Download Chapter 5 Diffusion Availability

Chapter 5
Diffusion Availability
The availability is the maximum work that can be extracted from the substance as it
undergoes a reversible process from a given state to the environmental state. When a
substance is in the environmental state, it is in thermal and mechanical equilibrium with
the environment and therefore has no work potential. In other words, a substance has a
zero availability at the environmental state.
There are many occasions in which the substance is in thermal and mechanical
equilibrium. Is there any work that theoretically can be developed as a substance comes
into chemical equilibrium with the environment? If work is developed, what is
maximum useful work that could be produced as the system and the environment reach
equilibrium? The equilibrium means only T=T0 and P=P0. The new condition that
justifies the nonflow availability question at this time is that equilibrium is not only
thermal and mechanical, but also chemical.
The work potential of this kind will be called either diffusion availability or chemical
availability. The objective of this chapter is to develop the concept of diffusion and
chemical availability and various engineering processes such as there in heating,
ventilating, and air conditioning (HVAC) and in combustion.
5-1 Diffusion availability
To determine the combination of thermomechanical and diffusion availability(combined
availability), we start with the equation for the available work of a substance in the
process between the state 1 and 2.
w ava  ( h  h  )  T (s  s  )
In this case, the state 2 is the dead state, the available work will become the combined
availability of the substance in state 1.
The combined availability is written as
af  ( h  h d )  Td (s  sd )
(5-1)
The dead state is defined as the state in which the substance is in thermal, mechanical,
and chemical equilibrium with the environment.
For simplicity the popular models for the environment are as follows :
Table 5-1 Environment ; T=25℃, 1atm
Components
Mole fraction (y0)
Nitrogen
0.7567
Oxygen
0.2035
Carbon dioxide
0.0003
Water vapor
0.0303
Other gases
0.0092
In the case of pure substance, the equilibrium means that the substance has a
temperature equal to the environmental temperature T0 and a pressure equal to the
partial pressure of the same component in the environment.
The temperature Td is the environmental temperature and Pd is the partial pressure of
the same component in environment.
That is
Td = T0
(1)
Pd = y0P0
(2)
Equation (1) is condition for the pure substance in chemical equilibrium with the
environment.
When the substance under consideration is an ideal gas, we can reduce Eq. (5-1) to

T
P 

af  c p (T  T )  T  c p ln
 R ln
T
y  P 

or
af  af ,th  af ,diff

T
P
where af ,th  c p (T  T )  T  c p ln
 R ln 
T
P 

(5-2)
(5-3)

(5-4)
y
Eq. (5-2) indicates that total availability of a substance at given state is the
thermomechanical availability a f ,th plus the diffusion availability a f ,diff at that state.
af ,diff  T R ln
We can imagine taking the reversible process between the given state and the dead
state and breaking it into two processes.
The first process will take the substance into thermal and mechanical equilibrium
with the environment. This process terminates at the environment state, where
temperature and pressure are respectively equal to T0 and P0. The second process starts
at the environment state (P0, T0) and eventually brings into chemical equilibrium with
the environment.
In the dead state the substance has temperature T0 and pressure equal to the partial
pressure of corresponding component in the environment.
The available work in the second process is defined as the diffusion availability for a
substance.

T
P 

af ,th  w ava  c p (T  T )  T  c p ln   R ln    T R ln
T
y  P 
y

The diffusion availability for pure substance does not depend on the thermodynamic
properties at a given state. Also, the diffusion availability is very much related to the
model of the environment.
We can write the total nonflow availability as the sum of the thermomechanical and
diffusion availability.
a  a th  adiff
(5-5)
The relationship between the total, physical, and chemical (diffusion) availability is
illustrated in Fig. 5-1(a) for nonflow systems and Fig. 5-1(b) for flow systems.
Note that the labeling in this figure is correct only if there are two or more
components in the mixture, n ≥2.
T , P,
*
i
T , P ,
i
ath
T , P ,
*
 ,i
adiff
a
(a) nonflow availability
(* : restricted dead state, μ: chemical potential)
T , P,
T , P ,
i
af ,th
*
i
T , P ,
*
 ,i
af ,diff
af
(b) flow availability
Fig. 5-1 The relationship between total, thermomechanical
(physical), and diffusion(chemical) availability
The thermomechanical or physical availability has been discussed in Chapter 3. The
diffusion nonflow availability is the available work between the environmental state (T0,
P0) and the dead state.
The dead state is the state at which substance is in chemical equilibrium with the
environmental state. For a pure substance the pressure of the substance at the dead state
is equal to the partial pressure of the same material in the environment.
The diffusion nonflow availability is

adiff  T R ln
(5-6)
y
The availability of steam :
The total availability of steam at any state (T, P) can be developed and expressed as
Pg (T )
af  h(T, P)  h g (T )  T s(T, P)  s g (T )  T R v ln
(5-7)
y ,v P
or
a f  a f , th  a f ,diff
where af ,th  h(T, P)  h g (T )  T s(T, P)  s g (T )
a f ,diff  T R v ln
Pg (T )
y ,v P
For the water only, Eq. (5-7) can be written in more convenient form.
h(T, P)  h f (T )  v f P  Pg (T )
(5-8)
(5-9)
(5-10)
where h(T, P)  u f (T)  Pv f (T)
h f (T )  u f (T )  Pg (T ) v f (T )
h(T, P)  h f (T )  v f (T )P  Pg (T )
or h(T, P)  h f (T)
s(T, P)  sf (T)
(5-11)
Substituting Eqs. (5-10) and (5-11) into Eq. (5-7), the availability of water at any
given state (T, P) as
Pg (T )
af  h f (T )  h g (T )  v f P  Pg (T )  T s f (T )  s g (T )  T R v ln
y ,v P
(5-12)
where subscript f denotes saturated water at the temperature T.
In Eq. (5-8), we set the enthalpy and entropy at the environmental state to be same as
those of saturated vapor at T0.
h g (T )  Ts g (T )  h f (T )  Ts f (T )
(5-13)
Eq. (5-8) can be reduced to
af ,th  h(T, P)  h f (T )  T s(T, P)  s f (T )
(5-14)
This equation is identical to the equation for thermomechanical availability
developed in Chapter 3.
Physically, the diffusion availability is the available work that can be extracted from
the substance as it undergoes a reversible process from the environment state (T0, P0) to
the dead state (T0, y0P0). In the case of dead state the substance is expected to have the
temperature T0 and the pressure equal to partial pressure of that substance in the
environment.
Example 5-1 In a long pipe, the steam experiences a temperature drop from 560℃ to
500℃ and a pressure drop from 12.5MPa to 10.0MPa. Calculate the combined flow
availability at both properties for both states.
(Sol.) Using the steam tube, we determine the steam properties for two states.
P1 = 12.5MPa
P2 = 10.0MPa
T1 = 560℃
T2 = 500℃
h  .kJ/kg
h     .k J / k g
s  .kJ/kgK
s   .k J / k g K
The total flow availability is given by the equation.
af  af ,th  af ,diff
where af , th  (h  h  )  T (s  s  )
a f ,diff  T R v ln
Pg (T )
y ,v P
1atm, 25℃ ; h0 = 104.89kJ/kg
s0 = 0.3674kJ/kgK
State 1 :
af ,th  (.  .)  (.  .)
 .kJ / kg
af ,diff     . ln
.
.  
 .kJ / kg
Total flow availability at state 1 is
af   af ,th  af ,diff 
 .kJ / kg
Similarly, the total flow availability at state 2 is
af ,th   (.  .)  (.  .)
 .kJ / kg
af ,diff     . ln
.
.  
 .kJ / kg
af   af ,th   af ,diff 
 .kJ / kg
The result of this problem, the diffusion availability for this steam flow is small
relative to the physical availability. Also, the diffusion availability remains unchanged
in the process, as expected. Therefore the change of total availability is exactly equal to
that of thermomechanical availability.
5-2 Availability of Mixtures
Consider a certain mixture containing n different constants. The volumetric analysis
as (y1, y2, …, yn). The mixture properties can be expressed as below :
mixture of n components :
n
u   yi ui
i 
n
h   yi hi
i 
n
s   yi si
i 
For ideal gas components the terms u i and h i are evaluated at the components
temperature, which is equal to the mixture temperature. The term s i is a function of
component temperature and pressure.
n
a f ,th   y i a fi ,th
or
i 

n


  y i h i (T )  h i (T )  T s i (T, y i P)  s i (T , y i P )

i 
Diffusion availability
In the same manner we can develop the equation for the diffusion availability of a
mixture.
n
a f ,diff   y i a fi ,diff
i 
n
= T R  y i ln
i 
yi
y ,i
Example 5-2 The combustion products have the volumetric analysis as following
table :
Table 5-2 Composition of combustion products
CO2
7.20%
H2O
7.80%
N2
76.12%
O2
8.88%
Calculate the diffusion availability and physical availability per mole of the mixture.
The products of combustion have 1atm and 800℃.
5-3 Moist air
Moist air is regarded as an ideal gas mixture of water vapor and dry air. Water vapor
is highly superheated in the moist air. Therefore water vapor can be approximately as an
ideal gas.
Composition of mixture ; water vapor + dry air
mole fraction yv, ya
na

ya 

n
na  n v
 v
na
mv
nv Mv
M m

 a v  cω
ma
na
M v ma
Ma
where c =
Ma
m
 . , ω  v ; humidity ratio
Mv
ma
∴ ya 

  cω
ya  y v  
y v  (  y a )   
∴ yv 

  cω
cω
  cω
Total availability


y P 
T
a f ,th  y a c pa T  T   T c pa ln
 R ln a 
T
y a , P 




y P 
T
 y v c pv T  T   T c pv ln
 R ln v 
T
y v , P 


……… thermomechanical availability

y
y 
a f ,diff  T R  y a ln a  y v ln v 
y ,a
y  ,v 

a f  a f ,th  a f ,diff
……… diffusion availability