PRESSURE I am teaching Engineering Thermodynamics to a class of 75 undergraduate students. These...
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Transcript of PRESSURE I am teaching Engineering Thermodynamics to a class of 75 undergraduate students. These...
PRESSURE
• I am teaching Engineering Thermodynamics to a class of 75 undergraduate students. • These slides follow closely my written notes (http://imechanica.org/node/288). • I went through these slides in three 90-minute lectures.
Zhigang Suo, Harvard University
The play of thermodynamics
2
energy space matter charge
ENTROPY
temperature pressure chemical potential electrical potential
heat capacity compressibility capacitanceHelmholtz function enthalpy
Gibbs functionthermal expansionJoule-Thomson coefficient
Plan
• A system with variable energy and volume• Graphic representations• Theory of co-existent phases• Theory of ideal gases• Theory of osmosis • Breed properties and equations of state• Basic algorithm of thermodynamics in
terms of free energy3
A half bottle of wine
4
• The reservoir of energy is a thermal system, modeled by a function SR(UR), and by a fixed temperature TR.
• The weight applies a fixed force fweight. Pressure Pweight = fweight/A. The
entropy of the weight Sweight is constant as the piston moves.
• The half bottle of wine is a closed system, modeled by a family of isolated systems of two independent variations, using function S(U,V).
liquid
fweight
vaporclosed system
reservoir of energy, TR
adiabatic
diathermal 2OModel the wine asa family of isolated systemsS(U,V)
liquid
vapor
1. Construct an isolated system with an internal variable, x.
2. When the internal variable is constrained at x, the isolated system has entropy S(x).
3. After the constraint is lifted, x changes to maximize S(x).
5
The basic algorithm of thermodynamicsEntropy = log (number of quantum states). Entropy is additive.
Construct an isolated system with internal variables
6
• Isolated system conserves space (kinematics): V = Ah • Isolated system conserves energy: Ucomposite = U + UR + fweighth = constant
• Entropy of the isolated system is additive:
• Entropy of the reservoir:
liquid
fweight
vaporclosed systemS(U,V)
reservoir of energy, TR
adiabatic
diathermal
Isolated system with internal variables U and V
Entropy of the reservoir of energy
7
reservoir of energy SR, UR,TR
A reservoir of energy is a thermal system, modeled by a function SR(UR) and by a fixed temperature, TR.
Clausius-Gibbs equation:
Integration:
Conservation of energy:
Entropy of the reservoir:
Entropy of the isolated system:
Calculus:
Calculus:
Maximize Scomposite to reach Thermodynamic equilibrium:
Temperature and pressure of the wine:
Gibbs equation:
Internal variables change to maximize the entropy of the isolated system
8
liquid
fweight
vapor
reservoir of energy, TR
wine reservoir weight
Count the number of quantum states of a half bottle of wine
Experimental determination of S(U,V)
9
• Measure V by geometry• Measure U by calorimetry (e.g., pass an electric current through a resistor)• Measure T by thermometry• Measure P by force/area
• Obtain S by integrating the Gibbs equation
2O
liquid
vapor A family of isolated systemsFive properties: U,V,S,P,T Two independent variables, U,V
10
liquid
fweight
vapor
reservoir of energy, TR
2O
liquid
vapor
• Each member in the family is a system isolated for a long time, and is in a state of thermodynamic equilibrium.
• Transform one member to another by fire (heat) and weight (work).• Five properties: U,V,S,P,T• Name all states of thermodynamic equilibrium by two independent variables (U,V)• Three functions (equations of state): S(U,V), T(U,V), P(U,V)• Once S(U,V) is known, two other equations of state are determined by the Gibbs equations:
isolated systemU,V,S,P,T
closed system
Model a closed system by a family of isolated systems
Plan
• A system with variable energy and volume• Graphic representations• Theory of co-existent phases• Theory of ideal gases• Theory of osmosis • Breed properties and equations of state• Basic algorithm of thermodynamics in
terms of free energy11
Calculus: a function of two variables
12
x
y
z z(x,y)
(x,y)
x
(x,y)y
A pair of values (x,y) corresponds to a point in the (x,y) plane.
The function z(x,y) corresponds to a surface in the (x,y,z) space.
Calculus: partial derivative
13
x
y
z
z(x,y)
slope
(x+dx, y)(x,y)
Each partial derivative corresponds to a slope of a plane tangent to the z(x,y) surface.
Partial derivatives:
Increment:
14
A pair of values corresponds toa point in the (U, V) plane.
S(U,V)
U
V
A state (U,V)
U
V
S S(U,V)
(U,V)
The function S(U,V) corresponds to a surface in the (U, V,S) space.
15
• Given a state (U,V), draw a plane tangent to the surface S(U,V)• The two slopes of the tangent plane give T(U,V) and P(U,V).
T(U,V) and P(U,V)
U
V
S
(U,V)
(P,T)
V
Entropy of the isolated system:
Internal variables change to maximize entropy:
Graphic derivation of the condition of equilibriumbetween the wine, reservoir and weight
16
S
liquid
fweight
vapor
reservoir of energy, TR
S(U,V) surfacewine
tangent plane
plane of slopes1/TR and Pweight/TR
Reservoir, weight
U
wine reservoir, weight
An isolated system of internal variables U,V
Plan
• A system with variable energy and volume• Graphic representations• Theory of co-existent phases• Theory of ideal gases• Theory of osmosis • Breed properties and equations of state• Basic algorithm of thermodynamics in
terms of free energy17
Three phases of a pure substance
18
u
v
liquid
solid
gas
intensive-intensive extensive-intensive extensive-extensive
T
liquid
solid
gas
criticalpoint
triplepoint
P
19
Equivalent statements•z(x,y) is convex•Each tangent plane touches the surface at a single point.•Roll the tangent plane with two degrees of freedom•One-to-one correspondence: (M,N) (x,y).
Calculus: convex function
x
y
z
A function of two variables:
Partial derivatives:
tangent plane of slopes M and N
(x,y)z(x,y)
20x
y
z
(M,N)
Calculus: an example of nonconvex function• A tangent plane touches the surface at two points.• Roll the tangent plane with one degrees of freedom• One-to-two correspondence: (M,N) (x’,y’) and (x’’,y’’).
z(x’,y’)
z(x’’,y’’)
tie line
21
• S(U,V) is smooth and convex.• Each tangent plane touches the surface at a single point.• Roll the tangent plane with two degrees of freedom.• point-to-point (P,T) (u,v).
u
v
s
Gibbs relations
tangent plane of slopes1/T and P/T
(u,v)s(u,v)
(P,T)
A phase of a pure substance
Two co-existent phases of a pure substance1. Each phase has its own smooth and convex s(u,v) function.
22
solid
s'(u’,v’)
liquid
u
v
s
s’’(u’’,v’’)
23
liquid
solid
isolated system of fixed u and v
u = (1 - x)u’ + xu’’v = (1 - x)v’ + xv’’s = (1 - x)s’ + xs’’
s'(u’,v’)
u
v
s
s’’(u’’,v’’)
(u’’,v’’)(u’,v’) (u,v)
Two co-existent phases of a pure substance2. Rule of mixture defines a line in the (s,u,v) space.
24
u
v
s
(u’’,v’’)(u’,v’) (u,v)
u
v
solidliquid
solid-liquid mixture
liquid
solid
(P,T)
tie line
tie line
Two co-existent phases of a pure substance3. Isolated system conserves energy and volume, but maximizes entropy.
Roll common tangent plane with one degree of freedom.One-to-many correspondence: (P,T) (all states on the tie line)
25u
v
s
s(u’’,v’’)
s(u’,v’)
liquid
gas
critical point
dome
(P,T)
liquid
gas
Two co-existent phases of a pure substance4. Liquid and gas share a smooth but non-convex surface s(u,v)
Roll common tangent plane with one degree of freedom.point-to-line (P,T) (u,v)
Three co-existent phases of a pure substanceEach phase has its own s(u,v) function. Common tangent plane cannot roll.
point-to-triangle (P,T) (u,v)
26
solid
s'(u’,v’)
u
v
s
s’’(u’’,v’’)
s’’’(u’’’,v’’’)
liquid
vapor
Rule of mixtureu = (1 – x - y)u’ + xu’’ + yu’’’v = (1 – x - y)v’ + xv’’ + yv’’’s = (1 – x - y)s’ + xs’’ + ys’’’
(P,T)
Experiment by Andrews (1869). Described by Gibbs (1873). A clay model built by Maxwell (1874)
Gibbs’s thermodynamic surface S(U,V)
Solid phase has its own smooth and convex s(u,v) function. Liquid and gas phases share a smooth but non-convex s(u,v) function.Use tangent planes to make S(U,V) surface convex (convexification)
entropy
en
erg
y
volu
me
gas
liquid
solid
critical point
liquid
solid
gas
volu
me
en
erg
y
28
Single phase •The tangent plane touches the s(u,v) surface of a single phase.•Roll the tangent plane with two degrees of freedom. •T and P change independently.•A single phase corresponds to a region in the T-P plane.•point-to-point (P,T) (u,v) Two coexistent phases•The tangent plane touches the s(u,v) surfaces of two phases.•Roll the tangent plane with one degree of freedom.•T depends on P. •Two coexistent phases correspond to a curve in the T-P plane. Phase boundary.•point-to-line (P,T) (u,v) Three coexistent phases•The tangent plane touches the s(u,v) surfaces of three phases.•The tangent plane cannot roll.•T and P are fixed.•Three coexistent phases correspond to a point in the T-P plane. Triple point. •point-to-triangle (P,T) (u,v)
T
liquid
solid
gas
criticalpoint
triplepoint
Gibbs’s phase rule for a pure substanceValues of (P,T) give the slopes of a plane tangent to s(u,v) surface
v
liquid
solid
gas
u
P
criticalpoint
extensive-extensive
Intensive-intensive
Plan
• A system with variable energy and volume• Graphic representations• Theory of co-existent phases• Theory of ideal gases • Theory of osmosis • Breed properties and equations of state• Basic algorithm of thermodynamics in
terms of free energy29
Ideal gas
When molecules are far apart, the probability of finding a molecule is independent of the location in the
container, and of the presence of other molecules.
Number of quantum states of the gas scales with VN
Definition of entropy S = kBlog
Gibbs equations:
Equations of state for ideal gases:
30
U,V,S,P,T,N,
31
Per mole Per unit mass
For water:For all substances:
Heat capacity of ideal gases
32
Entropy of ideal gas
33
Gibbs equation:
Laws of ideal gases:
Increment for ideal gas:
Approximation of constant specific heat
isentropic process:
Plan
• A system with variable energy and volume• Graphic representations• Theory of co-existent phases• Theory of ideal gases• Theory of osmosis • Breed properties and equations of state• Basic algorithm of thermodynamics in
terms of free energy34
Osmosis
35
mem
bran
e
solution
solvent
In a liquid, osmosis is balanced by gravity
Osmosis
36
In a bag (or a cell), osmosis is balanced by elasticity
http://antranik.org/movement-of-substances-across-cell-membranes/
Theory of osmosis
When solute particles are far apart, the probability of finding a particle is independent of the location in the container, and of
the presence of other particles.
Number of quantum states of the solution scales with VN
Definition of entropy S = kBlog
Gibbs equation:
Osmotic pressure:
37
mem
bran
e
solution
solvent
Plan
• A system with variable energy and volume• Graphic representations• Theory of co-existent phases• Theory of ideal gases• Theory of osmosis • Breed properties and equations of state• Basic algorithm of thermodynamics in
terms of free energy38
39
2O
liquid
vapor Isolated systemU,V,S,P,T
liquid
fweight
vapor
fire
S(U,V)
Model a closed system as a family of isolated systems•Each member in the family is a system isolated for a long time, and is in a state of thermodynamic equilibrium.•Transform one member to another by fire (heat) and weights (work).•Five thermodynamic properties: P, T, V, U, S.•Two independent variables, chosen to be U, V. •Three equations of state: S(U,V), P(U,V), T(U,V).•Determine S(U,V) by experimental measurement.•Obtain P(U,V) and T(U,V) from the Gibbs equations:•Eliminate U from P(U,V) and T(U,V) to obtain P(V,T).
closed system
40
Calculus: Notation
Function:
Partial derivative:
Name partial derivatives:
Derivatives are functions:
U(S,V)When V is fixed, S increases with U. Invert S(U,V) to obtain U(S,V).
41
Gibbs equation:
Another Gibbs equation
Calculus:
More Gibbs equations
U
V
S
S(U,V)U(S,V)V(U,S)
42
2O
liquid
vapor 2O
liquid
vapor
Isolated system Isolated systemstate U,V,S,P,T state U+dU, V+dV, S+dS. P+dP, T+dT
liquid
fweight
vapor
fire
Going between two states of thermodynamic equilibrium via any process
A special type of process to go from one state to another •Slow. Quasi-equilibrium process. •Reversible process•TdS reversible heat through a quasi-equilibrium process•PdV reversible work through a quasi-equilibrium process
Gibbs equations:
First law: dU = heat + work
43
Calculus: Legendre transform Turn a derivative into an independent variable
A function of two variables:
Partial derivatives:
Increment:
Define a Legendre transform:
Product rule:
Increment:
Partial derivatives:
A new function of two variables:
Enthalpy H(S,P)Define H = U + PV
44
Calculus:
Gibbs equation:
Combine the above:
Calculus:
More Gibbs equations:
liquid
fweight
vapor
reservoir of energy, TR
Include the weight in the system
Invert to obtain V(S,P). Convert U(S,V) to U(S,P). Obtain H(S,P)
Helmholtz function F(T,V)Define F = U - TS
45
Calculus:
Gibbs equation:
Combine the above:
Calculus:
More Gibbs equations:
Invert to obtain S(T,V). Convert U(T,V) to U(T,P). Obtain F(T,V)
Gibbs function G(T,P)Define G = U - TS + PV
G = H – TS = F - PV
46
Calculus:
Gibbs equation:
Combine the above:
Calculus:
More Gibbs equations:
Heat capacity under two conditions
47
liquid
fweight
vapor2O
liquid
vapor
fire fire
Name more partial derivatives
48
Coefficient of thermal expansion:
Isothermal compressibility:
Joule-Thomson coefficient:
Calculus: second derivatives
49
Maxwell relation
50
Gibbs equation:
Gibbs relations:
Maxwell relation:
Mathematical manipulation of dubious value
51
Find a reason to be unhappy with Gibbs equations:
Make up a excuse to study a different function:
Calculus:
Clausius-Gibbs equation:
Constant V:
Maxwell relation:
A pointless equation:
Another pointless equation
52
A pointless equation:
Gibbs equation:
Another pointless equation:
Breed thermodynamic properties like rabbitsA single function S(U,V) produces all other functions.
Inbreeding functions!
• Obtain T(U,V) and P(U,V) from the Gibbs equations,
• Eliminate U from T(U,V) and P(U,V) to obtain P(V,T). • Invert S(U,V) to obtain U(S,V).• Use Legendre transform of U(S,V) to define H(S,P), F(T,V), G(P,T).• Invert T(U,V) to obtain U(T,V).• Name lots of partial derivatives: CV(T,V), CP(T,P), (T,P), (T,P), (P,H),…
53
Plan
• A system with variable energy and volume• Graphic representations• Theory of co-existent phases• Theory of ideal gases• Theory of osmosis • Breed properties and equations of state• Basic algorithm of thermodynamics in
terms of free energy54
S(U,V,Y)
• Fix U and V, but let Y change.• The system is an isolated system with an internal variable Y.• Y changes to maximize S(U,V,Y).
55
liquid
vapor
Y = number of molecules in the vapor
Thermal system with constant T •One internal variable Y.
•Thermal equilibrium determines U(T,V,Y)
•Y changes to minimize U - TS.
Isolated system •Two internal variables U and Y.•U and Y change to maximize Scomposite (U,V,Y)
•Thermal equilibrium
•Y changes to maximize 56
liquid
vapor
reservoir of energy, T
Entropy vs. Helmholtz free energyY = number of molecules in the vapor. Fix T and V, but let Y change. Heat between wine and reservoir.
wine reservoir
liquid
vapor
reservoir of energy, T
Define F = U –TSF(T,V,Y)
liquid
Closed system with constant T and P •One internal variable Y.
•Thermal equilibrium
•Mechanical equilibrium
•U(T,P,Y), V(T,P,Y), S(T,P,Y)
•Y changes to minimize U-TS +PV.
Isolated system •Three internal variables U, V and Y.•U, V and Y change to maximize Scomposite (U,V,Y)
•Thermal equilibrium
•Mechanical equilibrium
•Y changes to maximize57
Entropy vs. Gibbs free energyY = number of molecules in the vapor. Fix P and T, but let Y change. Energy between wine, weight, reservoir.
reservoir of energy, T
Define G = U - TS + PVG(T,P,Y)
liquid
reservoir of energy, T
fweight
vapor
wine reservoir, weight
fweight
vapor
58
Represent states on (P,V) plane
liquid
fweight
vapor
fire
Van der Waals equation (1873)
59
Critical point
• b accounts for volume occupied by molecules.• a/v2 accounts for intermolecular forces.
https://en.wikipedia.org/wiki/Van_der_Waals_equation
Maxwell construction
60
P
vl
v
vg
Psat
T = constant
Gibbs equation:
Slope:
Non-monotonic P-v relation:
Rule of mixture:
Fix T and v, minimize f.
Coexistent phases:
Integrate Gibbs equation:
Maxwell construction:
vl
v
vgv
v
f
Non-convex f-v relation
flfg
f
Common tangent line
61Lu and Suo Large conversion of energy in dielectric elastomers by electromechanical phase transition. Acta Mechanica Sinica 28, 1106-1114 (2012).
Two co-existent phases of a pure substance
62
• Know the Gibbs function of each phase: g’(T,P), g’’(T,P).• Rule of mixture: g(T,P,x) = (1-x)g’(T,P) + xg’’(T,P)• The quality x is the internal variable.• For fixed (T,P), x changes to minimize g(T,P,x). • The condition for the two phases to equilibrate: g’(T,P) = g’’(T,P). • This condition determines the phase boundary as a curve in the (T,P) plane.
T
liquid
solid
gas
criticalpoint
triplepoint
P
Claypeyron equationAll quantities are for two coexistent phases of a pure substance
63
T
P
liquid
gas
Phase boundary:
Definition of the Gibbs function:
Increment along the phase boundary:
Gibbs equation:
Claypeyron equation:
Claypeyron equation for liquid-gas mixture
64
Claypeyron equation:
Latent heat variesslowly with temperature:
specific volume of the liquid is negligible:
Gas is nearly ideal:
Claypeyron equation:
Integration:
liquid
solid
gas
criticalpoint
triplepoint
P
T
65
liquid
solid
gas
criticalpoint
triplepoint
P
T
Claypeyron equation for liquid-solid mixtureWill pressure under a sharp blade cause ice to melt?
Claypeyron equation:
Latent heat varies slowly with temperature:
specific volume of water:
Specific volume of ice:
Melting point varies slowly with pressure:
Slope of phase boundary:
A person stands on a sharp blade:
66https://commons.wikimedia.org/wiki/File:Phase_diagram_of_water.svg
Summary• Model a closed system as a family of isolated systems.
• Gibbs’s thermodynamic surface, S(U,V).
• Tangent plane of S(U,V). Gibbs equations
• Theory of phases (rule of mixture. convexification. single phases, two coexistent phases, three coexistent phases, critical point).
• Ideal gas has two basic equations of state: U(T), PV = NkBT.
• For a closed system, S(U,V) generates all thermodynamic properties and equations of state (Alternative independent variables, Partial derivatives, Legendre transforms, Maxwell relations).
• Basic algorithm of thermodynamics in terms of Gibbs function, G = U - TS - PV. For a closed system of fixed P and T, the internal variable Y changes to minimize G(P,T,Y). 67
liquid
fweight
vapor
fire