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3. First Law of Thermodynamics and Energy Equation
3.1 The First Law of Thermodynamics for a Control Mass Undergoing a Cycle
The first law for a control mass undergoing a cycle can be written as
WQ
WQ )cycle(net)cycle(net
This statement is experimentally observed by James Joule in 19th century.
3.2 The First Law of Thermodynamics for a Change in State of a Control Mass Consider the following cycle
W Q
)2((1)
(2) W W QQ
(1) W W QQ
1
2
B
2
1
C
1
2
B
2
1
C
1
2
B
2
1
A
1
2
B
2
1
A
C
2
1
2
1
A
2
1
C
2
1
A
2
1
C
2
1
A
)W Q( )WQ(
W W QQ
Because A and C is arbitrary processes between state 1 and 2, we conclude that Q W must be a point function and must be the differential of a property. This property is the energy (E) [J] of the mass: WQ dE The property E represents all forms of the energy. It is convenient to separate E in this manner: E = U + KE + PE
Internal energy (U) [J]: a property representing all microscopic forms of energy.
Kinetic energy (KE) and Potential energy (PE) are in the macro scale.
Thus, the first law of thermodynamics for a change of state
WQ d(PE) d(KE) dU For kinetic energy and potential energy
)( d m2
1 )m
2
1( d d(KE) 22 VV
For potential energy
dZmg (mgZ) d d(PE) The first law of thermodynamics is
WQ dZ mg )d( m2
1 dU 2 V
Integrating the above equation yields
W Q )Z(Z mg
)( m2
1 )U(U
212112
21
2212
VV
Notes: This law cannot be mathematically
proven. Another name of this law is the
conservation of energy: two ways to change the energy of a system are heat and work crossing the system boundary.
As energy quantities, heat and work are not different.
The first law gives only the changes in U, KE, and PE not the absolute values.
Since KE and PE are extrinsic properties, U is also an extrinsic property.
3.3 Definition of Work
Work (W) [N-m or J]: a force acting through a displacement x
dx F W dx F W2
1
Sign convention: Work done by a system (+) Work done on a system () Work is a form of energy being transferred across a system boundary.
Work associated with a rotating shaft:
dT drF dx F W Power ( ) [W], time rate of doing work
W
T VF t
W W or t WW
2
1
Specific work (w) [J/kg]: work per unit mass
m
W w
For work SI unit 1 N-m = 1 J other units 1 kWh = 3600 kJ For power, SI unit 1 J/s = 1 W other units 1 hp (mechanical) = 745.7 W 1 hp (metric) = 735.5 W For specific work, SI unit 1 J/kg = 1 (m/s)2
3.4 Work Done at the Moving Boundary of a Simple Compressible System Consider a gas in a piston/cylinder device undergoing a quasi-equilibrium process
If P is the pressure of the gas, the work done by the system is
dV P W
dLA P W
In case that the relationship between P and V is available as a diagram:
We conclude that
dV P W W2
1
2
1
21
= area under the curve 1-2
= area a-1-2-b-a
However, we can choose different quasi-equilibrium processes from state 1 to state 2 (process A, B or C)
Thus, work done during each process is different. As a result, work is not only a function of the initial and end states. work depends on the path of each process. work is a path function. W is an inexact differential.
On the other hand, thermodynamics properties are point functions. Notes: symbolic convention For a point function (such as V) the differentials are exact by using symbol dV
12
2
1
VV dV For a path function (such as W) the differentials are inexact by using symbol W
21
2
1
W W To determine the work from the area under P-V curve, the relationship between P and V must be obtained.
The most common relationship between P and V is a polytropic process: onstantC PVn
n1
VPVP PdV 1122
2
1
This result is valid for any values of n except 1. For n =1,
1
211
2
1V
VlnVP PdV
which is equivalent to the isothermal process of an ideal gas
3.5 General Systems that Involve Work Other types of work are Stretched wire
dL W
2
1
21 L W
Where is tension [N]
Wire
dL
Surface line
dA W
2
1
21 A W
Where is surface tension [N/m]
Liquid Film
dA
Electric
dt i dZ W
where is electric potential [V] Z is electric charge [C] i is electric current [A]
2
1
21 t i W
i t
W W
Battery
+ E
dZ
Combining all types of work gives
...dZdAdL dV P W
...iA V P W V
Notes:
The process above contributes no work due to nonequilibrium process.
(a) Work done on the system (b) No work done on the system
3.6 Definition of Heat
Heat (Q) [J]: a form of energy which is transferred by the difference of temperature
Heat is a path function
21
2
1
Q Q
Rate of heat transfer (heat rate) ( ) [W]: heat transferred per unit time
Q
t
Q Q
or t QQ2
1
21
Specific heat transfer (q) [J/kg]: heat per unit mass
m
Q q
Adiabatic process: a process, in which there is no heat transfer
3.7 Heat Transfer Modes
Three modes of heat transfer: Conduction: heat transmitted by
diffusion of energy through a medium without the bulk motion Fourier's law of conduction
dx
dTAk Q
where k is thermal conductivity [W/m-K]
Convection: heat transferred by the motion (or flow) of a medium Newton's law of cooling )TT(A h Q surs where h is convective heat transfer coefficient [W/m2-K]
Radiation: heat transmitted by electromagnetic waves in space Stefan-Boltzmann's law )TT (A Q 4
sur4s
where is emissivity, is Stefan-Boltzmann constant [W/m2-K4]
Notes: Comparison between heat and work
Heat and work are both transient phenomena. Systems never possess heat or work
Heat and work are boundary phenomena. Both represent energy crossing the boundary of the system.
Both heat and work are path function and inexact differential
(a) Heat transfers across the boundary. (b) Work transfers across the boundary.
3.8 Internal Energy - a Thermodynamic Property U is an extensive property. Thus, we can define
Specific internal energy (u) [J/kg]:
m
U u
In general, the value of the specific internal energy must be given relative to a reference state (which is arbitrary). For water, uf = 0 at 0.01oC Note: the values of the specific internal energy can be negative. For the two-phase mixture,
fgf
fgf
gf
u x u
)u (u x u
ux ux)1( u
u represents the sum of all microscopic forms of energy.
moleculeintntranslatiomoleculeext uuu u
originates from the
intermolecular force between molecules.
moleculeextu
originates from the translation of molecules
ntranslatiou
originates from many
sources such as molecular rotation, molecular vibration, electron translation, electron spinning, atomic bond, nucleus bond.
moleculeintu
Sensible energy: all kind to kinetic energies as a function of temperature
Latent energy: intermolecular force for phase change process
Chemical energy: atomic bond
Nuclear energy: nucleus bond
3.9 Problem Analysis and Solution Technique 1. What is the control mass or control
volume? draw a diagram to illustrate heat/work flows
2. What do we know about the initial state?
3. What do we know about the final state?
4. What kind of the process takes place? Is anything constant or zero?
5. Is it helpful to draw p-v diagram? 6. What is the thermodynamics model
we use (steam tables, ideal gas, etc)? 7. What is our analysis (find work, the
first law, the second law, etc)? 8. How do we proceed to find what we
need?
3.10 The Thermodynamics Property Enthalpy Consider a quasi-equilibrium constant-pressure process:
)VP(U)VP(U
)VPVP( U U
)VV( P U U
PdV U U
W U U Q
111222
112212
1212
2
1
12
211221
U + PV is a property because it is a combination of other properties.
Enthalpy (H) [J]: PV U H Specific enthalpy (h) [J/kg]:
vP u m
H h
Thus, for a quasi-equilibrium constant-pressure process
1221 HH Q Notes: 1) Some thermodynamics tables does not contain data of u. Therefore, we can find u from u = h P v. 2) Some substance use enthalpy as a reference state instead of internal energy (such as R-12, Ammonia).
For the two-phase mixture,
fgf
fgf
gf
h x h
)h (h x h
hx hx)1( h
3.11 The Constant-Volume and Constant-Pressure Specific Heats From the first law:
PdV dU W dU Q
1. Constant volume ( 0 PdV W ) Specific heat (at a constant volume) (Cv) [J/(kg-K)]:
vvvv T
u
T
U
m
1
T
Q
m
1 C
2. Constant pressure ( dH Q ) Specific heat (at a constant pressure) (Cp) [J/(kg-K)]:
PPPp T
h
T
H
m
1
T
Q
m
1 C
Cv and Cp are also thermodynamics properties
For solid and liquid (incompressible substance: v = constant)
C C C Pv
u and h can be written as
dTC du
dP vdTC dPvdu dh
If C is a constant, we have
1212 TTC uu
121212 PPvTTC hh
In most cases, if 12 PP is not large, we can neglect the second term of the right side. As a result,
1212 TTC hh
C for various solid and liquid are listed in Tables A.3 and A.4
Notes: For some substances, C could changes significantly at a very high temperature.
3.12 The Internal Energy, Enthalpy, and Specific Heat of Ideal Gases
For internal energy, u is generally a function of two independent properties.
For a low-density gas
u depends strongly on T, but not much on P. Thus for an ideal gas
Pv = RT and u = f(T) only
For Cv, we mathematically write
General cases : v
v T
u C
Ideal gases : Td
du Cvo
subscript "o" denotes for the specific heat of an ideal gas
Thus, we can write
dTCdu vo
dTCuu
2
1
vo12
For enthalpy, from the definition of h = u + P v, in case of an ideal gas :
only f(T) h
RT u vP u h
For Cp, we mathematically write
General cases : P
P T
h C
Ideal gases : Td
dh CPo
dCdh Po T
dTChh
2
1
Po12
Because u = f(T) and h = f(T), Cvo = f(T) and CPo = f(T) as well. The values of Cvo and CPo at standard condition are given in Table A.5. The variations of CPo as a function of T are given in Table A.6.
The main factor causing Cvo and CPo to vary with temperature is molecular vibration.
Monoatomic gas: Cvo and CPo are a very weak function of T Diatomic gas: Cvo and CPo are somewhat a function of T Polyatomic gas: Cvo and CPo are strongly a function of T For an ideal gas
R C C
RdT dTC dTC
RdT du dh
RT u vP u h
voPo
voPo
on a molar basis
R C C voPo
To utilize the specific heat, there are three possible ways to find the enthalpy difference of an ideal gas
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