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Journal of Membrane Science 320 (20 08) 280–291

Contents lists available at ScienceDirect

Journal of Membrane Science

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / m e m s c i

Effects of multiple-stage membrane process designs on the achievableperformance of automatic control

Kevin W.K. Yee a, Alessio Alexiadis b, Jie Bao c, Dianne E. Wiley a,∗

a UNESCO Centre for Membrane Science and Technology, School of Chemical Sciences and Engineering, The University of New South Wales, Sydney, NSW 2052, Australiab Mechanical and Manufacturing Engineering, University of Cyprus, 75 Kallipoleos Avenue, P.O. Box 20537, Nicosia 1678, Cyprusc School of Chemical Sciences and Engineering, The University of New South Wales, Sydney, NSW 2052, Australia

a r t i c l e i n f o

Article history:

Received 10 January 2008

Received in revised form 3 April 2008

Accepted 5 April 2008

Available online 12 April 2008

Keywords:

Multiple-stage processes

Whey ultrafiltration

Automatic control

Process design

Dynamic operability

a b s t r a c t

Thereis limited information from literature on thedynamic operability of membrane processeswith mul-

tiple stagesor loops. Such information is usefulfor assessing the performance achievable by an automatic

controller proposed for a process design before the actual controller is implemented. Based on dynamic

modeling of an industrial whey ultrafiltration process with an increasing number of stages up to a max-

imum of 12, the dynamic responses of the flowrate and concentration of the retentate were obtained.

Features of the dynamic responses were used to determine the performance, in terms of quality and

speed, that can be achieved by automatic controllers. In particular, limitations on the performance are

indicatedby features of dynamic responses such as effective time delay andinverse responses. Changesin

effective time delay and inverse responses with the number of stages in the whey ultrafiltration process

demonstrate a trade-off between process performance and control performance. This trade-off should be

considered during process and controller design to maximize the economic return from the production

of whey protein concentrates.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

In many industrial membrane processes such as waste water

treatment and dairy processing, the quantity and the composition

of the final product stream are affected by theflowrateand compo-

sitionof the feed, which fluctuate based on the upstreamprocesses

thatgeneratethe feed.An automatic controller canbe implemented

to minimize the effects of the variations in feed flowrate and com-

position on the product stream. Such an action by an automatic

controller is called automatic control.

While it is common to separate process design and controller

design into two independent and sequential steps, many existing

studies [1,2] have noted that such an approach may lead to designs

thatare difficultto control,resulting in a pooreconomic return fromprocess operations. The optimal return from chemical processes is

more easily achieved by integrating process design with controller

design [3–5]. Dynamic operability analysis is one of the tools that

can be used to facilitate the integration of the two design steps [6].

Dynamic operability of a process refers to the best performance

that an automatic controller can achieve for the given design.

Dynamic operability analysis allows engineers to rapidly assess

∗ Corresponding author. Tel.: +61 2 9385 9843; fax: +61 2 9385 5456.

E-mail address: [email protected] (D.E. Wiley).

the achievable control performance of a process design before the

actual controller is implemented. However, its application in mem-

brane processes has not been thoroughly explored in the literature.

Dynamic operability analysis of membrane processes requires

an understanding of the dynamic behavior. Existing studies of the

dynamic behavior of membrane processes have mainly focused on

either the relationship between permeate flowrate and transmem-

brane pressure[7–9], or therelationshipbetween retentate concen-

tration and feed flowrate of single stage processes [10]. They pro-

videno information about howthe productstream froma multiple-

stage process is affected by variations in the flowrate and compo-

sition of the feed stream. In addition, the ability of an automatic

controller to minimize the effects of the variations in feed flowrate

andcompositionon theproductstreamhas received little attention.The aim of this paper is to analyze the dynamic operability

of a multiple-stage membrane process, using continuous ultra-

filtration of whey as a case study. Whey ultrafiltration is chosen

because existing studies only consider the optimization of pro-

cess and controller designs independently of one other [11–14].

Whey ultrafiltration is also an interesting large-scale process to

studybecause the potential interactions of thestages orloopsin the

process train might affect the performance and operability of the

process.A study of thedynamicoperabilityof whey ultrafiltration is

therefore a useful illustrationof howprocessand controller designs

can be integrated to improve productivity and process economics.

0376-7388/$ – see front matter © 2008 Elsevier B.V. All rights reserved.

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K.W.K. Yee et al. / Journal of Membrane Science 320 (2008) 280–291 281

In this paper, different process designs for the whey ultrafiltration

process are created by adding more stages or loops to the process

train. For each process design, features of the dynamic behavior

of the product stream and their effects on the performance of an

automatic controller are identified. The speed with which an auto-

matic controller could minimize the effects of the variations in feed

flowrate and composition on the product stream is also examined.

2. Case study: whey ultrafiltration process

2.1. Process description

Ultrafiltration (UF) is used for the production of whey protein

concentrates (WPC) from whey, the by-product of cheese making.

The retentate stream from the UF process is spray dried to obtain

WPC. Commercial WPC products consist of 35% WPC (containing

35%crude protein) or 80% WPC (containing 80% crude protein) [15].

They are produced from whey UF processes with 3 and 12 stages,

respectively [16]. UF processes with numbers of stages other than

3 or 12 produce retentate streams containing 35–80% crude pro-

tein which are not currently marketable. However, process designs

consisting of 1–12 stages are considered in this paper to determinethe generic performance of automatic control across all stages of

the process, andto gain additional insights into the practical impli-

cations of the process design.

Fig. 1 shows the schematic diagram of a general whey ultrafil-

tration process with any number of stages up to a maximum of

12. The last stage is labelled as ‘Stage N ’, which represents the total

number of stages beingconsidered. For example, N = 12 represents

a whey UF process with 12 stages. The whey UF process shown in

Fig. 1 without any controllers is called the open-loop process, in

contrast to the closed-loop process after an automatic controller is

implemented [17].

During the industrial production of 80% WPC, Stage 4 of the 12-

stage whey UF process is usually by-passed at start-up [18]. Stage

4 is brought into operation when the fouling of the membranes

reduces the permeation to a level where the active filtration area

from the other stages is too small to achieve the desired crude pro-

tein concentration in the retentate. In addition, diafiltration (DF)

water is usually supplied to Stage 10 of the 12-stage industrial pro-

cess. In order to preserve these characteristics, a whey UF process

consisting of four stages will not be considered in this paper. Stage

4 will also be by-passedfor whey UF processes consisting of five or

more stages. DF water is, however, mixed with the feed stream to

Stage 10 when N ≥ 10.

Fig.1. Schematic diagram of a wheyultrafiltrationprocess. Note: Flowrate symbols

used in this paper are labelled in the diagram.

2.2. Process variables

Theprocess operating parameters that are adjusted by the auto-

matic controller are called manipulated variables. These variables

are manipulated to ensure that the WPC product is maintained

within its desired specifications. Manipulated variables considered

in this paper are the diafiltration water flowrate, retentate recycle

andpermeaterecycle flowrates, which are expressedin dimension-less forms as shown below:

• Diafiltration ratio (Q DF): the ratio of diafiltration water flowrate

(qDF) (supplied toStage10 when there are more than 10 stages in

the process) to fresh feed flowrate (qF):

Q DF =qDF

qF(1a)

• Permeate recycle ratio (Q P,cyc): the proportion of the combined

permeate recycled and mixed with the fresh feed:

Q P,cyc =qP,cyc

qP,cyc + qP(1b)

• Retentate recycle ratio (Q R,cyc): the proportion of the retentatestream from each stage (i, where i = 1, 2, . . . , N ) being recycled

and mixed with the feed of that stage:

Q R,cyc =qR,i,cyc

qR,i,cyc + qR,i(1c)

The subscript i is omitted from Q R,cyc because the retentate recy-

cle ratio from each stage is assumed to be identical in this

paper.

The disturbance variables of the whey UF process consist of

the flowrate (qF) and the composition of the fresh sweet whey.

The latter is expressed in terms of the total solids concentra-

tion (TSF%) and true protein concentration (proteinF%). The total

solids in whey is the sum of all the non-aqueous components inthe whey, namely, true protein, non-protein nitrogen (NPN), lac-

tose, fat and ash. True protein and NPN are collectively known

as crude protein. WPC is sold on the basis of crude protein

content.

The desired specifications of the retentate stream for WPC pro-

duction are defined in terms of the following output variables (or

controlled variables):

• Total solids concentration in the retentate stream (TSR,N %):

TSR,N % =

5

n=1

C R,N,n =q(TS)R,N

qR,N (2a)

where n ranges from 1 to 5 to represent the five components

(true protein, NPN, lactose, fat and ash) of thetotal solidsof whey.q(TS)R,N is the total solids flowrate in the retentate and qR,N is the

retentate flowrate at the end of the process train.• Percentage of crude protein within the total solids on a dry basis

(WPCR,N %):

WPCR,N % =

C R,N of true protein and NPN

TSR,N %

=q(crude protein)R,N

q(TS)R,N (2b)

where q(crude protein)R,N is the flowrate of crude protein (true

protein and NPN) in the retentate. The retentate stream from the



282 K.W.K. Yee et al. / Journal of Membrane Science 320 (2008) 280–291

final stage of the ultrafiltration plant is spray dried and sold as

WPC powder.• Volume concentration ratio (VCR)—the ratio of fresh feedflowrate

(qF) tothat of the retentate stream(qR,N ) atthe end of the process

train:

VCR =qF

qR,N (2c)

3. Methodology and assumptions

3.1. Dynamic behavior of the whey UF process

Before the dynamic behavior of the entire whey UF process can

be determined,the effects of fluctuations in thefeed andchanges in

the recycle or diafiltration ratios on the concentration of the reten-

tate leaving a single UF membrane channel need to be identified.

Wiley and Fletcher [19] developed a computational fluid dynamics

(CFD) model to solve the coupled mass and momentum balances

that describe the behavior of the fluid in a single membrane chan-

nel and obtained an estimate of the concentration profiles. For a

membrane channel 2.7 m long and 1 mm high, around 3000 cells

were used in the computational mesh for solving the mass and

momentum balance equations. This approach is too computation-

ally intensive for the simulation of an entire whey UF process using

current computing technology and software.

The computation can be simplified by decoupling the mass

and momentum equations [20,21], if the viscosity of the fluid and

the concentration of the solutes at the membrane surface remain

approximately constant over the length of the channel. These con-

ditions can be justified for whey ultrafiltration because the solutes

have low diffusivities (in the order of 10−11 m2 s−1) [16] and a

constant composition gel-like layer is observed on the membrane

surface after initial start-up [22].

Even under such assumptions, however, the computation

involved is still too intensive if the decoupled mass and momen-

tum equations of every membrane channel of the entire whey

UF process are solved whenever the flowrate or concentration of the feed stream is changed. Instead, a transfer function model to

describe the dynamic relationship between the retentate and feed

concentrationsof a single membrane channelwas developed. Time-

dependent feed concentrations varying between two values (the

lower and upper bounds of their expected values) that are ran-

dom in duration, known as pseudo-random binary signals (PRBS),

were introduced to the membrane channel. Based on the PRBS, the

decoupled mass and momentum equations were solved to obtain

the time-dependent retentate concentrations. The transfer func-

tion between the input PRBS and the corresponding response in

retentate concentrationswas thendeterminedusing system identi-

ficationtools. In thispaper,MATLAB® SystemIdentification Toolbox

[23] was used to estimate the transfer function by minimizing the

sum of squares of the errors between the predicted response fromthe transfer functionand the actual response obtained by the input

PRBS.

An example of the PRBS for the total solids concentration in the

feed stream (C (TS)F) and the corresponding response of the reten-

tate concentration at the end of the first stage ( C (TS)R,1) is shown

in Fig. 2. The transfer function for this case is given by

C (TS)R,1(s) =1.03 e−0.72s

0.576s+ 1 C (TS)F(s) (3)

Once the transfer function model of a membrane channel was

developed, the dynamic models for the UF process shown in Fig. 1

with different numbers of stages (N = 1, 2, 3, 5, . . . , 11, 12) can be

obtained by connecting the single channel models according to

the network and configurations described in Yee et al. [18]. This

Fig. 2. The PRBS for the total solids concentration in the feed stream and the corre-

sponding responseof the retentate concentration froma singlemembrane channel.

is the approach used in this paper. If such a dynamic model is

too difficult to develop, an approximate dynamic model of the

UF process can be estimated based on the design flowsheet with

N = 1, 2, 3, 5, . . . , 11, 12, using the algorithm outlined in Weitz and

Lewin [24]. However, dynamic models estimatedfrom design flow-

sheets are often much less accurate than those from single channel

models.

The dynamic models of the UF process, whether they are

obtained by system identification tools or design flowsheets, are

linear models. The linear dynamic models obtained are valid in

the neighborhood of the nominal operating condition of the whey

UF process shown in Table 1 because the automatic controller

is designed to minimize the deviations of the output variables(defined by Eqs. (2a)–(2c)) fromtheir nominal operating conditions

following changes in feed flowrate and concentration. The effects

of long-term fouling on permeate flux were not considered in this

study because these effects are insignificant in relation to the time

frame being considered in this paper: after 6500s, the permeate

flowrate is only reduced by around 2% [18].

3.2. Qualitative dynamic operability analysis

Usingthe dynamic modelsof the UFprocesswithdifferent num-

bers of stages upto a maximum of12,thedynamicresponses of the

output variables were obtained after a step change in one of the

input or disturbance variables. While other input functions could

be used, step changes are widely used to evaluate the features of dynamic responses of output variables for many different types of

Table 1

Nominal operating condition and the step changes introduced to generate dynamic

responses (open-loop responses)

Parameters Unit Base case values Step change (% change from nominal)

Manipulated inputs

Q DF v/v 0.006 0.006 → 0.005 (− 17%)

Q P,cyc v/v 0.25 0.25 → 0.225 (−10%)

Q R,cyc v/v 0.96 No step changes

Disturbances

qF L/h 50,000 50,000 → 52,500 (+5%)

TSF% %, w/w 6.0 6.0 → 6.5 (+8%)

proteinF% %, w/w 0.65 0.65 → 0.75 (+15%)




processes [17,25]. The dynamic responses obtained after a change

in input or disturbance are called open-loop responses when no

automatic controller is implemented. In this study, the step change

was introduced after the whey UF process had achieved its steady

state under the nominal operating condition shown in Table 1.

A previous investigation of the steady state operation of the

whey UF process byYee et al.[18] has demonstrated thatthe effects

of the input and disturbance variables on the outputs vary signif-icantly throughout the feasible range of input variables and the

expected range of disturbances. Hence, small step sizes, as shown

in Table 1, were introduced to produce the dynamic responses of

the output variables in the neighborhood of the nominal operating

condition.

To compare the dynamic responses of the output variables, the

values of the output variables were normalized. For example, the

normalized WPCR,N % is expressed as

WPCN R,N %(t ) =

WPCR,N %(t )−WPCR,N %(t = 0)

WPCR,N %(t → ∞)−WPCR,N %(t = 0) (4)

where thenormalized variableis indicated by thesuperscript N .The

normalized variables are equal to 0 at the initial steady states (i.e.

t = 0), and approach 1 as the variables approach their final steady

states (i.e. t → ∞).

Features of the open-loop responses (in terms of the normal-

ized variables) were used to provide qualitative insights into the

performance of automatic control of the whey UF process con-

sisting of different numbers of stages (N = 1, 2, 3, 5, . . . , 11, 12).

Automatic controllers usually consist of feedback and feedforward

components. The feedback component can be found in virtually all

controllers, whereas the feedforward component is usually used

in combination with feedback [17]. The performance of feedfor-

ward control is limited only by the accuracy of the dynamic models

from which the controller is developed [6]. This paper therefore

focuses on the performance of feedback control, which is deter-

mined by the quality andthe speed of thedynamicresponses as the

output variables return to their desired specifications after adjust-

ments of the inputs (i.e. the closed-loop responses of the outputvariables). If the quality and speed of the closed-loop responses

are poor, the amount of retentate from the whey UF process

which meets the desired specifications will be reduced. The eco-

nomic return from the production of WPC will be decreased as a

result.

The quality of a closed-loop response is related to the extent

of oscillations of the output variables in the presence of automatic

controllers, while the speed is related to the time taken for the

output variables to return to their desired specifications under

automatic control. Features of the open-loop responses such as

effective time delay and inverse responses can be used to pre-

dict the quality and speed of the closed-loop responses, before any

automatic controller is implemented [17,26]. These key features are

illustrated in Fig. 3.Effective time delay (t d) is defined as the time before the

responses of thenormalized outputvariables can be observed after

a step change in input. It is often used in the analysis of dynamic

responses [27], especially when they are obtained from processes

with extensive recycle [28] such as the whey UF process in Fig. 1.

The longer the effective time delay of an open-loop response, the

longer it takes for the feedback controller to minimize the effects

of a disturbance by adjusting the manipulated variables. It will

also take longer for the output variables to return to their desired

specifications.

An inverse response is observed when the initial open-loop

response lies in the opposite direction to the final steady state,

or when the direction of short-term dynamics is opposite to that

of the long-term dynamics (as illustrated in Fig. 3). For an output

Fig.3. An illustrationof thetimedelay andinverseresponseof a normalized output

variable ( yN ) after a unit step change in input.

variable that shows an open-loop inverse response, the quality of

the closed-loop response is often poor because the output vari-

able will not return directly to its desired specification under

feedback control. There will be inverse response or extensive oscil-

lations before the output variable return to its desired specification

[29].

3.3. Quantitative dynamic operability analysis

While effective time delay enables engineers to gain qualita-

tive insights into the speed of closed-loop response of the output

variables, settling time can be used to quantitatively compare the

effects of effective time delay and inverse response on the best

achievable speed of automatic controlfor differentprocess designs.Settling time is the time required for the feedback controller to

restore the output variables to within a given tolerance of their

desired specifications after adjustments of inputs. Settling time is

thereforebased on theclosed-loop response of theoutput variables.

In this paper, the best achievable settling times ( t ∗s ) for process

designs consisting of different numbers of stages were estimated

from the open-loop dynamic responses. The tolerance with which

settling time is defined was set to 1% of the desired specifications

of the output variables.

In this paper, the settling time (t ∗s ) is called ‘the best-achievable’

because it represents the shortest settlingtime thatcan be achieved

by a feedback controller while delivering its best performance on a

theoretical basis. Hence there is only one t ∗s value for each process

design irrespective of the number of input and output variables of the process.

For each combination of input adjustments, one method of

determining the settling time for a particular process design

involves minimizing the effects of all disturbances on the outputs.

The value of t ∗s is then the smallest of the settling times achieved

from all possible combinations of input adjustments. However, this

method of determining t ∗s is time consuming when there are many

possible combinations of input adjustments to be considered. A

simpler approach for estimating t ∗s , based on the methodology of

dynamic operability analysis developed by Zhang et al. [2], was

therefore adopted in this paper. Details of the dynamic operability

analysis are given in Appendix A. The steps involved in estimat-

ing t ∗s canbe programmed (inMATLAB® for example) and executed

automatically by a computer.




4. Results and discussion

4.1. Qualitative dynamic operability analysis

To study the performance of automatic control of the whey

UF process consisting of different numbers of stages (N =1, 2, 3, 5, . . . , 11, 12), features of theopen-loop dynamicresponses,

such as effective time delay and inverse responses, were identifiedafter a step change in one of the input or disturbance variables.

Features of the open-loop responses can be used to predict the

quality and speed of automatic control before any controllers are

implemented.

4.1.1. Effective time delay

The responses of the normalized output variables for some

selected numbers of stages (N ) after a step decrease in permeate

recycle ratio (Q P,cyc) are shown in Fig. 4. When the UF process con-

sists of only one stage (i.e. N = 1), changes in the output variables

are almost instantaneous. However, as the total number of stages

increases to 12 (i.e. N = 12), for example, the effective time delay

is increased to around 700 s (approximately 12 min). Changes in

other input variables also result in an effective time delay in theresponse of the output variables that also increases with the num-

ber of stages.

The presence of effective time delay in the open-loop responses

of the output variables indicates that there is a limit to the

speed with which the output variables can return to their desired

specifications after input variables are adjusted by feedback con-

trollers. Effective time delay increases with the retentate recycle

ratio (Q R,cyc) of the whey UF process. Fig. 5 shows the normal-

ized dynamic responses of VCR (VCR N ) after a step decrease in

Q P,cyc when the whey UF process consisting of 12 stages is oper-ating under different retentate recycle ratios. While effective time

delay is only around 200 s for Q R,cyc = 0.50 and around 250 s for

Q R,cyc = 0.75, it increases to 700 s for Q R,cyc = 0.96.

The increase in effective time delay with Q R,cyc in Fig. 5 seems

counter-intuitive at first because the increase in the actual feed

flowrate (after mixing of the feed and recycled streams) to each

stage of the UF process with Q R,cyc reduces the average single-pass

residence time within each stage ( i), which is defined by Eq. (5):

i =V i

qF,i(total) (5)

where V i is the total free volume inside the membrane channels

within stage i (where i = 1, 2, . . . , N ), and qF,i(total)is thetotal feed

flowrate to stage i after mixing of the feed and recycle streams.A decrease in i as Q R,cyc increases should therefore reduce the

time for the effects of changes in Q P,cyc, for example, to propagate

Fig. 4. Dynamic responses of the normalized output variables for N = 1, 3, 7, 8, 10 and 12 after step decrease in Q P,cyc with Q P,cyc = −0.025 (10% decrease).




Fig.5. Normalizeddynamicresponsesof VCR(VCR N )for N = 12aftera stepdecrease

in Q P,cyc with Q P,cyc = −0.025 (10% decrease), when theprocessis operatingunder

Q R,cyc of 0.50, 0.75 and 0.96.

through the process, and hence decrease the effective time delay.

However, effective time delay is also dependent on the effect of

the recycle stream on the dynamics of the process. Based on the

dynamic process model for a general multiple-stage process with

recycle, AppendixB shows thateffective timedelay in factincreases

with recycle ratio.

Despite causing slow process dynamics, a high value of Q R,cyc

is often used in practice in order to maximize the feed flowrate to

each stage of the UF process and to enhance the recovery of crude

protein. A high feed flowrate also enhances the separation of crude

proteinand reduces membrane fouling. As the degree of separation

of crude protein increases, WPCR,N % increases along with the pro-

cess performance. However, the performance of automatic control,

in termsof the speed of closed-loop responses, is adverselyaffectedbecause of the increase in effective time delay. Hence, there is a

trade-off between the changes in process performance and control

performance on the economic return for the production of WPC.

Time delay in the open-loop responses of the output variables

increases not only with Q R,cyc, but also with the total number of

stages (N ) of the whey UF process. This is because the effects

of changes in input or disturbance variables need to propagate

through the intermediate stages of the process. This increase of t dwith N can also be confirmed using the dynamic process model as

shown in Appendix B. Due to the increase in effective time delay,

the speed of automatic control is reduced as N increases.

In addition to reducing the speed of automatic control, the

increase in effective time delay with retentate recycle ratio (Q R,cyc)

and numberof stages(N ) makes the design of automatic controllersdifficult for the whey UF process in Fig. 1. Chodavarapu and Zheng

[30] and del-Munro-Cuellar et al. [31] have shown that it is difficult

to design feedback controllers for processes with recycle streams

and effective time delay. In particular, it is difficult to guarantee

that the output variables are able to return to their desired spec-

ifications (or closed-loop stability). Although advanced controller

design techniques such as Smith predictors [32] and recycle com-

pensators [33] are able to improveclosed-loop stability, they do not

remove the limitations of the speed of closed-loop response due to

effective time delay.

4.1.2. Inverse responses

The largest inverse response (see Fig. 6) was observed for the

concentration of the total solids in the retentate stream (TSN

R,N %)

Fig. 6. Normalized dynamic responses of TSR,N (TS N R,N %) for N = 1, 6, 7, 8, 10 and 12

after a step increase in proteinF% with proteinF = 0.1% (15% increase).

after a step increase in true protein concentration of the feed

(proteinF%) when the UF process consists of seven stages ( N = 7).

Aninverseresponsein TSN PR % was also found forN = 6. The effective

time delay is also reduced significantly when inverse responses are

found in Fig. 6.

A step change in proteinF% also causes an inverse response in

the WPC content of the retentate (WPCN R,N %) (N = 7, 9 and 10 in

Fig. 7). Similarly, a step change in total solids concentration in the

feed (TSF%) produces an inverse response in TSN R,N % for N = 7 and 8

in Fig. 8.

The open-loop inverse responses in Figs. 6–8 indicate that the

output variables will show inverse responses or extensive oscil-

lations after automatic controllers are implemented. The bestachievable dynamic responses of the output variables with auto-

matic control (closed-loop responses) can be obtained basedon the

internal model control (IMC) framework (see Appendix A). With

the IMC controller (see Fig. A.1), the best achievable closed-loop

responsesof TSN R,N % after a step increase in proteinF% were obtained

in Fig. 9, when the whey UF process consists of 7 and 12 stages

Fig. 7. Normalized dynamic responses of WPCR,N % (WPCN R,N %) for N = 1, 3 ,7, 9, 10

and 12 after a step increase in proteinF% with proteinF = 0.1% (15% increase).




Fig. 8. Normalizeddynamic responses of TSR,N % (TSN R,N %)for N = 1,3, 7, 8,10and 12

after a step increase in TSF% with TSF = 0.5% (8% increase).

(i.e. N = 7 and 12). The large open-loop inverse response in TSN R,N %

when N = 7 corresponds to the initial oscillations and the large

closed-loop inverse response as TSN R,N % returns to its desired spec-

ification (i.e. TSN R,N % = 0). The closed-loop oscillations and inverse

responses are not desirable in the automaticcontrolof the whey UF

process, because the output variables will suffer from significant

deviations before returning to their desired specifications. These

transient deviations of the output variables fromtheir desired spec-

ifications will adversely affect product quality. Experience from the

operation of the industrial whey UF process confirms thatthe direc-

tion of closed-loop responses is sometimes opposite to that of the

steady state movement [16].

The open-loop inverse responses observed in TSN R,N % for N = 6

and 7 are due to interactions of two competing factors [17], namely,

the total solids flowrate (q(TS)R,N ) and the retentate flowrate at theend of the process train (qR,N ). An increase in total solids flowrate

increases TSR,N %, whereas an increase in the retentate flowrate

decreases TSR,N % (Eq. (2a)). If therelative increase of thetotal solids

flowrate is initially lower than that of the retentate flowrate after a

step increase in proteinF%, but becomes higher and exceeds that of

Fig. 10. Ratios of TS flowrates (q(TS)R,N ) and total flowrates (qR,N ) to their initial

steady states (t = 0) for N = 6, 7 and 8 after a step increase in protein F% with

proteinF = 0.1% (15% increase).

the retentate flowrate as time progresses, an inverse response will

be observed from TSN R,N %.

Fig. 10 demonstrates how the relative increases of the total

solids and retentate flowrates compare with each other by con-

sidering the ratios of q(TS)R,N and qR,N to their initial steady states

(i.e. q(TS)R,N /q(TS)R,N (0) and qR,N /qR,N (0), respectively) after a step

increase in proteinF%. For N = 6 and 7, the solid lines (qR,N /qR,N (0))

are initially above the dotted lines (q(TS)R,N /q(TS)R,N (0)). The two

lineslater cross eachother, thereby generatingthe inverse response

in TSN R,N %.

In order to investigate what happens within the UF process to

result in the inverse response of TSN R,N %, the gel-polarization model

and the transport equation of the total solids through the mem-brane are considered.

Retentate flowrate at the end of the process train (qR,N ) can be

evaluated from the overallmass balance of the wheyUF process and

the gel-polarization model, assuming thatthere is no accumulation

of wheywithin themembrane modules andan equilibrium gellayer

Fig. 9. Open-loop and closed-loop normalized dynamic responses of TS R,N % (TSN R,N %) for (a) N = 7 and (b) N = 12 after a step increase in protein F% with proteinF = 0.1%

(15% increase).




has been established on the membrane surface [34,35]:

qR,N = qF − (1− Q P,cyc)

N i=1

qP,i

− qDF (if N ≥ 10) (6a)

where

qP,i = ki Ai ln C g − C P,i

C F,i − C P,i

≈ ki Ai ln

C gC F,i

(6b)

For a feed stream to a particular stage i (where i = 1, 2, . . . , N ),

ki is the mass transfer coefficient determined by the Leveque cor-

relation [34], and C F,i is the total solids concentration of the feed.

Thetotal solidsflowrate at the endof the process train(q(TS)R,N )

is evaluated from the total solids mass balance and the transport

equation of the total solids through the membrane [36], again

assuming no accumulation:

q(TS)R,N = q(TS)F − (1− Q P,cyc)

N i=1

q(TS)P,i (7a)

where

q(TS)P,i = Ai

5n=1

P s,nC G, j + qP,i

5n=1

(1− n)C i,n (7b)

P s,n is the solute permeability (P s), and n the retention coefficient

of each total solids component n. In these equations, it is assumed

that the values of P s,n and n are properties of each solute, and

are independent of other operating conditions, such as pressure or

concentration. C i,n represents the average concentration difference

between the retentate and permeate for each component n at stage

i. Since the total solids in whey consists of five components (true

protein, NPN, lactose, fat and ash), the value of n ranges from 1 to

5.

Since true proteinhas the lowest permeability (P s) andthe high-

est retention ( ) ofall thetotalsolids components, a step increase intrueproteinconcentration of the freshwhey feed(proteinF%),while

maintaining TSF% constant, decreases the amount of total solids in

the permeate from each stage (q(TS)P,i) (Eq. (7b)), resulting in an

increase in the total solids flowrate at the end of the process train

(q(TS)R,N ) (Eq. (7a)). Thedecrease in q(TS)R,N increasesthe feed con-

centration to each stage (C F,i), and hence decreases the permeate

flowrate (qP,i) (Eq. (6b)). The decrease in qP,i from each stage cor-

responds to an increase in the retentate flowrate at the end of the

process train (qR,N ) (Eq. (6a)).

However, the relative increase of retentate flowrate (qR,N ) fol-

lowing a step increase in proteinF% is smaller than that of the total

solids flowrate (q(TS)R,N ) when the whey UF process consists of a

small number of stages (i.e. small values of N ). In addition, since it

takes time for ¯C N,n to reach itsnew steady state after a step increasein proteinF%, the relative increase of the total solids flowrate can

be lower than that of the retentate flowrate initially. The rela-

tive increase of the total solids flowrate then becomes higher and

exceeds that of the retentate flowrate as the process approaches

the final steady state. An inverse response is therefore observed,

e.g. when N = 6 and 7 in Fig. 6.

As N increases from 6 to 8, more permeate is removed and

the total solids concentration of the feed to the last stage ( C F,N )

increases. For a given relative change of C F,N (i.e. dC F,N /C F,N ), the

relative change of permeate flowrate from the last stage (qP,N )

increases with C F,N , and hence with N is

dqP,N

qP,N =

1

ln C g − ln C F,N dC F,N

C F,N (8)

On the other hand, the effect of the increase in C N,n (where

n = 1–5) on the total solids flowrate in the permeate from the last

stage (q(TS)P,N ) (Eq. (7b)) becomes less significant as the proportion

of crude protein within the total solids increases with N . The rel-

ative increase of retentate flowrate at the end of the process train

(qR,N ) thereforebecomeshigherthan thatof thetotal solids flowrate

q(TS)R,N as N increases from 6 to 8. In other words, the ratio of qR,N

to its initial steady state after a step increase in proteinF% increasesmore than that of q(TS)R,N . The ratio of qR,N to its initial steady state

therefore exceeds that of q(TS)R,N for N = 8 (see Fig. 10).

When N is large (e.g. N = 12), the relative increase of the reten-

tate flowrate (qR,N ) after a step increase in proteinF% is always

higher than that of the total solids flowrate (q(TS)R,N ) from the

initial to the final steady states. No inverse response in TSN R,N % is

therefore observed when N = 12.

Even though UF processes with numbers of stages other than 3

or 12 do not operate commercially, the above example illustrates

that the study of the effects of process design on the performance

of automatic control can be used to assess the possibility of oper-

ating the whey UF process with seven stages, for example, should

the need arise. For a whey UF process design with seven stages

(N =

7), a step increase in proteinF%, for example, produces aninverse response of TSR,N %. The open-loop inverse response in turn

indicates a poor quality of the closed-loop response, a conclusion

that can be drawn even before any automatic controller is imple-

mented.

4.2. Best achievable settling time, t ∗s

Toquantitativelycompare the speedof the closed-loop response

of the output variables before any feedback controller is imple-

mented in the whey UF process, the best achievable settling times

(t ∗s ) were estimated when the process train consists of different

numbers of stages (N = 1, 2, 3, 5, . . . , 11, 12). Since t ∗s is the short-

est settling time achievable for a given process design, there is only

one t

∗

s value for each N . The values of t

∗

s for some selected valuesof N are shown in Table 2. A settling time of 0 min for N = 1 is not

achievable in practice, but it shows that the closed-loop response

for a single stage is the fastest. The output variables from a single

stage process respond almost instantly after step changes in inputs,

as shown in Fig. 4.

In general, the values of t ∗s increase with the total number of

stages (N ), mainly due to increases in the effective time delay ( t d)

of theopen-loop responsesof the output variables withN . Although

the effective time delay of the responses when there are seven or

eight stages are generally longer than that when there are three

stages (seeFig.4), thevaluesof t ∗s for N = 7 and8 are slightly smaller

than that of N = 3. In particular, the value of t ∗s reaches a minimum

at N = 7.

When the whey UF process consists of seven stages (N = 7),

inverse responses in TSN R,N % and WPCN

R,N % are observed after step

increases in TSF% and proteinF% (Figs. 6–8). The presence of inverse

responsessignificantly reduces the effective timedelay of the open-

loopresponses. Since the estimationof t ∗s considersthe effects of all

Table 2

Best achievable settling times (t ∗s ) for the whey UF process

Total number of stages, N t ∗s (min)

1 0

3 35

7 27

8 30

10 41

12 60




input and disturbance variables on the outputs, the reduced effec-

tive time delay would increase the speed of feedback controllers

(i.e. the value of t ∗s decreases when N = 7). However, the open-

loop inverse responses for N = 7 also indicate poor closed-loop

responses.

While thespecifications of theretentatestream forWPC produc-

tion determines the process performance of the whey UF process,

the control performance is related to the effects of effective timedelay and inverse responses on t ∗s . When N increases from 1 to 12,

there exists a trade-off between process performance and control

performance that affects the economic return from the produc-

tion of WPC. For a UF process consisting of 12 stages, the feedback

controller is slower to respond to changes in feed flowrate and

concentration, and to restore the output variables to their desired

specifications than a process consisting of seven stages. However,

the output variables from the seven-stage UF process will suffer

from extensive oscillations after the feedback controller is imple-

mented.

5. Conclusions

Dynamic operability of multiple-stage membrane processes for

continuous whey ultrafiltration are studied in this paper. The per-

formance of automatic control for process designs consisting of

different numbers of stages can be assessed before an actual con-

troller is implemented.

Dynamic operability analysis firstly requires dynamic models to

describe the response of output variables after changes in input

and disturbance variables. The dynamic models can be obtained

from either system identification tools or design flowsheets. Once

the dynamic models are available, the dynamic operability analy-

sis provides qualitative insights into the performance of automatic

control for different process designs, and a quantitative mea-

sure of the time required to restore the output variables to their

desired specifications by the automatic controller. The combined

approaches enable a comparison of the performance of automatic

control for different process designs.Dynamic operability analysis of the whey ultrafiltration pro-

cess without any automatic controllers identifies the presence of

effective time delay and inverse responses of the output variables,

which indicates poor performance after automatic controllers are

implemented. While a high retentate recycle ratio is used to min-

imize membrane fouling, it also increases effective time delay

significantly. The longer the effective time delay, the slower the

automatic controller is in restoring the output variables to their

desired specifications. A trade-off therefore exists between the

process performance of minimizing fouling and the control per-

formance of minimizing the time required for the output variables

to return to their desired specifications.

Inverse responses observed in the output variables without any

automatic controllers reduce the quality of automatic control of the whey ultrafiltration process. The output variables will suffer

frominverse responsesor extensive oscillations before returning to

their desired specifications, when automatic controllers are imple-

mented. Since a whey ultrafiltration process design consisting of

seven stages shows the most significant inverse response in the

output variables, such a design is expected to experience extensive

oscillations when automatic controllers are implemented.

The quantitative approachof the dynamic operability study also

demonstrates the trade-offbetween process performance, in terms

of product specifications, and control performance of the whey

ultrafiltration process consisting of different number of stages. The

trade-off between process performance and control performance,

either with the retentate recycle ratio or the number of stages of

the process, should be considered during process and controller

designs. Sucha consideration is important to maximize the amount

of the final retentate stream that meets the desired specifications,

and hence the economic return from the production of whey pro-

tein concentrates.

The findings in this paper are based on a whey ultrafiltration

process under the given operating conditions. Since the effects of

process operating parameters, feed flowrate and composition on

the specification of the final retentate stream vary with differentoperating conditions, the study would need to be repeated for other

operating conditions. Although developed based on a whey ultra-

filtration process, the methodology in this paper can be adapted

to other membrane processes with multiple stages or loops. The

methodology can also be applied to an existing plant to evaluate

how well the existing controller is able to minimize the effects of

disturbance experienced by the plant, e.g. the best achievable set-

tling time (t ∗s ) from this paper could be compared to the actual

settling time achieved on the plant. Information on the perfor-

mance of the existingcontrolleris useful to determine if anyprocess

improvements are required,or if the existingplant is ableto achieve

a reasonable economic returnfrom the production of whey protein

concentrates, should there be any changes in product throughput

or specifications.

Acknowledgements

The authors gratefully acknowledge the support of an Australian

Research Council Discovery Grant. One of the authors (KWKY)

would also like to acknowledge the support of an Australian Post-

graduate Award (APA) and a Faculty of Engineering Scholarship

from the University of New South Wales.

Appendix A. Estimation of the best achievable settling time

(t ∗s )

1. The dynamic models of the output variables of the process with

respect to the input variables (G(s)) andthe disturbance variables

(Gd(s)) are obtained using either system identification tools [23]or the design flowsheet [24]. For a process with multiple input

and output variables, G(s) and Gd(s) are transfer function matri-

ces that contain the individual transfer functions between each

input and output variable [6]. Transfer functions are used to rep-

resent the relationship between the time-dependent variations

of the input and output variables in the Laplace domain (using

Laplace transforms) [17].

2. While the dynamic models determined by system identifica-

tion tools or design flowsheet are expressed in terms of transfer

functions, they need to be represented as state-space forms for

all-pass factorization. The state-space representation of G(s) can

be represented as follows:

d x(t )

dt = AG x(t )+ BGu(t )

y(t ) = C G x(t )+DGu(t )(A.1)

where x, u and y represent the state, input and output variables,

respectively [17,26]. The values of the matrices AG, BG, C G and DG

can be determined from the transfer function G(s) using a state-

space realization method (such as function tf2ss in MATLAB®

Control Toolbox.) The relationship between the transfer function

and the state-space representation is shown below:

G(s) = C G(sI − AG)−1BG +DG (A.2)

3. All-pass factorization of G(s) into N (s) and M (s) (i.e. G(s) =

N (s)M (s)) is performed. Details of how all-pass factorization is

performed are given in Zhang et al. [2]. The best achievable

closed-loop response of the output variables can be estimated




Fig. A.1. IMC structure based on all-pass factorization.

from N (s) and M (s) based on the internal model control (IMC)

framework [37]. The structure of IMC based on all-pass factor-

ization is shown in Fig. A.1.

4. Based on the IMC framework, the function T yd(s) is defined as

T yd(s) = Gd(s)−N (s)M (s)M (s)−1N (0)−1Gd(s) (A.3)

T yd(s) is called the disturbance sensitivity function, which indi-

cates how much the effects of disturbances on the output

variables can be minimizedby feedback control systems. For dis-

turbances fluctuating at a frequency ω , the function T yd( jω) isevaluated (e.g. using the MATLAB® command freqresp). (Note: j

represents the imaginary unit of complex numbers, i.e. j2= −1.)

5. The extent to which the effects of disturbances can be min-

imized (which is referred to as ‘disturbance rejection’ in the

context of process control) can be determined quantitatively

by the maximum singular value of T yd, or (T yd). The smaller

the value of (T yd), the better the effects of disturbance rejec-

tion. The maximum singular value of T yd at frequency ω, or

(T yd( jω)), is evaluated (e.g. using the MATLAB® command

norm). Values of (T yd( jω)) are then converted to units of decibel

(dB):

(T yd( jω)) (dB)= 20log10( (T yd( jω))) (A.4)

6. Thefrequency at which (T yd( jω)) = 0.7071dB, that is the cross-

over frequency (ωc) is determined. The best achievable settling

time (t ∗s ) can then be estimated from ωc as follows:

t ∗s =1

ωc(A.5)

Estimating t ∗s as an inverse of ωc is a rule-of-thumb that can be

used to compare the achievable speed of automatic control for

different process designs [38].

Steps 2–6 above can be programmed (inMATLAB® for example)

and executed automatically, once the dynamic models have been

obtained in Step 1.

Appendix B. Determination of effective time delay from the

dynamic process model

An input–outputrepresentation of a generalmultiple-stage pro-

cess with recycle is shown in Fig. B.1. The variables u and y are

the input and output variables of the process, respectively. Process

dynamics is represented in terms of the transfer function Gp(s). It

is assumed that the transfer function for the dynamics related tothe recycle stream (Gr(s)) is equal to the recycle ratio (r ). t D,p is the

time delay of the process, and t D,r the time delay of the recycled

stream.

Based on Fig. B.1, the transfer function for the first stage (G1(s))

is

G1(s) =Gp(s) e−t D,p s

1− rGp(s) e−(t D,p+t D,r)s (B.1)

A Taylor series expansionaround theterm e−t D,rs= 0 [28] gives:

G1(s) = Gp(s) e−t D,p s(1+ rGp(s) e−(t D,p+t D,r)s

+ r 2G2p(s) e−2(t D,p+t D,r)s

+ · · · ) (B.2)

Thetransfer functions forthe otherstages (i = 2, . . . , N ) are similar.

In order to evaluate the effect of recycle ratio on effective timedelay based on dynamic process model, a system with three stages

is considered. The overall transfer function of the system (G(s) =N

i=1Gi(s), where N = 3) can be approximated from Eq. (B.2). Asan

example, GN =3(s) is approximated to the 10th term (or truncated at

the G10p (s) term) as shown:

GN =3(s) ≈ Gp(s) e−t D,p s(1+ 3rGp(s) e−(t D,p+t D,r)s

+· · · + 55r 9G9p(s) e−9(t D,p+t D,r)s) (B.3)

Toevaluatethe effective time delayof thesystemwith3 stages, a

simple dynamic process modelof Gp(s) = 1/(s+ 1) and t D,p = t D,r =

5 s are considered. From Eq. (B.3), the dynamic response of the out-

putin termsof itsnormalized variable( yN

) canbe determined aftera step increase in u. Thedynamicresponses of yN withrecycle ratios

(r ) of 0.50, 0.75 and 0.96 are shown in Fig. B.2.

Fig. B.2 indicates that the effective time delay of yN increases

from 15s for r = 0.5 to around 40s for r = 0.96. As r increases,

the coefficients of all the terms in Eq. (B.3) increase. If r increases

from r i to r f , for example, the ratios of the coefficients of the

G2p(s) and the G10

p (s) terms are r f /r i and r 9f /r 9

i , respectively. Hence,

the coefficients of the higher order terms (e.g. G10p (s)) increase

more significantly than those of the lower order terms (e.g. G2p(s))

as r increases. Since the higher order terms are also associated

with higher orders for the exponential terms in Eq. (B.3), the nor-

malized variable will take a longer time to deviate significantly

from its initial value of 0, resulting in a longer effective time

delay.

Fig. B.1. An input–output representation of a general multiple-stage process with recycle.




Fig. B.2. Dynamic response of a normalized output variable ( yN ) after a unit step

increase in u, based on Eq. (B.3) when N = 3 and r = 0.50, 0.75 and 0.96.

A similar approach can be used to demonstrate the effect of the

number of stages (N ) on effective time delay. For N = 5, the transfer

function for the system can be approximated up to the 10th term

in the form:

GN =5(s) ≈ Gp(s) e−t D,ps(1+ 5rGp(s) e−(t D,p+t D,r)s

+· · · + 715r 9G9p(s) e−9(t D,p+t D,r)s) (B.4)

Based on Eq. (B.4), the dynamic response of yN after a step

increase in u when r = 0.96is shown in Fig. B.3. The assumptions of

Gp(s) = 1/(s+ 1)and t D,p = t D,r = 5 s areagainapplied inevaluating

the dynamic responses of yN .

As seen from Fig. B.3, the effective time delay is increased fromaround 40s for N = 3 to around 70s after two more stages are

added. The ratios of the coefficients of the G2p(s) and the G10

p (s)

terms in Eq. (B.4) to those in Eq. (B.3) are 1.67:1 and 13:1, respec-

tively. These ratios indicate that coefficients of the higher order

terms (e.g. G10p (s)) increase more significantly than those of the

lower order terms (e.g. G2p(s)) when the number of stages increases

Fig. B.3. Dynamic response of a normalized output variable ( yN ) after a unit step

increase in u, based on Eq. (B.4) when r = 0.96 and N = 3 and 5.

from 3 to 5. As N increases, therefore, the higher order terms

have a more significant effect on the dynamic response, corre-

sponding to a longer effective time delay from the normalized

output.

Nomenclature

A membrane area (m2)

AG, BG, C G, DG matrices for process dynamics in state-space

representation

C concentration (%, w/w)

C average concentration difference (%, w/w)

G(s) process dynamics model in terms of transfer func-

tions

k mass transfer coefficient (m s−1)

M (s), N (s) factors of Gp(s) obtained from all-pass factoriza-

tion (Appendix A)

N total number of stages (loops) in the whey UF pro-

cess (Fig. 1)

protein% true protein concentration (%, w/w)

P s solute permeability (m s−1)

q flowrate (L/h)q(crude protein) crude protein flowrate (L/h)

q(TS) total solids flowrate (L/h)

Q DF diafiltration ratio (v/v)

Q P,cyc permeate recycle ratio (v/v)Q R,cyc retentate recycle ratio for all the stages (v/v)

r recycle ratio (Appendix B)

t time elapsed after a change in input or disturbance

variable (s)t d effective time delay (s)

t ∗s best achievable settling time (s)

T yd(s) disturbance sensitivity function (Appendix A)

TS% total solids concentration (%, w/w)u input variable

V volume of membrane channel (m3)VCR volume concentration ratio

WPC% crude protein concentration of the total solids y output variable

Greek letters

retention coefficient

(.) maximum singular value

average single-pass residence time (s)

ω angular frequency (rad/s)

ωc cross-over frequency (rad/s)

Subscripts

c controller

d disturbanceDF diafiltration water

F feed

g gel layer

i stage number (i = 1, 2, . . . , N )n total solids component in whey (n = 1, 2, . . . , 5)

N the end of the process train

p process

P permeate

r recycle (Appendix B)

R retentate

Superscript

N normalized variable (Eq. (4))




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