This document discusses temperature and thermal equilibrium. It begins by explaining that temperature represents the average kinetic energy of atoms in a substance. When two substances are in thermal contact, heat will flow between them until they reach the same temperature and thermal equilibrium. Several temperature scales are then described, including Celsius, Fahrenheit, and Kelvin. The document also covers thermal expansion, explaining that all materials expand slightly when heated through the transfer of kinetic energy between atoms. Examples of applications that rely on thermal expansion, such as bimetallic strips and railway tracks, are provided.
The term phase transition (or phase change) is most commonly used to describe transitions between solid, liquid and gaseous states of matter, and, in rare cases, plasma (physics). A phase of a thermodynamic system and the states of matter have uniform physical properties. During a phase transition of a given medium certain properties of the medium change, often discontinuously, as a result of the change of some external condition, such as temperature, pressure, or others. For example, a liquid may become gas upon heating to the boiling point, resulting in an abrupt change in volume. The measurement of the external conditions at which the transformation occurs is termed the phase transition. Phase transitions are common in nature and used today in many technologies.
Understand the physical mechanism of convection and its classification.
Visualize the development of velocity and thermal boundary layers during flow over surfaces.
Gain a working knowledge of the dimensionless Reynolds, Prandtl, and Nusselt numbers.
Distinguish between laminar and turbulent flows, and gain an understanding of the mechanisms of momentum and heat transfer in turbulent flow.
Derive the differential equations that govern convection on the basis of mass, momentum, and energy balances, and solve these equations for some simple cases such as laminar flow over a flat plate.
Non dimensionalize the convection equations and obtain the functional forms of friction and heat transfer coefficients.
Use analogies between momentum and heat transfer, and determine heat transfer coefficient from knowledge of friction coefficient.
To download this lecture notes kindly visit website or contact me
Topics include: visit my website (www.mech-4u.weebly.com)
1) introduction to thermodynamics
2) basics concepts of thermodynamics
3) types of system
4) properties of system
5) zeroth law of thermodynamics
6) concept of heat and work
7) properties of steam
8) properties of ideal gas
The term phase transition (or phase change) is most commonly used to describe transitions between solid, liquid and gaseous states of matter, and, in rare cases, plasma (physics). A phase of a thermodynamic system and the states of matter have uniform physical properties. During a phase transition of a given medium certain properties of the medium change, often discontinuously, as a result of the change of some external condition, such as temperature, pressure, or others. For example, a liquid may become gas upon heating to the boiling point, resulting in an abrupt change in volume. The measurement of the external conditions at which the transformation occurs is termed the phase transition. Phase transitions are common in nature and used today in many technologies.
Understand the physical mechanism of convection and its classification.
Visualize the development of velocity and thermal boundary layers during flow over surfaces.
Gain a working knowledge of the dimensionless Reynolds, Prandtl, and Nusselt numbers.
Distinguish between laminar and turbulent flows, and gain an understanding of the mechanisms of momentum and heat transfer in turbulent flow.
Derive the differential equations that govern convection on the basis of mass, momentum, and energy balances, and solve these equations for some simple cases such as laminar flow over a flat plate.
Non dimensionalize the convection equations and obtain the functional forms of friction and heat transfer coefficients.
Use analogies between momentum and heat transfer, and determine heat transfer coefficient from knowledge of friction coefficient.
To download this lecture notes kindly visit website or contact me
Topics include: visit my website (www.mech-4u.weebly.com)
1) introduction to thermodynamics
2) basics concepts of thermodynamics
3) types of system
4) properties of system
5) zeroth law of thermodynamics
6) concept of heat and work
7) properties of steam
8) properties of ideal gas
This file contains slides on Transient Heat conduction: Part-I
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010. Contents: Lumped system analysis – criteria for lumped system analysis – Biot and Fourier Numbers – Response time of a thermocouple - One-dimensional transient conduction in large plane walls, long cylinders and spheres when Bi > 0.1 – one-term approximation - Heisler and Grober charts- Problems
Albania an important energy hub for the Southern Gas Corridor Realistic over...Albania Energy Association
Albania an important energy hub for the Southern Gas Corridor
Realistic overview of the Albanian and West Balkan
Erlet Shaqe
Co-Founder and Chairman of AEA
aea@aea-al.org
EU Gas Demand
Composition of the European gas supply
Existing and planned import pipelines to Europe
Natural gas demand on the Balkans
The Balkan Gas Hub
The Albania/Balkan region
Regional primary energy production WB
Natural Gas Role in the Albanian Energy Balance
Ionian – Adriatic Pipeline
TAP is considered important for Albania
Albania Natural Gas Forecast
Realistic potential area for developing the gas transmission and distribution in Albania
Underground Gas Storage potentials of Albania.
*Credits AEA-Albania Energy Association
Albania investments and Hydropower development 2017
The Albania Energy Association represents the Albania energy industry, covering all renewable, power, heat and fuels.
Albania Country of Opportunities .
CILESIA E AJRIT NE AMBIENTET E BRENDESHME
Studimi i mirëqënies termike nuk mund te jete i shkeputur nga ajoe e
cilësisë së ajrit në brendësi të ambienteve të rrethuara. Vëzhgimi i
sëmundjeve në dukje (sindroma të ndërtesës së sëmurë) të çuditshme në
kushte të caktuara ka çuar në lindjen e një linje tjetër studimi që i
referohet cilësisë kimike – fizike të ajrit qe, sot njihet si IAQ (Indoor Air Quality-Cilësia e Ajrit të Brendshëm). Materialet përshtatura për ndërtime, mobiljet, gazrat e përdorur në impiante, të gjithë këto prodhojnë substanca avulluese organike ( Volatile Organic Compound, VOC– Përzierje Organike Avulli) që, në përqëndrim të pakontrolluar mund të sjellin sëmundje të natyrave dhe rrezikshmërive të ndryshme.
TREGUESIT E CILËSISE SË AJRIT
PROÇESET PSIKROMETRIKE
RRL
Ngarkesa Ftohëse
CENTRALET FRIGORIFERIKE
Metodat e permasimit te tubave te ajrit
The revival and transformation of Europe’s largest onshore oilfield; the Pato...Albania Energy Association
Presentation: The revival and transformation of Europe’s largest onshore oilfield; the Patos-Marinza field
Leonidha Çobo, General Manager, Bankers Petroleum Albania Ltd
Presentation: Trans Adriatic Pipeline (TAP) – The European leg of the Southern Gas Corridor
Shkelqim Bozgo, Country Manager for Albania, Trans Adriatic Pipeline (TAP)
Presentation: Overall analysis of the onshore sector of Albania and current developments
Dritan Spahiu, Director of Policies and Development of Hydrocarbons, Ministry of Energy and Industry
How Albanian legislation facilitates the exploration and development of hydro...Albania Energy Association
Presentation: How Albanian legislation facilitates the exploration and development of hydrocarbons
Dael Dervishi, Executive Director, National Agency of Natural Resources (AKBN)
vercoming challenges in the exploration of Albania’s high potential carbonate...Albania Energy Association
Presentation: Overcoming challenges in the exploration of Albania’s high potential carbonate structures
Peter Sider, Vice President, Engineering, Operations and Business Development, Petromanas
Presentation: The role of Southern Gas Corridor for energy security of Europe
Ilkin Aslanov, Executive Assistant to SOCAR President Advisor, SOCAR, Azerbaijan
Presentation: Environmental services for the oil and gas industry
Giannis Karakolis, North Greece Director, POLYECO S.A.
Simon Geragthy, Drilling Waste Management Expert, POLYECO S.A
Presentation: Concessions and the legal framework in Energy
Etleva Kondi, Director of Concessions, Procurement and Privatisation, Ministry of Energy and Industry
Presentation: Strengthening integration on Energy Community and investment opportunities
Lorenc Gordani, Professor of EU Law and Project Director, ACERC
Presentation: The refining sector of Albania
Christophe Darbord, Chief Executive Officer, Armo Refinery
Presentation: What is the energy investment outlook in Albania for 2020?
Entela Çipa, Advisor to the Minister of Energy and Industry
Presentation: Extractive Industry Transparency Initiative - EITI in Albania
Dorina Çinari, Director, Extractive Industries Transparency Initiative (EITI) Albania
Presentation: The National Economic Council; Fostering Business-Government dialogue
Elona Varfi, COO, National Economic Council (NEC)
Keynote Presentation from the Ministry of Energy and Industry - Electricity Directorate
Agim Bregasi, Director of Policies and Development of Electricity, Ministry of Energy and Industry
Presentation: An overview of Albania’s greatest energy source - Hydropower
Aas Agnar, Director of Governmental Affairs, Devoll Hydropower, Albania
Presentation: An overview of the mining sector in Albania
Mehmet Hasalami, Policy and Development Directorate of Mines, Ministry of Energy and Industry
Transporti me litare, perfshine ato lloje te transporti ne te cilen objekti qe transportohet c’vendoset me ndihmen e nje litari. Kjo eshte konceptuar ne terheqjen e mjetit mbartes me anen e litarit, pergjate udhes se tij, ose shtriries se tij. Historikisht tipike, ky lloj transporti perfaqesonte levizjet e mjeteve lundruese, mbi lumenj ose kanale lundruese. Sot ky lloj transporti (lumenj lundruese) eshte gjeneruar ne forma te ndryshme, sic do ta shohim ne leksionet e meposhteme. Peshen kryesore ne kete lloj transporti e ze transporti me teleferik dhe ai me ashensore, (pa permendur makinat ngritese-transportuese). Transporti me terheqje rreshqanthi, pergjate tokes, eshte pak i perhapur. (Ekzistojne studime te cilat tentojne te perfshijne kete lloj transporti, te permendur ne fund, ne formen e linjave me shina, si nje forme e transportit urban ne qytetet e medha dhe me infrastrukture te dobed). Ne ne leksionin tone do te trajtojme transportin me teleferik, dhe ai me ashensore, si dy forma transporti me mundesi perhapje ne Shqiperi. Per transportin me makina peshengritese-transportuese, nuk do ti trajtojme, per specifikat e vecanta te tyre, si dhe per detyren qe ne kemi vene ne lenden tone, Transportin publik
Kerkesa per parkim
Karakteristika e problemeve te transportit
Parkimi eshte nje aspekt i rendesishem i transportit qe ndihmon (pranon) ne praktike levizjen e automjeit. Ne qendrat urbane politika e vendqendrimeve te inspirohet nga kritere dhe objektiva te ndryshme , por ne cdo rast p[ermban nje total racionale e bazuar ne njohjen e ofertes ne veprim dhe kerkeses me te gjitha kuptimet sasiore e cilesore. Vleresimi i ofertes eshte nje problem i nje vezhgimi te paster statistikore ndersa kerkesa varion ne hapsire dhe ne kohen dhe mbi gjithshka ne sjelljen e perdoruesave ne lidhjen me politiken e pergjitheshme dhe ne vecanti me vete politiken e vendqendrimeve. Ne kete kapitull do te mundohemi te ndertojme tek disa modele te vleresimit te kerkeses, ne vecanti tek modeli gravitacional , dhe do te dallohen disa parametra qe bejne matjen e kerkeses per parkim.
Po te levizjen per sherbim transporti si nje rryme fluidi (ajri/lengu), dhe infrastrukturen si nje labirinth tubash, me pike fillimi, mbarimi, si dhe interseksione te ndryshme, ku neper kete sistem tubash kalon fluidi ne fjale. Eshte e kuptueshme qe per kontrollin ose komandimin e rastit, te vendosen disa elemente konstruktive, te cilet disiplinojne kalimin e fluidit neper nje, ose disa tuba te vecante, ose te lidhur ne njefare sistemi. Keshtu me kete lloj perfytyrimi qe ne po krijojme, mund ta njehsojme levizjen kaotike neper infrastrukture, te perdoruesave te mjeteve (shofere transporti mallrash, ose pasagjeresh, ose qytetare perdorues te mjeteve private, te cilet levizin neper infrastrukturen ekzistuese, per te realizuar nje cvendosje nga nje pozicion ne nje tjeter). Ne q. se vazhdojme te krahasojme situacionin e mesiper, ne llidhje me nje fluid vleresohen parametrat cilesor te tij si:, temperaturat dhe vetit e fizike te tij, vizkoziteti, pasha specifike, shpejtesia etj. Te cilat vleresojne levizjen e fluidit ne keto tubacione. Ne nje sistem transporti levizja e perdoruesave te infrastruktures ne nje intinerar mund te vleresohet nga disa parametra ose karkateristikat e fluksit te perdoruesave, te cilat mund te grupojme 3 faktore qe influencojne direkt per kete fenomen, ose levizje. Mjeti-drejtuesi-ambjenti, te tre keto grupime jane elmente perberes dhe influencues ne cdo lloj levizje mjetesh ne te gjithe infrastrukturen.
Ne c’do projekt transporti, ne aspektin e trajtimit te skemes klasike, sic quhet teknologjia e transportit (ai i transportit te objektit nga nje vend ne nje tjeter),.kemi kete perberje ne procesin teknologjik :
Levizja e objektit (mall ose njerez) me mjetin mbartes. Ne kapitujt e mesiperm ne i kemi permendur keto lloj mjetesh te cilet percaktojne dhe llojin e transportit, konkretisht kane keto emertime:
• transporti automobilistik (mjeti eshte automobili)
• transporti detare e lumore (mjeti eshte lundra, varka)
• transporti hekurudhore (mjeti eshte vagoni i terhequr me lokomotive )
• transporti airore (mjeti eshte aeroplani, helikopteri, balona, )
• transporti me litare (mjeti eshte teleferiku, ashensori, ose c’do makine ngritese transportuese)
Nganjehere kerkohet te parashikohet nje perspektive e zhvillimit te transportit, ose te ristrukturohet ne drejtim te permiresimit te parametrave, nje sistemi transporti. Keshtu, mund dhe duhet te behen, ridimensionimet gjenerale te modeleve matematikore, ose te relacioneve matematikore te bashkangjitura keto projektit te sistemit te transportit. Sa me siper, mund te shprehim ne forme te pergjitheshme :
Tij* = ( fk1,fk2,...fkn )= f(A,B,C,...,M) ku :
T fluksi i spostimeve ndaj 2 zonave i-j dhe me karakteristikat e ketij fluksi fk1,..fkn. te cilat japin funksionet relative A,B,C,... per sistemin e transportit.
Eshte e domosdoshme te sqarojme se, ne evolucionin e kerkeses per transport, futen parametra me te dukshme, ndersa te tjerat nuk perfillen, per vete natyren diskrete, te ketyre rasteve, ky fakt e ben qe, here pas here te rishikohen vleresimet, per zgjidhjen e sistemit kzistues, (sepse faktoret ndryshojne, si koha/ kosto/ distanca e levizjes me kemb/ parkimi/ komoditeti etj.). Ne perdorimin e modelit matematik, per nje sistem transporti, praktikisht behet nje thjeshtim i realitetit, gje qe zgjidhjet optimale duhen te merren duke i permiresuar here pas here parametrat e nje sistemi transporti, ne perputhje me kushtet aktualehere.
Si duhet ta shikojme/studjojme rrealisht nje statistike ne fushen e transportitAlbania Energy Association
Elementi baze ne ndertimin e nje projekti transporti eshte analiza e kerkes-ofert per transport. Kjo shpesh vjen nga verejtjet, ose pamjat direkte ne terren, nga menyra se si sillen ose levizin njerezit apo mallrat. Ashtu si dhe ne vleresimet e tjera per terrenin, edhe ketu kemi nje logjike te perafert, e dalluar, si me poshte:
1) Niveli i sherbimit mbi infrastrukturen ekzistuese, kjo ka te beje me grumbullimin e te dhenave rreale, statistikore mbi infrastrukturen ekzistuese, si dhe kerkesen per transport, apo sherbimet e transportit, si dhe pyetjet ose ceshtjet te tjera, qe lindin ne kete bashkeveprim kerkese-oferte
2) Modelimi simulues i nje sistemi, ne te cilen jane parashikuar vleresimet tekniko-ekonomike, nder te tjerat dhe demografike, social-ekonomike, urbanistike, etj. Si kane ndikuar ose vijuar keto elemente, gjate shfrytezimit ?
3) Treguesit Financiare, si kane qene dhe si eshte bilanci financiare i ketij sherbimi, dinamika e tij, si dhe ndonje alternative e tij, ne pasqyren e kostove e te rendimentit
Per ndertimin e nje projekt transporti, duhet detyrimisht te merren ne konsiderate shume elemente, ose kushte, te cilat konvertohen ne nje bashkesi ndervaresie e quajtur Modeli i sistemit te transportit, ky model eshte i afte ti simuloje keto kushte apo elemente, te projektit duke ne dhene rezultatet respektive. Sa me siper, nje projekt per nje sistem transporti, duhet te ece neper keto faza logjike:
Individualizimi i objeketve qe kerkojne levizje (kerkesa per transport)
Analiza sasiore e sistemit te transportit aktual (ose per disa sisteme)
Ndertimi i nje model varesie, qe perfaqeson nje sitem transporti, i cili eshte ne gjendje te funksionoje me te dhenat (dati) aktuale, si edhe me parametrat e vlerave te prespektives, ne lidhje me kete sistem qe po projektojme, se bashkeu me sistemet e tjere konkurues
Pergjithesime
Nje nder problemet qe e veshtiresojne shperndarjen, shtririen e transportit te mallrave mbi rruge, eshte koha e madhe jo levizese e mallit, ne pikat e nderthuries se levizjeve (grumbullimin, shperndarjen, vazhdim levizjeje te mallit ne destinacion) e cila presupozon kohen e perpunimit te mallit. Si dhe siperfaqet e disponueshme per kete kohe ne keto pika te nderthurjes se levizjes e te perpunimit te mallrave. Keshtu mga viti 1930-40 filloj nje koncep i ri “unifikimi i njesive te ngarkesave” kjo do te thote qe menyra te ambalazhimit te jete e paraqykyar qe ne projektim si nje element ne teknologjine e organizimit te transportit te objektit, kjo gje solli qe:
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
2. Matter is very complex
A simple cup of coffee contains ~ 1023 atoms
Numerically impossible to follow trajectory of each
atom, use Newton’s laws for force, acceleration etc..
Two options
1. Statistical: probability distribution of molecule velocities
2. Macroscopic: a few variables (temperature, pressure,
volume…) characterize the bulk properties of matter
These variables are called state variables.
3. Temperature and thermal equilbrium
Temperature ~ average kinetic energy of atoms
Two pieces of matter in thermal contact exchange
energy until their temperatures (T ) are the same
(0th law of thermodynamics)
Example:
Molecules in coffee transfer energy to the air molecules.
Eventually all coffee molecules have the same average
kinetic energy as the molecules of air in this room
Reaching equilibrium (same T ) requires transfer of energy
This energy in transit is called heat Q.
4. ACT: Temperature
When a series of blocks are connected thermally, heat starts to flow
between the blocks as shown. What does this tell us about relative
temperatures of the blocks?
1
Q
A. T1 = T2 = T3 = T4
B. T1 > T2 > T3 > T4
C. T1 < T2 < T3 < T4
2
Q’
3
Q’’
4
This is how the sequence is
established empirically.
Also: You just applied the 2nd
law of thermodynamics....
5. Temperature scales: Celsius, Fahrenheit
Celsius
Based on the boiling and freezing points of water.
0°C = freezing point of water
100°C = boiling point of water
Fahrenheit
Based on the boiling and freezing points of alcohol.
Connection to Celsius: 0°C = 32°F
100°C = 212°F
9
TF = TC + 32
5
6. Constant volume gas thermometers
For gases, p is proportional toT for constant volume (Charles’ law).
p
p
= constant
T
T
gas
(He)
Liquid
(Hg)
7. Temperature scales: Kelvin
p(T) for different gases:
p
Extrapolation of ALL
lines points to
T = −273.15°C
T (°C)
Kelvin scale
0 K = −273.15°C (lowest energy state, quantum motion only)
2nd fixed point: 273.16 K (=0.01°C) is the triple-point for H 2O (ice, water,
steam coexist)
TK =TC + 273.15
With this choice, 1°C = 1K (equal increments)
8. Thermal expansion of a bar
A bar of length L0 expands ΔL when temperature is increased by ΔT.
Experimentally,
∆L = α L0 ∆T
α = coefficient of linear expansion (depends on material)
Basis for many thermometers
e.g. liquid mercury
Critically important in many engineering projects (expansion
joints)
9. In-class example: Bimetallic strips
Which of the following bimetallic strips will bend the furthest to the right
when heated from room temperature to 100°C?
α (µJ/m K)
Invar
Al
Al
Invar
Brass
C
Al
Brass
D
Ag
Au
E
Al/Invar have the largest difference in α. Aluminum
expands more than invar, so A will bend to the right.
1.3
Brass
B
A
24
Invar
Al
Al
21
Au
14
Ag
19
DEMO:
Bimetallic
strips
10. A couple of applications
Bimetallic strips can be used as mechanical thermometers.
About invar: Fe-Ni alloy with very small α
Train tracks have expansion joints (gaps) to prevent buckling in
hot weather.
(origin of the “clickety-clac, clickety-clac”)
High speed trains cannot afford the vibrations produced by
these gaps. Alloys with small α to build tracks are a key
development.
11. Area Thermal Expansion
b
b +Δb
ΔT
a
a +Δa
Afinal = (a + ∆a )(b + ∆b )
= ab + a ∆b + b ∆a + very small terms
≈ ab + a (bα∆T ) + b (a α∆T )
= ab + 2ab α∆T
∆
A
∆A = 2α A0 ∆T
12. ACT: Washer
DEMO:
Balls and rings
A circular piece of metal with a round hole is
heated so that its temperature increases. Which
diagram best represents the final shape of the
initial shape
metal?
A. Both inner, outer radii larger
B. Inner radius smaller, outer
radius larger
C. Same size
All lengths increase: a 2-D object grows in length and breadth
13. Opening tight jar lids
αglass = 0.4-0.9 × 10-5 K−1
αbrass = 2.0 × 10-5 K−1
If you place the jar top under the hot water faucet,
the brass expands more than the glass.
14. Volume Thermal Expansion
c
a
c +Δc
ΔT
b
a +Δa
b +Δb
Vfinal = (a + ∆a )(b + ∆b )(c + ∆c )
= abc + ab ∆c + ac ∆b + bc ∆a + smaller terms
≈ abc + 3abc (α∆T )
≈ V0 + 3 0α∆T
V
∆V
∆V = βV0 ∆T
β = 3α
Coefficient of volume expansion
15. The special case of water
Most materials expand when temperature increases.
Water between 0 and 4°C is the exception.
V
Maximum
density
Ice is less dense than
cold (< 4°C ) water.
Ice floats
α has an important dependence on T
0
4°C
T
This prevents lakes from freezing from the bottom up,
which would kill all forms of life.
16. Microscopic Model of Solids
Consider atoms on a lattice, interacting
with each other as if connected by a spring
movie
17. Microscopic Model of Expansion
(attempt 1)
Model potential
energy of atoms
U =
U
1
k (x − l0 )2 + const
2
vibration at high T
vibration at low T
l0
spacing l0
separation x
Vibrating atoms spend equal time at x > l0 as at x < l0
→ average separation of atoms = l0 , at both low and high T
→ material same size at low and high T
→ This spring model of matter does not describe expansion
18. Microscopic Model of Expansion (2)
Change model to non-ideal spring
U
U ≠
1
k (x − l 0 )2
2
asymmetric
parabola
vibration at high T
vibration at low T
spacing lT
lhighT
llowT
x
At high T, the average separation of atoms increases → solid expands
(Now we need another parameter in model, asymmetry of parabola)
19. Why two models?
Is this more useful compared to empirical ∆L = λ L0 ∆T ?
The empirical formula can be used for simple cases.
Example: for designing structures to take into account expansion.
The microscopic model of expansion (non-ideal spring) can be used
in wider number of cases.
Example: thermal expansion while also under high-pressure in a
chemical engineering plant
20. Thermal stress
A rod of length L0 and cross-sectional area A fits perfectly between
two walls. We want its length to remain constant when we increase the
temperature.
Constant length = zero net strain
∆L
∆L
∆L
=
+
=0
÷
÷
÷
L0 all L0 thermal L0 applied stress
α∆T +
F
=0
YA
F
= −αY ∆T
A
Thermal stress (stress
walls need to provide to
keep length constant)
21. Application of thermal stress
For very tight fitting of pieces (like wheels):
Wheel is heated
and then axle
inserted.
Wheel cools down and
shrinks around axle,
very tight.