Towards Efﬁcient Modularity in Industrial Drying A Combinatorial Optimization Viewpoint Alisina Bayati1a Amber Srivastava2 Amir Malvandi3 Hao Feng4and Srinivasa M. Salapaka1b

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Towards Efﬁcient Modularity in Industrial Drying: A Combinatorial

Optimization Viewpoint

Alisina Bayati1a, Amber Srivastava2, Amir Malvandi3, Hao Feng4and Srinivasa M. Salapaka1b

Abstract— The industrial drying process consumes approxi-

mately 12% of the total energy used in manufacturing, with

the potential for a 40% reduction in energy usage through

improved process controls and the development of new drying

technologies. To achieve cost-efﬁcient and high-performing dry-

ing, multiple drying technologies can be combined in a modular

fashion with optimal sequencing and control parameters for

each. This paper presents a mathematical formulation of this

optimization problem and proposes a framework based on the

Maximum Entropy Principle (MEP) to simultaneously solve

for both optimal values of control parameters and optimal

sequence. The proposed algorithm addresses the combinatorial

optimization problem with a non-convex cost function riddled

with multiple poor local minima. Simulation results on drying

distillers dried grain (DDG) products show up to 12% improve-

ment in energy consumption compared to the most efﬁcient

single-stage drying process. The proposed algorithm converges

to local minima and is designed heuristically to reach the global

minimum.

I. INTRODUCTION

Industrial drying is responsible for roughly 12% of the

total end-use energy used in manufacturing, equivalent to 1.2

quads annually [1]. The US Department of Energy estimates

that by implementing more efﬁcient process controls and new

drying technologies, it is possible to reduce this amount by

approximately 40% (0.5 quads/year), resulting in operating

cost savings of up to $8 billion per year [2]. Moreover, the

drying process has a signiﬁcant impact on the quality of

food products. Prolonged exposure to excessive heat can have

negative effects on the physical and nutritional properties of

the products [3].

In recent years, several more efﬁcient drying technologies

have been proposed in the literature, such as Dielectrophore-

sis (DEP) [4], ultrasound drying (US) [5][6], slot jet reattach-

ment nozzle (SJR) [7], and infrared (IR) drying [8]. These

technologies have helped improve product quality and energy

efﬁciency. Industrial drying units typically use one of these

technologies to achieve their drying goals. However, each

1Department of Mechanical Science and Engineering,

University of Illinois at Urbana-Champaign, 61801 IL, USA.

aabayati2@illinois.edu,bsalapaka@illinois.edu

2Automatic Control Laboratory, Swiss Federal Institute of Tech-

nology (ETH Zurich), Physicstrasse 3, 8092 Zurich, Switzerland.

asrivastava@ethz.ch

3Department of Agriculture and Biological Engineering Sciences,

University of Illinois at Urbana-Champaign, 61801 IL, USA.

3amirm2@illinois.edu

3North Carolina Agricultural and Technical State University, 27411 NC.

hfeng@ncat.edu

This work was supported by the U.S. Department of Energy under award

DE-EE0009125 and NCCR Automation (grant number 180545) funded by

the Swiss National Science Foundation.

Fig. 1: Schematics of the continuous smart dryer prototype with

seven buckets which accommodates multiple drying technologies

to achieve better performance. In the example shown above, two

DEP, two ultrasound, one IR, and two SJR modules are used in a

speciﬁc order.

technology performs with different efﬁciencies in different

settings. Depending on the operating conditions, some tech-

nologies may be more favorable than others. For example,

contact-based ultrasound technology is more effective in the

initial phase of the process, where the moisture content

of the food sample is relatively high, while pure hot air

drying consumes less energy and is more effective when the

moisture content is low. By combining these two processes,

it is possible to take advantage of both technologies and

compensate for their inefﬁciencies. Therefore, understanding

(a) the sequence in which different drying techniques should

be used, and (b) the operating parameters of each technology,

can help us maximize their capabilities. Dividing the drying

process into sub-processes that use different drying methods

and operating conditions can help alleviate their individual

limitations.

To illustrate, let us consider the continuous drying testbed

depicted in Fig.1, which includes several drying modules

such as ultrasound, DEP, SJR nozzle, and IR technologies.

Each drying module is controlled by a set of parameters

that inﬂuence the amount of moisture removal. For instance,

ultrasound power and duty cycle are the control parameters

of the ultrasound technology, while electric ﬁeld intensity

is the control variable of the DEP module. Additionally, the

control parameters of each dryer impact the amount of energy

consumed during the process, creating a tradeoff between

energy consumption and moisture removal. Therefore, to

minimize the total energy consumed by the testbed while

achieving the desired moisture removal, a combinatorial

optimization problem can be formulated to determine the

optimal order in which the drying modules should be placed

in the testbed and the optimal control parameters associated

with them.

Similarly, this approach can be extended to batch-process

drying with some adjustments. For example, in the testbed

arXiv:2210.01971v3 [eess.SY] 5 Apr 2023

Fig. 2: Schematics of the convective/ultrasound testbed for batch-

process drying which can be used for both pure hot air (HA)

and combined hot air and ultrasound (HA/US) processes. The

HA mechanism consists of a blower and a heater, whereas the

US mechanism is a vibrating sheet attached to the ultrasound

transducer. One can switch from HA/US process to pure HA process

by turning off the US transducer

shown in Fig.2, which is used for the batch drying process,

each technology can be used more than once. It includes an

ultrasonic module [5], a drying chamber with a rectangular

cross-section, a blower, and a heater. The food sample

is located on a vibrating sheet attached to the ultrasonic

transducer and exposed to the hot air coming from the heater,

allowing combined hot-air and ultrasound drying. In this

setup, the problem of interest is to reduce energy consump-

tion, if possible, by dividing the process into consecutive pure

hot-air (HA) and combined hot-air and ultrasound (HA/US)

sub-processes, each with different operating conditions.

Previous research in the ﬁeld of drying has largely focused

on improving the efﬁciency of existing drying methods or

developing new technologies [9], [5], [7]. Some studies

have used optimization routines such as the response surface

method (RSM), a statistical procedure, to optimize process

control variables using experimental data [10], [11]. How-

ever, there is limited literature that addresses optimization

problems related to integrating different drying technologies

through sequencing and parameter optimization. The primary

contribution of our work is the modular use of multiple

existing technologies to achieve cost efﬁciency with desired

performance levels, while also allowing for optimal operating

conditions that can vary over time, potentially improving

performance even further. In our simulation results, presented

in Section IV, we show up to a 12% reduction in energy

consumption compared to the most efﬁcient single-stage

hot-air/ultrasound drying process, as well as up to a 63%

improvement in energy efﬁciency compared to the commonly

used optimal hot air drying method. Similar optimization

problems can arise in various industrial processes that involve

using a sequence of distinct devices with similar functions

to form a uniﬁed process, such as the wood pulp industry

with drying drums varying in radius and temperature, route

optimization in multi-channel wireless networks with hetero-

geneous routers, and sensor network placement.

This paper introduces a framework based on the Maximum

Entropy Principle (MEP) to model and optimize the various

sub-processes in an industrial drying unit. These optimization

problems pose signiﬁcant challenges due to the combinato-

rially large number of valid sequences of sub-processes and

their discrete nature. To address these issues, we assign a

probability distribution to the space of all possible conﬁgura-

tions. However, determining the optimal operating conditions

of sub-processes alone is analogous to the NP-hard resource

allocation problem, with a non-convex cost surface contain-

ing multiple poor local minima. Traditional algorithms like

k-means often get trapped in these local minima and are

sensitive to initialization. To overcome this, our algorithm

uses a homotopy approach from an auxiliary function to the

original non-convex cost function. This auxiliary function is

a weighted linear combination of the original non-convex

cost function and an appropriate convex function, chosen as

the negative Shannon entropy of the probability distribution

deﬁned above. We start with weights that favor the negative

Shannon entropy term, making the function convex and

easily solvable. As the iteration progresses, the weight of

the original non-convex cost increases, and the obtained

local minima are used to initialize subsequent iterations.

The auxiliary function converges to the original non-convex

cost function at the end of the procedure. This approach is

independent of initialization and tracks the global minimum

by gradually transforming the convex cost function to the

desired non-convex cost function.

II. PROBLEM FORMULATION

We formulate the problem stated above as a parameterized

path-based optimization problem [12]. Such problems are

described by a tuple

M=hM,γ1,...,γM,η1,...,ηM,Di,(1)

where Mis the number of stages allowed, and γkdenotes the

sub-process chosen to be used in the k−th stage. In particular,

γk∈Γk:={fk1,..., fkLk} ∀1≤k≤M,(2)

where Γkis the set of all sub-processes permissible in the

k−th stage. Moreover,

ηk∈H(γk)⊆Rdγk∀1≤k≤M,(3)

where ηkand H(γk)denote the control parameters associated

with the k−th sub-process and its feasible set, respectively.

D(ω,η1,...,ηM)denotes the cost incurred along a path ω,

where ω∈Ω:={(f1i1,f2i2,..., fMiM):fkik∈Γk}represents

a sequence of sub-processes starting from the ﬁrst stage to

the terminal stage M. The objective of the underlying param-

eterized path-based optimization problem is to determine (a)

the optimal path ω∗∈Ω, and (b) the parameters η∗

kfor all

1≤k≤Mthat solves the following optimization problem

min

{ηk},ν(ω)∑

ω∈Ω

ν(ω)D(ω,η1,...,ηM),

subject to ∑

ω∈Ω

ν(ω) = 1,ν(ω)∈ {0,1}

ηk∈H(γk)∀1≤k≤M,

(4)

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TowardsEfcientModularityinIndustrialDrying:ACombinatorialOptimizationViewpointAlisinaBayati1a,AmberSrivastava2,AmirMalvandi3,HaoFeng4andSrinivasaM.Salapaka1bAbstractTheindustrialdryingprocessconsumesapproxi-mately12%ofthetotalenergyusedinmanufacturing,withthepotentialfora40%reductioninenergyusaget...

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Towards Efﬁcient Modularity in Industrial Drying A Combinatorial Optimization Viewpoint Alisina Bayati1a Amber Srivastava2 Amir Malvandi3 Hao Feng4and Srinivasa M. Salapaka1b

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