Journulof Food Engineering32 (1907) 26Y-2Y1 0 1997 Elsevier Science Limited

All rights reserved. Printed in Great Britain PII: SO260-8774(97)00010-l 0260-8774197 $17.00 + 0.00

ELSEVIER

Design of lkay Dryers for Food Dehydration

C. T. Kiranoudis,” Z. B. Marouliqa D. Marinos-Kouris,“” and M. Tsamparlis”

“Department of Chemical Engineering, National Technical University, GR-15780, Athens. Greece

“Department of Physics, University of Athens, GR-15783, Athens, Greece

(Received 9 August 1996; accepted 11 February 1997)

ABSTRACT

A mathematical model for the semi-batch operation of industrial dryers with trucks and trays is presented and analysed. Design aspects are discussed concerning problems involving both single dryer and systems of parallel dryers. In both cases, optimum flowsheet configuration and operation conditions arc sought and verified by appropriate formulation of design and optimization strategies. The optimization objective is the total annual cost of the plant, subjected to constraints imposed by the operation of the dryer; thermodynamics, and construction reasoning. The decision variables were the number of trucks and the drying air stream humidity for each dryer involved, as well as the total number of dryers. The MINLP nature of the design problem required mathematical programming techniques for its solution. The optimization was carried out for a wide range of production capacities, and the optimal points, where a new truck or a new dryer is introduced, were evaluated. The effect of market economic figures on the design results is illustrated. The analysis focused on the design of two commercial agricultural products - namely, raisins and currants. A characteristic case study is presented in order to demonstrate the effectiveness of the proposed approach. 0 1997 Elsevier Science Limited

NOTATION

Al,AZ,A.3 Constants of Antoine equation (eqn 23)

aw Water activity of air streams c* Unit total annual cost ($)

*Author to whom correspondence should be addressed.

269

270 C. T Kiranoudis et al.

CE

Ctl

CZ E

>A F A0

y h A0

ii:: hs

k, ko,k,

LM N nWflD,nE,nZ

ND

P

Pl P2 psat R Rw

f R

TA

T AC

T AM

T A0

top

Ts MAX

TA

UA

W XA

X AC

X AM

Unit total annual cost at infinite production level ($) Capital cost ($) Cost of electricity ($/kWh) Concentration of fuel in hydrogen (kg/kg) Operational cost ($/h) Specific heat of air (kJikg*K) Specific heat of dry solid (kJ/kg*K) Specific heat of vapor (kJ/kg*K) Specific heat of water &I/kg-K) Total annual cost ($) Cost of fuel ($/kg) Electrical energy consumed by the motor of the blowers (kWh) Percentage of capital cost on an annual rate Flowrate of inlet air stream (kg/h*db) Flowrate of fresh air stream (kg/h*db) Flowrate of drying air stream (kg/h-db) Specific enthalpy of wet air stream (kJ/kg) pecific enthalpy of fresh air stream (kJ/kg) Specific enthalpy of drying air stream &l/kg) Specific enthalpy of inlet air stream (kJ/kg) Specific enthalpy of humid product (kJ/kg) Truck of dryer Drying constant (l/h) Constants of drying constant equation (eqn 22) Heat of combustion of fuel &I/kg) Number of trucks Parameters of capital cost equation (eqn 29) Number of dryers Total pressure (kPa) Economic coefficient expressing capital cost inflation Economic coefficient expressing operating cost inflation Water vapor pressure (kPa) Ideal gas constant (kJ/mol-K) Production of water in combustion (kg/h) Time (h) Cycle period (h) Temperature of output air stream (“C) Temperature of drying air stream (“C) Temperature of inlet air stream (“C) Temperature of fresh air stream (“C) Operating time (h/year) Temperature of product (“C) Maximum temperature level for no thermal degradation observec

(“C) Volumetric heat transfer coefficient (kW/kg.K) Truck load (kg-db) Absolute humidity of output air stream (kg/kg-db) Absolute humidity of drying air stream (kg/kg*db) Absolute humidity of inlet air stream (kg/kg*db)

Design of tray driers for food dehydration 271

Greek letters

AP ATMAX

43

Absolute humidity of fresh air stream (kg/kg*db) Material moisture content (kg/kg*db) Equilibrium material moisture content (kg/kg*db) Desired material moisture content (kg/kgdb) Constants of the GAB (Guggenheim-Anderson-de Boer) equations (eqns (18-20)) Fuel flowrate (kg/h) Fuel consumed per cycle of operation (kg/cycle)

Parameters of capital cost equation (eqn 29)

Latent heat of vaporization of water (kJ/kg) Latent heat of vaporization of water at reference temperature (kJ/

kg) Mean pressure drop through the train of trucks (kPa) Maximum allowed temperature diminuation through the train of trucks (“C) Mass ratio of water and air molecules

INTRODUCTION

Dehydration operations are important activities of the chemical and food processing industries. The basic objective in drying food products is the removal of water in the solids up to a certain level, at which microbial spoilage is significantly minimized. The wide variety of dehydrated foods that are now available to the consumer (snacks, dry mixes and soups, dried fruits, etc.) and the increasing concern for meeting quality specifications and conserving energy emphasize the need for a thorough understanding of the drying operation and the problems related to the design and operation of dryers. In particular, process design principally involves structural and parametric optimization methodologies, carried out under flowsheet constraints, in order to formulate mathematical problems that tackle predefined construction and operational needs. Despite the explicit importance of developing a proper design methodology for dealing with these problems, empiricism still pre- vails. In the case of dryers, design has become an increasingly challenging problem aiming at the evaluation of the appropriate type of equipment, its associated flow- sheet structure, its optimum construction characteristics, and the operating conditions of each unit involved in the overall flowsheet. In addition, auxiliary equipment should be appropriately chosen together with its performance character- istics. However, most design efforts in this field face problems of extreme difficulty, related to the complex drying conditions. The latter include many interconnected and opposing phenomena, associated mainly with to the complex nature of drying (Kiranoudis et al., 1995). Although the modelling of drying processes is well developed, with adequate comprehension of the process itself, most models incor- porate a large number of thermophysical properties and transport coefficients,

272 C. T. Kiranoudis et al.

which in most cases are only imprecisely known, producing inaccurate or erroneous results on large-scale industrial applications.

Tray dryers constitute an important family of convective dryers, where the drying medium is hot air or combustion gases coming from a furnance. They are adaptable to the drying of almost any material that can be put in a tray. In general, batch or semi-batch operation is used. In all cases, design efforts in this field focus on the determination of the appropriate equipment configuration: i.e. number of parallel processor lines, trucks, and trays, determination of flowsheet sizing, and correspond- ing operational conditions within each processor. Despite the obvious importance of deriving design methodologies in this field, limited efforts are cited in the literature. Bertin and Blazquez (1986) presented a mathematical model for a tunnel dehy- drator, of the California type, for plum drying, and searched for the optimum capacity of the dryer. The model was obtained by heat and mass transfer balances at two levels: the product compartment and inside the tunnel itself. The optimum condition was given by the maximum production capacity of the dryer satisfying the final product quality. It was found that recirculation of a proportion of the exhaust air improved the dryer efficiency in terms of energy. Wentz and Thygeson (1988) tackle design through a list of short-cut equations. A simplified equation was used to calculate the total amount of tray area required for the specified production rates. The necessary data, usually obtained from laboratory tests, were the drying time for the given initial and final moisture content and the desired tray loading. The calculated tray area was then distributed in appropriate trucks and trays. The remaining process variables were calculated by means of the corresponding mass and energy balances over the dryer. Vagenas and Marinos-Kouris (1991) presented a mathematical model for the design and optimization of an industrial dryer for sultana grapes and applied it to the determination of size and optimal operating conditions of the dryer. The optimal conditions were evaluated by minimizing the heat consumption, expressed as the ratio of thermal load to production, with certain constraints regarding the production rate of the dryer and the maximum permissible air temperature. The optimization variables were temperature and humidity of the drying air and product loading thickness on the trays. The optimum conditions required the operation of the dryer at the maximum permissible air humidity, corresponding to a high degree of recirculation of exhaust air. This was achieved by automatic control of fresh air and humid air inlet and exhaust dampers along the length of dryer, during the entire drying cycle. Kiranoudis et al. (1996) presented a mathematical model for the design and operational optimization of an industrial semi-batch dryer for the dehydration of grapes. Its nominal conditions were evalu- ated by minimizing the fuel demand for the dryer per unit mass of dried product processed in each cycle period. The optimal operation of the dryer was evaluated by maximizing the total profit from the operation of the dryer. In this way, energy economy was achieved and the dryer operated at the maximum thermal efficiency, while the production rate was also maintained at a high level.

In the present work, the mathematical model for the semi-batch operation of a system of parallel industrial dryers with trucks and trays is presented and analysed. Design was carried out by appropriate optimizing of the total annual cost of the plant subject to constraints imposed by the operation of the dryer. Design was formulated as a mixed-integer non-linear optimization (MINLP) problem and was solved by using mathematical programming techniques. The optimization was carried out for a wide range of production capacities and the optimal points where

Design of tray driers for food dehydration 273

a new truck or a new dryer is introduced were evaluated. The effect of market economic figures on design results was presented.

MATHEMATICAL MODELING OF TRAY DRYERS

The industrial dryer examined here is a tunnel dryer with trucks and trays operating in semi-batch fashion. This configuration is typical for most industrial applications. The dryer contains a number of trucks, each of which behaves as a separate batch tray dryer. Material particles are placed uniformly on the surface of each tray. The air is blown over the trays containing the product, and the movement of the air is reversed for each truck, so that the product has uniform moisture content at the end of drying. The trucks enter through a batch operation and, at predefined intervals, move along the dryer counter-currently with the drying air. When the first truck of the train reaches the desired material moisture content, it is withdrawn from the dryer. The remaining trucks proceed towards the exit and a new truck is introduced at the end of the train. In this way a semi-batch procedure is introduced and a periodic operation is adopted. Recirculated air is heated by directly introducing combustion gases coming from a conventional burner. The burner operates with a hydrocarbon fuel and fresh air necessary for burning the fuel. A fraction of recircu- lated air, equal to the amount of combustion burner gases, leaves the dryer just after the train of trucks. It is common practice that temperature and humidity of the drying air stream entering the product are controlled. In this case, the final control elements are the fuel induction valve and the fresh air dumper, respectively. It is possible that, when the production capacity is high enough, more than one dryer operate in parallel. The interior of the dryer as well as the arrangements of its control facilities are included in Fig. 1.

The mathematical model of the dryer involves heat and mass balances of air streams and product trays in the dryer and the burner. In addition to the above, heat and mass transfer phenomena taking place during drying must also be taken into consideration. Moreover, the resulting equations are subject to quality and thermodynamic constraints. The arrangement of product and air streams within the dryer and burner compartments is also given in Fig. 1. The analysis of model components will commence with the corresponding heat and mass balances over each individual truck. We assume that the dryer involves N trucks and that each truck i (i = 1,. . . ,N) can be loaded to a maximum total load. The corresponding product dry mass is W and can be readily computed as a function of the maximum total load of the truck and the initial material moisture content of the product. Heat and mass transfer phenomena during drying are indeed complicated and their solution demands considerable computational time. They involve coupled transfer mechanisms within both the solid and the gas phase. A mathematical model expli- citly accounting for all transfer mechanisms should not be appropriate for optimization purposes since it demands considerable computational time. Computa- tional time is of utmost importance when a mathematical model is to be solved repeatedly in an optimization convergence procedure. In this case, a simplified model is considered. It involves first-order kinetics and contains a mass transfer coefficient of a phenomenological nature, which is usually called the drying constant. This constant chiefly accounts for mass diffusion within the solid phase, but also

214 C. ?: Kimzoudk et al.

embodies boundary layer phenomena when it is considered to be a function of all process variables affecting drying. Ample accuracy is combined with sufficient low computation time (Kiranoudis et al., 1995). On the basis of the above, moisture transfer mechanisms in the ith truck, are expressed by the following equation:

dX$ - - = k&&-X$,)

dt

Heat transfer is chiefly controlled by the heat transfer coeffient at the air boundary layer (Kiranoudis et al., 1995). For the purpose of developing the particular mathe- matical model, it is assumed that an overall phenomenological heat transfer coefficient embodying both conduction and boundary layer phenomena can be applied. On the basis of the above, heat transfer within the ith truck is expressed by means of the following equation:

Fan Burner I I

I

I

I

E

w Trucks

F AC

xi-1 A

Ti4 A

Fig. 1. Industrial dryer with trucks and trays.

Design of tray driers for food dehydration 275

dh$ dX; - = UA(7’&Tj;)+AHs-

dt dt (2)

The form of heat transfer implied by eqn (2) suggests that accumulation of energy within the truck, expressed by the product specific enthalpy derivative, is controlled by heat removed by the drying air stream and the corresponding moisture evapora- tion within the truck. Mass balance of air passing through the ith truck is given by the following relationship, suggesting that moisture evaporated within each truck is directly conveyed to the gas phase:

dx’, &-(X&X;‘)+W - = 0

dt

The corresponding heat balance for the drying air stream passing through the ith truck suggests that its specific enthalpy diminution through the product is equal to the amount of heat removed from the drying air stream, as expressed by the following equation:

F&hi - hc ‘) + WU,(Ti - 7-i) = 0 (4)

Equations (l-4) constitute the mathematical model of each drying truck for both product tray and drying air stream. Repetition of this model for all trucks participat- ing in the dryer form the overall mathematical model of the drying compartment. Clearly, the individual equations are connected with the intermediate drying stream variables as enter each truck. Thus, these variables are related to the dryer drying stream variables shown in Fig. 1 as follows:

The electrical energy following equation:

xi = X*c (5)

T”A = TAC (5)

xg = X* (7)

T; = TA (8)

consumed by the motor of the dryer blowers is given by the

E = APFAC

A simplified model of the actual burner is presented. We assume that the fuel is hydrocarbon and is burned using fresh ambient air to give the combustion products, solely carbon dioxide and water. The production rate of water vapor is given by the following equation, which expresses the elementary combustion reactions:

Rw = 9CHZ (10)

Based on eqn (lo), the total balance and the moisture balance over the burner are given by the following equations, which describe the combustion process:

FA = F,,+Z (11)

FAXAM = FAOXAO + Rw (14

276 C. I: Kirunoudis et al.

Assuming that the fuel has the same thermophysical properties as air, the corre- sponding energy balance over the dryer is given by the following relationship:

F&AM = FAohAo + XLM (13)

In the dryer point where the combustion gases are mixed with recirculation air, the mass and energy balances are as follows:

FACXAC=(FAC-FAWA+FAXAM (14)

F/&c = WAC - FAh + FAhAM (15) The mathematical model of the dryer involves thermophysical properties as well as transport coefficients. The specific enthalpy of the product, as a function of its temperature and moisture content, can be calculated by means of the following equation, assuming a reference temperature of 0°C (Pakowski et al., 1991):

hs = c&s + cpwTs& (16)

The specific enthalpy of an air stream, as a function of its temperature and absolute humidity, is given by the following equation, assuming a reference temperature of 0°C (Pakowski et al., 1991):

hA = CPATA+@HO+CPVTA)XA (17)

The equilibrium moisture content of desorption of product, as a function of water activity and temperature of the surrounding air, can be computed by using the well known GAB (Guggenheim-Anderson-de Boer) equation, which can be successfully applied in the case of most foods (Maroulis et al., 1988):

XSE xM,CKUw

(1 -Kaw>[l -(I -C)Kaw] (18)

c = C,[eXp(AH,/RT,)] (19)

K = K,[exp(AH,lRT,)] (20)

The drying air stream water activity used in eqn (18) can be evaluated by the following equation of the phychrometric chart (Pakowski et al., 1991):

x Ap

+ = (~,+xA)p”“t (21)

The drying constant used in eqn (1) as a function of drying air temperature is given by the following empirical equation, appropriately applied using regression analysis to the experiments carried out by Vagenas (1988):

kM = k,,Tk,: (22)

Water vapor pressure can be computed by means of the Antoine equation:

In pat = Al -Az/(A3+TA) (23)

The latent heat of water vaporization can be evaluated by means of the Clausius- Clapeyron equation:

Design of tray driers for food dehydration 217

(24)

The volumetric heat transfer coefficient is assumed to be constant for the drying air stream temperature and velocity range studied.

As mentioned before, the dryer operates in cycles. The exact transition period, tR, is determined by a critical constraint requiring that the material moisture content of the extreme truck reach the desired value:

X&,) = Xgrget (25)

Futhermore, the drying air temperature at the entrance of the truck train should not exceed an upper limit which affects product quality and prevent thermal degrada- tion:

T; 5 Tgax (26)

Thermodynamics dictate that the moisture content of the product should be greater than the corresponding equilibrium material moisture content imposed by the oper- ating conditions over the truck:

Xi; 2 XSE( Ti,ab) (27)

Uniform drying throughout the truck train is desirable so that the corresponding drying air temperature gradient can not create steep drying fronts within the dryer. This is guaranteed by not allowing the drying air stream temperature diminution through the truck train to exceed a predefined upper limit, at any time point of operation:

T; - T; I ATMAX (28)

Equations (l-28) constitute the mathematical model of the dryer. It involves five decision variables: the number of trucks, N, the cycle period of batches, tR; the drying air stream humidity, x,& the drying air stream temperature, T,+C.; the drying air stream flowrate, FAC. In this case, however, the cycle period of the operation is evaluated so that eqn (25) is satisfied. Drying air temperature is chiefly constrained by the relation of eqn (26). Drying air flowrate is also restricted by the relation of eqn (28), in the sense that minimum cost is reached when the equality holds. As a result of this, the only decision variables remaining are the number of utilized trucks and the drying air humidity. If parallel processors are used, then the total number of decision variables are the number of utilized trucks and the drying air humidity for each dryer plus the total number of dryers used.

On the basis of the above, the capital cost of a single dryer is expressed by means of the following equation:

ccp = cx&VW+ SIJP + CY(EEI” + C.+,.P (29

The first term is the cost of trucks, the second is the cost of the dryer and auxilliary equipment as a function of the number of trucks, the third term is the cost of blowers as a function of the motor electrical power, and the last term is the cost of the burner as a function of the fuel flowrate. All cost terms obey economy of scale laws. The corresponding operational cost is given by the following equation where the first term expresses the electricity cost and the second the cost of fuel consumed:

COP = c&+czZ cm

278 C. ‘I Kiranoudis et al.

The total annual cost of the plant can be expressed by means of the following equation:

CT = w&P + PdOPCOP (31)

In this expression the economic coefficients, p1 and p2, express the way in which each total cost component is affected by the current market economic situation for the purchased equipment and the utilities involved, respectively. The process description adopted formulates an MINLP problem subject to non-linear con- straints, for the determination of the optimal flowsheet structure and the operating conditions. In this case, the total annual cost objective function is minimized subject to model constraints. Such problems are solved using appropriate mathematical optimization techniques. Throughout the paper optimization was carried out by means of the successive quadratic programming algorithm implemented in the form of subroutine E04UCF/NAG. The periodic form of the problem requires iterative evaluation of the mathematical model until all variables converge at the end of the cycle time. All computations were performed on a SG Indy Workstation under UNIX.

DESIGN OF TRAY DRYERS

The single dryer case

In the discussion concerning the general design problem of tray dryers, it was clearly stated that, for a given production level, the objective of any design strategy should aim at the determination of the following:

(1) The optimum number of dryers and their corresponding number of pre- defined sized trucks (flowsheet structure)

(2) The optimal set points of controllers for each dryer involved (operating conditions)

When the production capacity of the plant is specified, a reasonable objective should be the minimization of the total annual cost of the plant. Taking into consideration that the mathematical model of the plant involves a significant number of design variables (i.e. integer and continuous), the problem is of an MINLP nature and therefore a cumbersome case to solve. Our first attempt to solve the general multi-dryer case is to consider the case of a single dryer. Our study will be focused on the processing of two very popular agricultural products (namely, raisins and currants), which are typical materials for this type of dryer. In this case, a maximum processing temperature of 75°C is taken into consideration so that thermal degradation of the product is prevented, a typical truck loading of 1 tn*wb is used, and a minimum temperature diminution of 10°C through the train of trucks is considered. The ambient temperature and air humidity are considered to be 25°C and 0.01 kg/kg*db, respectively. Raisins are dried from an initial material moiture content of 4 kg/kg.db to a desired one of O-2 kg/kg*db, while the corresponding values for currants are 3 kg/kg*db and 0.1 kg/kg*db.

In the single dryer case and for a specific product processed, the way in which the decision variables (i.e. the number of trucks and the drying air stream humidity) affect the fundamental process variables which determine cost is depicted in Fig. 2.

Design of tray driers for food dehydration 279

20

s 10

0 I _ 0

Raisins

__ ~- N=5 /

_ I I I 0.05 0.1 0.15 c

X, 0wks db)

0 0.05 0.1 0.15

X, (kg/kg db)

i --__ 7 T

0.05 0.1 0.15 012

X, (kg/kg db)

20

s 10

0

(a)

ClnTants

N=5

A;

III- 0.05 0.1 0.15 (

X,Gwks db)

2

0 0.05 0.1 0.15 0.2

X, kk db)

300

2 $ 200

z? N

100 I 0

\

I I I

0.05 0.1 0.15 1

X, Wk db)

Fig. 2. Effect of decision variables on process variables affecting cost. (a) Cycle period. (b) Recycle flowrate. (c) Fuel consumption.

C. 7: Kiranoudis et al.

Three variables are considered in this analysis: cycle period, flowrate of recircula- tion, and cycle fuel consumption. Figure 2(a) shows that, as drying air humidity increases, cycle period also increases. At high humidity levels, the drying conditions become less intense and, as a concequence, the product has to be treated for longer periods in the dryer. When the number of trucks participating in the dryer increases, the cycle period increases as well; although the drying conditions are more intense when the fewer trucks are used, the total residence time of a truck within the dryer involving more trucks is similar, since more cycles will be used to reach the desired material moisture content level. There is a maximum level of drying air humidity within the dryer, that for raisins being higher than that for currants. Obviously, when drying air humidity reaches this level, the cycle time approaches infinity, and beyond this value the dryer has to be equipped with more trucks so that a bigger production capacity is achieved. The corresponding recycle flowrate is determined when the equality holds for constraint (eqn (28)); the smaller the flowrate, the lower the cost of fans and electricity, but less intense drying conditions will prevail; therefore, the optimum is always located at the equality. Figure 2(b) implies that, as drying air humidity increases, recycle flowrate decreases. At high humidity levels the desired temperature diminuation through the train of trucks can be achieved will lower recirculation levels. When the number of trucks participating in the dryer increases, the recycle flowrate increases too; in this case, the drying conditions have to be more intense when more trucks are used, so that the desired temperature diminu- tion through the train of trucks can be achieved. The same effect is observed for fuel consumption within a cycle, as suggested by Fig. 2(c). We note that, as the drying air stream humidity increases, batch fuel consumption decreases. Again the reason is obvious: the higher the drying air stream humidity, the less the demand for fresh air and, subsequently, the less the rejection of hot recirculated air. In this way more energy is conserved and, therefore, the demand for fuel consumption is lower. Obviously, the number of trucks has no effect on fuel consumption. Clearly, at low values of drying air stream humidity energy consumption is higher, but more cycles can be performed within a production planning horizon. At high values of drying air stream humidity, it is the other way around. In other words, when striving for high production rates under considerable energy conservation, there arises a trade-off between fuel consumption and production capacity of the dryer.

The optimum structure and operating conditions of the dryer was determined by suitably optimizing the total annual cost of the dryer. Typical Greek market eco- nomic figures were taken into consideration for determining cost. Each 1 tn trucks costs 4000 $/truck, increased with a power law of 0.95. The dryer costs 3500 $/truck, increased with a power law of O-67. The cost of fans is 500 $/kWh of motor power, and for the burner 200 $/kg of fuel are utilized on an hour basis. The corresponding power laws are 0.3 and 0.4, respectively. The cost of the electrical unit is 0.08 $/kWh and of the fuel 0.2 $/kg. The fuel is a hydrocarbon with hydrogen content of 0.15 kg/ kg and corresponding heat of combustion 40000 kJ/kg. The capital cost economic coefficient pl is chosen to be 4 while the corresponding coefficient for the operating cost, p2, was fixed to a value of 2. For a given production capacity of the plant, the optimal unit total annual cost and its corresponding decision variables (i.e. drying air humidity and number of trucks) are given in Fig. 3 for the case of raisins. The unit total annual cost is presented in Fig. 3(a) as a function of plant production capacity. We note that it exhibits several local minima, while it varies considerably at low production capacity levels. For high production capacity levels the curves

Design of tray driers for food dehydration 281

0.25

0.23 --

5 0.21 --

z 0.19 --

5 0.17 -- \

0.15 --

0.13 1

0 100 200

Q &$wW

400 500 ml

(a)

Q k/h wb)

(b)

loo 200 300 400 500 600

Q (kg/h wb)

(cl Fig. 3. Objective and decision variables as a function of production capacity for the single

dryer case. (a) Unit total annual cost. (b) Drying air humidity. (c) Number of trucks.

C. T. Kiranoudis et al.

varies smoothly, although local minima still remain. This behavior is justified by the fact that the corresponding optimal number of trucks within the dryer increases as a function of production capacity. For a specific dryer configuration, where the number of trucks is fixed, there is only one global minimum for the unit total annual cost and this value corresponds to a fixed level of production capacity: i.e. for a given dryer there is always only one production capacity level where the dryer operates optimally with respect to cost. When this level is exceeded, then the production of more product units will cost more non-proportionally. There exists a production level where introduction of one more trucks to the train will result in lower cost. We note that introduction of a new truck in the dryer (i.e. enrichment of its structure) takes place on equally spaced intervals of production capacity, as suggested by the linear form of the corresponding curve of Fig. 3(c), where the optimal number of trucks is given as a function of the production capacity of the dryer. As implied by Fig. 3(b), within each production capacity width, where the structure of the dryer remains constant, the drying air stream humidity decreases as the production capacity increases, introducing in this way more extreme drying conditions for processing the product. The envelope of the unit total annual cost curve signifies the correponding cost curve if the number of trucks in the dryer was a continuous variable. This envelope decreases rapidly up to a certain level (around 300 kg/h*wb for the case of raisins) where a global minimum is reached. Clearly, processing more product with the specified dryer, no matter how many more trucks are introduced, will no longer produce products of even less unit cost. At this point, the need for introduction of a new dryer parallel to the one mentioned above is obvious. The situation will be further analysed in the following section.

Systems of parallel dryers

The design procedure adopted in the previous section is further expanded to the case of systems of parallel dryers. This problem is the one most frequently encoun- tered in industrial practice, since more processors are used to handle higher production capacity levels. Apart from the drying air humidity and number of trucks for each dryer utilized, a new decision variable is introduced in this case: i.e. the total number of parallel dryers. For a given production capacity, these variables are to be estimated by suitably optimizing the total annual cost of the entire plant. The problem is MINLP nature and more complex than the one tackled before.

In the case of systems of parallel dryers, the unit total annual cost of the plant for the two materials considered is given in Fig. 4. The local minima encountered in the previous section for the same variable studied are again present. In this case, however, the envelope of the curve does not exhibit a global optimum, but remains constant as the production capacity increases. In the case of raisins, the second dryer parallel to the first is introduced at a production capacity of around 300 kg/ h.wb, as predicted in the previous section, followed by an introduction of a third one at around 550 kg/h*wb and of a fourth at around 800 kg/h*wb. We note a systematic introduction of a new parallel dryer at intervals of production capacity of about 250 kg/h-wb. The behavior of the unit total annual cost curve for the case of currants is similar to the one for raisins. In this case, however, the cost is significantly lower and the production capacity intervals for the introduction of new trucks in a dryer and new dryers in the entire flowsheet are wider. The second dryer is introduced at

Design of tray driers for food dehydration 283

a production capacity level of around 450 kg/h*wb, while the corresponding produc- tion capacity interval for the introduction of the rest is around 330 kg/h.wb.

The variation of decision variables within each dryer as a function of production capacity is given in Fig. 5, for the case of raisins. The optimal production flowrate of each individual dryer is presented in Fig. 5(a). At low production levels only one dryer exists and its flowrate is identical to the production capacity of the plant. When a new dryer is introduced, the new flowrate profiles are again linear in shape, they coincide in certain periodic intervals and are different in some others. Clearly, the flowrates are the same when the dryers involve the same number of trucks. Otherwise, higher values of flowrates are computed for the dryers that have more trucks. The slopes of the interrupted lines are different for each dryer system. As the number of parallel dryers increases (and so does the production capacity of the plant), the slope of the linear flowrate curves of each individual dryer decreases due to multiple splitting of the entire production capacity. The situation becomes more

0.25

0.23 M- Single Dryer :

r:

z 0.17 --

3

3 5 0.15 --

cl

0.13 --

0.11 --

0.09 --

0.07 ~~

3 Dryers : 4 Dryers

Currants

Single Dryer 2 Dryers 3 Dryers

-I

0 100 200 300 400 500 600 700 800 900 1000

Q (kg/h wb)

Fig. 4. Unit total annual cost versus production capacity for the multiple dryer case.

284 C. 7: Kiranoudis et al.

p^ 250 3

100

50

0

z

0 200 400

Q O<g/h w;

800 1000

(4

12 -

10 --

8 --

6 --

4 --

2 Istdryer,

0

0 200 400

Q (kg/h w; 800 1000

(b)

0.16

s 3rddwr : &dryer ", 0.12

+

2 0.08

P

x 0.04

0 2lxl 400

Q (kg/h w;

800 1000

Cc)

Fig. 5. Decision variables as a function of production capacity for the multiple dryer case. (a) Dryer flowrate. (b) Number of trucks. (c) Drying air humidity.

Design of tray driers for food dehydration 2%

obvious when observing the corresponding curves of Fig. 5(b), where the number of trucks for each individual dryer are plotted against the production capacity of the plant. We note that each new truck is symmetrically placed in the cascade of dryers before the introduction of a new dryer. We imply that the design dictates that, for a given number of trucks, these are uniformly distributed to the rival parallel dryers. The drying air humidity for each dryer as a function of production capacity is presented in Fig. 5(c). It resembles the corresponding single dryer case, where the envelope’s amplitude decays as the production capacity increases. Clearly, at high levels of production capacity, the drying air humidity is restricted to values around 0.08 kg/kg.db. More process variables are given in Fig. 6 for the case of raisins. They involve the cycle period, the recycle flowrate, and the fuel consumption for each dryer involved in the flowsheet. Again, the impact of splitting is significant at all levels.

From the analysis presented above, it is obvious that there are three basic process variables that can express the entire design procedure discussed so far: the total number of parallel processors, the total number of trucks involved, and the unit total annual cost at infinite production capacity (this value is actually the optimal unit total annual cost at relatively high production capacity levels up to infinity). Clearly, for a given production capacity, the flowsheet configuration will be given in terms of the total number of dryers and the total number of trucks, uniformly distributed in the dryers. The unit total annual cost at infinite production capacity will give us a clear indication of the unit product cost of the process. The impact of the economic environment of these variables is quite significant for design. In this analysis, the economic environment is expressed by the ratio of capital to opera- tional expenses, in other words by the ratio of economic coefficients, p,/p2. This ratio is small when the cost of equipment is significantly lower than the one of the utilities, as in situations where materials and technology are cheaper than energy. In situations where energy is less expensive than materials and technology, it is the other way round. The effect of economic environment on these variables for a wide range of production capacities is given in Fig. 7, for both materials studied. The total number of dryers is greatly affected by the economic terms of the market. Clearly, as suggested by Fig. 7(a), for a fixed production capacity less expensive technology implies the use of more dryers. Unlike the total number of dryers used, the total number of trucks utilised is clearly unaffected by the economic figures of the market, being a function of solely production capacity, as suggested by Fig. 7(b). Furthermore, Fig. 7(c) implies that the unit total annual cost at infinite production capacity increases as capital costs become superior to operational expenses.

CASE STUDY

The effectiveness of the proposed approach can be illustrated by application to the design of a system of parallel tray dryer that can handle a production capacity of 750 kg/h.wb. Both cases involving raisins and currants were studied. The economic figures used are those mentioned in the previous paragraphs. The design procedure adopted, proposed for the case of raisins a configuration of three parallel dryers. Each dryer involved ten trucks. The product flowrate in each truck was 250 kgih.wb. The drying air humidity controller was set up to 0.081 kg/kg.db, while the corre- sponding cycle period was 4 h, the recycle flowrate was 157300 m”/h, and the fuel

286 C. T Kiranoudis et al.

0 200 450 600 800 loo0

Q O<g/h wb) (4

200

Y 100 r&

50

200

5 t e 3

150

N

loo

7

0 200 400 600 800 1000

Q (kg/h wb) (b)

0 200 400 600 800 loo0

Q Wh wb)

(4

Fig. 6. Process variables as a function of production capacity for the multiple dryer case. (a) Cycle period. (b) Recycle flowrate. (c) Fuel consumption.

Design of tray driers for food dehydration 287

150

100

z

50

0

0 500 1000 1500 2000 2500 3000 3500 0 500 1000 1500 2000 2500 3000 35(x)

L

Q &g/h wb) Q (kg/h wb)

----T I 1 I I

(a)

150

100

z

50

0

0 500 1000 1500 COCCI 2500 3000 3500

Q otti wb)

03

Fig. 7. Effect of market economics parallel processors. (b) Number of

0.3

0

P,lP2=1

1

500 1000 1500 2000 2500 3OOCl 3500

Q (kg/h wb)

0 1 2 3 4

(cl

on basic design variables of tray dryers. (a) Number of trucks. (c) Unit total annual cost at infinite production

capacity.

288 C. ?: Kiranoudis et al.

$

3 c o.oaot x

(a)

w

(4

Fig. 8. Variation of material and drying air process variables as a function of time. (a) Material moisture content (upper curve is for the last truck, lower curve is for the first truck). (b) Material temperature (upper curve is for the first truck, lower curve is for the last truck). (c) Drying air stream humidity (upper curve is for the last truck, lower curve is for the first truck). (d) Drying air stream temperature (upper curve is for the first truck, lower curve is

for the last truck).

Design of tray driers ,for food dehydration ‘8’)

4.

g

ij 2 XU

0 I 0

Truck

75

74 -

73

72

0

e

PI

71.

en 70

69

68 0 5 10

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(b)

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Fig. 9. Material and air process variables profiles within the process cycle. (a) Material moisture content. (h) Material temperature. (c) Drying air stream humidity. (d) Drying air

stream temperature.

290 C. i’Y Kiranoudis et al.

consumption was 118.4 kg/cycle. For the case of currants, the optimal configuration involved two parallel dryers. Each dryer involved seven trucks. The product flowrate in each truck was 375 kg/h*wb. The drying air humidity controller was set up to 0.069 kg/kg-db, while the corresponding cycle period was 2.7 h, the recycle flowrate was 116700 m3/h, and the fuel consumption was 115.6 kg/cycle.

The variation of material and drying air stream process variables within the cycle period for both cases are given in Fig. 8. They involve material moisture content and temperature and drying air stream humidity and temperature. They include varia- tion of these variables for each and every truck of the dryer. The curves are linear, except for the temperatures of material and drying air stream of the last truck which is introduced at the beginning of the change-over period. These particular curves exhibit the non-linear behavior of heat effects in the dryer.

The material and drying air stream profiles at the end of the change-over period within the dryer are given in Fig. 9. The material moisture content decreases through the trucks and so does its corresponding temperature which reaches the drying air stream temperature for the first truck of the train. Similar results are obtained for drying air stream humidity and temperature. Product temperature diminution is more abrupt than its corresponding drying air stream. Obviously, more intense drying conditions would result in greater fuel consumption but lower change-over time.

CONCLUSIONS

Design of tray dryers involve the determination of its optimum flowsheet configura- tion and operation conditions. Optimization seeks the total annual cost of the plant subjected to constraints imposed by the operation of the dryer, thermodynamics, and construction reasoning. The decision variables are the number of trucks and the drying air stream humidity for each dryer involved as well as the total number of dryers. The problem formulation is of an MINLP nature and involves implementa- tion of mathematical programming techniques for its solution. The introduction of new trucks in a dryer or a new dryer in the cascade, takes place for different production levels which are evaluated by the optimization procedure. The effect of market economic figures on design is critical for the number of dryers used and the overall cost of the plant.

REFERENCES

Bertin, R. & Blazquez, M. (1986). Modelling and optimization of a dryer. Dying Technology, 4, 45-66.

Kiranoudis, C. T., Maroulis, Z. B. & Marinos-Kouris, D. (1995). Heat and mass transfer model building in drying with multiresponse data. International Journal of Heat Mass Transfer, 38, 463-480.

Kiranoudis, C. T., Maroulis, Z. B., Marinos-Kouris, D. & Tsamparlis, M. (1996) Modeling and optimization of a tunnel grape dryer. DIying Technology, 14, 1695-1718.

Maroulis, Z. B., Tsami, E., Marinos-Kouris, D. & Saravacos, G. D. (1988). Application of the GAB model to the moisture sorption isotherms for dried fruits. JoumaE of Food Engineer- ing, 7, 63-75.

Design of tray driers for food dehydration 291

Pakowski, Z., Bartczak, Z., Strumillo, C. & Stenstrom, S. (1991). Evaluation of equations approximating thermodynamic and transport properties of water, stream and air for use in CAD of drying processes. Dlying Technology, 9, 753-773.

Vagenas, G. K. (1988) An application of heat and mass transfer principles to the drying of food materials. Ph.D. thesis, National Technical University, Athens.

Vagenas, G. K. & Marinos-Kouris, D. (1991). The design and optimization of an industrial dryer for sultana raisins. Dying Technology, 9, 439-461.

Wentz, T. H. & Thygeson, J. R. (1988) Drying of wet solids. In Handbook of Separation Techniques for Chemical Engineers, ed. P. A. Schweitzer, pp. 4-159. McGraw-Hill, New York.