atoms and molecules in a solid, for instance, constantly oscillate around its equilibrium point. This kind of excitation is called thermal motion. When a substance is heated, its constituent particles begin moving more, thus maintaining a greater average separation with their neighboring particles. The degree of expansion divided by the change in temperature is called the material’s coefficient of thermal expansion; it generally varies with temperature.

Fig 1: Thermal expansion of long continuous sections of rail tracks is the driving force for rail buckling. This phenomenon resulted in 190 train derailments during 1998–2002 in the US alone.

Thermal Expansion: A brief introduction to thermal expansion for students.

Expansion, Not Contraction

Why does matter usually expand when heated? The answer can be found in the shape of the typical particle-particle potential in matter. Particles in solids and liquids constantly feel the presence of other neighboring particles. This interaction can be represented mathematically as a potential curve. Fig 2 illustrates how this inter-particle potential usually takes an asymmetric form rather than a symmetric form, as a function of particle-particle distance. Note that the potential curve is steeper for shorter distance. In the diagram, (b) shows that as the substance is heated, the equilibrium (or average) particle-particle distance increases. Materials which contract or maintain their shape with increasing temperature are rare. This effect is limited in size, and only occurs within limited temperature ranges.

Fig 2: Typical inter-particle potential in condensed matter (such as solid or liquid).

Linear Expansion

To a first approximation, the change in length measurements of an object (linear dimension as opposed to, for example, volumetric dimension) due to thermal expansion is related to temperature change by a linear expansion coefficient. It is the fractional change in length per degree of temperature change. Assuming negligible effect of pressure, we may write:

[latex]alpha_text{L} = frac{1}{text{L}}frac{text{dL}}{text{dT}}[/latex],

where L is a particular length measurement and dL/dT is the rate of change of that linear dimension per unit change in temperature. From the definition of the expansion coefficient, the change in the linear dimension [latex]Delta text{L}[/latex] over a temperature range [latex]Delta text{T}[/latex] can be estimated to be:

[latex]frac{Delta text{L}}{text{L}} = alpha_text{L} Delta text{T}[/latex].

This equation works well as long as the linear-expansion coefficient does not change much over the change in temperature. If it does, the equation must be integrated.

Area Expansion

Objects expand in all dimensions. That is, their areas and volumes, as well as their lengths, increase with temperature.

Learning Objectives

Express the area thermal expansion coefficient in the form of an equation

Key Takeaways

Key Points

• The area thermal expansion coefficient relates the change in a material’s area dimensions to a change in temperature. It is defined as [latex]alpha_text{A} = frac{1}{text{A}} frac{text{dA}}{text{dT}}[/latex].
• The relationship between the area and linear thermal expansion coefficient is given as the following: [latex]alpha_text{A} = 2 alpha_text{L}[/latex].
• Just like the linear expansion coefficient, the area thermal expansion coefficient works as an approximation over a narrow temperature interval only.

Key Terms

• linear thermal expansion coefficient: The fractional change in length per degree of temperature change.

We learned about the linear expansion (in one dimension) in the previous Atom. Objects expand in all dimensions, and we can extend the thermal expansion for 1D to two (or three) dimensions. That is, their areas and volumes, as well as their lengths, increase with temperature.

Quiz

Before we look into details, here is an interesting question. Imagine that we have a rectangular sheet of metal with a circular hole in the middle. If the metal is heated, we can guess that the the piece, in general, will get larger due to thermal expansion. Now, what is going to happen with the circular hole in the middle? Is the hole going to be larger or smaller? Answer: Imagine that we have a similar metal sheet but without a hole. Draw an imaginary circular line representing the circular hole in our quiz. How does this imaginary circle change as the metal is heated? Yes. It will get bigger. Therefore, you can guess that the hole in our quiz will get larger.

Fig 1: In general, objects expand in all directions as temperature increases. In these drawings, the original boundaries of the objects are shown with solid lines, and the expanded boundaries with dashed lines. (a) Area increases because both length and width increase. The area of a circular plug also increases. (b) If the plug is removed, the hole it leaves becomes larger with increasing temperature, just as if the expanding plug were still in place.

Area thermal expansion coefficient

The area thermal expansion coefficient relates the change in a material’s area dimensions to a change in temperature. It is the fractional change in area per degree of temperature change. Ignoring pressure, we may write: [latex]alpha_text{A} = frac{1}{text{A}} frac{text{dA}}{text{dT}}[/latex], where is some area of interest on the object, and dA/dT is the rate of change of that area per unit change in temperature. The change in the linear dimension can be estimated as: [latex]frac{Delta text{A}}{text{A}} = alpha_text{A} Delta text{T}[/latex]. This equation works well as long as the linear expansion coefficient does not change much over the change in temperature [latex]Delta text{T}[/latex]. If it does, the equation must be integrated.

Relationship to linear thermal expansion coefficient

For isotropic materials, and for small expansions, the linear thermal expansion coefficient is one half of the area coefficient. To derive the relationship, let’s take a square of steel that has sides of length L. The original area will be A = L²,and the new area, after a temperature increase, will be [latex]begin{align} text{A} + Delta text{A} &= (text{L} + Delta text{L})^2 &= text{L}^2 + 2text{L}Delta text{L} + (Delta text{L} )^2 &approx text{L}^2 + 2text{L}Delta text{L} &= text{A} + 2 text{A} frac {Delta text{L}}{text{L}} end{align}[/latex]

The approximation holds for a sufficiently small [latex]Delta text{L}[/latex] campared to L. Since [latex]frac{Delta text{A}}{text{A}} = 2 frac{Delta text{L}}{text{L}}[/latex] from the equation above (and from the definitions of the thermal coefficients), we get [latex]alpha_text{A} = 2 alpha_text{L}[/latex].

Volume Expansion

Substances expand or contract when their temperature changes, with expansion or contraction occurring in all directions.

Learning Objectives

Compare the effects of the pressure on the expansion of gaseous and solid materials

Key Takeaways

Key Points

• Substances that expand at the same rate in every direction are called isotropic.
• In the case of a gas, expansion depends on how the pressure changed in the process because the volume of a gas will vary appreciably with pressure as well as temperature.
• For a solid, we can ignore the effects of pressure on the material, and the volumetric thermal expansion coefficient can be written as [latex]alpha_text{V} = frac{1}{text{V}} frac{text{dV}}{text{dT}}[/latex]. For isotropic materials, [latex]alpha_text{V} =3 alpha_text{L}[/latex].

Key Terms

• isotropic: Having properties that are identical in all directions; exhibiting isotropy.
• linear thermal expansion coefficient: The fractional change in length per degree of temperature change.

The volumetric thermal expansion coefficient is the most basic thermal expansion coefficient. illustrates that, in general, substances expand or contract when their temperature changes, with expansion or contraction occurring in all directions. Such substances that expand in all directions are called isotropic. For isotropic materials, the area and linear coefficients may be calculated from the volumetric coefficient (discussed below).

Volumetric Expansion: In general, objects expand in all directions as temperature increases. In these drawings, the original boundaries of the objects are shown with solid lines, and the expanded boundaries with dashed lines. (a) Area increases because both length and width increase. The area of a circular plug also increases. (b) If the plug is removed, the hole it leaves becomes larger with increasing temperature, just as if the expanding plug were still in place. (c) Volume also increases, because all three dimensions increase.

Thermal Expansion – Volume Expansion: A brief introduction to thermal expansion for students.

Mathematical definitions of these coefficients are defined below for solids, liquids, and gasses:

[latex]alpha_text{V}= frac{1}{text{V}} (frac{partial text{V}}{partial text{T}})_text{p}[/latex].

The subscript p indicates that the pressure is held constant during the expansion. In the case of a gas, the fact that the pressure is held constant is important, as the volume of a gas will vary appreciably with pressure as well as with temperature.

For a solid, we can ignore the effects of pressure on the material, thus the volumetric thermal expansion coefficient can be written:

[latex]alpha_text{V} = frac{1}{text{V}} frac{text{dV}}{text{dT}}[/latex],

where V is the volume of the material, and is dV/dT the rate of change of that volume with temperature. This means that the volume of a material changes by some fixed fractional amount. For example, a steel block with a volume of 1 cubic meter might expand to 1.002 cubic meters when the temperature is raised by 50 °C. This is an expansion of 0.2%. The volumetric expansion coefficient would be 0.2% for 50 °C, or 0.004% per degree C.

Relationship to Linear Thermal Expansion Coefficient

For isotropic material, and for small expansions, the linear thermal expansion coefficient is one third the volumetric coefficient. To derive the relationship, let’s take a cube of steel that has sides of length L. The original volume will be V = L³,and the new volume, after a temperature increase, will be:

[latex]begin{align} text{V}+ Delta text{V} &= (text{L} + Delta text{L})^3 &= text{L}^3 + 3text{L}^2Delta text{L} + 3text{L}(Delta text{L} )^2 +(Delta text{L})^3 &approx text{L}^3 + 3text{L}^2Delta text{L} &= text{V} + 3 text{V} frac {Delta text{L}}{text{L}} end{align}[/latex].

The approximation holds for a sufficiently small [latex]Delta text{L}[/latex] compared to L. Since:

[latex]frac{Delta text{V}}{text{V}} = 3frac{Delta text{L}}{text{L}}[/latex]

(and from the definitions of the thermal coefficients), we arrive at:

[latex]alpha_text{V} =3 alpha_text{L}[/latex].

Special Properties of Water

Objects will expand with increasing temperature, but water is the most important exception to the general rule.

Learning Objectives

Describe thermal expansion properties of water

Key Takeaways

Key Points

• Water expands with increasing temperature (its density decreases) when it is at temperatures greater than 4ºC (40ºF). However, it expands with decreasing temperature when it is between +4ºC and 0ºC (40ºF to 32ºF). Water is densest at +4ºC.
• Due to the peculiar thermal expansion property of water, a pond surface can completely freeze over, while the bottom may remain at 4ºC. Fish and other aquatic life can survive in 4ºC water beneath ice, due to this unusual characteristic of water.
• The solid form of most substances is denser than the liquid phase; thus, a block of most solids will sink in the liquid. However, a block of ice floats in liquid water because ice is less dense.

Key Terms

• hydrogen bond: A weak bond in which a hydrogen atom in one molecule is attracted to an electronegative atom (usually nitrogen or oxygen) in the same or different molecule.

Special Properties of Water

In general, objects will expand with increasing temperature. However, a number of materials contract on heating within certain temperature ranges; this is usually called negative thermal expansion, rather than “thermal contraction. ” Water is the most important exception to the general rule. Water has this unique characteristic because of the particular nature of the hydrogen bond in H₂O.

Density of Water as Temperature Changes

At temperatures greater than 4ºC (40ºF) water expands with increasing temperature (its density decreases). However, it expands with decreasing temperature when it is between +4ºC and 0ºC (40ºF to 32ºF). Water is densest at +4ºC.

Water Density vs. Temperature: The density of water as a function of temperature. Note that the thermal expansion is actually very small. The maximum density at +4ºC is only 0.0075% greater than the density at 2ºC, and 0.012% greater than that at 0ºC.

Perhaps the most striking effect of this phenomenon is the freezing of water in a pond. When water near the surface cools down to 4ºC it is denser than the remaining water and thus will sink to the bottom. This “turnover” results in a layer of warmer water near the surface, which is then cooled. Eventually the pond has a uniform temperature of 4ºC. If the temperature in the surface layer drops below 4ºC, the water is less dense than the water below, and thus stays near the top.

As a result, the pond surface can completely freeze over, while the bottom may remain at 4ºC. The ice on top of liquid water provides an insulating layer from winter’s harsh exterior air temperatures. Fish and other aquatic life can survive in 4ºC water beneath ice, due to this unusual characteristic of water. It also produces circulation of water in the pond that is necessary for a healthy ecosystem of the body of water.

Temperature in a Lake: Temperature distribution in a lake on warm and cold days in winter

Ice Versus Water

The solid form of most substances is denser than the liquid phase; thus, a block of most solids will sink in the liquid. However, a block of ice floats in liquid water because ice is less dense. Upon freezing, the density of water decreases by about 9%.

Thermal Expansion Of Steel

One man’s trash is another man’s treasure.

Sometimes the best thing you can do with an unused item is to trade it to someone else for something you value. To assist in this endeavor I present the Vending Machine.

When placed it looks like an unused crate that has been place upright on it’s side. The first player that access the Vending Machine after placement will take ownership and their name will be displayed on the exterior surface. In the image above “OreCruncher” placed the Vending Machine and accessed it.

The Vending Machine can be configured for up to six different trades. Each trade is comprised of three different slots: two ask slots and one offer slot. The ask slots define what a player would give in return for the item(s) in the offer slot. In the image above a player can give one Gold Ingot and 5 Gold Nuggets and recieve 16 Obsidian in return. Also, if a player wanted that Bookshelf they can give some Iron Ingots.

Once a Vending Machine is configured the contents of the trades will be displayed inside Vending Machine rendering. This view would give traders an idea of what is being offered for trade as well as what is wanted in return without having to open up the Vending Machine. The item stack display is sensitive to how far away the player is away from the Vending Machine. At certain ranges the quantity information and the item stacks will not render.

If a non-owner opens up the Vending Machine they will be presented with the trade dialog above. If an owner wants to access the trade dialog they can shift + right click with an empty hand.

Configuring Trades

Configuration is pretty straight forward. The trade slots are ghost slots. Clicking on them with an item stack will set the slot to that item type and quantity. When the mouse cursor is over a slot several other short cuts methods for manipulating the slot content are available:

• Wheel on the mouse. This will cause the quantity to go up/down depending on the direction the wheel is scrolled.
• Left Click. Reduce the stack by one.
• Shift+Left Click. Reduce the stack by half.
• Right Click. Increment the stack by one.
• Shift+Right Click. Double the stack size.

The region below the configuration slots is the Vending Machines inventory. Items that are to be handed out are placed into this inventory. Items that a player gives in return are also placed into this inventory. Make sure you leave empty space to hold incoming trade items. If there isn’t space a trade will not happen.

Thermal Expansion Mod

Trading Items

Trading is pretty easy. Just click on the item stack that is being offered for trade. The item(s) that are wanted in exchange are automatically taken from the player inventory and added to the Vending Machine inventory. The offered item will automatically be placed directly into the player inventory.

A trade could fail for several different reasons:

• The Vending Machine ran out of a particular item to give out.
• The Vending Machine does not have space in it’s inventory to accept items in return.
• The player does not have space in their inventory to get the item.
• The player does not have the right items to give in exchange.

Server Vending Machines

A Server Vending machine essentially has a bottomless inventory. It has infinite amount of items to hand out, and can accept an infinite amount of items in return. The operation is similar to Admin signs that are found on Bukkit servers.

To configure a Vending Machine for Server operation is pretty easy. As OP, equip a Signalum Security Lock from Thermal Expansion, and right click the Vending Machine with the lock. This will toggle the Vending Machine to admin mode. The name that is on the Vending Machine will change to “Server Shop”. Clicking on the Vending Machine again with the Signalum Security Lock will take it out of admin mode.

Setting Title Colors

As a Vending Machine owner, the color of the name displayed on the Vending Machine exterior can be customized. To change the color equip a piece of dye (as defined by the Forge OreDictionary “dye” entry, which includes the standard Minecraft dye items) and right click the Vending Machine. This will change the foreground color of the name to match the dye that was used.

Setting the background color is a little more complicated. By default the background color is black. To change this click on the Vending Machine twice with a piece of dye. First time will set the name foreground color, the second time will set the background color. You will then have to reset the foreground color to the desired color.

(Shift click is not possible with dye. That action is blocked down in the Minecraft code. :)

Other Notes

Expansion Slots Computer

• A player can give items away for free by not setting the ask slots. The Vending Machine will keep giving out item stacks until it runs out of those items from it’s internal inventory.
• A Server Vending Machine will give out an infinite supply of items if no ask slots are configured, so be careful.
• Once ownership is established, the only player that can break a Vending Machine is the owner or someone that is OP.
• By default transport pipes cannot connect to a Vending Machine. This is to prevent siphoning of items in unprotected regions. If a server has region protection in place transport pipe connection could be enabled to allow for automated inventory management.
• Regardless of the transport pipe connection, Vending Machines that are configured for server operation, or are owned by [Thermal Recycling] will automatically block pipe connections.
• The Vending Machine will not drop it’s inventory when in server mode or are owned by [Thermal Recycling].
• The Vending Machine works well with mods that add types of currency, such as Minecraft Money.

Thermal Flux Meter

Configuration Settings