Epstein N., Grace J.R. Spouted and Spout-Fluid Beds: Fundamentals and Applications

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11 Drying of particulate solids

Maria Laura Passos, Esly Ferreira da Costa Jr., and Arun Sadashiv Mujumdar

This chapter focuses on analyzing the design of spouted bed dryers for particulate solids

whose drying curve is characterized by the falling rate period only (Figure 11.1a), as

well as those displaying both constant and falling-rate drying periods (Figure 11.1.b).

Because compromise among costs, efﬁciency, product quality, and a clean environ-

ment is required in the design and operation of any process equipment, simulation and

optimization of the drying process are the best ways to obtain the appropriate dimensions

and operating conditions for a dryer and its ancillary equipment. This chapter starts with

a brief review of recent developments in the drying of particulate solids in spouted beds

(SBs). A concise analysis of various possible dryer design models follows. Three dif-

ferent model levels, which have been used for modeling SB drying of particulate solids,

are considered.

11.1 Various spouted bed dryers for particulate solids

The SB technique, originally developed by Mathur and Gishler

for drying wheat,

has found numerous applications, not only f or drying of particulate solids, but also in

combined operations, such as drying–powdering, drying–granulation, drying–coating,

and drying–extraction. Therefore this chapter also provides information for the design

of such combined operations, which are considered elsewhere in this book.

The conventional spouted bed (CSB), characterized by a conical-cylindrical column

geometry, has its use for drying essentially limited to pilot-scale operation because of

two serious disadvantages, which restrict its application on a large scale:

1. The CSB dryer capacity is limited by the bed depth (H) and the column diameter (D),

as H must be less than H

to ensure stable spouting and D  1 m to avoid dead zones

inside the bed of particles.

2. The CSB dryer efﬁciency is restricted by the technique itself, as the gas ﬂowrate is

limited by the requirements of the spouting regime rather than those of heat and mass

transfer.

These limitations are conﬁrmed by simulated data for drying wheat in optimized CSB

dryers, shown in Figure 11.2. The feed rate of wheat is up to 1000 kg/h for an acceptable

dryer efﬁciency.

Spouted and Spout-Fluid Beds: Fundamentals and Applications, eds. Norman Epstein and John R. Grace.

Published by Cambridge University Press.



Cambridge University Press 2011.

188 Maria Laura Passos, Esly Ferreira da Costa Jr., and Arun Sadashiv Mujumdar

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

Barley in CCSB at 45 °C [1]

Beans in CSB at 85 °C [2]

Soybeans in 2DSB at 150 °C [3]

Groundnut in CSB at 85 °C [4]

Wheat in PBSB at 80 °C [5]

Cork stoppers in CSB at 60 °C [6]

Annatto seeds in CCSB at 72 °C [7]

0.1

0.0

0 20 40 60 80 100

Dryin

time (min)

Moisture content ratio (-)

120 140 160

Figure 11.1a. Spouted bed drying curves for different particulate solids with low initial moisture

content (in falling-rate drying period). Moisture content ratio = Y

(ratio of the solids

moisture content at speciﬁc drying time to initial solids moisture content); temperature

displayed = T

g|in

(inlet air temperature).

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

0204060

Carrot cubes in SFB at 60 °C [8]

Shrimp in JSB at 100 °C [9]

Guava cubes in CSB (two stages) at 60 °C

Fresh yeasts in JSB at 40 °C [11]

80 100

Dryin

time (min)

Moisture content ratio (-)

120 140 160

Figure 11.1b. Spouted bed drying curves for shrinkable solids with high initial moisture content

(in constant-rate and falling-rate drying periods). Moisture content ratio = Y

(ratio of solids

moisture content at speciﬁc drying time to initial solids moisture content); temperature

displayed = T

g|in

(inlet air temperature).

Drying of particulate solids 189

60.0

55.0

50.0

45.0

40.0

35.0

200 400 600 800 1000 1200

Feed rate (k

/h)

Maximum dryer efficiency (%)

= 59.5 °C

= 54 °C

= 44 °C

D/D

= 6.6

D/D

= 7.8

Figure 11.2. Simulated dryer efﬁciency cur ves vs. wheat feed rate as function of particle

temperature, T

, in an optimized CSB unit. Drying conditions: Y

p|in

= 0.25 and Y

p|out

= 0.15;

ambient air temperature = 32

◦

C; relative humidity = 51%. Dashed curves: locus of constant

D/D

with dryer efﬁciency deﬁned as amount of energy required to heat solids and evaporate

water per total energy to heat and pump air (after Passos et al.

To overcome the dryer capacity limitation of CSBs, numerous alternative SB con-

ﬁgurations have been proposed in the literature.

13–17

The most practical and useful

conﬁgurations for drying particulate solids are summarized in Table 11.1.

Mathur and Epstein

were the pioneers in organizing all information on the SB

technique available by the early 1970s. They analyzed the ﬂuid ﬂow and solids motion

in the spout, annulus, and fountain regions, as well as the basic problems associated with

spout stability in the CSB geometry. Using this analysis and updating information from

the literature, Passos et al.

13,16

presented, proposed, and discussed the basic principles

and criteria for design of SB grain dryers, based on the CSB and the two-dimensional

spouted bed (2DSB) conﬁgurations, and the simpliﬁed diffusion drying model proposed

by Becker and Sallans.

Their conclusions show that the maximum dryer capacity and

efﬁciency depend not only on the type of grain and column geometry, but also on the

particle properties such as size, shape, porosity, and density.

Based on the Becker and Sallans

model for drying of wheat in a SB and the Zahed

and Epstein

review of mathematical models, Freitas

simulated the drying of annatto

seeds in a batch conical spouted bed (CcSB) to subsequently help apply this technology to

extraction of bixin powder from these s eeds.

20,21

Kmie

c and Szafran

presented a good

review of CcSB applications, emphasizing the use of an internal draft tube to stabilize the

spouting regime for small particles (d

< 1 mm). These authors combined the constant

drying rate model with the diffusion model to simulate the drying of microspherical

particles (d

= 0.22 mm) produced from the dust emitted by an electrical power station

190 Maria Laura Passos, Esly Ferreira da Costa Jr., and Arun Sadashiv Mujumdar

Table 11.1. Some spouted bed conﬁgurations applied to drying of particulate solids.

Conﬁguration Main characteristics Solids dried (typical cases)

CSB

• Easy to process coarse particles

• High inlet gas temperature without

thermal damage of product

• Low investment cost

• Limitation of the bed depth and the

column diameter (without draft tube)

• Gas ﬂowrate dictated by spouting

• Seeds and grains

1,2,4,12−19,21,34

• Seeds and grains

38,47,49,58

• Shrinkable foodstuffs

10,11,17

• Cork stoppers

• Polyvinyl choride

CcSB

• No limitation of bed depth

• Easy to handle equidensity particles

with wide size distribution

• Low minimum pressure drop

• Gas ﬂowrate dictated by spouting

• Seeds and grains

7,16,20

• Coal microspheres

• Fine sand particles

23,24

2DSB

• Increase in capacity by adding extended

units

• High gas ﬂowrate and low pressure

drop at minimum spouting (ms)

• Draft plates: no limitation of bed depth,

controlled solids motion, decrease in

gas ﬂowrate at ms and in downcomer

• Seeds and grains

3,13,25−30,49

• Spice capsules

• Grain popping

TSB

• Increase in capacity by inserting units

in a hexagonal conﬁguration with a

gas injector at the hexagon center

• Insertion of draft plate is required for

stable spouting

• Seeds and grains

30−32

PBSB

• More stable spouting regime

• Lack of data and information

• Seeds and grains

SFB

SPOUT

ANNULUS

AERATION

• Greater operating ﬂexibility

• High annular gas ﬂowrate

• ms condition dependent on the inlet

annular gas ﬂowrate

• More complex grid design

• Seeds and grains

2,34

• Shrinkable foodstuffs

RJSB

CENTRAL

SPOUT

ROTATING

SPOUT

• Greater operating ﬂexibility

• Different intermittancy of spouting to

attend to drying needs

• Low thermal and mechanical damage

• High costs (nozzle: construction and

maintenance)

• Seeds and grains

41−43

• Heat-sensitive materials

JSB • Similar to CSB or CcSB but with

larger D

• Foodstuffs

9,35−37

• Sawdust

Drying of particulate solids 191

ﬁred by coal. Even with a draft tube (or plates, in the case of 2DSBs), which can improve

spouting s tability and solids circulation, as well as decrease the pressure drop and gas

ﬂowrate at minimum spouting, caution is needed to design such a modiﬁed bed, as

the amount of gas ﬂowing into the annular or downcomer region is always reduced by

such internal elements. Moreover, particle abrasion can be enhanced by impacts on the

walls of the draft tube or plates, causing mechanical damage of fragile particles. On

the other hand, wall–particle collisions can improve powder extraction from particles,

as shown by Cunha et al.

for the case of bixin production. Because of low particle

size segregation in CcSBs,

Altzibar et al.

have proposed insertion of a porous or

open-sided draft tube in a CcSB dryer for sand with a wide size distribution. Their

results conﬁrm that improvement of annulus aeration reduces the drying time for sand

when using an open-sided draft tube instead of a nonporous one.

Two-dimensional spouted beds, as well as triangular spouted beds (TSBs), work better

with draft plates to ensure stable spouting. Because of the severe bed depth limitation

in a 2DSB geometry,

Kalwar et al.

proposed and developed a pilot scale 2DSB

= 0.75 m, L

= 0.06 m, L

= 0.05 m, 30

◦

≤ θ ≤ 180

◦

, and 1.35 m depth) with

adjustable draft plates to dry grains, such as shelled corn, soybeans, and wheat. Their

results conﬁrmed the high quality of dried grains. An empirical model described the

grain-drying kinetics as a function of various geometrical and operating parameters.

Dias et al.

demonstrated that black beans dried in a 2DSB (L

= 0.40 m, L

= L

0.10 m, θ = 60

◦

, and 1 m depth) at 50

◦

C were of high quality without damage in

their physical and cooking properties. Wiriyaumpaiwong et al.

3,27

analyzed the drying

of soybeans in a pilot-scale 2DSB with draft plates (L

= 0.60 m, L

= 0.15 m, L

0.04 m, θ = 60

◦

and 2 m depth) for posterior use in the food industry. They considered

the quality of the dried grain (percentage of cracking and breakage, as well as urease

enzyme activity) and recycled part of the exit air to improve the dryer efﬁciency. They

concluded that the best conditions for drying soybeans and for inactivating the urease

enzyme, thus preserving the protein content, were inlet air temperature not lower than

140

◦

C with 80 percent to 90 percent of outlet air recycled. These conditions ensure

energy consumption of 30 to 50 MJ/kg water evaporated. Earlier works

28,29

had already

conﬁrmed the technical and economic feasibility of the 2DSB for drying of rice paddy

at 150

◦

C, with recycling of 70 percent of the outlet air. In their pilot-scale study, the

volumetric capacity of the 2DSB dryer was easily improved by connecting several units

together, thereby simplifying scaleup.

In Thailand, the capacity of these dryers has

been increased up to 3500 kg/h of grains (soybean, corn, and rice paddy) with energy

consumption comparable to that of other commercial dryers.

The triangular spouted bed (TSB) with draft plate was ﬁrst implemented in Australia

for drying rice paddy from 25 percent to 15 percent wet basis (w.b.) as part of a project

with Thailand, China, and Vietnam.

With f avorable results obtained on the laboratory

scale,

a pilot-scale unit was built to dry 85 kg of rice paddy

= 0.314 m, θ =

◦

, 2.4 m depth, 1.2 m of draft plate inserted at 0.0025 m from the column base). To

avoid mechanical and thermal damage of the grains, this unit was operated using two

successive inlet air temperatures: 150

◦

C during the ﬁrst 2 hours and then 80

◦

C until

the end of drying. The main characteristic of this geometry, as noted in Table 11.1, is

192 Maria Laura Passos, Esly Ferreira da Costa Jr., and Arun Sadashiv Mujumdar

the possibility of increasing the volumetric dryer capacity by inserting similar units in

a novel hexagonal design. The insertion of six units to form a hexagon yields a batch

processing capacity of 510 kg of rice paddy.

Another SB conﬁguration proposed more recently is the parabolic-cylindrical spouted

bed (PBSB),

whose shape mimics the particle trajectory at the base of the annulus

region. Because this conﬁguration stabilizes spouting, reducing the minimum spouting

pressure drop and airﬂow rate, this should ensure an increase in dryer capacity. However,

its impact on drying particulate solids is not as expected intuitively when compared to a

2DSB and a TSB; hence the complexity of constructing a parabolic-cylindrical column

does not appear to be justiﬁed.

The insertion of mechanical stirrers in a bed of cohesive particles can ensure good air–

particle mixing and spout stability.

17,33

Modiﬁcations of the CSB to improve dryer per-

formance for high-moisture-content particles (Figure 11.1b) include combinations of two

or more techniques of ﬂuid–solid contact, such as spout-ﬂuid beds,

2,8,34

ﬁxed-spouted-

ﬂuidized bed combinations,

10,11

and jet-spouted beds.

9,35,36

Passos et al.

demonstrated

by computer simulation the potential for using a spout-ﬂuid bed (SFB) dryer to control

the airﬂow rate required to dry different grains. Lima and Rocha

found that a SFB can

dry black beans; however, a ﬁxed-bed dryer was shown to dry batches of these grains

more efﬁciently.

With the aid of a simulation model for the drying kinetics of carrot cubes in a SFB,

Bia

lobrzewski et al.

demonstrated that SFBs can effectively dry foodstuffs of high

moisture content, maintaining their quality; this is attributed to the additional aeration of

the annular region. Grabowski et al.

showed the advantages of SBs for predrying fresh

yeast particles of high moisture content (70% w.b.). Almeida et al.

proposed different

airﬂow regimes (ﬁxed bed, ﬂuidization, spouting, and slugging) to dry guava cubes, as

this material is very cohesive and shrinks greatly during drying. For drying cohesive

particles of high moisture content, Kudra and Mujumdar

suggested jet-spouted-beds

(JSBs), because their diluted annular region contributes to reducing particle contacts

and, consequently, the potential f or particle agglomeration. This type of SB has an inlet

nozzle diameter, D

, larger than that in CSBs. Wachiraphansakul and Devahastin

used a

JSB (D/D

= 2, D = 0.20 m, θ = 40

◦

, and 0.192 m depth) for drying of okara (a ﬁbrous

and insoluble by-product of soy milk left after the soy milk has been extracted from

the ground soybean pur

ee) from 75 percent to 80 percent to 9 percent w.b. Adsorbing

particles (silica gel) were added to the JSB to improve the drying operation, lowering the

drying temperature and improving the product quality (reduced oxidation level, faster

rehydration, and greater protein solubility). Niamnuy et al.

dried shrimp in a JSB and

developed a model to describe the shrimp temperature, moisture content, internal stress,

and shrinkage during drying to optimize the JSB for product quality.

For preservation of food quality, Feng et al.

proposed to combine, in the same

equipment, microwave and spout-bed techniques for drying diced apples and showed

that these combined techniques can accelerate drying when optimized for quality (color

and texture, in the case of apples). Jumah and Raghavan

modeled the drying kinetics

of wheat in a microwave-spouted bed, assuming that the heat generated by microwaves is

homogeneously distrib uted in the spouted bed of particles, making the dr ying operation

Drying of particulate solids 193

more uniform and favoring its application to dry heat-sensitive solids. Working in a pilot-

scale unit, Berghel

demonstrated the feasibility of drying sawdust in a superheated

steam-spouted bed for producing granulated fuel. The advantages of superheated steam

drying are well known.

Parallel to these modiﬁed SBs, an important innovation that effectively improves t he

SB dryer efﬁciency is i ntermittency of the spouting ﬂow regime. Jumah and Mujumdar

proposed and successfully developed a rotating jet-spouted bed (RJSB), whose inlet ﬂuid

nozzle consisted of two injectors: a central one, which produced a conventional spout in

the central zone of the bed, and a rotating one, which produced a single s pout in the annu-

lus region of the bed as it rotated slowly (Table 11.1). Thus, in RJSBs, the entire annulus

is not spouted continuously. Periodic spouting of the annular region reduces the net con-

sumption of air and power required for spouting. This device thus spouts part of the bed

locally in an inter mittent fashion, providing ﬂexibility to match the drying requirements

of any particulate material. RJSBs can be operated in different modes, such as with two

spouts (one central and another rotating radially at an angular velocity as required by

the drying objectives), with only one central and intermittent spout, or with only one

lateral and rotating spout. Jumah

conﬁrmed the technical and economic feasibility of

this RJSB for drying g rains, with energy savings up to 40 percent and a higher quality

of dried grains (reduction in grain ﬁssures and breakage induced by thermal-mechanical

damage). Jumah

and Jumah and Mujumdar

developed a diffusion-based model to

describe grain drying in a batch RJSB under different operating conditions (continuous

spouting and heating, continuous spouting and sinusoidal heating, intermittent spout-

ing and heating, continuous spouting and intermittent heating, continuous spouting and

sawtooth heating, etc.).

Although the investment and maintenance costs of RJSBs should be higher than

those of CSBs, the results conﬁrm the wide and ﬂexible conditions under which this

type of dryer can work, especially for heat-sensitive solids.

41–42

RJSBs improve the

dryer performance and product quality, reducing the operating cost by saving energy.

Devahastin et al.

simpliﬁed the RJSB design to obtain only an annular periodic

spout for drying of wheat. Devahastin and Mujumdar

demonstrated that a pulsed

spouted bed, operated at a speciﬁc pulse frequency range, could dry heat-sensitive

grains in a manner similar to that of the RJSB, but with lower construction and operating

costs.

The concept of spout intermittency has also been applied in SB dryers. A batch 2DSB

with intermittent spout was used for drying cardamom

(an aromatic Indian spice),

saving 25 percent of thermal energy and improving product quality with respect to color

and texture, as well as oleoresin extraction. A CcSB with inert particles and intermittent

feed of the liquid suspension improved the drying–powdering process when the control

of the inter mittent feed intervals was linked to changes in outlet air temperature.

Two

different types of intermittency were used in a batch CSB dryer of black beans.

One

involved a periodic reduction in the airﬂow rate for a cyclic change in ﬂow regimes

(spouting to a ﬁxed bed and vice versa), and the other periodic injection (on/off) of

airﬂow. The latter mode resulted in the best dryer efﬁciency (more than twice as high as

the efﬁciency achieved in a continuous air injection CSB).

194 Maria Laura Passos, Esly Ferreira da Costa Jr., and Arun Sadashiv Mujumdar

11.2 Dryer design models

Published models for drying particulate solids in a SB can be divided into three levels,

according to the complexity of the model equations. Based on Oakley,

zero-level

models are composed of simple mass and energy balances for gas and solid phases,

at the entrance and exit of the dryer. They result in simple algebraic equations, which

are easy to solve. Knowing the inlet conditions of both phases and the outlet (or ﬁnal)

conditions of the solid phase, the outlet air conditions are immediately determined.

One-level models involve overall mass and energy balances for the gas and solid phases,

at the entrance and exit of the dr yer, as well as the sorption curve for the particulate

solid. Besides the outlet air conditions, these models can also provide quickly the outlet

or ﬁnal solid moisture content in the continuous or batch SB dryer. These two classes of

models are unable to provide data or information for scaleup, design, or control of SB

dryers.

Two-level models, subdivided into types A and B, consider that the gas mixture (air

and water vapor) behaves as a continuous medium, described by the Eulerian approach,

whereas the particulate solids behave as a discrete phase, described by the Lagrangian

approach. Consequently, these models are more complex and require numerical solutions

of the governing equations. The 2A models simplify the gas and particle motion inside

the SB dryer by assuming the statistical mean particle path to be the standard for all

particles and by incorporating all signiﬁcant changes into the main gas ﬂow direction.

Because of these simpliﬁcations, the time for solving their equation system is compatible

with one required for a drying control application. The 2B models are signiﬁcantly more

complex, as they incorporate the CFD package into their solution program for describing

the gas ﬂow in three directions and for tracking large numbers of individual particles.

These models require many hours of advanced computer time for solving the equations.

They can be used to explore a new SB design, and for generating data to validate criteria

for spout stability and scaleup.

The schematic ﬂowchar t in Figure 11.3 represents a generic model used to determine

the best SB dryer dimensions for continuously drying a speciﬁc particulate solid. The

discrete motion of the solid phase is considered in the residence time distribution (RTD)

of the particles, which is incorporated into the drying kinetics model. Souza

developed

a similar model for deﬁning optimal design of CSB and 2DSB units for drying grains.

11.3 Drying model formulation

11.3.1 Mass and energy balances

A generic formulation of the 2A models is based on the assumption that the gas ﬂow and

solid circulation are predominantly longitudinal,

29,50,51

so the annular and spout regions

can be divided into n-inﬁnitesimal segments of depth H/n and unit cross-sectional area.

Mass and energy balances can be written for each segment, considering that the solid

and gas properties vary from one segment to the next, as shown in Table 11.2. These

properties are, in principle, independent of the two other spatial coordinates; their values

Drying of particulate solids 195

INPUT DATA:

1 - LIBRARY: Solids

2 - LIBRARY: air-vapor

4 - SUBROUTINE STABILITY

3 - SUBROUTINE DRYING

5 - SUBROUTINE EQUATIONS

1. Solids processing rate (kg/h)

2. Outlet solid moisture content

3. Input the necessary data in

Drying temperature

(based on product quality)

Selecting T

g in

Determining the bed

holdup required

Possible combinations of

parameters dimensions

for designing dryer

Calculating the dryer

efficiency

data output

Ye s

minimum spouting conditions: air

velocity and pressure drop

for each particulate solid

physical, thermal and diffusion

properties as function of T and Y

sorption curves as function of T

parameters of product quality as

function of T and Y

diffusion model

psychrometric equations

physical, thermal, and diffusion

properties as function of T and Y

criteria for spout stability

recommended range for geometric

column parameters

residence time distribution model

for solids (default - well-mixing)

overall mass and energy balances

(at dryer inlet and outlet)

g in

– T

g out

= 0.5)

Library of Solids

the highest one?

Figure 11.3. Computer program ﬂowchart for optimizing SB dryer based on its efﬁciency and on

the quality of solids (after Souza

in each inﬁnitesimal segment represent averages along these two coordinates. This

approach has been used to model the ﬂuid ﬂow behavior in a CcSB,

the solid-stress

distributioninCSBs,

and bed failure mechanisms in CcSBs.

In the mass and energy balances (Table 11.2), the superﬁcial mass ﬂux of ﬂuid, G, and

the mass ﬂow rate of particles, W

, are expressed in terms of the dry gas and dry solids,

whereas the gas humidity, Y

, and solids moisture content, Y

, are in d.b. (dry basis).

The continuous gas phase, composed of air and water vapor, moves in plug ﬂow in

the SB dryer. Hot air is injected continuously into the bed of particles at a known mass

ﬂow rate, G

s|in

, temperature, T

g|in

, and humidity, Y

g|in

. If the annulus is aerated from

below with a volumetric ﬂow rate, Q

, and at the same temperature and humidity as

the inlet spouting air, then the only change that enters the present analysis is that Q

(z)

at z = 0 becomes Q

instead of zero. The discrete phase, formed by wet particles,

has its properties (ρ

, d

, sphericity, ε

, c

p-p

, λ

) known as a f unction of temperature,

moisture content, and, for the case of sorption Y

curves, the air relative humidity. The

spouting regime is stable (U > U

) and particles move cyclically, entering the spout at

the base of the column and the annulus (or downcomer) at the top of the bed. Bypassing

or short-circuiting of particles is neglected. Therefore, the mass ﬂow rate of particles,

196 Maria Laura Passos, Esly Ferreira da Costa Jr., and Arun Sadashiv Mujumdar

Table 11.2. Mass and energy balance equations in annulus, spout and fountain regions.

ANNULUS REGION

(i) A

(i) Y

g|a

(i) + W

p|a

(i − 1)

p|a

(i) + G

(i − 1) A

(i − 1) Y

g|a

(i − 1)

(i) A

(i) Y

g|it

(11.1)

(i) A

(i)c

p−g|a

(i){T

g|a

(i) − T

}

p−p|a

(i − 1){T

p|a

(i − 1) − T

}+H

×{Y

p|a

(i) − Y

p|a

(i − 1)}

+ W

p−p|a

(i){T

p|a

(i) − T

}

(i − 1) A

(i − 1)c

p−g|a

(i − 1)

×{T

g|a

(i − 1) − T

}

(i) A

(i)c

p−g|it

(i){T

g|it

(i) − T

}−q

lost

(11.2)

SPOUT REGION

(i) A

(i) Y

g|s

(i) + W

p|s

(i)

p|s

(i −1)+G

(i −1) A

(i −1)Y

g|s

(i −1)

−G

(i) A

(i) Y

g|it

(11.3)

(i) A

(i)c

p−g|s

(i){T

g|s

(i) − T

}

p−p|s

(i){T

p|s

(i) − T

}

+H

p|s

(i − 1) − Y

p|s

(i)}

p−p|s

(i − 1){T

p|s

(i − 1) − T

}

(i − 1) A

(i − 1)c

p−g|s

(i − 1)

×{T

g|s

(i − 1) − T

}−G

(i) A

(i)c

p−g|it

(i)

×{T

g|it

(i) − T

}−q

(11.4)

FOUNTAIN REGION

p|s

(N ) − Y

p|a

(N )}=G

s|in

g|out

− Y

g| f

)

(11.5)

s|in

p−g|out

g|out

− T

)

p−p|a

(N ){T

p|a

(N ) − T

}

+H

p|s

(N ) − Y

p|a

(N )}

s|in

p−g| f

g| f

− T

)

p−p|s

(N ){T

p|s

(N ) − T

} (11.6)

with

g| f

(N ) A

(N )Y

g|a

(N )+G

(N ) A

(N )Y

g|s

(N )

s|in

g| f

(N ) A

(N )c

p−g|a

(N ){T

g|a

(N )−T

}+G

(N ) A

(N )c

p−g|s

(N ){T

g|s

(N )−T

}

s|in

p−g| f