The computer aided design of plate heat exchanger with mixed grouping of plates is considered. It is formulated as the mathematical problem of finding the minimal value for implicit nonlinear discrete/continues objective function with inequality constraints. The optimizing variables include the number of passes for both streams, the numbers of plates with different corrugation geometries in each pass, the plate type and its size. To estimate the value of objective function in a space of optimizing variables the mathematical model of plate heat exchanger is developed. To account for thermal and hydraulic performance of channels between plates with different geometrical forms of corrugations, the exponents and coefficients in formulas for heat transfer coefficients and friction factors calculation are used as model parameters. The procedure and software for numerical experiment to identify model parameters by comparing the calculation results with those obtained with free available in web computer programs of plate manufacturers is developed. The sets of such parameters are obtained for a number of industrially manufactured plates. The described approach is implemented as software for plate heat exchangers calculation

Arsenyeva, O. P.

Kapustenko, P. O.

Khavin, G. L.

Tovazhnyansky, L. L.

Library of National Technical University "Kharkiv Polytechnic Institute"

20th European Symposium on Computer Aided Process Engineering – ESCAPE20 
S. Pierucci and G. Buzzi Ferraris (Editors)  
© 2010 Elsevier B.V.  All rights reserved.  
Computer Aided Design of Plate Heat Exchangers 
Olga Arsenyevaa, Leonid Tovazhnyanskya, Petro Kapustenkoa, Gena Khavinb 
aNational Technical University “Kharkiv Polytechnic Institute”,  
21 Frunze Str., 61002, Kharkiv, Ukraine, kap@kpi.kharkov.ua 
bAO SODRUGESTVO-T, 2 Krsnoznamenny Per., 61002, Kharkiv, Ukraine, 
sodrut@gmail.com 
Abstract 
The computer aided design of plate heat exchanger with mixed grouping of plates is 
considered. It is formulated as the mathematical problem of finding the minimal value 
for implicit nonlinear discrete/continues objective function with inequality constraints. 
The optimizing variables include the number of passes for both streams, the numbers of 
plates with different corrugation geometries in each pass, the plate type and its size. To 
estimate the value of objective function in a space of optimizing variables the 
mathematical model of plate heat exchanger is developed. To account for thermal and 
hydraulic performance of channels between plates with different geometrical forms of 
corrugations, the exponents and coefficients in formulas for heat transfer coefficients 
and friction factors calculation are used as model parameters. The procedure and 
software for numerical experiment to identify model parameters by comparing the 
calculation results with those obtained with free available in web computer programs of 
plate manufacturers is developed.  The sets of such parameters are obtained for a 
number of industrially manufactured plates. The described approach is implemented as 
software for plate heat exchangers calculation.  
 
Keywords: Plate Heat Exchanger, Design, Mathematical Model, Model Parameters 
1. Introduction 
Plate heat exchangers (PHEs) are one of the most efficient types of heat transfer 
equipment. The principles of their construction and design methods are sufficiently well 
described elsewhere, see e.g. Hesselgreaves (2001), Wang, Sunden and Manglik (2007),  
Tovazshnyansky et al (2004). This equipment is much more compact and requires much 
less material for heat transfer surface production, much smaller footprint, than 
conventional shell and tubes units. PHEs have a number of advantages over shell and 
tube heat exchangers, such as compactness, low total cost, less fouling, flexibility in 
changing the heat transfer surface area, accessibility. Due to differences in construction 
principles from conventional shell and tube heat exchangers, PHEs require substantially 
different methods of thermal and hydraulic design. Such methods should be based on 
accurate enough mathematical models.  
2. Mathematical model of PHE 
The PHE consists of a set of corrugated heat transfer plates clamped together between 
fixed and moving frame plates and tightened by tightening bolts, see e.g Hesselgreaves 
(2001). The plates are equipped with the system of sealing gaskets, which are also 
separate the streams and organizing their distribution over the inter plate channels. In 
multi-pass PHE plates are arranged in such way, that they are forming groups of parallel 
channels. Such group is corresponding to one pass and the stream is going consequently 
  O. Arsenyeva et al. 
through the passes. The temperature distributions in passes are different, and in different 
groups of channels both counter-current and co-current flows may occur. 
The mathematical model of PHE was presented by Arsenyeva et al.(2009). The PHE is 
regarded as a system of one-pass blocks of plates. The conditions for all channels in one 
such block are equal. The total number of blocks is nb=X1X2 and the number of heat 
transfer units in one block, counted for hot stream: 
2 1 1( )b b bNTU U F X G c= ⋅ ⋅  (1) 
where Ub – overall heat transfer coefficient in block, W/(m2K); G1- mass flow rate of 
hot stream, kg/s; c1 – specific heat of hot stream, J/(kg·K); X1 and X2 – the number of 
passes for hot and cold stream, respectively. 
When G1c1/X2 <G2 c2/X1 block heat exchange effectiveness bε  for counter current flow: 
[ ] [ ]1 exp( ) 1 exp( )b b b b b b b bNTU R NTU R NTU R NTUε = − ⋅ − − ⋅ ⋅ −          (2) 
where 1 1 1 2 2 2( )bR G c X G c X= ⋅ ⋅ ⋅ ⋅ - the ratio of heat capacities of streams going 
trough block; G2 and c2 mass flow rate [kg/s] and specific heat  [J/(kg·K)] of cold 
stream. 
If Rb=1, then (1 )b b bNTU NTUε = + . 
In case of co-current flow directions [ ] ( )1 exp( ) 1b b b b bNTU R NTU Rε = − − ⋅ − +             (3) 
On the other hand the heat exchange effectiveness of block i : 1bi i it tε δ= Δ ,    
where 1itδ - temperature drop in block i; itΔ - the temperature difference of streams 
entering block i. The temperature change of cold stream: 2 1i i bt t Rδ δ= ⋅   
These relations can be regarded as mathematical model of block, which describes the 
dependence of temperature changes from characterising block values of Fb and Ub .  
For every block we can write the equation which describes the link of temperature 
change in this block to temperature changes in all other blocks of PHE.  
For any number of passes such system presented in matrix form:  
[ ][ ] [ ]i biZ tδ = ε Δ ,                                                                    (4) 
where [ ]itδ - vector-column of temperature drops in blocks; [ ]biε Δ - vector-column of 
the right hand parts of the system; [ ]Z - matrix of system coefficients.  
The numerical solution of linear algebraic equations system (7) easily made on PC. 
After that the total temperatures change in PHE calculated as: 
( )
1 2
21 1
11 1
1X X
i X ii
i ii
t t
XΣ − += =
⎛ ⎞δ = δ⎜ ⎟⎜ ⎟⎝ ⎠∑ ∑ ;   
( )
( )
1 1
2 1
2 2
G c
t t
G cΣ ∑
δ = δ .                        (5) 
The total heat load of PHE:                           
1 1 1Q t G cδ Σ= ⋅ ⋅  ,                                                              (6) 
This system should be accompanied by equations for calculation of overall heat transfer 
coefficient U ,  W/(m2K), as below. ( )1 21 1 1 w w fU R= α + α + δ λ +              (7) 
Computer Aided Design of Plate Heat Exchangers   
where 1 2,α α  - film heat transfer coefficients for hot and cold streams, respectively, 
W/(m2K); wδ  - the wall thickness, m; wλ - heat conductivity of the wall material, 
W/(mK); 
1 2f f fR R R= + - the sum of fouling thermal resistances for streams, m2K/W. 
For plate heat exchangers the film heat transfer coefficients are usually calculated by 
empirical correlations: 
0.4 0.14(Re, Pr) *Re Pr ( / )n wNu f A μ μ= =                               (8) 
Here μ  and wμ  dynamic viscosity at stream and at wall temperatures, respectively; 
Nusselt number is:                                                   /Nu deα λ= ⋅ ,                     
The streams velocities are calculated as                   /( )chw g f ρ=                       
Where  g  is flowrate of stream through one channel, kg/s.              
The pressure drop in one PHE channel ( ) 2 2p ep L d wζ ρΔ = ⋅ ⋅ ⋅ ,             (9) 
where pL -effective plate length; ζ – friction factor, which is usually determined by 
empirical correlations of following form: 
/ RemBζ =                           (10) 
For multi pass PHE the pressure drop in one pass multiplied by number of passes X. 
In modern PHEs plates of one type are usually made with two different corrugation 
angles, that can form three different channels, when assembled in PHE. Plates of H type 
have corrugations with bigger angles (about 600) that form the H channels with higher 
efficiency of heat transfer and hydraulic resistance. Plates of L type have a lower angle 
(about 300) and form the L channels with lower heat transfer and hydraulic resistance. 
Combined, these plates form channels MH or ML with intermediate characteristics. 
Such principle of design enables to change thermal and hydraulic performance of plates 
pack with the level of discreteness equal to one plate in a pack.  
In one PHE two groups of channels are usually used. One is of higher hydraulic 
resistance and heat transfer (x-channel), another of lower characteristics (y-channel). 
When the stream is going through set of such channels, the temperature changes in 
different channels are differ. After mixing in collector part of PHE block, the 
temperature is determined by heat balance. Then the heat exchange effectiveness of 
plates block with different channels:  
)()( yyxxyyyxxxb ngngngng ⋅+⋅⋅⋅+⋅⋅= εεε ,        (11) 
where xn and yn are the numbers of x and y channels in a block of plates, respectively; 
 , ,x y x y chg w fρ= ⋅ ⋅ - the mass flow rates through one channel of type x or y. These 
 flow rates should satisfy equation x yp pΔ = Δ  and material balance: 
x x y y bg n g n G⋅ + ⋅ = ,            (12) 
where bG - flow rate of the stream through the block of plates. 
The principle of mixing plates in one heat exchanger gives the best results with 
symmetrical arrangement of passes (X1=X2) and Gbis equal to total flow rate of 
respective stream. The unsymmetrical arrangement X1≠X2 is usually used when all 
channels are the same (any of the three available types).  
  O. Arsenyeva et al. 
When the numbers of channels are determined, the numbers of plates calculated as: 
( ) ( )1 21 1 2 2
1 1
1
X X
pl x i y i x i y i
i j
N n n n n
= =
= + + + +∑ ∑                       (13) 
The total heat transfer area of PHE (two end plates not included), m2: 
( 2)PHE pl plF N F= − ⋅ ,            (14) 
where plF - heat transfer area of one plate, m2. 
The above algebraic equations (1)-(14) describe the relationship between variables 
which characterize PHE and heat transfer process in it.  They can be regarded as a 
mathematical model of PHE, which solution enables to calculate pressure and 
temperature change of streams entering the heat exchanger. It is a problem of PHE 
rating (analysis).  
The problem of PHE design (synthesis) require to find its characteristics (such as plate 
type, numbers of passes, numbers of plates with different corrugations) which will in 
the best way satisfy the required process conditions. Here the optimal design with 
pressure drop specification is considered, in a sense as described by Wang and Sunden 
(2003).  The objective function is total heat transfer area of PHE with conditions that 
specified heat load 0Q  and allowable pressure drops for both streams 0 01 2,p pΔ Δ must 
be satisfied. These conditions can be regarded as partial inequality constraints: 
0Q Q≥ ;               01 1p pΔ ≤ Δ ;                 02 2p pΔ ≤ Δ                       (15) 
Analysis of relations (1) – (15) lets to conclude, that we have the mathematical problem 
of finding the minimal value for implicit nonlinear discrete/continues objective function 
with inequality constraints.  It does not permit analytical solution without considerable 
simplifications. To solve it by numerical methods, the software is developed for IBM 
compatible PC. The mathematical model contains some parameters, namely coefficients 
and powers in empirical correlations that are not easily available.             
 
3. Identification of mathematical model parameters  
As a rule the empirical correlations for design of industrially manufactured PHEs are 
obtained during tests on such heat exchangers at specially developed test rigs. Such tests 
are made for every type of new developed plates and inter plate channels. The results 
are property of manufacturing company and usually not published. 
Based on described above mathematical model it was developed the technique of 
numerical experiment that enables to identify model parameters by comparison with 
results obtained for the same conditions with the use of PHE calculation software, 
which  is now available in internet for most of PHE manufacturers. The results for some 
plates manufactured by Alfa Laval presented in Table 1. The geometrical parameters of 
plates, for which correlations were acquired are given in Table 2. 
The comparison of results obtained with our software to those of Alfa Laval free 
available software has shown good agreement (discrepancies not more then 4% on 
surface area). We should note, that obtained correlations and developed software can be 
used only for preliminary calculations, when optimizing PHEs or heat exchanger 
network. The final calculations when ordering the PHE must be performed by its 
manufacturer.  
Computer Aided Design of Plate Heat Exchangers   
 
Table 1. Parameters in correlations for some Alfa Laval PHEs (Re>250)    
Plate 
 type  
Channel 
type  
A n Re B m Re B m 
Н 0.265 0.7 <520 33.0 0.25 ≥520 10.7 0.07 
L 0.12 0.7 <1000 18.8 0.33 ≥1000 8.8 0.22 
 
М3 
ML/MH1 0.18 0.7 <1000 44.0 0.4 ≥1000 5.1 0.10 
Н 0.25 0.7 <1250 10.0 0.2 ≥1250 2.4 0.0 
L 0.12 0.7 <1500 5.1 0.3 ≥1500 1.7 0.15 
 
М6 
ML/MH 0.165 0.7 <930 9.3 0.3 ≥930 2.72 0.12 
Н 0.27 0.7 <1300 11.7 0.13 ≥1300 4.55 0.0 
L 0.11 0.71 <2200 4.23 0.23 ≥2200 1.88 0.12 
 
М6M 
ML/MH 0.14 0.73 <2100 5.61 0.16 ≥2100 1.41 0.0 
Н 0.24 0.7 <2000 11.1 0.15 ≥2000 3.5 0.0 
L 0.11 0.7 <1500 12 0.36 ≥1500 2.42 0.14 
 
М10B 
ML/MH 0.12 0.74 <2700 6.2 0.2 ≥2700 1.9 0.05 
Н 0.26 0.7 <2000 5.84 0.05 ≥2000 5.84 0.05 
L 0.085 0.74 <2900 5.2 0.28 ≥2900 1.57 0.13 
 
М15B 
ML/MH 0.13 0.74 <3500 4.3 0.15 ≥3500 1.25 0.0 
 
Table 2. Geometrical parameters for some Alfa Laval PHE plates 
Plate 
type 
δ , 
mm 
ed , 
mm 
b ,
mm 
Fpl ,  
m2 connection
D , 
mm 
chf ·103,  
m2 
pL , 
mm 
M3 2.4 4.8 100 0.032 36 0.240 320 
M6 2.0 4.0 216 0.15 50 0.432 694 
M6M 3.0 6.0 210 0.14 50 0.630 666 
M10B 2.5 5.0 334 0.24 100 0.835 719 
M15B 2.5 5.0 449 0.62 150 1.123 1381 
 
4. Case study 
It is required to heat 5 m3/h of distillery wash fluid from 28 to 90 oC by hot water 
coming with temperature 95 oC and flow rate 15 m3/h. The pressure of both fluids is 5 
bar. Allowable pressure drop for hot stream 1.5 bar. For cold stream 1 bar. The 
properties of wash fluid are taken constant as follows: density – 978,4 kg/m3; heat 
capacity – 3,18 kJ/(kg·K); сconductivity – 0.66 W/(kg·m). Dynamic viscosity at 
temperatures 25; 60; 90t C= o is taken as 19,5;16,6;9 cPμ= .  
The results of calculations for different passes numbers X1 and X2 and optimal for those 
passes plates arrangements are presented in Table 3. The analysis show that the global 
optimum (38 plates) is achieved at X1=2 and X2=4 with all medium channels (19 H and 
19 L plates in PHE). The closest other option (41 plates) is at X1=X2=2 with mixed 
                                                          
1 In our study we neglected by small differences in ML and MH channels parameters 
  O. Arsenyeva et al. 
Table 3. The influence of passes and plate arrangement on number of plates in M6M PHE 
X1 X2 
1 2 3 4 
1 56 plates 
1*28H / 1*27H 
235 plates 
2*59H/1*116H 
157 plates 
3*26H/1*78H 
184 plates 
4*23H/1*91H 
2 72 plates 
1*35H / 2*18H 
41 plates 
2*(7H+3*ML)/ 
2*(7H+3*MH) 
60 plates 
3*10MH / 
2*14ML 
64 plates 
4*8L / 
1*15L+1*16L 
3 44 plates 
1*21H / 
2*7H+1*8H 
49 plates 
2*12MH / 
3*8ML 
43 plates 
3*(4MH+3*L)/ 
3*(4ML+3*L) 
50 plates 
3*6L+1*7L / 
3*8L 
4 63 plates 
1*31MH / 
1*7ML+3*8ML 
38 plates 
2*9MH / 
1*4ML+3*5ML 
44 plates 
3*7L / 
2*5L+2*6L 
50 plates 
3*6L+1*7L / 
4*6L 
channel arrangement in one pass. If we would have only one plate type in PHE, the 
minimal number of plates would be 44 for both H and L plates, or 15% higher than with 
mixed channels. 
5. Conclusions 
The algorithm and software for computer aided design of multi-pass PHE assembled 
with plates of different corrugation patterns is developed. The model parameters 
corresponding to some industrially manufactured plates are obtained. The examples of 
calculation results for case study show the possibility with such method to obtain 
optimal solutions with exact satisfaction of constraints for total heat load and pressure 
drop of one stream. It gives the considerable reduction in heat transfer surface area of 
PHE.  
6. Acknowledgements  
The financial support of EC Project ECOPHOS (contract № INCO-CT-2005-013359) is 
sincerely acknowledged. 
References  
O. Arsenyeva, L. Tovazhnyansky, P. Kapustenko, G. Khavin, 2009, Mathematical 
Modelling and Optimal Design of Plate-and-Frame Heat Exchangers. Chemical 
Engineering Transactions, 18: 791-796. 
A. Gogenko, O. Anipko, O. Arsenyeva, P. Kapustenko, 2007,  Accounting for fouling 
in plate heat exchanger design. Chemical Engineering Transactions, 12: 207-213. 
J. Hesselgreaves, 2001, Compact Heat Exchangers. Selection, Design and Operation. 
Elsevier, Amsterdam. 
L. Tovazshnyansky, P. Kapustenko, G. Khavin, O. Arsenyeva., 2004, PHEs in Industry. 
NTU KhPI, Kharkiv (in Russian). 
L. Wang, B. Sunden, R. Manglik , 2007, PHEs. Design, Applications and Performance. 
WIT Press, Southhampton. 
L. Wang, B. Sunden, 2003, Optimal design of PHEs with and without pressure drop 
specifications. Applied Thermal Engineering. 23, 295-311. 
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Computer Aided Design of Plate Heat Exchangers

Electronic National Technical University &quot;Kharkiv Polytechnic Institute&quot; Institutional Repository (eNTUKhPIIR)

20th European Symposium on Computer Aided Process Engineering – ESCAPE20 S. Pierucci and G. Buzzi Ferraris (Editors)  © 2010 Elsevier B.V.  All rights reserved.  Computer Aided Design of Plate Heat Exchangers Olga Arsenyevaa, Leonid Tovazhnyanskya, Petro Kapustenkoa, Gena Khavinb aNational Technical University “Kharkiv Polytechnic Institute”,  21 Frunze Str., 61002, Kharkiv, Ukraine, kap@kpi.kharkov.ua bAO SODRUGESTVO-T, 2 Krsnoznamenny Per., 61002, Kharkiv, Ukraine, sodrut@gmail.com Abstract The computer aided design of plate heat exchanger with mixed grouping of plates is considered. It is formulated as the mathematical problem of finding the minimal value for implicit nonlinear discrete/continues objective function with inequality constraints. The optimizing variables include the number of passes for both streams, the numbers of plates with different corrugation geometries in each pass, the plate type and its size. To estimate the value of objective function in a space of optimizing variables the mathematical model of plate heat exchanger is developed. To account for thermal and hydraulic performance of channels between plates with different geometrical forms of corrugations, the exponents and coefficients in formulas for heat transfer coefficients and friction factors calculation are used as model parameters. The procedure and software for numerical experiment to identify model parameters by comparing the calculation results with those obtained with free available in web computer programs of plate manufacturers is developed.  The sets of such parameters are obtained for a number of industrially manufactured plates. The described approach is implemented as software for plate heat exchangers calculation.   Keywords: Plate Heat Exchanger, Design, Mathematical Model, Model Parameters 1. Introduction Plate heat exchangers (PHEs) are one of the most efficient types of heat transfer equipment. The principles of their construction and design methods are sufficiently well described elsewhere, see e.g. Hesselgreaves (2001), Wang, Sunden and Manglik (2007),  Tovazshnyansky et al (2004). This equipment is much more compact and requires much less material for heat transfer surface production, much smaller footprint, than conventional shell and tubes units. PHEs have a number of advantages over shell and tube heat exchangers, such as compactness, low total cost, less fouling, flexibility in changing the heat transfer surface area, accessibility. Due to differences in construction principles from conventional shell and tube heat exchangers, PHEs require substantially different methods of thermal and hydraulic design. Such methods should be based on accurate enough mathematical models.  2. Mathematical model of PHE The PHE consists of a set of corrugated heat transfer plates clamped together between fixed and moving frame plates and tightened by tightening bolts, see e.g Hesselgreaves (2001). The plates are equipped with the system of sealing gaskets, which are also separate the streams and organizing their distribution over the inter plate channels. In multi-pass PHE plates are arranged in such way, that they are forming groups of parallel channels. Such group is corresponding to one pass and the stream is going consequently   O. Arsenyeva et al. through the passes. The temperature distributions in passes are different, and in different groups of channels both counter-current and co-current flows may occur. The mathematical model of PHE was presented by Arsenyeva et al.(2009). The PHE is regarded as a system of one-pass blocks of plates. The conditions for all channels in one such block are equal. The total number of blocks is nb=X1X2 and the number of heat transfer units in one block, counted for hot stream: 2 1 1( )b b bNTU U F X G c= ⋅ ⋅  (1) where Ub – overall heat transfer coefficient in block, W/(m2K); G1- mass flow rate of hot stream, kg/s; c1 – specific heat of hot stream, J/(kg·K); X1 and X2 – the number of passes for hot and cold stream, respectively. When G1c1/X2 <G2 c2/X1 block heat exchange effectiveness bε  for counter current flow: [ ] [ ]1 exp( ) 1 exp( )b b b b b b b bNTU R NTU R NTU R NTUε = − ⋅ − − ⋅ ⋅ −          (2) where 1 1 1 2 2 2( )bR G c X G c X= ⋅ ⋅ ⋅ ⋅ - the ratio of heat capacities of streams going trough block; G2 and c2 mass flow rate [kg/s] and specific heat  [J/(kg·K)] of cold stream. If Rb=1, then (1 )b b bNTU NTUε = + . In case of co-current flow directions [ ] ( )1 exp( ) 1b b b b bNTU R NTU Rε = − − ⋅ − +             (3) On the other hand the heat exchange effectiveness of block i : 1bi i it tε δ= Δ ,    where 1itδ - temperature drop in block i; itΔ - the temperature difference of streams entering block i. The temperature change of cold stream: 2 1i i bt t Rδ δ= ⋅   These relations can be regarded as mathematical model of block, which describes the dependence of temperature changes from characterising block values of Fb and Ub .  For every block we can write the equation which describes the link of temperature change in this block to temperature changes in all other blocks of PHE.  For any number of passes such system presented in matrix form:  [ ][ ] [ ]i biZ tδ = ε Δ ,                                                                    (4) where [ ]itδ - vector-column of temperature drops in blocks; [ ]biε Δ - vector-column of the right hand parts of the system; [ ]Z - matrix of system coefficients.  The numerical solution of linear algebraic equations system (7) easily made on PC. After that the total temperatures change in PHE calculated as: ( )1 221 111 11X Xi X iii iit tXΣ − += =⎛ ⎞δ = δ⎜ ⎟⎜ ⎟⎝ ⎠∑ ∑ ;   ( )( )1 12 12 2G ct tG cΣ ∑δ = δ .                        (5) The total heat load of PHE:                           1 1 1Q t G cδ Σ= ⋅ ⋅  ,                                                              (6) This system should be accompanied by equations for calculation of overall heat transfer coefficient U ,  W/(m2K), as below. ( )1 21 1 1 w w fU R= α + α + δ λ +              (7) Computer Aided Design of Plate Heat Exchangers   where 1 2,α α  - film heat transfer coefficients for hot and cold streams, respectively, W/(m2K); wδ  - the wall thickness, m; wλ - heat conductivity of the wall material, W/(mK); 1 2f f fR R R= + - the sum of fouling thermal resistances for streams, m2K/W. For plate heat exchangers the film heat transfer coefficients are usually calculated by empirical correlations: 0.4 0.14(Re, Pr) *Re Pr ( / )n wNu f A μ μ= =                               (8) Here μ  and wμ  dynamic viscosity at stream and at wall temperatures, respectively; Nusselt number is:                                                   /Nu deα λ= ⋅ ,                     The streams velocities are calculated as                   /( )chw g f ρ=                       Where  g  is flowrate of stream through one channel, kg/s.              The pressure drop in one PHE channel ( ) 2 2p ep L d wζ ρΔ = ⋅ ⋅ ⋅ ,             (9) where pL -effective plate length; ζ – friction factor, which is usually determined by empirical correlations of following form: / RemBζ =                           (10) For multi pass PHE the pressure drop in one pass multiplied by number of passes X. In modern PHEs plates of one type are usually made with two different corrugation angles, that can form three different channels, when assembled in PHE. Plates of H type have corrugations with bigger angles (about 600) that form the H channels with higher efficiency of heat transfer and hydraulic resistance. Plates of L type have a lower angle (about 300) and form the L channels with lower heat transfer and hydraulic resistance. Combined, these plates form channels MH or ML with intermediate characteristics. Such principle of design enables to change thermal and hydraulic performance of plates pack with the level of discreteness equal to one plate in a pack.  In one PHE two groups of channels are usually used. One is of higher hydraulic resistance and heat transfer (x-channel), another of lower characteristics (y-channel). When the stream is going through set of such channels, the temperature changes in different channels are differ. After mixing in collector part of PHE block, the temperature is determined by heat balance. Then the heat exchange effectiveness of plates block with different channels:  )()( yyxxyyyxxxb ngngngng ⋅+⋅⋅⋅+⋅⋅= εεε ,        (11) where xn and yn are the numbers of x and y channels in a block of plates, respectively;  , ,x y x y chg w fρ= ⋅ ⋅ - the mass flow rates through one channel of type x or y. These  flow rates should satisfy equation x yp pΔ = Δ  and material balance: x x y y bg n g n G⋅ + ⋅ = ,            (12) where bG - flow rate of the stream through the block of plates. The principle of mixing plates in one heat exchanger gives the best results with symmetrical arrangement of passes (X1=X2) and Gbis equal to total flow rate of respective stream. The unsymmetrical arrangement X1≠X2 is usually used when all channels are the same (any of the three available types).    O. Arsenyeva et al. When the numbers of channels are determined, the numbers of plates calculated as: ( ) ( )1 21 1 2 21 11X Xpl x i y i x i y ii jN n n n n= == + + + +∑ ∑                       (13) The total heat transfer area of PHE (two end plates not included), m2: ( 2)PHE pl plF N F= − ⋅ ,            (14) where plF - heat transfer area of one plate, m2. The above algebraic equations (1)-(14) describe the relationship between variables which characterize PHE and heat transfer process in it.  They can be regarded as a mathematical model of PHE, which solution enables to calculate pressure and temperature change of streams entering the heat exchanger. It is a problem of PHE rating (analysis).  The problem of PHE design (synthesis) require to find its characteristics (such as plate type, numbers of passes, numbers of plates with different corrugations) which will in the best way satisfy the required process conditions. Here the optimal design with pressure drop specification is considered, in a sense as described by Wang and Sunden (2003).  The objective function is total heat transfer area of PHE with conditions that specified heat load 0Q  and allowable pressure drops for both streams 0 01 2,p pΔ Δ must be satisfied. These conditions can be regarded as partial inequality constraints: 0Q Q≥ ;               01 1p pΔ ≤ Δ ;                 02 2p pΔ ≤ Δ                       (15) Analysis of relations (1) – (15) lets to conclude, that we have the mathematical problem of finding the minimal value for implicit nonlinear discrete/continues objective function with inequality constraints.  It does not permit analytical solution without considerable simplifications. To solve it by numerical methods, the software is developed for IBM compatible PC. The mathematical model contains some parameters, namely coefficients and powers in empirical correlations that are not easily available.              3. Identification of mathematical model parameters  As a rule the empirical correlations for design of industrially manufactured PHEs are obtained during tests on such heat exchangers at specially developed test rigs. Such tests are made for every type of new developed plates and inter plate channels. The results are property of manufacturing company and usually not published. Based on described above mathematical model it was developed the technique of numerical experiment that enables to identify model parameters by comparison with results obtained for the same conditions with the use of PHE calculation software, which  is now available in internet for most of PHE manufacturers. The results for some plates manufactured by Alfa Laval presented in Table 1. The geometrical parameters of plates, for which correlations were acquired are given in Table 2. The comparison of results obtained with our software to those of Alfa Laval free available software has shown good agreement (discrepancies not more then 4% on surface area). We should note, that obtained correlations and developed software can be used only for preliminary calculations, when optimizing PHEs or heat exchanger network. The final calculations when ordering the PHE must be performed by its manufacturer.  Computer Aided Design of Plate Heat Exchangers    Table 1. Parameters in correlations for some Alfa Laval PHEs (Re>250)    Plate  type  Channel type  A n Re B m Re B m Н 0.265 0.7 <520 33.0 0.25 ≥520 10.7 0.07 L 0.12 0.7 <1000 18.8 0.33 ≥1000 8.8 0.22  М3 ML/MH1 0.18 0.7 <1000 44.0 0.4 ≥1000 5.1 0.10 Н 0.25 0.7 <1250 10.0 0.2 ≥1250 2.4 0.0 L 0.12 0.7 <1500 5.1 0.3 ≥1500 1.7 0.15  М6 ML/MH 0.165 0.7 <930 9.3 0.3 ≥930 2.72 0.12 Н 0.27 0.7 <1300 11.7 0.13 ≥1300 4.55 0.0 L 0.11 0.71 <2200 4.23 0.23 ≥2200 1.88 0.12  М6M ML/MH 0.14 0.73 <2100 5.61 0.16 ≥2100 1.41 0.0 Н 0.24 0.7 <2000 11.1 0.15 ≥2000 3.5 0.0 L 0.11 0.7 <1500 12 0.36 ≥1500 2.42 0.14  М10B ML/MH 0.12 0.74 <2700 6.2 0.2 ≥2700 1.9 0.05 Н 0.26 0.7 <2000 5.84 0.05 ≥2000 5.84 0.05 L 0.085 0.74 <2900 5.2 0.28 ≥2900 1.57 0.13  М15B ML/MH 0.13 0.74 <3500 4.3 0.15 ≥3500 1.25 0.0  Table 2. Geometrical parameters for some Alfa Laval PHE plates Plate type δ , mm ed , mm b ,mm Fpl ,  m2 connectionD , mm chf ·103,  m2 pL , mm M3 2.4 4.8 100 0.032 36 0.240 320 M6 2.0 4.0 216 0.15 50 0.432 694 M6M 3.0 6.0 210 0.14 50 0.630 666 M10B 2.5 5.0 334 0.24 100 0.835 719 M15B 2.5 5.0 449 0.62 150 1.123 1381  4. Case study It is required to heat 5 m3/h of distillery wash fluid from 28 to 90 oC by hot water coming with temperature 95 oC and flow rate 15 m3/h. The pressure of both fluids is 5 bar. Allowable pressure drop for hot stream 1.5 bar. For cold stream 1 bar. The properties of wash fluid are taken constant as follows: density – 978,4 kg/m3; heat capacity – 3,18 kJ/(kg·K); сconductivity – 0.66 W/(kg·m). Dynamic viscosity at temperatures 25; 60; 90t C= o is taken as 19,5;16,6;9 cPμ= .  The results of calculations for different passes numbers X1 and X2 and optimal for those passes plates arrangements are presented in Table 3. The analysis show that the global optimum (38 plates) is achieved at X1=2 and X2=4 with all medium channels (19 H and 19 L plates in PHE). The closest other option (41 plates) is at X1=X2=2 with mixed                                                           1 In our study we neglected by small differences in ML and MH channels parameters   O. Arsenyeva et al. Table 3. The influence of passes and plate arrangement on number of plates in M6M PHE X1 X2 1 2 3 4 1 56 plates 1*28H / 1*27H 235 plates 2*59H/1*116H 157 plates 3*26H/1*78H 184 plates 4*23H/1*91H 2 72 plates 1*35H / 2*18H 41 plates 2*(7H+3*ML)/ 2*(7H+3*MH) 60 plates 3*10MH / 2*14ML 64 plates 4*8L / 1*15L+1*16L 3 44 plates 1*21H / 2*7H+1*8H 49 plates 2*12MH / 3*8ML 43 plates 3*(4MH+3*L)/ 3*(4ML+3*L) 50 plates 3*6L+1*7L / 3*8L 4 63 plates 1*31MH / 1*7ML+3*8ML 38 plates 2*9MH / 1*4ML+3*5ML 44 plates 3*7L / 2*5L+2*6L 50 plates 3*6L+1*7L / 4*6L channel arrangement in one pass. If we would have only one plate type in PHE, the minimal number of plates would be 44 for both H and L plates, or 15% higher than with mixed channels. 5. Conclusions The algorithm and software for computer aided design of multi-pass PHE assembled with plates of different corrugation patterns is developed. The model parameters corresponding to some industrially manufactured plates are obtained. The examples of calculation results for case study show the possibility with such method to obtain optimal solutions with exact satisfaction of constraints for total heat load and pressure drop of one stream. It gives the considerable reduction in heat transfer surface area of PHE.  6. Acknowledgements  The financial support of EC Project ECOPHOS (contract № INCO-CT-2005-013359) is sincerely acknowledged. References  O. Arsenyeva, L. Tovazhnyansky, P. Kapustenko, G. Khavin, 2009, Mathematical Modelling and Optimal Design of Plate-and-Frame Heat Exchangers. Chemical Engineering Transactions, 18: 791-796. A. Gogenko, O. Anipko, O. Arsenyeva, P. Kapustenko, 2007,  Accounting for fouling in plate heat exchanger design. Chemical Engineering Transactions, 12: 207-213. J. Hesselgreaves, 2001, Compact Heat Exchangers. Selection, Design and Operation. Elsevier, Amsterdam. L. Tovazshnyansky, P. Kapustenko, G. Khavin, O. Arsenyeva., 2004, PHEs in Industry. NTU KhPI, Kharkiv (in Russian). L. Wang, B. Sunden, R. Manglik , 2007, PHEs. Design, Applications and Performance. WIT Press, Southhampton. L. Wang, B. Sunden, 2003, Optimal design of PHEs with and without pressure drop specifications. Applied Thermal Engineering. 23, 295-311. View publication stats

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Computer Aided Design of Plate Heat Exchangers

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Computer Aided Design of Plate Heat Exchangers

Abstract

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Library of National Technical University "Kharkiv Polytechnic Institute"

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