משתמש:Hanenr/התפשטות תרמית

התפשטות תרמית היא הנטייה של החומר לשינוי הנפח כתגובה לשינוי טמפרטורה.

כאשר החומר עובר חימום, החלקיקים מתחילים לנוע מהר יותר, ולכן בדרך כלל מתקבל מרחק ממוצע גדול יותר ביניהם. חומרים שמתכווצים עם עליית הטמפרטורה הם נדירים. האפקט הזה מתרחש בטווחי טמפרטורה מוגבלים (ראו דוגמאות בהמשך). אם נחלק את עוצמת ההתפשטות בשינוי בטמפרטורה, נקבל את המקדם של החומר להתפשטות תרמית. מקדם התפשטות תרמית משתנה עם הטמפרטורה.

סקירה כללית

חיזוי התפשטות

אם המשוואה של המצב זמינה, ניתן להשתמש בה כדי לחזות את הערכים של התפשטות תרמית עבור כל הטמפרטורות הנדרשות והלחצים , עם הרבה פונקציות המצב אחרות.

תופעות התכווצות

מספר חומרים מתכווצים עם חימום בטווחי טמפרטורה מסוימים, זה נקרא בדרך כלל התפשטות תרמית שלילית , ולא "התכווצות תרמית". כך, לדוגמה, מקדם התפשטות תרמית של המים יורד לאפס כאשר הוא מקורר בערך ל- 4 ° C, ואז הופך לשלילי מתחת לטמפרטורה זו, זה אומר למים יש צפיפות מקסימלית בטמפרטורה זו, וזה מוביל גופי מים להחזיק בטמפרטורה זו בעומקים נמוכים יותר במהלך תקופות ארוכות של מזג אוויר מתחת לאפס. כמו כן, לסיליקון טהור ל יש מקדם התפשטות תרמי שלילי בטווח טמפרטורות בין 18 קלווין ועד 120 קלווין. .

גורמים המשפיעים על התפשטות תרמית

בניגוד גזים או נוזלים, חומרים מוצקים נוטים לשמור על צורתם כאשר עובר התפשטות תרמית.התפשטות תרמית בדרך כלל יורדת עם הגידול באנרגית קשר, שיש לה גם השפעה על קשיות של מוצקים, ולכן, לחומרים קשחים יש התפשטות תרמית נמוכה. באופן כללי, נוזלים מתפשטים יותר ממוצקים אבל גם הם במידה מועטה. ההתפשטות התרמית של זכוכית גבוהה יותר מזו של גבישים. בסידור מחדש בטמפרטורת המעבר זגוגית, אשר מתרחשים בחומר אמורפי, מתרחשים שיבושים באיפיון של מקדם התפשטות תרמית או חום סגולי. שיבושים אלה מאפשרים זיהוי של טמפרטורת המעבר הזגוגית בהם הופך נוזל בקירור לזכוכית. . ספיגה או שחרור של מים (או ממיסים אחרים) יכול לשנות את הגודל של הרבה חומרים נפוצים, חומרים אורגניים משנים את הגודל במידה רבה יותר בגלל האפקט הזה מאשר ההשפעה של ההתפשטות התרמית. פלסטיק נפוץ שנחשף למים , לטווח הארוך, יכול להתפשט במספר אחוזים רבים.

מקדם התפשטות תרמית

מקדם התפשטות תרמית , מתאר כיצד הגודל של  האובייקט משתנה  עם שינוי הטמפרטורה.  באופן ספציפי, זה מודד את השינוי בגודל בכל טמפרטורה בלחץ קבוע.  מספר סוגים של מקדמים  פותחו: 

מקדם נפחי, מקדם של שטח ומקדם ליניארי. כל מקדם תלוי ביישום מסוים שבו הממדים נחשבים חשובים. עבור מוצקים, אפשר לצפות לשינוי אורך, או שטח.

מקדם התפשטות תרמית נפחי הוא מקדם התפשטות תרמית הבסיסי ביותר.  באופן כללי, חומרים מתפשטים או מתכווצים כאשר הטמפרטורה שלהם משתנה, ההתפשטות או ההתכווצות מתרחשת לכל הכיוונים.  חומרים המתפשטים באותו קצב לכל כיוון נקראים איזוטרופיים .  עבור חומרים איזוטרופיים, ניתן לחשב מקדם שטח או מקדם לנארי באמצעות מקדם נפח.

מקדם התפשטות תרמית נפחי כללי

במקרה הכללי של גז, נוזל, או מוצק, מקדם נפחי של התפשטות תרמית ניתנת על ידי: 

$\alpha _{V}={\frac {1}{V}}\,\left({\frac {\partial V}{\partial T}}\right)_{p}$

p מציין כי הלחץ מוחזק קבוע במהלך התפשטות, ו "V" מדגיש כי מדובר במקדם נפחי (לא ליניארי).  במקרה של גז, העובדה שהלחץ מוחזק קבוע חשוב, כי נפח הגז משתנה באופן ניכר עם  שינוי לחץ, וטמפרטורה.  עבור גז בעל צפיפות נמוכה ניתן לראות בחוק  הגז האידאלי  .

Expansion in solids

Materials generally change their size when subjected to a temperature change while the pressure is held constant. In the special case of solid materials, the pressure does not appreciably affect the size of an object, and so, for solids, it's usually not necessary to specify that the pressure be held constant.

Common engineering solids usually have coefficients of thermal expansion that do not vary significantly over the range of temperatures where they are designed to be used, so where extremely high accuracy is not required, practical calculations can be based on a constant, average, value of the coefficient of expansion.

Linear expansion

The linear thermal expansion coefficient relates the change in a material's linear dimensions to a change in temperature. It is the fractional change in length per degree of temperature change. Ignoring pressure, we may write:

$\alpha _{L}={\frac {1}{L}}\,{\frac {dL}{dT}}$

where $L$ is the linear dimension (e.g. length) and $dL/dT$ is the rate of change of that linear dimension per unit change in temperature.

The change in the linear dimension can be estimated to be:

${\frac {\Delta L}{L}}=\alpha _{L}\Delta T$

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

Effects on strain

For solid materials with a significant length, like rods or cables, an estimate of the amount of thermal expansion can be described by the material strain, given by $\epsilon _{\mathrm {thermal} }$ and defined as:

$\epsilon _{\mathrm {thermal} }={\frac {(L_{\mathrm {final} }-L_{\mathrm {initial} })}{L_{\mathrm {initial} }}}$

where $L_{\mathrm {initial} }$ is the length before the change of temperature and $L_{\mathrm {final} }$ is the length after the change of temperature.

For most solids, thermal expansion is proportional to the change in temperature:

$\epsilon _{\mathrm {thermal} }\propto \Delta T$

Thus, the change in either the strain or temperature can be estimated by:

$\epsilon _{\mathrm {thermal} }=\alpha _{L}\Delta T$

where

$\Delta T=(T_{\mathrm {final} }-T_{\mathrm {initial} })$

is the difference of the temperature between the two recorded strains, measured in degrees Celsius or kelvin, and $\alpha _{L}\,$ is the linear coefficient of thermal expansion in inverse kelvin.

Area expansion

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:

$\alpha _{A}={\frac {1}{A}}\,{\frac {dA}{dT}}$

where $A$ 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:

${\frac {\Delta A}{A}}=\alpha _{A}\Delta T$

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

Volumetric expansion

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

$\alpha _{V}={\frac {1}{V}}\,{\frac {dV}{dT}}$

where $V$ is the volume of the material, and $dV/dT$ is 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%. If we had a block of steel with a volume of 2 cubic meters, then under the same conditions, it would expand to 2.004 cubic meters, again an expansion of 0.2%. The volumetric expansion coefficient would be 0.2% for 50 °C, or 0.004% per degree C.

If we already know the expansion coefficient, then we can calculate the change in volume

${\frac {\Delta V}{V}}=\alpha _{V}\Delta T$

where $\Delta V/V$ is the fractional change in volume (e.g., 0.002) and $\Delta T$ is the change in temperature (50 C).

The above example assumes that the expansion coefficient did not change as the temperature changed. This is not always true, but for small changes in temperature, it is a good approximation. If the volumetric expansion coefficient does change appreciably with temperature, then the above equation will have to be integrated:

${\frac {\Delta V}{V}}=\int _{T_{0}}^{T_{0}+50}\alpha _{V}(T)\,dT$

where $T_{0}$ is the starting temperature and $\alpha _{V}(T)$ is the volumetric expansion coefficient as a function of temperature T.

Isotropic materials

For exactly isotropic materials, and for small expansions, the linear thermal expansion coefficient is one third the volumetric coefficient.

$\alpha _{V}=3\alpha _{L}$

This ratio arises because volume is composed of three mutually orthogonal directions. Thus, in an isotropic material, for small differential changes, one-third of the volumetric expansion is in a single axis. As an example, take a cube of steel that has sides of length L. The original volume will be $V=L^{3}$ and the new volume, after a temperature increase, will be

$V+\Delta V=(L+\Delta L)^{3}=L^{3}+3L^{2}\Delta L+3L\Delta L^{2}+\Delta L^{3}\approx L^{3}+3L^{2}\Delta L=V+3V{\Delta L \over L}$

We can make the substitutions $\Delta V=\alpha _{V}L^{3}\Delta T$ and, for isotropic materials, $\Delta L=\alpha _{L}L\Delta T$. We now have:

$L^{3}+L^{3}\alpha _{V}\Delta T=L^{3}+3L^{3}\alpha _{L}\Delta T+3L^{3}\alpha _{L}^{2}\Delta T^{2}+L^{3}\alpha _{L}^{3}\Delta T^{3}\approx L^{3}+3L^{3}\alpha _{L}\Delta T$

Since the volumetric and linear coefficients are defined only for extremely small temperature and dimensional changes (that is, when $\Delta T$ and $\Delta L$ are small), the last two terms can be ignored and we get the above relationship between the two coefficients. If we are trying to go back and forth between volumetric and linear coefficients using larger values of $\Delta T$ then we will need to take into account the third term, and sometimes even the fourth term.

Similarly, the area thermal expansion coefficient is 2/3 of the volumetric coefficient.

$\alpha _{A}={\tfrac {2}{3}}\alpha _{V}$

This ratio can be found in a way similar to that in the linear example above, noting that the area of a face on the cube is just $L^{2}$. Also, the same considerations must be made when dealing with large values of $\Delta T$.

Anisotropic materials

Materials with anisotropic structures, such as crystals (with less than cubic symmetry) and many composites, will generally have different linear expansion coefficients ${\frac {}{}}\alpha _{L}$ in different directions. As a result, the total volumetric expansion is distributed unequally among the three axes. If the crystal symmetry is monoclinic or triclinic, even the angles between these axes are subject to thermal changes. In such cases it is necessary to treat the coefficient of thermal expansion as a tensor with up to six independent elements. A good way to determine the elements of the tensor is to study the expansion by powder diffraction.

Expansion in gases

For an ideal gas, the volumetric thermal expansion (i.e., relative change in volume due to temperature change) depends on the type of process in which temperature is changed. Two known cases are isobaric change, where pressure is held constant, and adiabatic change, where no work is done and no change in entropy occurs.

In an isobaric process, the volumetric thermal expansivity, which we denote $\beta _{p}$, is given by the ideal gas law:

$PV=nRT\,$

$\ln \left(V\right)=\ln \left(T\right)+\ln \left(nR/P\right)$

$\beta _{p}=\left({\frac {1}{V}}{\frac {dV}{dT}}\right)_{p}=\left({\frac {d(\ln V)}{dT}}\right)_{p}={\frac {d(\ln T)}{dT}}={\frac {1}{T}}.$

The index $p$ denotes an isobaric process.

Expansion in liquids

Theoretically, the coefficient of linear expansion can be found from the coefficient of volumetric expansion (β≈3α). However, for liquids, α is calculated through the experimental determination of β.

Apparent and absolute expansion

When measuring the expansion of a liquid, the measurement must account for the expansion of the container as well. For example, a flask, that has been constructed with a long narrow stem filled with enough liquid that the stem itself is partially filled, when placed in a heat bath will initially show the column of liquid in the stem to drop followed by the immediate increase of that column until the flask/liquid/heat bath system has thermalized. The initial observation of the column of liquid dropping is not due to an initial contraction of the liquid but rather the expansion of the flask as it contacts the heat bath first. Soon after, the liquid in the flask is heated by the flask itself and begins to expand. Since liquids typically have a greater expansion over solids the liquid in the flask eventually exceeds that of the flask causing the column of liquid in the flask to rise. A direct measurement of the height of the liquid column is a measurement of the Apparent Expansion of the liquid. The Absolute expansion of the liquid is the apparent expansion corrected for the expansion of the containing vessel.

Examples and applications

The expansion and contraction of materials must be considered when designing large structures, when using tape or chain to measure distances for land surveys, when designing molds for casting hot material, and in other engineering applications when large changes in dimension due to temperature are expected.

Thermal expansion is also used in mechanical applications to fit parts over one another, e.g. a bushing can be fitted over a shaft by making its inner diameter slightly smaller than the diameter of the shaft, then heating it until it fits over the shaft, and allowing it to cool after it has been pushed over the shaft, thus achieving a 'shrink fit'. Induction shrink fitting is a common industrial method to pre-heat metal components between 150 °C and 300 °C thereby causing them to expand and allow for the insertion or removal of another component.

There exist some alloys with a very small linear expansion coefficient, used in applications that demand very small changes in physical dimension over a range of temperatures. One of these is Invar 36, with α approximately equal to 0.6תבנית:E/°C. These alloys are useful in aerospace applications where wide temperature swings may occur.

Pullinger's apparatus is used to determine the linear expansion of a metallic rod in the laboratory. The apparatus consists of a metal cylinder closed at both ends (called a steam jacket). It is provided with an inlet and outlet for the steam. The steam for heating the rod is supplied by a boiler which is connected by a rubber tube to the inlet. The center of the cylinder contains a hole to insert a thermometer. The rod under investigation is enclosed in a steam jacket. One of its ends is free, but the other end is pressed against a fixed screw. The position of the rod is determined by a micrometer screw gauge or spherometer.

The control of thermal expansion in ceramics is a key concern for a wide range of reasons. For example, ceramics are brittle and cannot tolerate sudden changes in temperature (without cracking) if their expansion is too high. Ceramics need to be joined or work in consort with a wide range of materials and therefore their expansion must be matched to the application. Because glazes need to be firmly attached to the underlying porcelain (or other body type) their thermal expansion must be tuned to 'fit' the body so that crazing or shivering do not occur. Good example of products whose thermal expansion is the key to their success are CorningWare and the spark plug. The thermal expansion of ceramic bodies can be controlled by firing to create crystalline species that will influence the overall expansion of the material in the desired direction. In addition or instead the formulation of the body can employ materials delivering particles of the desired expansion to the matrix. The thermal expansion of glazes is controlled by their chemical composition and the firing schedule to which they were subjected. In most cases there are complex issues involved in controlling body and glaze expansion, adjusting for thermal expansion must be done with an eye to other properties that will be affected, generally trade-offs are required.

Heat-induced expansion has to be taken into account in most areas of engineering. A few examples are:

• Metal framed windows need rubber spacers
• Rubber tires
• Metal hot water heating pipes should not be used in long straight lengths
• Large structures such as railways and bridges need expansion joints in the structures to avoid sun kink
• One of the reasons for the poor performance of cold car engines is that parts have inefficiently large spacings until the normal operating temperature is achieved.
• A gridiron pendulum uses an arrangement of different metals to maintain a more temperature stable pendulum length.
• A power line on a hot day is droopy, but on a cold day it is tight. This is because the metals expand under heat.
• Expansion joints that absorb the thermal expansion in a piping system.

Thermometers are another application of thermal expansion — most contain a liquid (usually mercury or alcohol) which is constrained to flow in only one direction (along the tube) due to changes in volume brought about by changes in temperature. A bi-metal mechanical thermometer uses a bimetallic strip and bends due to the differing thermal expansion of the two metals.

Thermal expansion coefficients for various materials

This section summarizes the coefficients for some common materials.

In the table below, the range for α is from 10⁻⁷/°C for hard solids to 10⁻³/°C for organic liquids. α varies with the temperature and some materials have a very high variation.

For isotropic materials the coefficients linear thermal expansion α and volumetric thermal expansion β are related by β = 3α. For liquids usually the coefficient of volumetric expansion is listed and linear expansion is calculated here for comparison.

(The formula β≈3α is usually used for solids.)

Material	Linear coefficient, α, at 20 °C (10−6/°C)	Volumetric coefficient, β, at 20 °C (10−6/°C)	Notes
Aluminium	23	69
Benzocyclobutene	42	126
Brass	19	57
Carbon steel	10.8	32.4
Concrete	12	36
Copper	17	51
Diamond	1	3
Ethanol	250	750
Gallium(III) arsenide	5.8	17.4
Gasoline	317	950
Glass	8.5	25.5
Glass, borosilicate	3.3	9.9
Gold	14	42
Indium phosphide	4.6	13.8
Invar	1.2	3.6
Iron	11.8	33.3
Kapton	20	60	DuPont Kapton 200EN
Lead	29	87
MACOR	9.3
Magnesium	26	78
Mercury	61	182
Molybdenum	4.8	14.4
Nickel	13	39
Oak	54		Perpendicular to the grain
Douglas-fir	27	75	radial
Douglas-fir	45	75	tangential
Douglas-fir	3.5	75	parallel to grain
Platinum	9	27
PVC	52	156
Quartz (fused)	0.59	1.77
Quartz	0.33	1
Rubber	77	231
Sapphire	5.3		Parallel to C axis, or [001]
Silicon Carbide	2.77	8.31
Silicon	3	9
Silver	18	54
Sitall	0.15	0.45
Stainless steel	17.3	51.9
Steel	11.0 ~ 13.0	33.0 ~ 39.0	Depends on composition
Titanium	8.6
Tungsten	4.5	13.5
Water	69	207
YbGaGe	≐0	≐0

See also

• Autovent
• Grüneisen parameter
• Apparent molar property

References

שגיאות פרמטריות בתבנית:הערות שוליים
פרמטרים [ טורים ] לא מופיעים בהגדרת התבנית

External links

• Glass Thermal Expansion Thermal expansion measurement, definitions, thermal expansion calculation from the glass composition
• Water Expansion Calculator Water thermal expansion calculator
• DoITPoMS Teaching and Learning Package on Thermal Expansion and the Bi-material Strip
• Engineering Toolbox – List of coefficients of Linear Expansion for some common materials
• Article on how β is determined
• MatWeb: Free database of engineering properties for over 79,000 materials
• USA NIST Website – Temperature and Dimensional Measurement workshop
• Hyperphysics: Thermal expansion
• Understanding Thermal Expansion in Ceramic Glazes

• קטגוריה:Thermodynamics
• קטגוריה:Heat transfer
• קטגוריה:Physical quantities
• קטגוריה:Building defects