Phase Equilibria and Drying
Rates
1
Wet solids
Liquid
Hot air
Material
Moisture
Hot air
Material
LiquidNecessary condition: pw  p
Material
Hot air
Liquid
Driving force：
( p  p)  0
w
Hot air
•Questions: Will (pw-p) remain constant or change during
drying process?
What is the final result(equilibrium) of drying process?
•In addition to air conditions, the equilibrium is related to the
property of wet solids.
2
1.Phase Equilibria[Reading: pp.779~782]
•Types of water in wet solids:
1)Adsorption water: Characteristic: pw=ps
pw=water vapor pressure on the surface of wet solids;
ps=saturated vapor pressure exerted by liquid water at the
same temperature with pw.
2)Capillary water:
a)Non-hygroscopic （ 非 吸 湿 ） substances(or in large
capillary): Characteristic: pw=ps
b)Hygroscopic substances: Characteristic: pw<ps
3)Swelling water（溶涨水）: In cell walls.
3
(1)Equilibrium water(moisture) and Free
water(moisture); Equilibrium-moisture curves
Wet solids
Liquid
Hot air
Material
Moisture
Hot air
Material
For air of definite humidity(definite vapor
Liquidpressure), because of fine capillary effect,
Material
liquid water exerts an abnormally low
LiquidHot air
vapor pressure because of the highly
Hot air concave凹面 curvature(曲率) of the surface, pw
getting lower an lower.
When pw=p, N=0，X=X*. Here, X* is the equilibrium water
content
4
Water corresponding to concentrations lower than X*
can not be removed by drying.
•Free water(moisture) :
The difference between the total water content of the
solid and the equilibrium water content.
•Free water X=total water XT - equilibrium water X*
•Equilibrium-moisture curves: Relationship of X*~ ,
under certain temperature (Figure24.3 or figure 5-10).
5
•Factors influencing X*:
1)Kinds of solids: X*hygroscopic > X* non-hygroscopic
2)Air conditions: ——X*
t —— X* 
(2)Bound and unbound water结合水分与非结合水分
Bound water：
Characteristic: pw<ps (Water in wet solids exerts a vapor
pressure less than that of liquid water at the same
temperature.)
Bound water may exist in several conditions. 1)In fine
capillaries, because of the highly concave curvature of the
surface; 2)In cell or fiber walls, because of solids dissolved in
water; 3)In natural organic substances,
6
•3)In natural organic substances, because water is in
physical and chemical combination, the nature and
strength of which(combination) vary with the
nature and moisture content of the solid
•Unbound water. Characteristic: pw=ps. Water is
largely held in the voids of the solid(nonporous
particles, etc.), strength of combination of water and
solid is weak.
•The distinction区别 between bound and unbound
water only depends on the material itself.
7
•Determining the bound and unbound water (Refer to
Figure 5-11)
If an equilibrium curve is continued to its intersection交点 with
the axis for 100% percent humidity, water corresponding to
concentrations lower than that indicated by the intersection of
the curve with the line for 100 percent humidity is bound water,
water corresponding to concentration greater than that
indicated by the intersection is unbound water. Why?
Because, = 100% means p=ps, corresponding to pw=ps when
in equilibrium;
•If <1, pw=p<ps when in equilibrium;[bound water]
•If =1, pw=p=ps when in equilibrium.[unbound water]
8
•Relationship among total water, equilibrium and free
water, bound and unbound water(Fig.5-11)
Total water content
Bound waterBound water
Bound water
Bound
water water
Unbound water
Unbound
Unbound water
Equilibrium water
Unbound
water
Equilibriu
m
water
Equilibrium water
Free water
Equilibriu
m
water
Free water
Free water
Free water
9
2.Drying curves and drying rate curves under
constant drying conditions [Reading: pp.782~788]
(1)Constant drying conditions恒定干燥条件.
Assume that the temperature, humidity, and velocity and
direction of flow of the air across the drying surface are
constant. This is called drying under constant drying conditions.
•For example, when small amount of materials(wet solids) is
dried by large amount of air, conditions in the air-stream can
be considered as constant.
(2)Drying experiment and drying curves
10
热电偶（测）
干燥室
天平（测W）
湿空气
物料（纸浆板）
湿空气 天平（测W）
秒表（测）
预热器
干燥室
预热器 秒表（测）
t0 , H 0
热电偶（测
）
(2)Drying
experiment
and
drying
curves
天平（测
W）
热电偶（测
Therm
ocoup
le
(
for

)
t0 , H0）
QP ocoup
Therm
le( for
物料（纸浆板）
Moist
air
秒表（测


）
物料（纸浆板）
Moist air
Therm
ocoup
leu( for( for
 ) W
QP
Balance
( for
W
) Balance
气
干燥室
preheater
t0 , H 0
Moist
air
干燥室
preheater
Balance
ule
Therm
ocoup
(
for

)
气
Stopwatch(u for (for
) WW ) ( for 
Stopwatch
天平（测W）
QP
preheater
air
天平（测

W
）
u
Stopwatch
( for
W
( for W )air
t1 , H1  )
Moist air
Moist
air
秒表（测
Balance
）
u气
u airMaterial
秒表（测
H） Material
t
,
t2 , H 2
1
1  )
Stopwatch
preheater ( for
tpreheater
0, H0
Dryer
W
t0 , H 0 t , H Dryer Material
2
2
u air
u air
Q
P
Dryer t1 , H1
QP
Material
Material
u
气
t2 , H 2
u气
Dryer
Dryer
W
W
dW  W 
U

[kg water /(m 2  s)]
t1 , H1 rate U:
Drying
t1 , H1

dWSd SW
2
t2 , H 2
U


[
kg
水
/(
m
 s)]
t 2 , Hunder
2 t  t constant

How to keep the dryer
drying conditions
dW 2 
W
1
Sd
U
H  H S[kg水 /(m 2  s)]
t 2  t21S1
Sd

large amount of air versus small amount of materials:
t 2  t1 H 2  H1
H 2  H1
11
•Drying curves: X~, ~  (Figure 5-12)
[Here, =drying time ,h]
(3)Drying rate curves and drying process of solid materials
Definition of drying rate U:
dW  W 
U

[kg water /(m 2  s)]
Sd
S
t 2  t1
GdX
 dW   GdX ,U  
 (5  46)
Sd
H 2  H1
Drying rate curves: X~U curve (Figure 5-13)
12
•Drying rate curve: X~U curve (Figure 24.6)
Falling rate period Constant-rate
period
Falling rate
Falling
period
rate period
UC
Three periods of
drying process:
•AB；
•BC；
•CDE
UC
U
U
13
•AB—Preheating period
•BC——Constant-rate period
•CDE——Falling-rate period. CD: First falling-rate
period(some fraction of solid surface become dry); DE:
Second falling-rate period(air-water interface recedes).
Falling rate
period
Constant-rate period
Falling rate
Falling
period
rate
UC
UC
U
U
14
period
(4)Drying mechanism of wet solids and the influencing factors
1)Constant-rate period (Period of controls of surface water
vaporization[BC]
•Characteristics: drying rate Uc unchanged, solid very wet(a
continuous film of liquid exists over the entire external
surface), solid surface temperature tw,
t、H unchanged（t  t w）
, and（H s ,tw  H）const

dW 
UC  K（
 （

50s）
[Ut t w）
（k5H( H
H H s，tw  H）
,tw  H )
rtw
Sd
dW 
dQ S (t  t w )  d
[U 
 k H ( H s ,tw  H ) dW  

]
Sd
rtw
rtw
S (ttoEqs.(24.9),(24.10),(24.13).
t w )  d
dQ
Eq.(5-50)
is equivalent
dW  

] 15
rtw
rtw
[1)Constant-rate period]
Water vaporized is unbound water.
Factors influencing drying rate:
(a)Air conditions(t，H);
(b)Air velocity;
(c)Patterns of air-solid interaction in dryers(method
of contacting the solids and air).
Three basic patterns of air-solid interaction in
dryers: (a)air flow is parallel to surface of solids;
(b)air flow is perpendicular to surface of solids;
(c)solids (particles) suspended in air.
16
2)Drying in the falling-rate period (period of
controls of water diffusing from interior to solid
surface)
•Characteristics: Water diffusing rate from the solid
interior to surface is less than rate of surface vaporization,
therefore, rate of drying U decreases gradually.
•Factors influencing drying rate U:
Mainly the structures and sizes of materials.
Air conditions don’t influence much, but air temperature
has certain effect, such as (t-) is heat transfer driving
force. If t is very high, material is deformed easily.
17
(5)Critical water(moisture) content XC and its
influencing factors临界含水量XC
•The point XC at which the constant-rate period ends is
called the critical water content.
•If the initial moisture content of the solid is below the
critical moisture content, there will be no constant-rate
period.
•XC is influenced by the property and thickness of material,
and the drying rate。.
1) X C ,hygroscopi c  X C ,nonhygrosc opic
2）Thickness of m aterial X C 
3) Drying rate U  X C 
( falling  rate period com es early).
18
•Methods for increasing rate of drying U:
、H不变，（
t  t w）、（
H sConstant-rate
•1)Drying
in the
period
,tw  H）恒定。

干燥速率U C  K （
 （t  t w）

（5  50）
H H s，tw  H）
rtw
a) , k H  (uair )  U 
)  Ub）（
 t  t ）, ( H  H )  ( such as t , H )  U 
w
s ,tw
( H s ,tw  H
)  ( suchmethod
as t of
, Hcontacting
)  U solids and air(such as particles
c)Improve
ir
3)改善物料与空气接触状态（如颗粒状、悬浮态）   ，k 
与空气接触状态（如颗粒状、悬浮态）   ，k 
suspended in air)
2)Drying in the falling-rate period
a)Decreasing thickness of material;
b）t (before material is deformed).
19
UC
X
3.Calculation of drying time under constant
XC
drying
conditions [Reading:
788~790]
O
U [kg /(m 2  s )]
Falling-rate period
O
A
X [kg水 / kg干物料
]BConstant-rate period
O
A
A
O A
B OO
C
UC
BB
A

C AA
D
C
X
UBC C
D BB
E
D
XC C D
C
X
E
X
C
2
1
U
E
E
C
D
U [kg水
/(m  s )]
D
X
X1 E D
X 2
X11
U CC X
X
X [kg水
]
E / kg干物料
2
X 2 X 1E
/(mX恒速干燥阶段
 s )]
O UX[ kg
2
XX2水
C
X1
X 2X
2
恒速干燥阶段
恒速干燥阶段
A X
降速阶段
water
/
kg
bone

dry
solid
X [Ckg
U
[
kg
水
/(
m

s
)]
1 ]
恒速干燥阶段
X2
恒速干燥阶段
2
降速阶段
B U [kg
X2
水 /(m X [skg
)] 水 /20kg干物料]降速阶段
降速阶段
降速阶段
恒速干燥阶段
O
UC
(1)Drying time of constant-rateXperiod 1

XC
GdX
G
UC  
 d  
dX
Sd
UC S
G
Integration :  1   
dX
UC S
X1
XC
O
O
A
O
X [kg水 / kg干物料
]B Constant rate period A
O
A
O A
B OO
C
UC
B
B
A
C AA
D
C
X
BC C
U
D BB
E
X C C D
D
X
E CC
X1
2
E
U
E
C
D
U [kg水
D/(m  s )]
X
X

X
C
1
X11
UC X
ED
X 2
X [kg水
干物料
]
E / kg
2
X 2 X 1E
/(mX恒速干燥阶段
 s )]
O U
X水
X[ kg
X
2
C
X1 2
X 2X ]
2
恒速干燥阶段
恒速干燥阶段
A X
X [Ckg
water
/ kg
bone
solid
U降速阶段
[kg
水 /(m
 s )] dry
1
恒速干燥阶段
X2
恒速干燥阶段
2
降速阶段
降速阶段
X
B U [降速阶段
X
[
kg
水
/
kg
干物料
]
kg水 /(m  s )]
2
降速阶段
恒速干燥阶段
C X [kg水 / kg干物料]
恒速干燥
降速阶段
D
降速阶段
U [kg /(m 2  s )]
Falling-rate period
G
1 
( X 1  X C )  (5  51) [Similar to Eq.(24.16)]
UC S
E
X1
UC is determined by drying rate curve or
by the following
X
恒速干燥阶段
equation:

2
UC 
rtw
(t  t w )  k H ( H s降速阶段
,tw  H )
 is determined by Eqs. (5-52),(5-53),(5-54) (p.271).
21
GdX
G
(2)Drying time of falling-rate period

X
2
U 
 d  
X
UC

Sd
G dX
  
S
U
C
GdX
G
U 
 d  
dX
Sd
SU
GdX
G
X
U   G 2 
d  
dX
dX
SU
 2   SdX
S XX C dX
U
G
G
dX
 22 


X 关系非线性，
U
U
1)若U SS~ X
X
2
2
SU
O
O
A
O
2 period
X [kg水 / kg干物料
]B ConstantXrate
O
A
A
O A
B OO
C
UC
B
2B
A
C AA
D
C
X
X
BC C
U
C
D BB
E
X C C D
D
X
E CC
X1
2
E
U
E
D
U [kg水
D/(m  s )] C
X
X1 E D
X11
U CC X
XX 2
X [kg水
]
E / kg干物料
2
X 2 X 1E
恒速干燥阶段
/(
m

s )]
O U
X水
X
X[ kg
X
2
C
X1 2
X 2X ]
2
恒速干燥阶段
恒速干燥阶段
A X
X [Ckg
water
/ kg
bone
solid
U降速阶段
[kg
水 /(m
 s )] dry
1
恒速干燥阶段
X2
恒速干燥阶段
2
降速阶段
X2
B U [降速阶段
kg水 /(m X [skg
)] 水 / kg干物料]降速阶段
降速阶段
恒速干燥阶段

C X [kg水 / kg干物料]
恒速干燥阶段
降速阶段
X
D
降速阶段
U [kg /(m 2  s )]
Falling-rate period
1)若U ~ X关系非线性，
用图解积分法（参见例5 
C
C
1)If the relation
of U~X is non-linear,
drying
2）若
U  time
k ( X of
 X )[线性
1
))若
用图解积分法（参见例
 10determined
）。 E
1falling-rate
若U
U~
~X
X关系非线性，
关系非线性，
period 2 5is
by graphical
X2
G
dX
X
 5-10).
integration
(Example
用图解积分法（参见例
5

10
）。
用图解积分法（参见例
5

10
）。
2）若U  k X ( X  X )[线性关系]，则：
X 2  



恒速干燥阶段
S
k
(
X

X
)
Here,
 )[linear]，
X
X
2
）
If
U

k
(
X

X
X
C
2
2）若U  k X X( X  X )[线性关系]，则：
降速阶段
1
2
G X
dX
 2  G X

dX

G
dX
S
k
(
X

X
)
 2 
XC X



UC  0
kX 
XC  X 
2
2
2
S
S

kk XX (( X
X
X
X  ))
UC  0
其中, k  U

0
U

0
C
C
其中
,
k

X

X
其中, k  C


X
X CC
22
G X C  X  X C 
 2  
ln

GdX
GU
U 
 d   X dX


Sd
SUX U U   G dX  d   G d
SU
X2
(2)Drying time
of falling- U [kg /(Xm  s)]
OSd
G dX
X
Falling-rate
O
AGO X dX
O
rate
period

 2 G



2
dX
G
period
X
[
kg
水
/
kg
干物料
]

U
[
kg
/(
m

s
)]
GdXS  dU  G  dX
O 2 
B Constant
rate O
A
A 
U
A
O
U



d



dX
XC
X [kg
O
O水A/ kg干物料
S]
U
C

C
C

2
2
C
2
Falling-rate period
period
A
B XC
Sd
SU
B
Sd
SU
UC
OO
BB O AA C
B
C
B
C OAOA
D
B
1)若GU XX~22 dX
X关系非线性， X  UA C1C)A若
C AA
X
G
dX
BC UB CC U ~DX关系非线性，
U
B
D
E
 22 

D B BB
E

D
XC C
D
D

X
C
C D
用图解积分法（参见例
5  10）。
S
X
X  用图解积分法（参见例
E C CC 5 
E
S XC U
U
EE 2 EE UX UX
Constant rate period
C

C
C

DD
U [/(
kgm
水
D/(
U [kg水
m
s )] sC)]1 X
D
X
X
C
1
X1
U X
X
X 1 EEDD
X 2X 2]

X [CCkg
/
kg
干物料
X水
U
E
X
X
1
2
1）若
2
U]恒速干燥阶段
 k ( X XXX2 X E)[线性
X [kg水
kg
1
U
2）若
 k X ( X  X )[线性关系
U
[ kg
X 水 /(
EO/]，则：
X干物料
X
1))若
若
U~
~X
XU关系非线性，
关系非线性，
2 mX  s )] X
2
X
恒速干燥阶段
水
/(
m

s
)]
O UX[Akg
X
2 XX 1E
X
2X [1kg
X 2X
恒速干燥阶段
C U降速阶段
X1 ]
恒速干燥阶段
water
/
kg
bone

dry
solid
[
kg
水
/(
m

s
)]
恒速干燥阶段
X1 X 2
X2 2
恒速干燥阶段
X
X
恒速干燥阶段
恒速干燥阶段
用图解积分法（参见例
降速阶段
X2X
water
/Xkg
bone
dry
solid
[skg
/ kg干物料
]降速阶段
[m
kg
/(m
s )]dX
U [降速阶段
kg
水U
/(降速阶段
)] 水
水
2 1 ]
G
用图解积分法（参见例
 10
10）。
）。 A XXXB[Ckg
恒速干燥阶段
降速阶段
G 2
dX 55 
恒速干燥阶段
恒速干燥阶
2 降速阶段
水
2X/kg[skg
kg
水
干物料
] 干物料]降速阶段
恒速干燥
X
B UC
水 / kg
2X [
[
kg
/(
m
)]
2



降速阶段
降速阶段

2
降速阶段

S
k
(
X

X
)
2
）
If
U

k
(
X

X
)[
linear
]
，
D
恒速干燥阶段
Here,
降速阶段
2）若U  kSX X( X kXX( X)[
线性关系
]，则：
C X [kg水 / kg干物料X]C X
X
)
恒速干
XC
E
降速阶段
X
2
D
降速阶
X1
UC  0
 X 2
G
dX
G
dX
UC  0 
X 2k X 
 22 

E




其中
,
k

X

X
S
k
(
X

X
)
恒速干燥阶段
C
S XX CC k XX ( X  X  )
X1
XC  X
降速阶段
X2
X 
G X  X 
2
XC
C
1

C

1
2
1
C
2
C


2
2


U
C
C
U CC 
0
0






ln
其中
,
k

恒速干燥阶段
2
其中, k  G  X C  X
X

X
C
S57a ) U C
X2 
 2  X
 X
ln


(
5

X CC 

X
降速阶段

S
U
X

X


 C X
2
 X

X

X
G
X

X
X

X
G
C
C

 C
ln

 22 

ln C

 ((5
5
57
57a
a))   （连续操作）


23
S
U
X
1
2
总
S
U CC
X 22 
X
X
XC C  X
U CG 0 X C  X   X C  X  
其中,
k  2  G   X C  X ln X C X (5  57a)
S X  UC
X 2  X   (5  57a )
X




ln
C
2
(3)Total time S
of drying
 T：
U
X

X
C
2



XC  X
G XC  X
 2      ln(Continuous
 (5  57a)
 operation)
TS
1U 2
X2  X
C
 1（连续操作）
总
 总
（间歇操作）
1
2 2  装卸
 T   1   2   l &d Batch operation
 总   1  （连续操作）
2
 T   1Where,
  2   l &d
=loading and discharging time
24
XC
U 0

X

X
C
用图解积分法（参见例
5  10）。
C
1)若
U ~, X
其中
k 关系非线性，


X drying
*4.Calculation of drying2）若
timeUunder
constant

X

X C ]，
X
G
C X )[线性关系

k
(
X

X
 2 

ln
conditions
XS
UC
X2  X
G 2
dX
 2 Problem

Assignments: Problem 24.2;
5-8 or 24.4?
S XC k X ( X  X  )
Key to Problem 24.2:
 T  13U
.5Ch  0
其中, k 

X

X
 总  1 C  2   装卸
（间歇操作）
Hint of Problem 5-8:
U  G(t Xt wC) X  X C  X 
 2 rtw 
ln


S
UC
X2  X
  1.17( L)0.37
(5  53)
 T  4.0h
 1   2 
 装卸
（间歇操作）
Hint of Problem 24.4: 总
1)Assume
constant
enthalpy
Key to Problem 24.4:
drying process; 2)U is linear in X in falling-rate period.
25
•Exercises：
1.Is critical water(moisture) content XC the
distinction point between bound and unbound water?
2.In order to increase the rate of drying of materials
mainly containing bound water, must the air velocity
be increased？
3.When water in the wet solids is in equilibrium with
the air, what is the relationship between the water
vapor pressure of the solid surface and the partial
pressure of water vapor in the air?
4. What are the equilibrium water contents of kaolin
and wool X* when relative humidity is 60%?
26
5.The air temperature leaving the dryer t2 is
_____________ greater than the entrance air
adiabatic saturation temperature of the dryer tas,
for the purpose of
.
6.Would you explain the relationships among t, tw,
tas, and td?
7.Are the water vapor pressures of different wet
solids all the same when in equilibrium with the air
in the same room? ______. Are their water contents
all the same? _____. Are their temperatures all the
same? _______.
27
8.Some wet solids are dried by air
with temperature t and humidity U C
H. The simplified drying rate
X
UC
curve is showed by the attached X C
figure. Please draw the new dryingU [kgU水C /(mX2  s)]

X
rate curves under following drying X [kg水 / kgX干物料
C
]
X C U [kg水 /(m 2  s)]
conditions relative to the position
2
X [/(kg
kg干物料]
U [kg水
m水
 s/ )]
of original drying rate curve:
X [kg水 / kg干物料]
28
(1)Air velocity increases and air temperature t and
humidity H keep unchanged.
(2)Air velocity and temperature unchanged, and
humidity increased; or air velocity and humidity
unchanged, and temperature decreased.
(3) Air velocity and air conditions keep unchanged,
but thickness of materials increases.
(4) Air velocity and temperature unchanged, and
humidity decreased; or air velocity and humidity
unchanged, and temperature increased.
29