KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY
CHEMICAL ENGINEERING DEPARTMENT
CHE 354: SIMULTANEOUS HEAT AND MASS TRANSFER
INSTRUCTOR: Dr. (Mrs.) Mizpah A. D. Rockson
Lecture 6: DRYING II
Learning Objectives
At the end of the lecture the student is expected to
•
The use of psychrometry and psychrometric chart in drying.
•
Explain equilibrium-moisture content of solids.
•
Explain types of moisture content used in making dryer calculations.
•
Describe the four different periods in direct-heat drying.
•
Calculate drying rates for different periods.
•
Apply models for a few common types of dryers.
6.1 PSYCHROMETRY
The capacity of air for moisture removal depends on its humidity and its temperature. The study
of relationships between air and its associated water is called psychrometry. For moisture to be
evaporated from a wet solid, it must be heated to a temperature at which the vapor pressure of
the moisture exceeds the partial pressure of the moisture in the gas in contact with the wet solid.
Calculations involving the properties of moisture-gas mixtures for application to drying are
most conveniently carried out with psychrometric charts, which have been presented in Lecture
4 with definition of terms involved. An example of use of psychrometry is shown in Example
6.1
Example 6.1
At a certain location in a dryer where benzene is being evaporated from a solid, nitrogen gas
at 50°F and 1.2 atm has a relative humidity for benzene of 35%.
Determine:
(a) Partial pressure of the benzene if the vapor pressure of benzene at 50°F = 45.6 torr.
(b) Humidity of the nitrogen-benzene mixture.
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(c) Saturation humidity of the mixture.
(d) Percentage humidity of the mixture.
Solution
𝐻𝑅 = 35%, 𝑃 = 1.2 𝑎𝑡𝑚 = 912 𝑡𝑜𝑟𝑟,
𝐴 = 𝑏𝑒𝑛𝑧𝑒𝑛𝑒,
𝐵 = 𝑁2 ,
𝑀𝐴 = 78.1,
𝑀𝐵 = 28,
𝑃𝐴𝑆 = 45.6 𝑡𝑜𝑟𝑟
(a) From Equation 4.4
𝐻𝑅 = 100
𝑃𝐴
𝐻𝑅 𝑃𝐴𝑆 (35)(45.6)
⇒ 𝑃𝐴 =
=
= 16 𝑡𝑜𝑟𝑟
𝑃𝐴𝑆
100
100
(b) From Equation 4.1
𝐻=
𝑀𝐴 𝑃𝐴
78.1
16
lb benzene
)
=(
= 0.050
𝑀𝐵 (𝑃 − 𝑃𝐴 )
28 (912 − 16)
lb dry nitrogen
(c) From Equation 4.2
𝐻𝑠 =
𝑀𝐴 𝑃𝐴𝑠
78.1
45.6
lb benzene
)
=(
= 0.147
𝑀𝐵 (𝑃 − 𝑃𝐴𝑠 )
28 (912 − 45.6)
lb dry nitrogen
(d) From Equation 4.3
𝐻𝑃 = 100
𝐻
0.050
) = 34%
= 100 (
𝐻𝑆
0.147
6.2 EQUILIBRIUM-MOISTURE CONTENT OF SOLIDS
6.2.1 Forms of Moisture in Wet Solids
The moisture content of a material is usually expressed in terms of its water "liquid" content as
a percentage of the mass of the dry material, though moisture content is sometimes expressed
on a wet basis. If a material is exposed to air at a given temperature and humidity, the material
will either lose water "if the air have lower humidity than that corresponding to the moisture
content of the solid " or gain water "if air has more humid than the solid in equilibrium with it,
the solid absorbs moisture from the air" until an equilibrium condition is established.
Moisture may be present in the following forms:Bound Moisture.
This is the moisture "water" “contained by a substance that it exerts a vapor pressure less than
that of free water at the same temperature. Bound water may be existing in several conditions
2
such as water retained in small capillaries, adsorbed on surfaces, or as a solution in cell walls
and in organic substance in the physical and chemical combination. Bound moisture can also
be defined as moisture that is held chemically as, e.g., water of hydration of inorganic crystals.
This is one example of bound moisture dissolved in the solid.
Unbound Moisture.
This is the moisture "water" contained by a substance which exerts a vapor pressure as high as
that of free water at the same temperature and is largely held in the voids of solid.
Equilibrium Moisture Content X*.
Is the portion of the water in the wet solid which cannot be removed by the inlet air.
Free Moisture X.
This is water which is in excess of the equilibrium moisture content. Where X = XT - X*,
where XT is the total moisture content (See figure 6.1).
6.2.2. Classification of Wet Solids
Wet solids classified into two categories according to their drying behaviour:
I.
Granular or crystalline solids that hold moisture in open pores between particles.
These are mainly inorganic materials, examples of which are crushed rocks, sand,
catalysts, titanium dioxide, zinc sulphate, and sodium phosphates. During drying, the
solid is unaffected by moisture removal, so selection of drying conditions and drying
rate is not critical to the properties and appearance of the dried product. Materials in
this category can be dried rapidly to very low moisture contents.
II.
Fibrous, amorphous, and gel-like materials that dissolve moisture or trap moisture
in fibres or very fine pores. These are mainly organic solids, including tree, plant,
vegetable, and animal materials such as wood, leather, soap, eggs, glues, cereals, starch,
cotton, and wool. These materials are affected by moisture removal, often shrinking
when dried and swelling when wetted. With these materials, drying in the later stages
can be slow. If the surface is dried too rapidly, moisture and temperature gradients can
cause checking, warping, case hardening, and/or cracking. Therefore, selection of
drying conditions is a critical factor. Drying to low moisture contents is possible only
when using a gas of low humidity.
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6.2.3 Equilibrium Moisture Content
In a direct-heat drying process, the extent to which moisture can be removed from a solid is
limited, particularly for the second category, by the equilibrium-moisture content of the solid,
which depends on factors that include temperature, pressure, and moisture content of the gas.
Even if the drying conditions produce a completely dry solid, subsequent exposure of the solid
to a different humidity can result in an increase in moisture content.
A non-porous insoluble solid, such as sand or china clay, has an equilibrium moisture content
approaching zero for all humidities and temperatures, although many organic materials, such
as wood, textiles, and leather, show wide variations of equilibrium moisture content.
Figure 6.1 Typical isotherm for equilibrium moisture content.
A number of terms have been defined to describe and utilize the equilibrium-moisture content.
These are shown in Figure 6.1 with reference to a hypothetical equilibrium isotherm. The
moisture content, X, is expressed as mass of moisture per 100 mass units of bone-dry solid.
This is the most used way to express moisture content of a solid and is equivalent to weight %
moisture on a dry-solid basis. This is analogous to expressions for humidity and is most
4
convenient in drying calculations where the mass of bone-dry solid and bone-dry gas remain
constant while moisture is transferred from the solid to the gas. Less common is weight %
moisture on a wet-solid basis, W. The two moisture contents are related by the expression
𝑋=
100𝑊
100 − 𝑊
[6.1]
100𝑋
100 + 𝑋
[6.2]
or
𝑊=
In Figure 6.1, the equilibrium-moisture content, X* 1, is plotted for a typical solid of the second
category, for a given temperature and pressure against the relative humidity, HR, where the
limit is 100%. In some cases, humidity, H, is used with a limit of the saturation humidity, Hs.
At HR = 100%, the equilibrium-moisture content is called bound moisture, XB. If the wet solid
has a total moisture content, XT, greater than XB, the excess, XT - XB, is called unbound
moisture. At an intermediate relative humidity less than 100%, the excess of XT over the
equilibrium-moisture content, i.e., XT - X*, is called the free-moisture content.
In the presence of a saturated gas, only the unbound moisture can be removed during the drying
process. For a partially saturated gas, only the free moisture can be removed. But if HR = 0, all
solids, given enough time, may be dried to a bone-dry state. Solid materials that can contain
bound moisture are hygroscopic. Equilibrium-moisture isotherms at 25°C and 1 atm are shown
for several materials of the second category in Figure 6.2. Such curves must be determined
experimentally.
Figure 6.2 Equilibrium-moisture content at 25oC and 1 atm.
5
Temperature has a significant effect on the equilibrium moisture content.
Example 6.2
One-kilogram blocks of wet Borax laundry soap with an initial water content of 20.2 wt% on
the dry basis are dried with air in a tunnel dryer at 1 atm. In the limit, if the soap were brought
to equilibrium with the air at 25oC and a relative humidity of 20%, determine the kg of moisture
evaporated from each block.
Solution
The initial moisture content of the soap on a wet basis is obtained from
𝑊=
100𝑋
100(20.2)
=
= 16.8 𝑤𝑡%
100 + 𝑋
100 + 20.2
Initial weight of moisture = 0.168 (1.0) = 0.168 kg H2O
Initial weight of dry soap = 1 - 0.168 = 0.832 kg dry soap
From Figure 6.2, for soap at HR = 0.20, X* = 0.037
Final weight of moisture = 0.037(0.832) = 0.031 kg
Moisture evaporated = 0.168 - 0.031 = 0.137 kg H2O/kg soap block
6.3. RATE OF DRYING
The time required for drying of a moist solid to final moisture content can be determined from
a knowledge of the rate of drying under a given set of conditions The drying rate of a solid is
a function of temperature , humidity , flow rate and transport properties ( in terms of Reynolds
number and Schmidt number) of the drying gas .
In drying, it is necessary to remove free moisture from the surface and also moisture from the
interior of the material. If the change in moisture content for a material is determined as a
function of time (see figure 6.3a) a smooth curve is obtained from which the rate of drying at
any given moisture content may be evaluated. The form of the drying rate curve varies with
the structure and type of material.
If that curve is differentiated with respect to time and multiplied by the ratio of the mass of dry
solid to the interfacial area of contact between the mass of wet solid and the gas, a plot can be
made of drying-rate flux, R, as a function of moisture content, as shown in Figure 6.3b.
𝑅=
𝑑𝑚𝑣
𝐿𝑠 𝑑𝑋
=−
𝐴𝑑𝑡
𝐴 𝑑𝑡
[6.3]
where: 𝑚𝑣 = 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑚𝑜𝑖𝑠𝑡𝑢𝑟𝑒 𝑒𝑣𝑎𝑝𝑜𝑟𝑎𝑡𝑒𝑑,
6
𝐿𝑠 = 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑏𝑜𝑛𝑒 − 𝑑𝑟𝑦 𝑠𝑜𝑙𝑖𝑑
Figure 6.3 Drying curves for constant drying conditions.
In both plots, the final equilibrium-moisture content is X*. Although both plots can exhibit four
drying periods, the periods are more distinct in the drying-rate curve. For some wet materials
and for some hot gas conditions, less than four drying periods may be observed. In the period
from A to B, the wet solid is being preheated to an exposed-surface temperature equal to the
wet-bulb temperature of the gas. Some moisture is evaporated in this preheat period, at an
increasing rate, as the surface temperature increases. At the end of the preheat period, if the
wet solid is of the granular character of the first category, a cross section has the appearance of
Figure 6.4a, where the exposed surface is still covered by a film of moisture. A wet solid of the
second category is covered on the exposed surface by free moisture. The drying rate now
becomes constant during the period from B to C, which prevails as long as free moisture still
covers the exposed surface.
This surface moisture may be part of the original moisture that covered the surface or it may
be moisture brought to the surface by capillary action in the case of wet solids of the first
category or by liquid diffusion in the case of wet solids of the second category. In either case,
the rate of drying is controlled by external mass and heat transfer between the exposed surface
of the wet solid and the bulk gas. The migration of moisture from the interior of the wet solid
to the exposed surface is not a rate-affecting factor. This period is referred to as the constantrate drying period. It terminates at point C, referred to as the critical moisture content. When
drying wet solids of the first category under agitated conditions, as in a direct-heat rotary dryer,
fluidized-bed dryer, flash dryer, or agitated batch dryer, such that all particle surfaces are in
direct contact with the gas, the constant-rate drying period may extend all the way to X*
7
Figure 6.4 Drying stages for granular solids.
At the beginning of the period from C to D, the moisture just barely covers the exposed surface.
From then until point D is reached, as shown in Figure 6.4b, the surface tends to a dry state
because the rate of liquid travel by diffusion or capillary action to the exposed surface is not
sufficiently fast. In this period, the exposed-surface temperature remains at the wet-bulb
temperature if heat conduction is adequate, but the wetted exposed area for mass transfer
decreases. Consequently, the rate of drying decreases linearly with decreasing average moisture
content. This period is referred to as the first falling-rate drying period. It is not always
observed with wet solids of the second category.
During the period from C to D, the liquid in the pores of wet solids of the first category begins
to recede from the exposed surface. In the final period from D to E, as shown in Figure 6.4c,
evaporation occurs from liquid surfaces in the pores, where the wet-bulb temperature prevails.
However, the temperature of the exposed surface in the solid rises to approach the dry-bulb
temperature of the gas. During this period, called the second falling-rate drying period, the
8
rate of drying may be controlled by vapor diffusion for wet solids of the first category and by
liquid diffusion for wet solids of the second category. The rate falls exponentially with
decreasing moisture content.
6.4 DRYING TIME
6.4.1 Constant Rate Drying Period
In the constant rate period of drying, the surface of grains of solid in contact with drying air
flow remain completely wetted. Drying of material occurs by mass transfer vapor from the
saturated surface of the material through the air film to the bulk gas phase. The rate of moisture
movement within the solid is sufficient to keep the surface saturated. The rate removal of water
vapor (drying) is controlled by the rate of heat transfer to the evaporating surface which
furnishes the latent heat of evaporation for the liquid. At steady state the rate of mass transfer
balances the rate of heat transfer.
To estimate the time of drying for a given batch of material, based on actual experimental data.
The time required is determined directly from the drying curve of free moisture content vs.
time.
Rearranging Equation 6.3 and integrating over the time interval to dry from X1 at t1=0 to X2 at
t2=t gives
𝑡2 =𝑡
𝑋2
𝐿𝑠
𝑑𝑋
𝑡 = ∫ 𝑑𝑡 = − ∫
𝐴
𝑅
𝑡1 =0
[6.4]
𝑋1
When drying takes place within the constant rate period, R = constant = RC.
𝑡=
𝐿𝑠
(𝑋 − 𝑋2 )
𝐴𝑅𝐶 1
[6.5]
9
Assuming
a) only heat transfer to the solid surface by convection from the hot gas to the surface of
the solid and mass transfer from the surface to the hot gas.
b) no heat transfer by conduction from metal pans or surface
c) heat transfer by radiation is neglected.
Note: Refer to Geankoplis (2003) page 548 for derivation of the rate of drying during the
constant rate period that considers convection, conduction, and radiation.
The rate of heat transfer by convection (q) from the gas at T to the surface of the solid at Tw is:
𝑞 = ℎ𝐴(𝑇 − 𝑇𝑤 )
[6.6]
The flux of water vapour from the surface is:
𝑁𝐴 = 𝑘𝑦 (𝑦𝑤 − 𝑦)
𝑁𝐴 = 𝑘𝑦
[6.7]
𝑀𝐵
(𝐻 − 𝐻)
𝑀𝐴 𝑤
[6.8]
𝐻⁄
𝑀𝐴
𝑦=
, 𝐻⁄𝑀 𝑖𝑠 𝑛𝑒𝑔𝑙𝑒𝑐𝑡𝑒𝑑 𝑑𝑢𝑒 𝑡𝑜 𝐻 𝑏𝑒𝑖𝑛𝑔 𝑠𝑜 𝑠𝑚𝑎𝑙𝑙
1⁄ + 𝐻⁄
𝐴
𝑀𝐵
𝑀𝐴
𝑦=
𝑀𝐵 𝐻
𝑀𝐵 𝐻𝑤
⇉ 𝑦𝑤 =
𝑀𝐴
𝑀𝐴
The amount of heat needed to vaporize NA kmol/s·m2 (neglecting sensible heat changes) water
is
𝑞 = 𝑀𝐴 𝑁𝐴 𝜆𝑤 𝐴
[6.9]
Where 𝜆𝑤 is the latent heat of vaporization at Tw in J/kg. Equating equation [6.6] and [6.9] and
substituting into [6.8]:
𝑅𝐶 =
𝑞
ℎ(𝑇 − 𝑇𝑤 )
=
= 𝑘𝑦 𝑀𝐵 (𝐻𝑤 − 𝐻)
𝐴𝜆𝑤
𝜆𝑤
Therefore equation [6.5] becomes:
10
[6.10]
𝑡=
𝐿𝑠 𝜆𝑤
(𝑋 − 𝑋2 )
𝐴ℎ(𝑇 − 𝑇𝑤 ) 1
𝑡=
[6.11]
𝐿𝑠
(𝑋 − 𝑋2 )
𝐴𝑘𝑦 𝑀𝐵 (𝐻𝑤 − 𝐻) 1
[6.12]
To predict RC, the heat transfer coefficient h must be known.
a) if air is flowing parallel to the drying surface:h = 0.0204 G0.8
h = 0.0128 G0.8
SI unit
English unit
at Tair = 45-150oC, G=2450-2950 kg/m2·h
or velocity of 0.61-7.6 m/s
b) if air is flowing perpendicular to the drying surface :h = 1.17 G0.37 SI unit
h = 0.37 G0.37 English unit
G=3900-19500 kg/m2·h or velocity of 0.9-4.6 m/s
Where gas mass velocity G=ρair.νair
On the basis of the above equations, effects of some important parameters on the rate of drying
can be determined as:Effect of gas velocity: If conduction through the solid and radiation are neglected, the drying
rate becomes proportional to G0.8 for parallel flow and to G0.37 for perpendicular flow. If
conduction and radiation are present, effect of air rate will be less important.
Effect of gas temperature: Increased air temperature increases the quantity (𝑇−𝑇𝑤) increases
drying rate. Neglecting radiation and, RC becomes directly proportional to (𝑇−𝑇𝑤).
Effect of gas humidity: At moderate temperatures the rate of drying varies directly as yw - y
hence increasing gas humidity reduces drying rate.
Effect of thickness of drying solid: If conduction through solid occurs, RC decreases as solid
thickness increases, but rate of drying may sometimes increase due to higher conduction of
heat through the edges.
6.4.2 Falling Rate Drying Period
During drying, when moisture travels from the interior of a wet solid to the surface, a moisture
profile develops in the wet solid. The shape of this profile depends on the nature of the moisture
movement. If the wet solid is of the first category, where the moisture is not held in solution or
11
in fibres, but is held as free moisture in the interstices of powders and granular solids such as
paint pigments, minerals, clays, soil, and sand, or is moisture above the fibre-saturation point
in textiles, paper, wood, and leather, then moisture movement occurs by capillary action. For
wet solids of the second category, the internal moisture is bound moisture, as in the last stages
of the drying of clay, starch, flour, textiles, paper and wood, or soluble moisture, as in soap,
glue, gelatine, and paste. This type of moisture migrates to the surface by liquid diffusion.
Moisture can also migrate by gravity, external pressure, and by vaporization-condensation
sequences in the presence of a temperature gradient. In addition, vapor diffusion through the
solid can occur in indirect-heat dryers when heating and vaporization occur at opposed
surfaces.
A typical moisture profile for capillary flow is shown in Figure 6.5a. The profile is concave
upward near the exposed surface, concave downward near the opposed surface, and with a
point of inflection in between. For flow of moisture by diffusion, as shown in Figure 6.5b, the
profile is concave downward throughout. If the diffusivity is independent of the moisture
content, the solid curve applies. If, as is often the case, the diffusivity decreases with decreasing
moisture content, due mainly to shrinkage, the dashed profile applies.
Figure 6.5 Moisture distribution in wet solids during drying
During the falling-rate period of drying, idealized theories for capillary flow and diffusion can
be applied to estimate drying rates. Alternatively, the estimate could be made by a strictly
empirical approach that ignores the mechanism of moisture movement, but instead relies on
12
the experimental determination of drying rate as a function of average moisture content for a
particular set of drying conditions. The empirical approach is examined first.
6.4.2.1 Graphical Integration Method.
In falling rate period, the rate of drying R is not constant but decreases when drying proceeds
past the critical free moisture content Xc. The time for drying for any region between X1 and
X2 is given by: 𝑋2
𝐿𝑠
𝑑𝑋
𝑡=− ∫
𝐴
𝑅
[6.13]
𝑋1
In falling rate period, R varies for any shape of falling rate drying curve, equation [6.13] can
be integrated graphically by plotting 1/R vs. X and determining the area under the curve.
Calculation Method for special cases in Falling Rate Period.
1. Rate is a linear function of X.
Both X1 and X2 < Xc and the rate R is linear in X over this region.
R=aX+b
[6.14]
Where a is a slope of the line and b is a constant. Differentiating equation [6.14] gives 𝑑𝑅 =
𝑎 𝑑𝑋. Substituting this into Eq. [6.13],
𝑅1
𝐿𝑠
𝑑𝑅 𝐿𝑠 𝑅1
𝑡=
∫
=
𝑙𝑛
𝑎𝐴
𝑅
𝑎𝐴 𝑅2
[6.15]
𝑅2
Since 𝑅1 = 𝑎𝑋1 + 𝑏 and 𝑅2 = 𝑎𝑋2 + 𝑏
𝑎=
𝑅1 − 𝑅2
𝑋1 − 𝑋2
[6.16]
Substituting Eq. [6.16] into [6.15]
𝑡=
𝐿𝑠 (𝑋1 − 𝑋2 ) 𝑅1
𝑙𝑛
𝐴(𝑅1 − 𝑅2 ) 𝑅2
13
[6.17]
2. Rate is a linear function through origin.
In some cases, a straight line from the critical moisture content passing through the origin
adequately represents the whole falling rate period. Then the rate of drying is directly
proportional to the free moisture content.
R=aX
[6.18]
Differentiating, 𝑑𝑋 = 𝑑𝑅 ⁄𝑎. Substituting into Eq. [6.13]
𝑅1
𝐿𝑠
𝑑𝑅 𝐿𝑠 𝑅1
𝑡=
∫
=
𝑙𝑛
𝑎𝐴
𝑅
𝑎𝐴 𝑅2
[6.19]
𝑅2
The slope a of the line is 𝑅𝑐 ⁄𝑋𝑐 , and for 𝑋1 = 𝑋𝑐 𝑎𝑡 𝑅1 = 𝑅𝑐 ,
𝑡=
𝐿𝑠 𝑋𝑐 𝑅𝑐
𝑙𝑛
𝐴𝑅𝑐 𝑅2
[6.20]
Noting also that 𝑅𝑐 ⁄𝑅2 = 𝑋𝑐 ⁄𝑋2 ,
𝑡=
𝐿𝑠 𝑋𝑐 𝑋𝑐
𝑙𝑛
𝐴𝑅𝑐 𝑋2
[6.21]
Or
𝑅 = 𝑅𝑐
𝑋
𝑋𝑐
[6.22]
6.4.2.2 Drying Time in Falling Period According Liquid Diffusion Theory.
When liquid diffusion of moisture controls the rate of drying in the falling rate period. The
equation for diffusion used Fick’s second law for unsteady state diffusion using the
concentrations as X kg free moisture/kg dry solid instead of concentrations kmol moisture /m3,
𝜕𝑋
𝜕 2𝑋
= 𝐷𝐿 2
𝜕𝑡
𝜕𝑥
[6.23]
Where DL is liquid diffusion coefficient in m2/h and x is distance in the solid in m.
This type of diffusion is often characteristic of relatively slow drying in non-granular materials
such as soap, gelatine, and glue and in the later stages of drying of bound water in clay, wood,
textiles, leather, paper, food, starches, and other hydrophilic solids.
14
During diffusion type drying, the resistance to mass transfer of water vapour from the surface
is usually very small and the diffusion in the solid controls the rate of drying. Then the moisture
content at the surface is at the equilibrium value X*. This means that the free moisture content
X at the surface is zero.
Assuming that initial moisture distribution is uniform at 𝑡 = 0, equation [6.23] may be
integrated to give:𝑋𝑡 − 𝑋 ∗
𝑋
8 −𝐷 𝑡(𝜋⁄2𝑥 )2 1 −9𝐷 𝑡(𝜋⁄2𝑥 )2
1 −25𝐷 𝑡(𝜋⁄2𝑥 )2
𝐿
1
𝐿
1
𝐿
1
=
=
[𝑒
+
𝑒
+
𝑒
+⋯]
𝑋𝑡1 − 𝑋 ∗ 𝑋1 𝜋 2
9
25
[6.24]
Where
𝑋 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑓𝑟𝑒𝑒 𝑚𝑜𝑖𝑠𝑡𝑢𝑟𝑒 𝑐𝑜𝑛𝑡𝑒𝑛𝑡 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡.
𝑋1 = 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑓𝑟𝑒𝑒 𝑚𝑜𝑖𝑠𝑡𝑢𝑟𝑒 𝑐𝑜𝑛𝑡𝑒𝑛𝑡 𝑎𝑡 𝑡𝑖𝑚𝑒 = 0.
X* = equilibrium free moisture content.
x1 = 0.5 thickness of the slab when drying occurs from the top and the bottom parallel faces,
and x1= total thickness of slab if drying only from the top face.
Equation [6.24] assumes that DL is constant, but DL is rarely constant; it varies with moisture
content, temperature, and humidity. For long drying times, only the first term in the equation
[6.24] is significant, and the equation becomes
𝑋
8
2
= 2 𝑒 −𝐷𝐿𝑡(𝜋⁄2𝑥1)
𝑋1 𝜋
[6.25]
Solving for the time of drying,
𝑡=
4𝑥12
8𝑋1
𝑙𝑛 2
2
𝜋 𝐷𝐿 𝜋 𝑋
[6.26]
In this equation if diffusion mechanism starts at 𝑋 = 𝑋𝑐 , then 𝑋1 = 𝑋𝑐 . Differentiating
Equation [6.26] with respect to time and rearranging,
𝑑𝑋
𝜋 2 𝐷𝐿 𝑋
=−
𝑑𝑡
4𝑥12
[6.27]
Multiplying both sides by −𝐿𝑠 /𝐴,
𝑅=−
𝐿𝑠 𝑑𝑋 𝜋 2 𝐿𝑠 𝐷𝐿
=
𝑋
𝐴 𝑑𝑡
4𝑥12 𝐴
15
[6.28]
Equation [6.27 and 6.28] state that when internal diffusion controls for long times, the rate of
drying is directly proportional to the free moisture X and the liquid diffusivity and that the rate
of drying is inversely proportional to the thickness squared.
Or, stated at the time of drying between fixed moisture limits, the time varies directly as the
square of the thickness. The drying rate should be independent of gas velocity and humidity.
6.4.2.3 Drying Time in Falling Period According Capillary Theory.
Water can flow from region of high concentrations to those of low concentrations as a result
of capillary action rather than by diffusion if the pore size of granular materials are suitable.
The capillary theory assumes that a packed bed of nonporous spheres contains a void space
between the spheres called pores. As water is evaporated, capillary forces are set up by the
interfacial tension between the water and solid. These forces provide the driving force for
moving the water through the pores to the drying surface. If the moisture movement follows
the capillary theory rate of drying R will vary linearly with X. Since the mechanism of
evaporation during this period is the same as in the constant-rate period, the effects of the
variables of the drying gas of gas velocity, temperature of the gas, humidity of the gas and so
on will be the same for the constant-rate drying period.
The defining equation for the rate of drying is,
𝑅=−
𝐿𝑠 𝑑𝑋
𝐴 𝑑𝑡
[6.3]
For the rate R varying linearly with X given previously,
𝑡=
𝐿𝑠 𝑋𝑐 𝑋𝑐
𝑙𝑛
𝐴𝑅𝑐 𝑋2
[6.21]
Or
𝑅 = 𝑅𝑐
𝑋
𝑋𝑐
[6.22]
We define t as the time when 𝑋 = 𝑋2 and
𝐿𝑠 = 𝑥1 𝐴𝜌𝑠
[6.29]
16
Where 𝜌𝑠 = 𝑠𝑜𝑙𝑖𝑑 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑘𝑔 𝑑𝑟𝑦𝑠𝑜𝑙𝑖𝑑/𝑚3 . Substituting Equation [6.29] and 𝑋 = 𝑋2 into
Equation [6.21],
𝑡=
𝑥1 𝜌𝑠 𝑋𝑐 𝑋𝑐
𝑙𝑛
𝑅𝑐
𝑋
[6.30]
Substituting Equation [6.10] for Rc,
𝑡=
𝑥1 𝜌𝑠 𝜆𝑤 𝑋𝑐 𝑋𝑐
𝑙𝑛
ℎ(𝑇 − 𝑇𝑤 ) 𝑋
[6.31]
Hence Equations [6.30] and [6.31] state that when capillary flow controls in the falling-rate
period, the rate of drying is inversely proportional to the thickness. The time of drying between
fixed moisture limits varies directly as the thickness and depends upon the gas velocity,
temperature, and humidity.
Note: Mathematical models for estimating drying rates and moisture profiles have been
developed with applications different types of dryers. These can be found in the reference texts
and other books on drying when needed.
6.5 Material and Heat Balance for Continuous Dryers.
Figure 6.6 below shows a flow diagram for a continuous type dryer where the drying gas flows
counter currently to the solids flow.
Figure 6.6 General configuration for a continuous, direct-heat dryer
The solid enters at a rate of Ls kg dry solid/h, having a free moisture content X1 and a
temperature Ts1. It leaves at X2 and Ts2. The gas enters at a rate G kg dry air/h, having a humidity
H2 kg water vapor/ kg dry air and a temperature of TG2. The gas leaves at TG1 and H1.
For material balance on the moisture,
𝐺𝐻2 + 𝐿𝑠 𝑋1 = 𝐺𝐻1 + 𝐿𝑠 𝑋2
17
[6.32]
For heat balance a datum of To oC is selected. A convenient temperature is 0 oC (32oF). The
enthalpy of wet solid is composed of the enthalpy of the dry solid plus that of the liquid as free
moisture. The heat of wetting is usually neglected.
The enthalpy of gas 𝐻𝐺′ in kJ/kg dry air is
𝐻𝐺′ = 𝑐𝑆 (𝑇𝐺 − 𝑇𝑜 ) + 𝐻𝜆𝑜
[6.33]
where
λo is the latent heat of vaporization of water at To oC, 2501 kJ/kg at 0 oC.
𝑐𝑆 is the humid heat, given as kJ/kg dry air·K,
𝑐𝑠 = 1.005 + 1.88𝐻
[4.5]
The enthalpy of the wet solid 𝐻𝑠′ in kJ/kg dry solid, where (𝑇𝑠 − 𝑇𝑜 )℃ = (𝑇𝑠 − 𝑇𝑜 )𝐾, is
𝐻𝑠′ = 𝑐𝑝𝑠 (𝑇𝑠 − 𝑇𝑜 ) + 𝑋𝑐𝑝𝐴 (𝑇𝑠 − 𝑇𝑜 )
[6.34]
where
𝑐𝑝𝑠 𝑖𝑠 𝑡ℎ𝑒 ℎ𝑒𝑎𝑡 𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑟𝑦 𝑠𝑜𝑙𝑖𝑑 𝑖𝑛 𝑘𝐽⁄𝑘𝑔 𝑑𝑟𝑦 𝑠𝑜𝑙𝑖𝑑 ∙ 𝐾 𝑎𝑛𝑑
𝑐𝑝𝐴 𝑖𝑠 𝑡ℎ𝑒 ℎ𝑒𝑎𝑡 𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦 𝑜𝑓 𝑙𝑖𝑞𝑢𝑖𝑑 𝑚𝑜𝑖𝑠𝑡𝑢𝑟𝑒 𝑖𝑛 𝑘𝐽⁄𝑘𝑔 𝐻2 𝑂 ∙ 𝐾
The heat of wetting or adsorption is neglected.
A heat balance on the dryer is
′
′
′
′
𝐺𝐻𝐺2
+ 𝐿𝑠 𝐻𝑠1
= 𝐺𝐻𝐺1
+ 𝐿𝑠 𝐻𝑠2
+𝑄
[6.35]
where Q is the heat loss in the dryer in kJ/h. For an adiabatic process 𝑄 = 0, and if heat is
added, Q is negative.
Air recirculation in dryers
In many dryers it is desired to control the wet bulb temperature at which the drying of the solid
occurs. Also, since steam costs are often important in heating the drying air, recirculation of
the drying air is sometimes used to reduce costs and control humidity. Part of the moist air
leaving the dryer is recirculated and combined with fresh air. This is shown in Figure 6.7. Fresh
air having a temperature TG1 and H1 is mixed with air at TG2 and H2 to give air at TG3 and H3.
This mixture is heated to TG4 with H4= H3. After drying, the air leaves at a lower temperature
TG2 and a higher humidity H2.
18
Figure 6.7 Process flow air recirculation in drying.
The following material balances on the water can be made. For water balance on the heater,
noting that 𝐻6 = 𝐻5 = 𝐻2 ,
𝐺1 𝐻1 + 𝐺6 𝐻2 = (𝐺1 + 𝐺6 )𝐻4
[6.36]
Making a water balance on the dryer,
(𝐺1 + 𝐺6 )𝐻4 + 𝐿𝑠 𝑋1 = (𝐺1 + 𝐺6 )𝐻2 + 𝐿𝑠 𝑋2
[6.37]
In a similar manner heat balances can be made on the heater and dryer and on the overall
system.
6.6 Rate of Drying for Continuous Direct Heat Driers.
Drying continuously offers a number of advantages over batch drying. Smaller sizes of
equipment can often be used, and the product has more uniform moisture content. In a
continuous dryer the solid is moved through the dryer while in contact with a moving gas
stream that may flow parallel or counter-current to the solid.
In Figure 6.8 typical temperature profiles of the gas TG and the solid TS are shown for a
continuous counter-current dryer. In the preheat zone, the solid is heated up to the wet bulb or
adiabatic saturation temperature. Little evaporation occurs here, and for low temperature drying
this zone is usually ignored. In the constant-rate zone, I, unbound and surface moisture are
evaporated and the temperature of the solid remains essentially constant at the adiabatic
saturation temperature if heat is transferred by convection. The rate of drying would be constant
here, but the gas temperature is changing and also the humidity. The moisture content falls to
the critical value Xc at the end of this period.
19
Figure 6.8 Temperature profiles for a continuous counter-current dryer
In zone II, unsaturated surface and bound moisture are evaporated and the solid is dried to its
final value X2. The humidity of the entering gas entering zone II is H2 and it rises to Hc. The
material balance equation [6.32] may be used to calculate Hc as follows.
𝐿𝑠 (𝑋𝑐 − 𝑋2 ) = 𝐺(𝐻𝑐 + 𝐻2 )
[6.38]
where 𝐿𝑠 is kg dry solid/h and G is kg dry gas/h.
Equation for constant-rate period.
The rate of drying in the constant-rate region in zone I would be constant if it were not for
varying gas conditions. The rate of drying in this section is given by an equation similar to
equation [6.10]
𝑅 = 𝑘𝑦 𝑀𝐵 (𝐻𝑤 − 𝐻) =
ℎ(𝑇𝐺 − 𝑇𝑤 )
𝜆𝑤
[6.39]
The time for drying is given by 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 [6.4] using the limits X1 and Xc
𝑋1
𝐿𝑠
𝑑𝑋
𝑡=
∫
𝐴
𝑅
[6.40]
𝑋𝑐
where A/LS is the exposed drying surface m2/kg dry solid. Substituting Eq. [6.39] into [6.40]
and (𝐺/𝐿𝑆 )𝑑𝐻 for dX,
𝐻1
𝐺 𝐿𝑠
1
𝑑𝐻
𝑡= ( )
∫
𝐿𝑠 𝐴 𝑘𝑦 𝑀𝐵
𝐻𝑤 − 𝐻
[6.41]
𝐻𝑐
where G = kg dry air/h, 𝐿𝑆 = 𝑘𝑔 𝑑𝑟𝑦 𝑠𝑜𝑙𝑖𝑑 ⁄ℎ , 𝑎𝑛𝑑 A/LS = m2/kg dry solid. This can be
integrated graphically.
For the case where 𝑇𝑤 or 𝐻𝑤 is constant for adiabatic drying, Eq. [6.34] can be integrated.
20
𝑡=
𝐺 𝐿𝑠
1
𝐻𝑤 − 𝐻𝑐
( )
𝑙𝑛
𝐿𝑠 𝐴 𝑘𝑦 𝑀𝐵 𝐻𝑤 − 𝐻1
[6.42]
The above can be modified by use of log mean humidity difference.
∆𝐻𝐿𝑀 =
(𝐻𝑤 − 𝐻𝑐 ) − (𝐻𝑤 − 𝐻1 )
𝐻1 − 𝐻𝐶
=
ln[(𝐻𝑤 − 𝐻𝑐 ) /(𝐻𝑤 − 𝐻1 )] 𝑙𝑛[(𝐻𝑤 − 𝐻𝑐 ) /(𝐻𝑤 − 𝐻1 )]
[6.43]
Substituting Eq. [6.43] into [6.42], an alternative equation is obtained.
𝑡=
𝐺 𝐿𝑠
1 𝐻1 − 𝐻𝐶
( )
𝐿𝑠 𝐴 𝑘𝑦 𝑀𝐵 ∆𝐻𝐿𝑀
[6.44]
From Eq. [6.38], 𝐻𝐶 can be calculated as follows,
𝐻𝐶 = 𝐻2 +
𝐿𝑠
(𝑋 − 𝑋2 )
𝐺 𝑐
[6.45]
Equation for falling-rate period.
For the situation where unsaturated surface drying occurs, 𝐻𝑤 is constant for adiabatic drying,
the rate of drying is directly dependent upon X as in Eq. [6.22] and Eq. [6.39] applies.
𝑅 = 𝑅𝑐
𝑋
𝑋
= 𝑘𝑦 𝑀𝐵 (𝐻𝑤 − 𝐻)
𝑋𝑐
𝑋𝑐
[6.46]
Substituting Eq. [6.46] into [6.4]
𝑋𝑐
𝐿𝑠 𝑋𝑐
𝑑𝑋
𝑡=( )
∫
(𝐻𝑤 − 𝐻)𝑋
𝐴 𝑘𝑦 𝑀𝐵
[6.47]
𝑋2
Substituting G dH/LS for dX and (𝐻 − 𝐻2 )𝐺 ⁄𝐿𝑆 + 𝑋2 𝑓𝑜𝑟 𝑋,
𝐻𝑐
𝐺 𝐿𝑠 𝑋𝑐
𝑑𝐻
𝑡= ( )
∫
(𝐻𝑤 − 𝐻)[(𝐻 − 𝐻2 )𝐺 ⁄𝐿𝑆 + 𝑋2 ]
𝐿𝑠 𝐴 𝑘𝑦 𝑀𝐵
[6.48]
𝐻2
𝑡=
𝐺 𝐿𝑠 𝑋𝑐
1
𝑋𝑐 (𝐻𝑤 − 𝐻2 )
( )
𝑙𝑛
𝐿𝑠 𝐴 𝑘𝑦 𝑀𝐵 (𝐻𝑤 − 𝐻2 )𝐺 ⁄𝐿𝑆 + 𝑋2 𝑋2 (𝐻𝑤 − 𝐻𝑐 )
[6.49]
Again, to calculate 𝐻𝑐 , Eq. [6.45] can be used.
These equations for the two periods can also be derived using the last part of Eq. [6.39] and
temperature instead of humidities.
21