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Advances in Information Science and Applications - Volume I
A method for optimization of plate heat
exchanger
Václav Dvořák
a method for configuration optimization.
Fábio et al. in work [3] presented an algorithm for the
optimization of heat exchange area of plate heat exchangers.
The algorithm was based on the screening method. For each
kind of plate, subject to certain constraints, optimal
configurations were found which presented the smallest area.
Each of these found configurations had local optima
characteristics.
Similarly Arsenyeva et al. [4] discussed the developments in
design theory of plate heat exchangers, as a tool to increase
heat recovery and efficiency of energy usage. The optimal
design of a multi-pass plate-and-frame heat exchanger with
mixed grouping of plates was considered. The optimizing
variables included the number of passes for both streams, the
numbers of plates with different corrugation geometries in
each pass, and the plate type and size. The mathematical
model of a plate heat exchanger was developed to estimate the
value of the objective function in a space of optimizing
variables. To account for the multi-pass arrangement, the heat
exchanger was presented as a number of plate packs with coand counter-current directions of streams, for which the system
of algebraic equations in matrix form was readily obtainable.
The exponents and coefficients in formulas to calculate the
heat transfer coefficients and friction factors were used as
model parameters to account for the thermal and hydraulic
performance of channels between plates with different
geometrical forms of corrugations. These parameters were
reported for a number of industrially manufactured plates. The
described approach was implemented in software for plate heat
exchangers calculation.
In another work Gut et al. [5] presented a screening method
for selecting optimal configurations for plate heat exchangers.
The optimization problem was formulated as the minimization
of the heat transfer area, subject to constraints on the number
of channels, pressure drops, flow velocities and thermal
effectiveness, as well as the exchanger thermal and hydraulic
models. The configuration was defined by six parameters,
which are as follows: number of channels, numbers of passes
on each side, fluid locations, feed relative location and type of
channel flow. The proposed method relied on a structured
search procedure where the constraints were successively
applied to eliminate infeasible and sub-optimal solutions. The
method can be also used for enumerating the feasible region of
the problem; thus any objective function can be used.
Examples showed that the screening method was able to
Abstract— Research of devices for heat recovery are currently
focused on increasing the temperature and heat efficiency of plate
heat exchangers. The goal of optimization is not only to increase the
heat transfer or even moisture but also reduce the pressure loss and
possibly material costs. During the optimization of plate heat
exchangers using CFD, we are struggling with the problem of how to
create a quality computational mesh inside complex and irregular
channels. These channels are formed by combining individual plates
or blades that are shaped by molding, vacuum forming, or similar
technology. Creating computational mesh from the bottom up
manually is time consuming and does not help later optimization. The
paper presents a method of creating meshes based on dynamic mesh
method provided by software Fluent. Creating of mesh by pulling is
similar to the own production process, i.e. it is perpendicular to the
plates. The advantages of this method are: The ability to change
quickly the whole geometry of the plate, possibility to use
optimization algorithms, ability to control the size of the wall
adjacent cells and similarity of meshes even in completely different
geometries. The paper discusses the problems with very narrow gaps
and distortions of the mesh. Using this method, a row of cases with
waves and ridges were created. The resulting dependence of
efficiency and pressure loss on the ridges count are replaced by
mathematical relationships. An objective function is suggested and
verified to optimize heat transfer surface of the exchanger.
Keywords—Dynamic mesh, heat exchanger, optimization.
T
I. INTRODUCTION
he development of recuperative heat exchangers in recent
years focused on increasing efficiency. Another challenge
is the development of so-called enthalpy exchangers for
simultaneous heat and moisture transport, i.e. transport of both
sensible and latent heat, as presented by Vít et al. in work [1].
A lot of others researchers dealt with design and
optimization of plate heat exchangers. For example Gut et al.
[2] developed a mathematical model in algorithmic form for
the steady simulation of plate heat exchangers with generalized
configurations. The configuration is defined by the number of
channels, number of passes at each side, fluid locations, feed
connection locations and type of channel-flow. The main
purposes of this model were to study the configuration
influence on the exchanger performance and to further develop
Author gratefully acknowledges financial support by Czech Technological
Agency under the project TACR TA01020313.
V. Dvořák is with the Technical university of Liberec, Faculty of
mechanical engineering, Department of Power Engineering Equipment.
Address: Studentska 2, 46117, Liberec. Phone: +420 485 353 479; e-mail:
[email protected].
ISBN: 978-1-61804-236-1
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Advances in Information Science and Applications - Volume I
successfully determine the set of optimal configurations with a
reduced number of exchanger evaluations.
However the optimal design of plates itself is not under
investigation in these mentioned studies, while several
researchers tried to optimize even the shape of heat exchanger
area.
E.g. multi-objective optimization of a cross-flow plate fin
heat exchanger (PFHE) by means of an entropy generation
minimization technique was described by Babaelahi et al. in
study [6]. Entropy generation in the PFHE was separated into
thermal and pressure entropy generation as two objective
functions to be minimized simultaneously. The Pareto optimal
frontier was obtained and a final optimal solution was selected.
By implementing a decision-making method, here the
LINMAP method, the best trade-off was achieved between
thermal efficiency and pumping cost. This approach led to a
configuration of the PFHE with lower magnitude of entropy
generation, reduced pressure drop and pumping power, and
lower operating and total cost in comparison to singleobjective optimization approaches.
Kanaris et al. [7] suggested a general method for the optimal
design of a plate heat exchanger (PHE) with undulated
surfaces that complies with the principles of sustainability.
They employed previously validated CFD code to predict the
heat transfer rate and pressure drop in this type of equipment.
The computational model was a three-dimensional narrow
channel with angled triangular undulations in a herringbone
pattern, whose blockage ratio, channel aspect ratio,
corrugation aspect ratio, angle of attack and Reynolds number
are used as design variables. An objective function that
linearly combines heat transfer augmentation with friction
losses, using a weighting factor that accounts for the cost of
energy, was employed for the optimization procedure. New
correlations were provided for predicting Nusselt number and
friction factor in such PHEs. The authors stated that the results
were in very good agreement with published data.
Han et al. in article [8] numerically investigated the thermalhydrodynamic characteristics of turbulent flow in chevron-type
plate heat exchangers with sinusoidal-shaped corrugations.
The computational domain contained a corrugation channel,
and the simulations adopted the shear-stress transport κ-ω
model as the turbulence model. The numerical simulation
results in terms of Nusselt number and friction factor were
compared with limited experimental data and existing
correlations in order to verify the accuracy of the numerical
model. The corrugation depth, corrugation angle, corrugation
pitch, and fluid inlet velocity were identified as design
variables, and 200 samples were selected using the maximum
entropy design method to build the metamodel for obtaining
the heat transfer coefficient as well as the pressure drop per
unit length. A multi-objective genetic algorithm was utilized as
the optimizer. The optimization results were presented in the
form of Pareto solution set, which clearly showed its
dominance over the entire design space and the tradeoff
between the two optimization objectives: maximizing heat
ISBN: 978-1-61804-236-1
transfer coefficient and minimizing drop per unit length. Also,
the Pareto optimal designs were validated against the values
directly obtained from numerical simulations. The
approximation-assisted optimization shows that all optimal
designs have largest enlargement factor values inside the
design space, and the optimal corrugation angle increases with
the increase of maximum heat transfer coefficient.
This work is motivated by the need to optimize heat transfer
area of plate heat recovery heat exchangers. Heat exchangers
are assembled from plates. Plates are made of metal, paper or
plastic and are shaped by press molding. Ridges and grooves
are supposed to increase the heat (and mass) transfer and also
determine the plate pitch and carry the heat exchanger.
To optimize a heat exchanger, we have to create a model
and a computational mesh and use computational fluid
dynamic (CFD) software. By assembling the heat exchanger,
complicated and irregular narrow channels are created.
Disadvantages of repeated generation of computational meshes
are: It is slow, meshes made in different models are not similar
and parameterization of the model is problematic. Further,
even a small change of geometry requires to go through the
whole process of model creation and mesh generation again.
As a result, there is high probability of creation errors of
model and low quality of mesh cells. It is necessary to setup
the solver, boundary conditions and all models for all
computed variants. Furthermore meshes are not similar, i.e. the
size, shape, height of wall adjacent cells are not the same for
different topologies.
E.g. Novosád in work [9], investigated the influence of
oblique waves on the heat transfer surface. The biggest
problem in this work was the creation of custom geometry.
Each option had to be modeled separately and a meshed. Each
model had to be loaded into the solver, set the boundary
conditions and subsequently evaluated by calculation.
Therefore, a new method for generation computational
variants was developed and is presented in this work.
II. METHODS
A. Method for rapid generation of computational model
In this paper, we discuss the case of a counter flow heat
exchanger, which has symmetrical heat transfer area. Processes
in such heat exchanger can be investigated by modeling the
flow around only one plate using symmetrical boundary
conditions. How such a model appears can be seen from Fig.
1. The heat transfer surface is divided into two parts. Input and
output portions (reported as wall) is fixed, and serve to
develop the velocity profiles before the central portion (main
wall) which will be deformed. Input boundary conditions are
specified by mass flow rates (mass flow inlets), the output
boundary conditions are specified as pressure outlets with
static pressure 0 Pa.
In this study, we used turbulence model SST κ-ω, medium
was air considered as ideal gas. As a results, we obtained
pressure, velocity, turbulence and temperature fields inside the
computational domain for average inlet velocity of air 3 m/s.
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Advances in Information Science and Applications - Volume I
method for model generation, we calculated and evaluated two
successions of geometries, each with a pitch plates of δ = 3
(mm). In the first case, the geometry of corrugation was
defined as waves by function

y −y
π ⋅ n  ,
z = cos 0
2
⋅
y
0


(1)
where is relative vertical coordinate of the wall,
(m) is
width of the model, (m) lateral coordinate and
number of waves. In the second case, the geometry of
corrugation was defined as ridges by function
Fig. 1 Model of heat exchange surface of counter flow recuperative
heat exchanger.


y −y
z = arcsin − cos 0
π ⋅ n 
y
2
⋅
0



The new method is based on dynamic mesh method
provided by software Fluent. First a simple model with straight
wall, see Fig. 1 is created. This model is fully functional, i.e.
read into Fluent, boundary conditions are set and it is possible
to calculate the flow and heat transfer. In a subsequent step,
the computational mesh is deformed using the so-called UDF
(User defined functions). Actually, individual nodes of
computational mesh are manipulated with. The result is a
transformation of the mesh, see Fig. 2 The figure also shows
that during the deformation, it is possible to change the
spacing of the plates, to create dents or corrugations and also
define the size of the cells adjacent to the wall.

(2 ⋅ π ) ,
(2)

were created on the tops
while tabs of the width
of the ridges, as it is obvious from Fig. 3.
B. Theory of counter flow heat exchangers
Most of the recuperative heat exchangers in air conditioning
works in the isobaric mode, where mass flow rates of warm
and cold air are equal, i.e. m
c = m
 h . Assuming equality of
specific heat capacities,
c p c = c p h , we can write the
coefficient of efficiency as
η=
th i − th o
t h i − tc i
⋅ 100 (%) ,
(3)
Where t h i (℃) is the inlet temperature of hot air.
Furthermore index c denotes a cold stream, index i inlet into
the heat exchanger and index o the outlet of the heat
exchanger.
For the pressure drop assessment, it is used local loss
coefficient ξ . It is the ratio of total pressure los between the
Fig. 2 Computational mesh after deforming.
It is obvious that, as for commonly created mesh, the
refinement of the mesh defines minimal size of geometrical
details which can be described by the mesh. It can be seen
from Fig. 3 that shapes of peaks and valleys of each ridge vary
according the grid and ridge spacing.
inlet and outlet ∆p and dynamic pressure pd , i.e.
ξ=
∆p p0 i − po
=
pd
pd
(4)
where ∆p is the pressure difference between average total
pressure in the pressure inlets
a)
pressure at the pressure outlets
p0 i (Pa) and average static
po (Pa).
The dependence between the heat balance and efficiency η
is expressed as
b)
Fig. 3 Cut of the computational mesh, a) – initial mesh, b) – mesh
after deforming and ridges creating.
 ⋅ cp ⋅
k⋅A=m
For this initial study, no direct optimization method were
used to control the deforming of the surface. Using mentioned
ISBN: 978-1-61804-236-1
η
1 −η
where Pkg/s) is the mass flow rate,
195
(5)
c p (J/(kg·K)) is isobaric
Advances in Information Science and Applications - Volume I
specific heat capacity, k (W/(m2·K)) heat transfer coefficient
and A (m2) is the area of heat transfer surface.
III. RESULTS
The results of computations of both cases are plotted in Fig.
4. It is clearly evident from the figure that with the increasing
number of waves in the model the efficiency of the heat
exchanger increases. The course for both designs of the heat
exchanger are similar, but it is evident for smaller count of
waves that wall with ridges has higher efficiency than wall
with waves. The course of the pressure loss in contrast, is such
that for the first few waves, the pressure loss decreases,
reaches a minimum for n = 3 ÷ 3.5 and then begins to grow.
For comparison, the diagram also shows the results plotted for
the wave number n = 0, i.e. for the flat plate. There is clearly
apparent advantages of such an arrangement. Variant with
waves or ridges needs at least n = 15 to achieve the same
efficiency, but the pressure loss is higher comparing to the flat
plates.
Fig. 5 Surface heat transfer coefficient – pressure loss coefficient for
all calculated variants.
Pressure loss coefficient ξ1 similarly counted the pressure
loss only in the deformed part of the heat exchanger surface
and is calculated from the relationship
ξ1 = ξ − ξ 0
L0 ,
L0 + L1
(7)
Where ξ (1) is the total pressure loss coefficient of the case
and ξ 0 (1) is the total loss coefficient for the case of flat
plates.
Fig. 4 Diagram efficiency – pressure loss for all variants.
The same results are also plotted in a diagram in Fig. 5,
where the heat transfer is represented by heat transfer
coefficient k1 (W/(m2·K)), which takes into account the heat
transfer only in the middle deformed part of the exchanger
model and is calculated from the relationship
 ⋅ cp ⋅
B ⋅ (k1 ⋅ L1 + k 0 ⋅ L0 ) = m
η ,
1 −η
(6)
Fig. 6 Dependency of heat exchanger efficiency on count of ridges.
where k0 (W/(m2·K)) is the heat transfer coefficient of a flat
plate, B (m) is the width of the model, L0 and L1 (m) are
Generally, as we can see from Fig. 4 and Fig. 5, the higher
efficiency is obtained for cases with higher pressure loss.
However, in practice, pressure loss and efficiency are required
when optimizing the heat exchanger. Usually, the maximal
pressure loss is specified and for that given pressure restrain, a
solution with maximal efficiency is sought. We used obtained
dependence of efficiency on count of ridges, which is
lengths of flat and deformed parts of the wall respectively, see
Fig. 1. Let us mention yet that the heat transfer coefficient is
evaluated without respect for the increase in heat transfer area
heat caused by deforming. This approach is fully in line with
the practice, for which the area of used material is critical.
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presented in Fig. 6, while the dependency of pressure loss is in
Fig. 7.
function for required values of pressure loss of the heat
exchanger is in Fig. 8.
Fig. 7 Dependency of pressure loss of the heat exchanger on count of
ridges.
Fig. 9 Course of objective function during optimization for required
pressure loss ∆p R .
The courses of objective function during optimization for
given required pressure loss, while a simple gradient method
was used, are in Fig. 9.
IV. CONCLUSIONS
Method has been developed for the rapid generation of
computational models. The method simulates the forming
process, wherein the heat exchanger plates are made, and
allows arbitrarily shaped heat exchanging surface. Among its
advantages are the ability to change the pitch of heat
exchanger plates and control the size of the cell adjacent to the
wall. Another advantage is the similarity of calculating variant
and the ability to use the method in the optimization. It is also
advantageous to use obtained calculated data for initialization
of the next computed variant.
The method was tested on calculation of dependencies of
efficiency and pressure drop for two variants of the
corrugation of the heat exchange surfaces of heat exchanger.
The flow and heat transfer in the heat exchanger with
symmetrical heat exchange surface were studied. For forming
walls of the heat exchanger, waves and ridges, which proved to
be more advantageous, were used. The obtained dependencies
were replaced by polynomial functions and an objective
function was designed for possible optimization. The function
is based on the practice that maximum efficiency of the heat
exchanger is required for a given pressure drop constraints.
Suitable constants of the function were found and the functions
were tested by optimization using simple gradient methods.
Only one parameter – number of ridges – was optimized.
The method for rapid generation of computational models
and suggested objective function will be used in further work
to optimize the shape of the heat transfer surface dependent on
more than one parameter.
Fig. 8 Course of objective function for various required pressure loss
∆p R .
Because in practice we optimize for maximum efficiency for
a given pressure drop, was designed a target function as
 ∆p − ∆p R
F = η − cη 
 cE ⋅ ∆p R
N
 ,



(8)
where ∆p R (Pa) is the reference or required pressure loss, cE
(1) is maximal relative deviation of the required pressure drop
∆p R , N is exponent and cη is penalization of the objective
function if the actual pressure ∆p loss differs from required.
The shape of the objective function was optimized, and it was
found that the optimum parameters for the fastest convergence
are N = 1, cη = 2% and cE = 0.01 . The course of the target
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