Basic process calculations and simulations in drying

Design—a selection of a suitable dryer type and size for a given product to optimize the capital and operating costs within the range of limits imposed—this case is often termed pro-cess

3 Basic Process Calculations and Simulations in Drying

Zdzisław Pakowski and Arun S Mujumdar

CONTENTS

3.1 Introduction 54

3.2 Objectives 54

3.3 Basic Classes of Models and Generic Dryer Types 54

3.4 General Rules for a Dryer Model Formulation 55

3.4.1 Mass and Energy Balances 56

3.4.1.1 Mass Balances 56

3.4.1.2 Energy balances 56

3.4.2 Constitutive Equations 57

3.4.2.1 Characteristic Drying Curve 58

3.4.2.2 Kinetic Equation (e.g., Thin-Layer Equations) 58

3.4.3 Auxiliary Relationships 59

3.4.3.1 Humid Gas Properties and Psychrometric Calculations 59

3.4.3.2 Relations between Absolute Humidity, Relative Humidity, Temperature, and Enthalpy of Humid Gas 60

3.4.3.3 Calculations Involving Dew-Point Temperature, Adiabatic-Saturation Temperature, and Wet-Bulb Temperature 60

3.4.3.4 Construction of Psychrometric Charts 61

3.4.3.5 Wet Solid Properties 61

3.4.4 Property Databases 62

3.5 General Remarks on Solving Models 62

3.6 Basic Models of Dryers in Steady State 62

3.6.1 Input–Output Models 62

3.6.2 Distributed Parameter Models 63

3.6.2.1 Cocurrent Flow 63

3.6.2.2 Countercurrent Flow 64

3.6.2.3 Cross-Flow 65

3.7 Distributed Parameter Models for the Solid 68

3.7.1 One-Dimensional Models 68

3.7.1.1 Nonshrinking Solids 68

3.7.1.2 Shrinking Solids 69

3.7.2 Two- and Three-Dimensional Models 70

3.7.3 Simultaneous Solving DPM of Solids and Gas Phase 71

3.8 Models for Batch Dryers 71

3.8.1 Batch-Drying Oven 71

3.8.2 Batch Fluid Bed Drying 73

3.8.3 Deep Bed Drying 74

3.9 Models for Semicontinuous Dryers 74

3.10 Shortcut Methods for Dryer Calculation 76

3.10.1 Drying Rate from Predicted Kinetics 76

3.10.1.1 Free Moisture 76

3.10.1.2 Bound Moisture 76

3.10.2 Drying Rate from Experimental Kinetics 76

3.10.2.1 Batch Drying 77

3.10.2.2 Continuous Drying 77

3.11 Software Tools for Dryer Calculations 77

3.12 Conclusion 78

Nomenclature 78

References 79

3.1 INTRODUCTION

Since the publication of the first and second editions of

this handbook, we have been witnessing a revolution in

methods of engineering calculations Computer tools

have become easily available and have replaced the old

graphical methods An entirely new discipline of

com-puter-aided process design (CAPD) has emerged

Today even simple problems are solved using

dedi-cated computer software The same is not necessarily

true for drying calculations; dedicated software for this

process is still scarce However, general computing

tools including Excel, Mathcad, MATLAB, and

Mathematica are easily available in any engineering

company Bearing this in mind, we have decided to

present here a more computer-oriented calculation

methodology and simulation methods than to rely on

old graphical and shortcut methods This does not

mean that the computer will relieve one from thinking

In this respect, the old simple methods and rules of

thumb are still valid and provide a simple

common-sense tool for verifying computer-generated results

3.2 OBJECTIVES

Before going into details of process calculations we

need to determine when such calculations are

neces-sary in industrial practice The following typical cases

can be distinguished:

. Design—(a) selection of a suitable dryer type

and size for a given product to optimize the

capital and operating costs within the range of

limits imposed—this case is often termed

pro-cess synthesis in CAPD; (b) specification of all

process parameters and dimensioning of a

selected dryer type so the set of design

param-eters or assumptions is fulfilled—this is the

com-mon design problem

. Simulation—for a given dryer, calculation of

dryer performance including all inputs and

out-puts, internal distributions, and their time

de-pendence

. Optimization—in design and simulation an

op-timum for the specified set of parameters is

sought The objective function can be

formu-lated in terms of economic, quality, or other factors, and restrictions may be imposed on ranges of parameters allowed

. Process control—for a given dryer and a speci-fied vector of input and control parameters the output parameters at a given instance are sought This is a special case when not only the accuracy of the obtained results but the required computation time is equally important Al-though drying is not always a rapid process, in general for real-time control, calculations need

to provide an answer almost instantly This usu-ally requires a dedicated set of computational tools like neural network models

In all of the above methods we need a model of the process as the core of our computational problem A model is a set of equations connecting all process parameters and a set of constraints in the form of inequalities describing adequately the behavior of the system When all process parameters are determined with a probability equal to 1 we have a deterministic model, otherwise the model is a stochastic one

In the following sections we show how to construct

a suitable model of the process and how to solve it for a given case We will show only deterministic models of convective drying Models beyond this range are im-portant but relatively less frequent in practice

In our analysis we will consider each phase as a continuum unless stated otherwise In fact, elaborate models exist describing aerodynamics of flow of gas and granular solid mixture where phases are considered noncontinuous (e.g., bubbling bed model of fluid bed, two-phase model for pneumatic conveying, etc.)

3.3 BASIC CLASSES OF MODELS AND GENERIC DRYER TYPES Two classes of processes are encountered in practice: steady state and unsteady state (batch) The differ-ence can easily be seen in the form of general balance equation of a given entity for a specific volume of space (e.g., the dryer or a single phase contained in it):

Inputs outputs ¼ accumulation (3:1)

For inst ance, for mass flow of mois ture in a solid

phase being dried (in kg/s) this equati on reads:

WS X1  W S X2  w D A ¼ m S

dX

dt (3 : 2)

In steady -state proc esses, as in all continuou sly ope

r-ated dryers , the accumul ation term van ishes and the

balance e quation assum es the form of an algebr aic

equati on W hen the pro cess is of batch type or when a

continuous process is be ing started up or shut down,

the accumul ation term is nonzero an d the balance

equati on becomes an ord inary different ial equ ation

(ODE) with respect to time

In writing Equation 3.1, we have assum ed that

only the inpu t an d output pa rameters coun t Inde ed,

when the volume unde r con siderati on is perfec tly

mixed , all pha ses inside this volume will have the

same pr operty as that at the output This is the

prin-ciple of a lumped pa rameter model (LPM)

If a pr operty varies c ontinuousl y along the flow

direction (in one dimens ion for sim plicity), the

bal-ance equatio n can only be writt en for a different ial

space elemen t Here Equat ion 3.2 will now read

As we can see for this case, which we call a dist ributed

parame ter mod el (DPM) , in steady state (in the one

-dimens ional case) the model beco mes an ODE wi th

respect to space coordinat e, and in uns teady state it

becomes a partial different ial eq uation (PDE) Thi shas a far-reachi ng influen ce on methods of solvin g themodel A corres pondi ng equati on will have to bewritten for y et another phase (gaseo us), and the equ a-tions will be co upled by the drying rate exp ression.Befo re star ting with constru cting and solvin g aspecific dryer mod el it is reco mmended to class ifythe methods , so typic al cases c an easily be identifie d

We will classify typic al cases when a soli d is co ntactedwith a he at carri er Three facto rs will be co nsidered :

1 Operation type—we will co nsider eithe r batch

or co ntinuous process wi th respect to givenphase

2 Flow g eometry type—w e will consider onlyparallel flow, cocurrent , countercur rent, andcross-flow cases

3 Flow type—w e will con sider two lim iting cases,either plug flow or perfectly mixe d flow

These three assum ptions for two pha ses present resul t

in 1 6 generic cases as sho wn in Figure 3.1 Beforeconstructing a model it is de sirable to identi fy theclass to whi ch it be longs so that writing appropriatemodel equations is facilitated

Dryers of type 1 do not exist in industry; fore, dryers of type 2 are usually called batch dryers as

there-is done in ththere-is text An additional uous —will be us ed for dry ers descri bed in Secti on 3.9.Their principle of operation is different from any ofthe types shown in Figure 3.1

term—semicontin-3.4 GENERAL RULES FOR A DRYER MODEL FORMULATION

When trying to derive a model of a dryer we first have toidentify a volume of space that will represent a dryer

countercurrent

Continuous cocurrent

Continuous cross-flow

If a dryer or a whol e system is composed of many such

volume s, a separat e submod el will have to be built for

each vo lume and the mod els co nnected toget her by

streams e xchanged between them Each stre am

enter-ing the volume must be identified wi th parame ters

Basical ly for syst ems unde r c onstant pressur e it is

enough to describe e ach stream by the na me of the

compon ent (humi d gas, wet solid, conden sate, etc.),

its flowra te, moisture content , and tempe ratur e All

heat an d other energy fluxe s must also be identified

The followi ng five parts of a determ inistic mo del

can usually be dist inguishe d:

1 Balance equati ons—the y repres ent Natur e’s

laws of con servation and can be wri tten in

the form of Equation 3.1 (e.g , for mass and

energy)

2 Constitu tive e quations (also called kinetic

equatio ns)—they conn ect fluxe s in the syst em

to respect ive drivi ng forces

3 Equilib rium relationshi ps—neces sary if a pha se

bounda ry exist s somew here in the system

4 Property equ ations—som e propert ies c an be

consider ed constant but, for exampl e, satura ted

water vap or pre ssure is strong ly dependen t on

temperatur e even in a narrow tempe rature

range

5 Geometri c relationsh ips—they a re usually ne

-cessary to co nvert flowra tes present in balance

equatio ns to flux es present in consti tutive eq

ua-tions Bas ically they include flow cross-sect ion,

specific area of phase contact , etc

Typical form ulation of ba sic mod el eq uations will be

summ arized late r

3.4.1 MASS AND ENERGY BALANCES

Input–out put balance equ ations for a typical case of

convecti ve drying and LPM assum e the foll owing

WB Y 1  W B Y 2 þ wDm A ¼ mB

dY

dt (3 : 6)

3.4.1.2 Energy balancesSolid pha se:

WS i m1  W S i m2 þ (S qm  wDm hA )A ¼ m S

dim

dt (3 : 7)Gas pha se:

In the case of DPMs for a given pha se the balanceequati on for prop erty G reads:

div [G u]  div[ D  grad G]  b aV DG G

@G

where the LH S terms are, respectivel y (from the left):convecti ve term, diffusion (or axial disper sion) term ,interfaci al term, source or sink (prod uction or de-struction) term, an d accumul ation term

Thi s eq uation can now be writt en for a single pha sefor the case of mass an d energy transfer in the foll owingway:

div[ r X  u]  div[ D  grad( r X ) ]  kX aV DX @r X

@t ¼ 0(3 : 10)

Note that density here is related to the whole volume

of the phase: e.g., for solid phase composed of lar material it will be equal to rm(1 «) Moreover,the interfacial term is expressed here as kXaVDXforconsistency, although it is expressed as kYaVDYelse-where (see Equat ion 3.27)

granu-Now, consider a one-dimensional parallel flow oftwo phases either in co- or countercurrent flow, ex-changing mass and heat with each other Neglectingdiffusional (or dispersion) terms, in steady state thebalance equations become

carry the positive sign for cocurrent and the negative

sign for cou ntercurren t ope ration Both heat and

mass fluxes, q and wD , are ca lculated from the con

sti-tutive equ ations as explai ned in the follo wing sectio n

Havin g in mind that

dig

dl ¼ ( cB þ c A Y )dtg

dl þ ( cA tg þ D hv0 )dY

dl (3 : 16)and that enthal py of steam eman ating from the solid

is

hAv ¼ c A t m þ Dhv0 (3 : 17)

we can now rewrite (Eq uation 3.12 throu gh Equation

3.15) in a more con venient worki ng version

operati on

For a monoli thic soli d phase conv ective and inter

-facial terms disappea r and in uns teady stat e, for the

one-dim ensional case, the eq uations beco me

These equati ons are named Fick ’s law and Fourier’ s

law, respect ively, and can be solved with suita ble

bounda ry and initial condition s Li terature on solving

these eq uations is ab undan t, and for diff usion a sic work is that of Crank (1975) It is wort h mentio n-ing that, in view of irreversi ble thermod ynamics, massflux is also due to therm odiffu sion and barod iffusion.Formula tion of Equation 3.22 an d Equation 3.23contai ning terms of therm odiffu sion was favore d byLuikov (1966)

clas-3.4.2 CONSTITUTIVE EQUATIONSThey are ne cessary to estimat e eithe r the local non-convecti ve flux es caused by co nduction of heat ordiffusion of mois ture or the inter facial fluxes ex-changed eithe r betw een two phases or through syst embounda ries (e.g , heat losse s throu gh a wall) The firstare usu ally express ed as

moisture, Y* will result from Equation 3.46 and asorption isotherm This is in essence the so-calledequilibrium method of drying rate calculation.When the drying rate is controlled by diffusion inthe solid phase (i.e., in the falling drying rate period),the conditions at solid surface are difficult to find,unless we are solving the DPM (Fick’s law or equiva-lent) for the solid itself Therefore, if the solid itselfhas lumped parameters, its drying rate must be repre-sented by an empirical expression Two forms arecommonly used

3.4.2.1 Characteristic Drying Curve

In this approach the measured drying rate is

repre-sented as a function of the actual moisture content

(normalized) and the drying rate in the constant

dry-ing rate period:

The f function can be represented in various forms to fit

the behavior of typical solids The form proposed by

Langrish et al (1991) is particularly useful They split

the falling rate periods into two segments (as it often

occurs in practice) separated by FB The equations are:

f ¼ FacB for F # FB

f ¼ Fa for F > FB

(3:30)

Figure 3.2 shows the form of a possible drying rate

curve using Equation 3.30

Other such equations also exist in the literature

(e.g., Halstro¨m and Wimmerstedt, 1983; Nijdam and

Keey, 2000)

3.4.2.2 Kinetic Equation (e.g., Thin-Layer

Equations)

In agricultural sciences it is common to present drying

kinetics in the form of the following equation:

F¼ f (t, process parameters) (3:31)

The function f is often established theoretically, for

example, when using the drying model formulated by

agricul-The volumetric drying rate, which is necessary inbalance equations, can be derived from the TLE inthe following way:

dF

dt(Xc X *) (3:37)while

mS¼ V (1  «)rS (3:38)and

f ¼ (1  «)rS (X c  X *)

kY f( Y *  Y ) aV

d F

dt (3 : 40)

To be able to calcul ate the volume tric drying rate

from TLE, one needs to know the voidage « and

specific contact area aV in the dryer

W hen dried soli ds are monoli thic or grain size is

overly large , the above lumped parame ter approxim

a-tions of drying rate woul d be una cceptable , in which

case a DPM repres ents the entire soli d phase Such

models are shown in Se ction 3.7

3.4.3 AUXILIARY RELATIONSHIPS

3.4.3.1 Humid Gas Properties and Psychrometric

Calculations

The ability to perfor m psychro metric calcul ations

forms a basis on which all drying models are

built One princi pal prob lem is how to determine the

solid tempe rature in the constant drying rate co

ndi-tions

In psychrom etric calculati ons we co nsider therm

o-dynami cs of three pha ses: inert gas pha se, mois ture

vapor phase, and mois ture liquid pha se Two gaseou s

phases form a solution (mixtur e) call ed humid gas To

determ ine the degree of co mplex ity of our approach we

will make the follo wing assum ptions :

. Inert gas componen t is insol uble in the liquid

phase

. Gaseous phase be havior is close to ideal gas;

this limits our total pressur e ran ge to less than

2 bar

. Liquid phase is incompr essi ble. Comp onents of both phases do not chemi callyreact with thems elves

Before writi ng the psyc hrometric relat ionship s wewill first present the ne cessary approxim ating equ a-tions to descri be phy sical propert ies of syst em com-ponen ts

Depe ndence of satur ated vapor pressure on peratur e (e.g , Antoin e eq uation):

for gases These data can be found in specializedbooks (e.g., Reid et al., 1987; Yaws, 1999) and com-puterized data banks for other liquids and gases

TABLE 3.1

Coefficients of Approximating Equations for Properties of Selected Liquids

3.4.3.2 Relations between Absolute Humidity,

Relative Humidity, Temperature,

and Enthalpy of Humid Gas

With the above assumptions and property equations

we can use Equation 3.45 through Equation 3.47 for

calculating these basic relationships (note that

mois-ture is described as component A and inert gas as

component B)

Definition of relative humidity w (we will use here

wdefined as decimal fraction instead of RH given in

dry gas):

ig¼ (cAYþ cB)tþ Dhv0Y (3:47)

Equation 3.46 and Equation 3.47 are sufficient to

find any two missing humid gas parameters from Y,

w, t, ig, if the other two are given These calculations

were traditionally done graphically using a

psychro-metric chart, but they are easy to perform numerically

When solving these equations one must remember that

resulting Y for a given t must be lower than that at

saturation, otherwise the point will represent a fog

(supersaturated condition), not humid gas

3.4.3.3 Calculations Involving Dew-Point

Temperature, Adiabatic-Saturation

Temperature, and Wet-Bulb Temperature

Dew-point temperature (DPT) is the temperature

reached by humid gas when it is cooled until it

becomes saturated (i.e., w ¼ 1) From Equation3.46 we obtain

Adiabatic-saturation temperature (AST) is thetemperature reached when adiabatically contactinglimited amounts of gas and liquid until equilibrium.The suitable equation is

Equa-. For other systems with higher Lewis numbersthe deviation of WBT from AST is noticeableand can reach several degrees Celsius, thus caus-ing serious errors in drying rate estimation Forsuch systems the following equation is recom-mended (Keey, 1978):

TABLE 3.2Coefficients of Approximating Equations for Properties of Selected Gases

Typically in the wet-bulb calculations the

fol-lowing two situations are common:

. One searches for humidity of gas of which both

dry- and wet-bulb temperatures are known: it is

enough to substitute relationships for Ys, Dhv,

and cHinto Equation 3.52 and solve it for Y

. One searches for WBT once dry-bulb

tempera-ture and humidity are known: the same

substi-tutions are necessary but now one solves the

resulting equation for WBT

The Lewis number

Le¼ lg

is defined usually for conditions midway of the

con-vective boundary layer Recent investigations (Berg

et al., 2002) indicate that Equation 3.52 needs

correc-tions to become applicable to systems of high WBT

approaching boiling point of liquid However, for

common engineering applications it is usually

suffi-ciently accurate

Over a narrow temperature range, e.g., for water–

air system between 0 and 1008C, to simplify

calcula-tions one can take constant specific heats equal to

cA¼ 1.91 and cB ¼ 1.02 kJ/(kg K) In all calculations

involving enthalpy balances specific heats are averaged

between the reference and actual temperature

3.4.3.4 Construction of Psychrometric Charts

Construction of psychrometric charts by computer

methods is common Three types of charts are most

popular: Grosvenor chart, Grosvenor (1907) (or the

psychrometric chart), Mollier chart, Mollier (1923)

(or enthalpy-humidity chart), and Salin chart (or

deformed enthalpy-humidity chart); these are shown

schematically in Figure 3.3

Since the Grosvenor chart is plotted in undistortedCartesian coordinates, plotting procedures are simple.Plotting methods are presented and charts of high ac-curacy produced as explained in Shallcross (1994) Pro-cedures for the Mollier chart plotting are explained inPakowski (1986) and Pakowski and Mujumdar (1987),and those for the Salin chart in Soininen (1986)

It is worth stressing that computer-generated chrometric charts are used mainly as illustration ma-terial for presenting computed results or experimentaldata They are now seldom used for graphical calcu-lation of dryers

psy-3.4.3.5 Wet Solid PropertiesHumid gas properties have been described togetherwith humid gas psychrometry The pertinent data forwet solid are presented below

Sorption isotherms of the wet solid are, from thepoint of view of model structure, equilibrium rela-tionships, and are a property of the solid–liquid–gas system For the most common air–water system,sorption isotherms are, however, traditionally consid-ered as a solid property Two forms of sorption iso-therm equations exist—explicit and implicit:

where aw is the water activity and is practicallyequivalent to w The implicit equation, favored byfood and agricultural sciences, is of little use indryer calculations unless it can be converted to theexplicit form In numerous cases it can be done ana-lytically For example, the GAB equation

(1 bw)(1 þ cw) (3:56)can be solved analytically for w, and when the wrongroot is rejected, the only solution is

Numerous sorption isotherm equations (of

approxi-mately 80 available) cannot be analytically converted

to the explicit form In this case they have to be solved

numerically for w* each time Y* is computed, i.e., at

every drying rate calculation This slows down

com-putations considerably

Sorptional capacity varies with temperature, and

the thermal effect associated with this phenomenon is

isosteric heat of sorption, which can be numerically

calculated using the Clausius–Clapeyron equation

Dhs¼  R

MA

d ln wd(1=T)

X ¼ const

(3:58)

If the sorption isotherm is temperature-independent

the heat of sorption is zero; therefore a number of

sorption isotherm equations used in agricultural

sci-ences are useless from the point of view of dryer

calculations unless drying is isothermal It is

note-worthy that in the model equations derived in this

section the heat of sorption is neglected, but it can

easily be added by introducing Equation 3.59 for the

solid enthalpy in energy balances of the solid phase

Wet solid enthalpy (per unit mass of dry solid) can

now be defined as

im¼ (cSþ cAlX )tm DhsX (3:59)

The specific heat of dry solid cSis usually presented as

a polynomial dependence of temperature

Diffusivity of moisture in the solid phase due to

various governing mechanisms will be here termed as

an effective diffusivity It is often presented in the

Arrhenius form of dependence on temperature

Deff ¼ D0exp Ea

RT

(3:60)

However, it also depends on moisture content

Vari-ous forms of dependence of Deff on t and X are

available (e.g., Marinos-Kouris and Maroulis, 1995)

3.4.4 PROPERTYDATABASES

As in all process calculations, reliable property data

are essential (but not a guarantee) for obtaining sound

results For drying, three separate databases are

neces-sary: for liquids (moisture), for gases, and for solids

Data for gases and liquids are widespread and are

easily available in printed form (e.g., Yaws, 1999)

or in electronic version Relatively good property

prediction methods exist (Reid et al., 1987) However,when it comes to solids, we are almost always con-fronted with a problem of availability of propertydata Only a few source books exist with data forvarious products (Nikitina, 1968; Ginzburg andSavina, 1982; Iglesias and Chirife, 1984) Some dataare available in this handbook also However, numer-ous data are spread over technical literature and re-quire a thorough search Finally, since solids are notidentical even if they represent the same product, it isalways recommended to measure all the required prop-erties and fit them with necessary empirical equations.The following solid property data are necessaryfor an advanced dryer design:

. Specific heat of bone-dry solid. Sorption isotherm

. Diffusivity of water in solid phase. Shrinkage data

. Particle size distribution for granular solids

3.5 GENERAL REMARKS ON SOLVING MODELS

Whenever an attempt to solve a model is made, it isnecessary to calculate the degrees of freedom of themodel It is defined as

where NVis the number of variables and NEthe ber of independent equations It applies also to modelsthat consist of algebraic, differential, integral, or otherforms of equations Typically the number of variablesfar exceeds the number of available equations In thiscase several selected variables must be made constants;these selected variables are then called process vari-ables The model can be solved only when its degrees

num-of freedom are zero It must be borne in mind that notall vectors of process variables are valid or allow for asuccessful solution of the model

To solve models one needs appropriate tools.They are either specialized for the specific dryer de-sign or may have a form of universal mathematicaltools In the second case, certain experience in hand-ling these tools is necessary

3.6 BASIC MODELS OF DRYERS

IN STEADY STATE 3.6.1 INPUT–OUTPUTMODELSInput–output models are suitable for the case whenboth phases are perfectly mixed (cases 3c, 4c, and 5c

in Figure 3.1), which almost never hap pens On the

other ha nd, this mo del is very often used to repres ent

a case of unmixe d flows when there is lack of a DP M

Input–out put modeli ng consis ts basica lly of

balan-cing all inputs and outp uts of a dryer and is often

perfor med to iden tify, for exampl e, heat losse s to the

surroundi ngs, calculate performan ce, and for dryer

audits in general

For a steady-st ate dryer balanci ng can be made

for the whol e dryer only, so the system of Equation

3.5 through Equat ion 3.8 now c onsists of only two

equati ons

WS ( X 1  X2 ) ¼ W B ( Y2  Y 1 ) (3: 62)

WS ( im2  i m1 ) ¼ W B ( ig1  i g2 ) þ qc  ql þ Dqt þ qm

(3 : 63)where sub scripts on heat fluxes indica te: c, indir ect

heat inp ut; l, heat losses; t, net he at carried in by

transp ort devices; and m, mechani cal en ergy input

Let us assum e that all q, WS , W B , X1, im1 , Y 1, i g1 are

known as in a typic al design case The remaining

variab les are X2, Y2, i m2, and i g2 Sin ce we have two

equati ons, the syst em ha s two degrees of freedom and

cannot be solved unless two other varia bles are set as

process pa rameters In design we can assum e X2 since

it is a design specifica tion, but then one ex tra

param-eter must be assum ed This of co urse ca nnot be done

rationa lly, unless we are su re that the process runs in

constant drying rate pe riod—th en im2 can be

calcu-lated from WBT Othe rwise, we must look for oth er

equati ons, whi ch could be the foll owing:

3.6.2 DISTRIBUTED PARAMETER MODELS3.6.2.1 Cocurrent Flow

For cocurrent operatio n (case 3a in Figure 3.1) boththe case design and sim ulation are simple The fou rbalance equ ations (3.1 8 through 3.21) supp lemented

by a suit able drying rate and heat flux equ ationsare solved starting at inlet end of the dr yer, whereall bounda ry conditio ns (i.e., all parame ters of incom-ing streams) are defin ed Thi s situ ation is shown inFigure 3.4

In the case of design the calcul ations are term ated when the design parame ter, usually final mois -ture con tent, is reached Dis tance at this point is therequir ed dryer length In the case simu lation the cal-culation s are terminat ed onc e the dryer lengt h isreached

in-Par ameters of both gas and solid pha se (repr sented by gas in eq uilibrium with the so lid surfa ce)can be plotted in a psychrom etric ch art as pro cesspaths Thes e phase diagra ms (no timescal e is availab lethere) show schema tically how the pr ocess goes on

e-To illustrate the case the mo del compo sed ofEquation 3.18 through Equation 3.21, Equation3.26, an d Equation 3.27 is solved for a set oftypical con ditions and the resul ts are sh own in

3.6.2.2 Countercurrent Flow

The situati on in countercur rent case (case 4a in

Fig-ure 3.1) design a nd sim ulation is shown in Fi gure 3.6

In both cases we see that bounda ry conditio ns are

defined at oppos ite en ds of the integ ration domain

It leads to the split boundar y value problem

In design this prob lem can be avoided by using the

design parame ters for the solid specified at the exit

end Then, by writin g input–out put balances ov er the

whole dryer, inlet parame ters of gas can easil y be

found (unles s local heat losses or other distribut ed

parame ter phe nomena need also be consider ed)

Howev er, in sim ulation the split bounda ry value

problem exist s and must be solved by a suit able merical method , e.g., the shooting met hod Basical lythe method consis ts of assum ing certain parame tersfor the exit ing gas stre am and perfor ming integ rationstarting at the so lid inlet end If the gas parame ters atthe other en d conve rges to the known inlet gasparame ters, the assum ption is sati sfactory; otherwis e,

nu-a new nu-assump tion is mnu-ade The process is repenu-at edunder co ntrol of a suit able convergence co ntrolmethod, e.g , Wegst ein Figure 3.7 co ntains a samplecountercur rent c ase calcul ation for the same mate rial

as that used in Figu re 3.5

0 20 40 60 80

% of dryer length

100 0.0 25.0 50.0

50.0

75.0 100.0 125.0 150.0

150.0

175.0 200.0 225.0

t ⬚C

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0

0.0 0.0

0.0

0.0

20.0 20.0

40.0 60.0 80.0 100.0

80.0 90.0 100.0

Calculated profile graph for cocurrent contact of sand containing water with air

Y, g/kg X, g/kg

⬚C Continuous cocurrent contact of sand and water in air

20 30 50 70 90

tg

y x

3.6.2.3 Cross-Flow

3.6.2.3.1 Solid Phase is One-Dimensional

This is a sim ple case corresp onding to case 5b of

per-fectly mixe d in the direction of gas flow, the solid

phase beco mes one dimen sional This situ ation oc

-curs with a co ntinuous plug-flow fluid be d dryer

Schemat ic of an elem ent of the dr yer length wi th finite

thickne ss D l is sho wn in Figure 3.8

The balance e quations for the solid pha se can be

derive d from Equation 3.12 and Equat ion 3.14 of the

1S

dWB (Y 2  Y 1 )

dl ¼ wD aV (3 : 68)energy balance

1S

In other models (CDC an d TLE) the drying rate will

be modified as sh own in Sectio n 3.4.2.Since the heat and mass coeffici ents can be defined

on the basis of eithe r the inlet drivi ng force or themean logari thmic driving force, DYm and D tm arecalculated respect ively as

DYm ¼ (Y *  Y 1 ) (3: 72)

0 0.0 0.0 0.0

10.0 10.0

10.0

20.0 20.0

20.0

30.0 30.0

70.0

70.0 80.0

80.0

80.0 90.0

90.0

Y, g/kg x, g/kg

100.0 110.0

50 70 100

Calculated profile graph for countercurrent contact of sand containing water with air Continuous countercurrent contact of sand and water in air

kJ/kg

@101.325 kPa

10

tm y

x tg

FIGURE 3.7 Process paths and longitudinal distribution of parameters for countercurrent drying of sand in air

dl

Y1 Y2

To solve Equat ion 3.68 and Equat ion 3.69 one need s

to assum e a unifor m dist ribution of gas over the

whole lengt h of the dryer, an d therefore

dWB

dl ¼WB

When the algebr aic Equation 3.68 and Equat ion 3.69

are solved to obtain the exiting gas parame ters Y2 and

ig2 , one c an plug the LH S of these equati ons into

starting at the soli ds inlet In Figu re 3.9 sample pr

o-cess pa rameter profiles alon g the dryer are sho wn

Cros s-flow drying in a plug-flow , continuou s fluid

bed is a case when axial disper sion of flow is often

consider ed Let us briefly present a method of solving

this case First, the governi ng ba lance equ ations forthe soli d pha se will have the followi ng form de rivedfrom Equat ion 3.10 and Equat ion 3.11

These equations are supplemented by equations for

wDand q according to Equation 3.70 and Equation3.71 It is a common assumption that Em ¼ Eh,because in fluid beds they result from longitudinalmixing by rising bubbles Boundary conditions(BCs) assume the following form:

FIGURE 3.9 Longitudinal parameter distribution for a cross-flow dryer with one-dimensional solid flow Drying of amoderately hygroscopic solid: (a) material moisture content (solid line) and local exit air humidity (broken line): (b) materialtemperature (solid line) and local exit air temperature (broken line) tWBis wetbulb temperature of the incoming air