Heat transfer analysis of a non-Newtonian fluid flowing through a Plate Heat Exchanger using CFD

Accepted Manuscript Title: Heat transfer analysis of a non-newtonian fluid flowing through a plate heat exchanger using CFD Author: Erika Y. Rios-Iribe, Maritza E. Cervantes-Gaxiola, Eusiel RubioCastro, Oscar M. Hernández-Calderón PII: DOI: Reference:

S1359-4311(16)30237-X http://dx.doi.org/doi: 10.1016/j.applthermaleng.2016.02.094 ATE 7823

To appear in:

Applied Thermal Engineering

Received date: Accepted date:

16-10-2015 25-2-2016

Please cite this article as: Erika Y. Rios-Iribe, Maritza E. Cervantes-Gaxiola, Eusiel RubioCastro, Oscar M. Hernández-Calderón, Heat transfer analysis of a non-newtonian fluid flowing through a plate heat exchanger using CFD, Applied Thermal Engineering (2016), http://dx.doi.org/doi: 10.1016/j.applthermaleng.2016.02.094. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Heat transfer analysis of a non-Newtonian fluid flowing through a Plate Heat Exchanger using CFD Erika Y. Rios-Iribe1, Maritza E. Cervantes-Gaxiola2, Eusiel Rubio-Castro2, Oscar M. Hernández-Calderón2* 1

Programa de Posgrado en Biotecnología, Facultad de Ciencias Químico Biológicas, Universidad Autónoma de

Sinaloa, Av. de las Américas y Blvd. Universitarios, Ciudad Universitaria, CP 80013, Culiacán, Sinaloa, México. 2

Chemical Engineering Department, Facultad de Ciencias Químico Biológicas, Universidad Autónoma de

Sinaloa, Av. de las Américas y Blvd. Universitarios, Ciudad Universitaria, CP 80013, Culiacán, Sinaloa, México. *

Corresponding author: Tel.+52 667 713 7860; e‐mail: [email protected]

Highlights A CFD analysis is applied to a non-Newtonian flow through plate heat exchangers. The effect of plates number, distance between plates and flow regime is analyzed. An empirical correlation for the friction factor is proposed. The relation between Nu and Pe is described by a modified Sieder-Tate equation. The relation between the heat transfer rate and the pumping power is reported. Abstract In this work, the moment and heat transfers of a non-Newtonian fluid flowing in steady laminar regime through a plate heat exchanger at constant wall temperature is studied using CFD. Single-pass U-type plate heat exchangers with multiple flat plates with and without baffles are used. The effect of plates number and distance between plates on the heat transfer and the pressure drop are investigated. An empirical correlation for the friction factor as a function of the generalized Reynolds number and the ratio between the friction characteristic length and the flow trajectory length is developed. It is found that the dependence of the Nusselt number on the Peclet number can be described by the modified Sieder-Tate equation. Although the flow pattern is highly complex, under an adequate definition of the characteristics parameters, it is possible to establish simple correlations between the dimensionless numbers that characterize the thermal-hydraulic behavior of plate heat exchangers. Keywords: CFD analysis, non-Newtonian flow, plate heat exchanger, baffles, moment and heat transfers. 1. Introduction Page 1 of 33

Heat exchanger can be classified as direct contact-type or indirect contact-type. In the direct contact-type, heat is transferred by complete or partial physical mixing of two or more streams. In an indirect contact type, the fluid streams remain separate, and the heat transfer takes place continuously through a separating wall. Plate Heat Exchanger (PHE) is an indirect-contact type system. PHE is normally built by a pack of thin corrugated metal plates with port holes at their corners providing a passage for the two fluids between which the heat transfer will take place. The stack of plates is placed in a bolted frame with solid end plates that hold the remaining plates together. PHEs can be fabricated in a gasketed or welded model, in which the flow space for the two heat exchanging fluids is sealed. The seal between the fluid streams is achieved by means of gaskets placed in the side grooves of the plates. The corrugation of the plates facilitates turbulence inside the channels and improves the mechanical strength of the plate pack [1]. PHEs are commonly used in diverse industrial process such as chemical, pharmaceutical, biochemical industries, power plants and central cooling systems, being predominantly in the food industry due to several reasons such as: suitability in hygienic applications, ease of cleaning and the thermal control required for sterilization and pasteurization, low space requirement, low fouling tendency and high efficiency. Also, PHEs possess exceptional hydro-thermal characteristics which allow more compact designs than achievable with conventional shell and tube heat exchangers, and have a very high surface/volume ratio and can easily adapted for different requirements simply by increasing or decreasing the number of plates needed [2]. However, the main disadvantage of PHEs is the limitation of their operational range with the maximum operating pressure being limited to 20.4 bar and the operating temperature about 150 °C [1]. As a result of the wide range of application of this heat transfer device, several experimental and modelling works have been performed in order to study the thermal-hydraulic performance of different plate patterns, fouling tendency, corrosion mechanisms, fluid flow distribution, and the optimization of different arrangements and configurations, using Newtonian or non-Newtonian fluids [3]. In this sense, Afonso et al. [4] conducted an experimental investigation to obtain a correlation for the determination of convective heat transfer coefficients of stirred yoghurt in a PHE, the obtained correlation reveals the large effects of the thermal entry length due to the high Prandtl numbers and to the short length of the PHE. Cabral et al. [5] carried out a study about the pressure drop of pineapple juice in a PHE with 50° chevron plates, which the power law model described well the rheological behavior. The friction factor for non-isothermal flow of pineapple juice in the PHE was Page 2 of 33

obtained for diagonal and parallel/side flow, and experimental results were well correlated with the generalized Reynolds number (20 ≤

Re

≤ 1230) and were compared with predictions from equations

from the literature. Khan et al. [6] investigated experimentally the heat transfer characteristics of a commercial PHE with different chevron angles, corrugation depths and configurations under turbulent flow conditions, Reynolds number was varied in the range of 500–2500 and it was found that Nusselt number increases with increasing Reynolds number and chevron angle. Muthamizhi and Kalaichelvi [7] derived Nusselt number correlations using dimensional analysis in terms of all the parameters to determine the heat transfer coefficients in a PHE for various concentrations of carboxymethyl cellulose solution and these correlations were also compared with the available models in literature, showing a very good agreement. Akturk et al. [8] performed experimentally an hydrodynamic analysis of a gasketed-PHE with different number of plates. As a result of the performance analysis made by using the experimental data, new empirical correlations for Nusselt number and friction factor coefficients were found as a function of Reynolds number in the range of 450-5250. Kumar et al. [1] presented a review about the application of nanofluid which improves the heat transfer performance of PHE. Crespí-Llorens et al. [9] presented an experimental study of the generalization method of the Reynolds number and the viscosity of non-Newtonian fluid flow in ducts of non-uniform cross-section. Specifically, the procedure was developed for two models of scraped surface heat exchanger with reciprocating scrapers. Basically, their proposed generalization method is a extended version of Delplace and Leuliet [10] relationship with more parameters Heat transfer in a PHE is strongly dependent on geometrical properties of plates, specifically on corrugation angle, area enlargement factor and channel aspect ratio. However, when dealing with a non-isothermal flow of a non-Newtonian fluid, the rheological behavior of the fluid will enhance local variations of physical properties in the PHE, in this sense Computational Fluid Dynamics (CFD) is useful in order to understand the flows in the complex geometries of PHEs [3]. A broad literature is available concerning modeling heat transfer in plates heat exchanger using CFD [11]. Taking this approach, Gut and Pinto [12] developed a mathematical model in algorithmic form for the steadystate simulation of gasketed PHEs with generalized configurations, where 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. Fernandes et al. [13] performed simulations of stirred yoghurt processing in a PHE with two corrugated plates using CFD and the results compared with experimental data showed a very good agreement. Fernandes et al. [3] analyzed the thermal behavior of yoghurt in a PHE using Page 3 of 33

CFD. They studied the influence of Reynolds number on local Nusselt numbers as well as the influence of entry effects on average Nusselt number, and the obtained thermal correlations were compared with the experimental ones and a very good agreement was found. Also, they performed simulations with a non-Newtonian fluid having lower Prandtl numbers than yoghurt in order to analyze the influence of this variation on the Reynolds number exponent of the thermal correlation. Carezzato et al. [14] developed a thermal algebraic model of a PHE for generalized configurations, which was adjusted to fit experimental data obtained from non-Newtonian flow and heat transfer by use of the generalized Reynolds and Prandtl numbers for a power-law. Fernandes et al. [15] analyzed fully developed laminar flows in double-sine chevron-type PHEs passages using CFD. They proposed relations to predict the tortuosity coefficient and shape factor. Fernandes et al. [16] studied numerically laminar flows of Newtonian and power-law fluids through cross-corrugated chevron-type PHEs in terms of the geometry of the channels. They proposed Single friction curves

f Re  K

for

both Newtonian and non-Newtonian fluids for each the corrugation angle by developing an adequate definition of the generalized Reynolds number good concordance between the coefficient

Re K

. For all values of corrugation angle, they found a and experimental data, and that coefficient

K

depends on the tortuosity coefficient. Tsai et al. [17] investigated the hydrodynamic characteristics and distribution of flow in two cross-corrugated channels of PHEs using CFD. They solved a 3D model with the real-size geometry of the two cross-corrugated channels provided by chevron plates and taking into account the inlet and outlet ports. Also, they found that distribution of the fluid from the inlet port into two channels is not uniform: the flow rate of the first channel is higher than that of the second channel. Kanaris et al. [18] suggested a general method for the optimal design of a PHE with undulated surfaces that complies with the principles of sustainability, for which they employed a previously validated CFD in order to predict the heat transfer rate and pressure drop in this type of equipment. Abu-Khader [2] carried out a review about the advances in PHEs both in theory and application, which emphasized on the heat transfer performance and pressure drop characteristics, general models (highlighting the CFD study) and calculations change of phase (boiling and condensation), fouling and corrosion, and welded type PHEs and finally other related areas. Clearly, it is important to study the effect of corrugated plate geometry (corrugation angle, area enlargement factor and channel aspect ratio) on the thermal-hydraulic characteristics (efficiency of heat transfer and flow resistance) in PHEs, because the flow channel cross section between the corrugated plates changes in a very complicated way, likely to cause turbulence. In case of nonPage 4 of 33

Newtonian fluids, the fluid flow is highly viscous that no turbulence develops in wide range of operation conditions, therefore predominates generally a fluid flow in laminar regime [4, 19-21]. In addition to PHEs, triple pipe heat exchangers (TTHEs) are used for non-Newtonian fluids, as they offer a large heat transfer area and present a better flow characteristic, because the non-Newtonian fluid flows through the small annular gap between two concentric tubes [22]. In this sense, PHE with flat plates possess a configuration similar to TTHE. However, PHE has a greater area/volume ratio than TTEH. On the other hand, the fluid flow through parallel flat plates is characterized by the formation large dead flow regions. This undesirable characteristic can be reduced by the introduction of baffles to guide the flow through the parallel flat plates, which can increase the pressure drop and the heat transfer. In summary, there is a great interest to modify the plates shape and configuration with the objective of achieving outstanding heat transfer results with a reasonable pressure drop, as well as to obtain an in-depth understanding of the momentum and heat transfer mechanisms and finally to enhance the performance of the heat exchangers. However, non-Newtonian fluids have been considered in very few CFD investigations, therefore, it is necessary to increase knowledge about the thermo-hydraulic performance in these devices. Thus, the main objective of this CFD study is to determine the moment and heat transfer characteristics of a non-Newtonian fluid flowing through a PHE at constant wall temperature. A parallel flat plates configuration with and without baffles is used. The effect of plates number and distance between plates on the heat transfer and the friction factor are investigated over a wide generalized Reynolds number range (0.2 < with baffles and 0.2 <

Re

Re

< 140 for configurations

< 1800 for configurations without baffles).

2. Mathematical model In this investigation, a 3D numerical simulation of the heat transfer was conducted using the CFD software FLUENTTM 6.3. The CFD modeling involves numerical solutions of the mass, momentum and energy conservation equations [23]. The mass conservation equation (or continuity equation) for a single phase fluid is written as follows:  t

  v



 0

(1)

The conservation of momentum in an inertial (non-accelerating) reference frame is described as follows:

Page 5 of 33

 t

where

p

v

is the static pressure,

Meanwhile, the stress tensor

τ

    vv



  p    τ   g

(2) g

is the stress tensor, and

τ

is the gravitational body force.

is given by:

t τ    v   v   

where  is the molecular viscosity,

Ι

2 3

  v Ι 

(3)

is the unit tensor, and the second term on the right hand side

is the effect of the volume dilation. For incompressible Newtonian fluids, the shear stress is proportional to the rate of deformation tensor

D

:

τ  D

where

D

(4)

is defined by: D  v  v

(5)

t

This is because for incompressible Newtonian fluids the density is constant, therefore Eq. (1) is simplified to

 v  0

, which reduces the Eq. (3) to the Eq. (4). For some non-Newtonian fluids, the

shear stress can similarly be written in terms of a non-Newtonian viscosity  : τ   D  D

(6)

Specifically, for a non-Newtonian fluid that obeys the power-law model, the viscosity is described by:   k

where

k

is the consistency index,

n

n 1

is the behavior index, and

(7) 

is the shear rate, which is defined as

follows:  

1 2

(8)

D :D

The conservation of energy is described by:  t

where

kt

E 

   v

E

 p       k t  T    v



(9)

is the thermal conductivity and for incompressible flows, the total energy per mass unit ( E )

is defined as follows:

Page 6 of 33

E  H 

where

H

1 2

v

2

(10)

is the sensible enthalpy.

3. Simulation In the current simulation, the 3D laminar model is applied to describe the non-Newtonian flow through flat PHEs with and without baffles under steady conditions in single-pass U-type arrangement. The governing equations and constitutive relations are discretized and solved by using the commercial software FLUENTTM 6.3. The material of the exchanger is stainless steel, and the fluid under study is tomato puree at 12 °Brix, whose temperature-dependent rheological and thermophysical properties were reported by Carrazco-Escalante [24] (see Table 1 in the supplementary material). All the three-dimensional grids were created in a CAD program called GAMBIT TM 2.2 and exported to FLUENTTM. The geometry dimensions for the heat exchangers used are as follows: plate size of 200 mm plates of

t 



200 mm (



), outlet and inlet diameter of

3, 4 and 5 mm, plates number of

passage; baffle length of

b



np 

d in  d o u t 

20 mm, distance between

2, 4, 8 and 16; baffles number of

160 mm, and distance between baffles of

s 

nb 

4 per flow

40 mm (see Figure 1).

Each base mesh contains 618’000, 1’245’000, 2’499’000 and 5’007’000 hexahedral cells for heat exchanger with 2, 4, 8 and 16 heating flat plates, respectively. Figure 1 An U-type arrangement of a simple pass in counter-flow was utilized in PHE with multiple plates (see Figure 2). In this case hot water flow was not simulated between the plates, because the operation condition at constant wall temperature was utilized, which is valid when a high hot water flow is considered. Figure 2

Page 7 of 33

The main inlet of the non-Newtonian flow is designated as a velocity-inlet in FLUENTTM, where the direction of fluid flow is normal to the surface boundary. The main inlet-fluid velocities ( U in ) used are as follows:

U in



1 2

np

without baffles (where

-1   0.05, 0.10, 0.25, 0.50, 1.00, 1.50, 2.50, 3.50, 5.00 and 10.00 m s for PHEs

np

is the plates number used);

U in



1 2

np

  0.05, 0.10, 0.25, 0.50, 0.75, 1.00,

1.25 and 1.50 m s-1 for PHEs with baffles; and the inlet-fluid temperature is 20 °C. Atmospheric pressure is imposed at the outlet of exchanger (i.e., it is designated as pressure-outlet in FLUENTTM). All the cell walls are static, the velocity tangent to the walls is assumed to be zero, and the plate wall temperature is set as 60 °C. The fluid flow is thermally coupled to the walls. The pressure–velocity coupling is obtained using a SIMPLE algorithm. The solver specifications for the discretization of the domain involve the following procedures: Standard for pressure and First Order Upwind for momentum and energy. A convergence criterion of 10−7 was used for the pressure, 10−8 for the velocity components, and 10−6 for the energy. The grid independence was ensured by meshing the geometry with different spacing between the cells. The meshes were refined successively by the temperature gradient adaption approach (i.e. each mesh was refined only in those cells that exhibited high temperature gradient) to achieve friction factor and Nusselt number variations under 5 % between two successive refinements of cells. Figures 1-2 in the supplementary material shows the effect of cells number of the mesh on the friction factor and the Nusselt number for configurations with two plates. In fact, it is possible to observe that for configurations with baffles at a main inletfluid velocity greater than 1.50 m s-1 require high computational effort to ensure the mesh independence, therefore, in this study configurations with multiple plates at

U

in



1 2

np

 greater than

1.50 m s-1 were not considered. Data reduction The use of dimensionless numbers allows reducing the need of expensive experimental measurements required for determining process performance. In this sense, the Reynolds, fanning friction factor, Prandtl and Nusselt dimensionless numbers are commonly applied in the investigation of the heat and momentum transfer processes. In this context, the Reynolds number is the ratio of inertia and viscous forces. For non-Newtonian fluids, the generalized Reynolds number ( R e ) is defined as follows:

Page 8 of 33

Re 

U LH

(11)

'

where  is the density, U is the average velocity, L H is the hydraulic characteristic length and 

'

is

the non-Newtonian viscosity, which is given by: n

 3 n  1   8U   ' k     4n   LH 

where

k

is the consistency index and

n

n 1

(12)

is the behavior index. Note that due to the fluid viscosity is

not constant (see Eq. (7)), it is necessary to use a characteristic viscosity of the process (  ' ) defined by the geometry of the flow region and rheological nature of the fluid (behavior index and consistency index) which is given by Eq. (12). This characteristic viscosity is linked to the macroscopic behavior characteristics, and is equal to Newtonian viscosity which would give the same relationship between the volumetric flow rate and the pressure drop [25]. For the fluid flow in ducts, the friction factor is described by a dimensionless number, this is the Fanning friction factor (

f

), which is defined as

follows: f 

where

Δp

is the pressure drop and

1   Δ p   LH    2   4  12  U   L 

L

(13)

is the flow trajectory length. In the heat transfer at a surface

within a fluid, the Nusselt number ( N u ) is the ratio of the convective to the conductive heat transfer across normal to the boundary and it is given by: Nu 

hLh

(14)

kt

where h is the convective heat transfer coefficient and L h is the thermal characteristic length. The Prandtl number ( P r ) is the ratio of the momentum and the thermal diffusivities, and it describes the static properties of the fluid substance. The generalized Prandtl number is defined by: Pr 

 'c p

(15)

kt

where c p is the specific heat. Also, the Peclet number is defined as: (16)

Pe  Re Pr

The average pressure at the flow inlet and outlet,

p in

and

p out

, can be obtained by the following

relationship: Page 9 of 33

p 

where





p( x ) n  dS S





(17)

n  dS S

is the pressure field over inlet or outlet regions, and

p(x)

n

is the normal unitary vector over

the inlet or outlet surfaces. The pressure drop was calculated by:   p  p in  p o u t

(18)

With respect to the average heat transfer coefficient ( h ), it was calculated by: Q

h 

where

Q

is the heat transfer rate,

(19)

 tm S h

Sh

is the heating wall surface area, and

 tm

is the

logarithmic mean temperature difference which is given by:  tm 

where

tw

 tw

 tout

   tw

ln   t w  t o u t

 t in

 /  tw

is the heating wall temperature, and

t in



(20)

 t in  

and

tout

are the inlet and outlet temperatures,

respectively. The heat transfer rate was determined by: Q 

 k T t

 ndS

(21)

S

where

S

is the heating surface and

is an unit vector normal to the heating surface. Alternatively,

n

in order to validate the numerical simulations, the heat transfer rate was also determined by the following energy balance: Q  m out H

out

 m in H

(22)

in

in which the specific enthalpies of the inlet and outlet flows ( H in and

H

out

) were determined by the

next mass-weighted average integral:

 H (x )   v  dS  i

Hi 

S

  v  dS

(23) i

S

where (t

i  in

 t o u t  t in

and

out

. This is because at high flow-rate, the achieved temperature increase

) is very low and the accuracy of the temperature obtained by the following mass-

weighted average integral:

Page 10 of 33

 t(x )   v  dS  i

ti 

S

  v  dS

(24) i

S

it does not guarantee that the heat transfer rate given by Eq. (25) be equivalent to that obtained by the Eq. (21). Q  m

here

m  m in  m o u t



to u t t in

(25)

c pdt

.

Notice that for the PHEs without baffles, U is defined as the average velocity of inlet flow ( U in ), the hydraulic length

LH

whose dimensions are

2

length

L

is determined using the average cross-section area of the flow space, and t , the friction characteristic length is

b  t

is equal to the distance between the inlet and outlet positions (here,

, the flow trajectory L  2 2 .6 3

mm). For

the PHEs with baffles U is defined as the average velocity of the flow region confined by the plates and the baffles, the hydraulic length whose dimensions are L

s

LH

is determined using the cross-section area of the flow space

and t , the friction characteristic length is

b  LH

, the flow trajectory length

is equal to the flow trajectory length between the inlet and outlet positions, i.e.

(here,

nb

= 4 and

L  1000

mm). The hydraulic length was evaluated as follows:

is the flow cross-sectional area and Lh  LH

P

L  ( n b  1)

LH  4 A P

where

A

is the wetted perimeter. For both cases, the thermal length is

.

In contrast to previous works [9], here the typical generalized dimensionless number are used, and the characteristic parameters as such

U

,

LH

,

L

and

b

are defined in order to the flow

behavior. 4. Results and Discussion Figures 3 show the typical behavior of normalized velocity field (with respect to inlet velocity) in two parallel flat plates without and with baffles at different inlet velocities, respectively. The plates arrangement without baffles exhibit large dead flow regions, which decrease when the inlet velocity increase, this due to the fluid inertia induces a fluid flow in all space between the two plates when the inlet flow is high (see Figure 3a). In contrast, the plates arrangement with baffles, exhibit small dead

Page 11 of 33

flow regions, which increase when the inlet velocity increase, because the baffles arrangement causes that the flow changes its direction at 180°, inducing a centrifugal force that generates a dense flow core adjacent to only one of two baffles (see Figure 3b). Figure 3 Figure 4 shows the behavior of the friction factor as funcion on the Reynolds number in heat exchangers with two flat plates with and without baffles. For all heat exchangers with two plates, it is possible to observe a good agreement between the calculated generalized Reynolds number and friction factor and the theoretical relationship (

f  16 R e

) over the Reynolds number range of 0.2-

20. This is because, in both cases (with and without baffles), the flow pattern is similar to a fluid flowing through of a conduit. For PHEs without baffles at a low flow-rate, the flow shape between heating plates is similar to the flow in a conduit with: a constant flow cross-section area which is defined approximately by the geometrical shape of flow inlet; and a flow length equal to the distance between the positions of inlet and outlet flow. In other words, the behavior of velocity profile is strongly dependent on the shape of the inlet flow, because the velocity profile is mainly developed in the inlet region. For PHEs with baffles at a low flow rate, the flow shape is similar to a fluid flowing through a rectangular conduit of constant flow cross-section area whose dimensions are the distance between plates and the distance between baffles and, a flow length equal to the flow trajectory length showed in Figure 1a. On the other hand, for PHE without baffles at

R e  20

, the velocity field

tends to be uniform over almost all the flow cross-section areas, therefore friction factor tends to be independent on the flow regime, but strongly dependent on the geometrical shape of the heat exchanger. Regarding to PHE with baffles at

R e  20

, the baffles arrangement causes that the flow

changes its direction at 180°, inducing a centrifugal force that produces a dense flow core adjacent to only one of two baffles, and whereby large dead-flow zones are generated. In this sense, the flow shape is not similar to a fluid flowing through a rectangular conduit but this is defined by centrifugal forces. Thus, at a high flow-rate the flow shape tends to a limiting flow core, which is independent on the flow regime, but strongly dependent on the geometrical arrangement of the baffles and plates, therefore, also in this case, the friction factor tends to be independent on the flow regime, and it is defined uniquely by the geometrical arrangement of the plates and baffles. For the Reynolds number range investigated in this work, the following empirical correlation of the friction factor as a function

Page 12 of 33

of the generalized Reynolds number and the ratio between the friction characteristic length ( b ) and the flow trajectory length ( L ) is proposed: 9 /5   16   f    0 .0 8 0 7   R e  

with a correlation coefficient of

R

2

 1  3 L   9 .8 9 8  1 0    Re b   

 0 .9 9 9 2

5

   

7 /20

   

5/9

(26)

. Figure 4 exhibits a good fit between the empirical

correlation and CFD data for all the numerical experiments with two plates, which is due to a proper definition of the characteristic lengths ( b ,

LH

and

L

) used. Notice that for configurations with

baffles, the main inlet-fluid velocities ( U in ) used are as follows:

U in 

0.05, 0.10, 0.25, 0.50, 0.75,

1.00, 1.25, 1.50, 2.00, 3.00, 4.00 and 5.00 m s-1. It is noted that in the velocity interval of 2.00 – 5.00 m s-1, the simulations exhibited mesh independence with respect to the friction factor, but not with respect to the Nusselt number (see Tables 2-5 in the supplementary material). Also, in the Eq. (26) the exponent

5

is positive for configurations without baffles and negative for configurations with

baffles. This is due to the way that it was defined the characteristic velocity ( U ) of both configurations. Figure 4 Figure 3a in the supplementary material shows the effect of the flow regime and the geometry on the friction and dynamic losses for the configurations with two plates. It is observed that there is a power law relationship between the Reynolds number and the friction and dynamic losses in both configurations (with and without baffles) at low Reynolds number. This behavior deviates from the power law at high Reynolds number observing larger desviations in the configurations with baffles. Besides, it is observed that the calculated friction and dynamic losses are greater in the configurations with baffles than those observed in the configurations without baffles at similar Reynods number; finally it is noticed that the distance between plates affects in a more significant way the friction and dynamic losses in the configurations without baffles The variation of the Nusselt number as a function of the Peclet number ( P e

= R ePr

) for all the

case studies with two flat plates is presented in Figure 5, in which it is possible to observe that the increase of the Peclet number results in an increase of the Nusselt number as well. The increase of

Page 13 of 33

the Nusselt number indicates an enhancement in the heat transfer coefficient due to the convection increases. For the heat transfer of Newtonian fluid flows in a circular cross-sectional area, the Nusselt number can be appropriately described by the modified Sieder and Tate [26] equation:

N u  1 .8 6 P e

where

D

y

1/3

 D     L 

1/3

(27)

are the tube diameter and lenght, respectively. In this work a similar correlation for

L

the Nusselt number is proposed and it is defined by: N u   (P e , LH

where

 (P e , L H

)

  1/3  L )  1 .8 6 P e  H    

1/3

  

(28)

is a correction factor dependent of the geometrical configuration and the Peclet

number. However, two different behavior patterns can be observed depending on the configuration of PHE used. For the PHE without baffles, the Nusselt number exhibits a power-law behavior as a function on the Peclet number, which is given by the following factor de corrección:  (P e , LH

     L  5  L 6  )   -5 .7 6 7 5  1 0  H  + 3 .5 5 0 1  1 0  P e   -8 .1 5 0 0  H  + 1 .2 7 8 2         

with a correlation coefficient of

R

2

 0 .9 9 7 0

(29)

. It is found that Eq. (29) describe properly the behavior

of Nusselt number over all Peclet number range investigated. In additon, in Eq. (29), it is possible to observe that at low  (P e , L H

)

LH

values (i.e., at shorter distance between plates) the correction factor

increases faster with the increase of

Pe

than with respect to large

LH

values. For the

PHEs with baffles, the relationship between the Peclet number and the Nusselt number requires the following correction factor:  (P e , LH

with a correlation coefficient of

)  6 .5 8 0 3  1 0

R

2

 0 .9 9 5 2

6

P e + 7 .3 5 6 3  1 0

1

(30)

. It is noticed that Eq. (30) exhibits a greater slope than

Eq. (29), but a smaller intercept value, indicating that configurations without baffles presents higher values for the correction factor at low Peclet number than the configurations with baffles at similar Peclet numbers (the effect of the intercept value); on the other hand, for configurations with baffles the correction factor increases faster with the increase of the Peclet number than the configurations

Page 14 of 33

without baffles (the effect of the slope). This behavior of the correction factor (the effect of the slope) indicates a strong enhancement in the heat trasfer process for the PHEs with baffles, which is due to the baffles arrangement changes abruptly the flow direction at 180°, generating a dense flow core adjacent only to one of two baffles. This flow behavior generates large dead-flow zones, causing that the effective convective heat transfer surface decreases, but not as fast as the hydrodynamic layer thickness decreases. Therefore, the formation of this dense flow core produces an important improvement in the heat transfer process. Figure 5 Figure 6a shows the variation of heat transfer rate as a function on the pumping power for plate exchangers with two plates as a result of the variation of flow regime. For the PHEs with baffles, CFD data of heat transfer rate and pumping power can be described by a unique relationship, which is expressed by: Q  1 4 6 8 .2 9 W

with a correlation coefficient of

R

2

 0 .9 9 9 1

0 .3 9 2 2

(31)

. This interesting behavior can help to simplify the design

of the PHEs, because once the geometrical dimensions and the flow regime are established, the pressure drop can be calculated, therefore the pumping power is obtained, and finally through Eq. (31) the heat transfer rate is determined. In these configurations, the dead-flow zones are increased due to the formation of a flow core when flow regime is increased. In this sense, the flow regime defines the flow shape and, it also defines the effective friction and heat transfer surfaces between the flow and the walls, and the thermal and hydraulic layer thicknesses, which decrease when increasing flow-rate. Therefore, both transfer surfaces and both layer thicknesses must decrease at the same rate when flow rate increases; otherwise, it is not possible to explain the behavior observed in Figure 6a, which indicates that for different operating conditions with the same pumping power, the heat transfer rate is approximately the same. On the other hand, for the PHEs without baffles, the relation between the heat transfer rate and the pumping power is strongly dependent on the distance between plates. In fact, at low flow-rate, the behavior of the heat transfer rates as a function on the pumping power at different distance between plates is very similar to each other; and at high flowrate, the heat transfer rate increases faster with the pumping power when the distance between plates is smaller. In these configurations there are large dead-flow regions, which decrease when

Page 15 of 33

increasing flow-rate, because the velocity field tends to be uniform over almost all the flow cross section areas. However, the spreading of the velocity profile is different in each geometrical configuration. In fact, the hydrodynamic layer thickness decreases faster with decreasing distance between plates, which in turns increases the heat transfer rate. Additionally, for these configurations, the relationships between the heat transfer rate and the power pumping are fitted as follows: Q  1 8 9 .0 3 ln W  8 1 2 .9 0 ln W

 2 4 7 2 .9

(32)

Q  1 4 8 .6 1 ln W  8 1 2 .2 2 ln W

 2 4 3 1 .5

(33)

Q  1 2 2 .2 1 ln W  7 8 5 .8 0 ln W

 2 3 8 5 .4

(34)

2

2

2

for the PHEs without baffles with

t 

3, 4 and 5 mm, respectively, and a correlation coefficient ( R 2 )

of 0.9992, 0.9996 and 0.9998 for each case respectively. Furthermore, Figure 6b shows the dependence of the heat transfer rate with respect to the volumetric flow-rate. It is noted that at low volumetric flow-rate the configurations with and without baffles with the same distance between plates exhibit similar heat transfer rates; and in each of the PHEs configurations studied (with and without baffles) at shorter distance between plates a higher heat transfer rate is obtained. Also, in the PHEs configurations with baffles the heat transfer rate increases more rapidly with increase of the volumetric flow-rate than the PHEs configurations without baffles. Figure 6

Concerning the effect of plates number on the flow pattern in a PHE with multiple plates, in Figure 7 is shown the normalized velocity field over the middle planes in each flow region for PHEs with eight flat plates without and with baffles. Here flow region represents the flow space between two plates in which non-Newtonian fluid flows. In this work the flow region between the plates

2 k -1

and

2k

, therefore a PHE with

M

flat-plates has

are the flow spaces between the plates 1 and 2, 3 and 4, ...,

M 1

and

M 2 M

k

is the flow space

flow-regions, which

. Also, velocity field is

normalized using the average inlet velocity bewteen two plates corresponding to a homogeneous

Page 16 of 33

flow distribution in a PHE, this is

U in



1 2

np

 where

np

is the plates number and

U in

is main inlet

velocity in PHE. In fact, U

k

=

1

M 2

 M/2

k =1

where

Uk

U

k



U in 1 2

(35)

np

is the average-inlet velocity for flow region

k

and

U

k

the is the average of

Uk

values. For

PHE with eight flat plates without baffles, in Figure 7a it is possible to obseve that the inlet and outlet effects are significant in each flow region. Also, the fed flow rate is distributed according to the path with less resistance to flow, which corresponds to the flow region more close to the main feed. Therefore, there is a non-homogeneous distribution of average-inlet velocity for each flow region, which is caused by inlet effects. This same behavior is found in all the configurations of PHE with multiple flat plates and without baffles for the different inlet velocities tested in this work. For PHE with eight flat plates and with baffles, Figure 7b exhibits a similar flow pattern in each flow region. This is due to the baffles produce a similar resistance to the flow, causing a more homogeneous distribution of average-inlet velocity for each flow region. An identical behavior is found in the different configurations of PHE with multiple flat plates and with baffles for the different inlet velocities used in this work. Figure 7 Figure 8 shows the distribution of normalized average-inlet velocity for each flow region in a PHE with 16 flat plates without and with baffles at different inlet velocities. Figure 8a shows that there is a non-homogeneous average-inlet velocity distribution for each flow region in a PHE without baffles, which prove that the fed flow rate is distributed according to the path with less resistance to flow, which corresponds to the flow region more close to the main feed. Also, it is found that average-inlet velocity distribution is more non-homogeneous at high flow-rate. In contrast, for PHE with baffles the average-inlet velocity distribution is practically homogeneous, because the baffles produce a high flow resistance, which promotes a more uniform average-inlet velocity distribution for each flow region (see Figure 8b). Figure 8

Page 17 of 33

Regarding the effect of the plates number on the average-inlet velocity distribution in each flow region, Figure 9 depicted the standard deviation (  U ) of distribution of normalized inlet velocity for each flow region, as a function on the plates number ( n p



4, 8 and 16) and the main inlet velocity

( U in ) for PHEs with and without baffles. The standard deviation  U is calculated by: 1 2

np

 k 1

U 

 U U k k  1  U in  2 n p 1 2

   

2

(36)

np

Substituting Eq. (35) into Eq. (36), 1 2

np

 U 

k 1

  Uk  1 2 np    1  U in    1 2

2

(37)

np

It is possible to observe that PHEs without baffles have a more non-homogeneous average-inlet velocity distribution than PHEs with baffles. Also, for PHEs without baffles, the average-inlet velocity distribution inhomogeneity is higher for PHEs with a high plate number and a high inflow-rate. For PHEs with baffles, the standard deviation value of normalized average-inlet velocity distribution is small, and is practically constant in each configuration, being only affected by the plates number, therefore when the plates number increases, the standard deviation value of normalized averageinlet velocity distribution increases as well. Figures 3b-d in the supplementary material presents the effect of the flow regime and geometry on the friction and dynamic losses for configurations with multiple plates. It is possible to observe in the Figures 3a-d in the supplementary material that the flow distribution in the PHEs without baffles affects very significantly the relationship between the friction and dynamic losses and the Reynolds number. At low Reynolds number the behavior of the friction and dynamic losses as a function of the Reynolds number is independent of the plates number, however, at higher Reynolds number the friction and dynamic losses are greater for a higher plates number. In fact, the friction and dynamic losses of the configurations without baffles approach to those obtained for the configurations with baffles as the Reynolds number and plates number increase. Regarding the PHEs

Page 18 of 33

configurations with baffles significant effects of the plates number on the friction and dynamic losses are not observed. Figures 4a-c in the supplementary material shows the behavior between the Reynolds number and the friction factor for all configurations with multiple plates (4, 8 and 16 plates). The comparison of these results and those obtained for the configurations with two plates (Figure 4) shows that the configurations with baffles not exhibit an important effect of the plates number on the behavior of the friction factor as a function of the Reynolds number; in contrast, for the configurations without baffles the effect of the plates number on the friction factor is relevant; in fact, this is due to a greater plates number, larger asymptotic values of the friction factor are achieved at greater flow regime, which cause that the behavior of the friction factor as a function of the Reynolds number deviates from the theoretical relationship (

f  16 R e

) at low Reynolds number.

Regarding the effect of the plates number on the Nusselt number (see Figure 5, and Figures 5a-c in the supplementary material), it is observed that for configurations with baffles the plates number does not has an important effect on the behavior of the Nusselt number, and for the configurations without baffles the plates number has a little significant effect on the Nusselt number due to the higher the plates number the Nusselt number increases slightly at high Peclet numbers. Figure 9 In Figure 10, the effect of plates number on the relation between the heat transfer rate and the pumping power for a non-Newtonian fluid flowing in PHEs with and without baffles at different flow-rate and at different distance between plates is presented. For the PHEs with baffles, at low flow-rate the relation between the heat transfer rate and the pumping power is properly represented by Eq (31), in contrast at high flow-rate, CFD data of transfer rate and the pumping power is slightly different from that given in Eq. (31); strictly, the higher the number of plates used (specifically at n p  16

), the lower the ratio

Q W

. These differences observed in the ratio

Q W

could be due to the

non-homogeneous distribution of average-inlet velocity, which is significant in this configuration (see Figure 9). For the PHEs without baffles, at high flow-rate the average-inlet velocity distribution is more non-homogeneous than in the heat PHEs with baffles when a greater plates number is used, therefore the thermo-hydraulic efficiency decreases when the plate number increases. In Figure 10 is

Page 19 of 33

possible to observe than the PHEs without baffles exhibits a better thermo-hydraulic performance than the PHEs with baffles only at a low pumping power; therefore, the use of baffles is only recommended at high flow-rates, due to enhance the heat transfer process. Figure 10 Besides, in Figure 6 in the supplementary material is presented the behavior of the heat transfer rate as a function of the volumetric flow-rate for the PHEs configurations with 4, 8 and 16 plates, in which is observed that the correlations obtained for the configurations with two plates described appropriately the behavior of similar PHEs configurations with 4, 8 and 16 plates. In this sense, it is possible to indicate that the observed differences in the ratio

Q W

for the case of the PHEs

configurations without baffles are due to the effect of the plates number over the friction and dynamic losses, as a result of the non-homogenous distribution of the flow In general, for all devices in this work at a high flow-rate, the greater the plates number, the lower the ratio between the heat transfer rate and pumping power, i.e. the lower the thermohydraulic performance, and this effect is more pronounced when baffles are not used. Finally, the accuracy of each numerical simulation is evaluated and it is observed that for all the PHEs configurations without baffles in the interval of error for the mass conservation is in the interval of

U in

7 .5 6 2 1  1 0



1 2

12

np

-1   0.05 - 5.00 m s , the relative

 1 .9 4 5 4  1 0

6

(see Table 2 in the

supplementary material), and for the energy conservation, the relative error is in the interval of 1 .3 1 3 4  1 0

2

 1 .9 7 0 8  1 0

4

(see Table 3 in the supplementary material); and besides, for all the PHEs

configurations with baffles in the interval of mass conservation is in the interval of

U in

1 .2 1 5 4  1 0



1 2

11

np

-1   0.05 – 1.50 m s , the relative error for the

 7 .5 9 5 5  1 0

4

(see Table 4 in the supplementary

material), and for the energy conservation, the relative error is in the interval of 1 .6 5 4 2  1 0

2

 1 .0 1 6 1  1 0

3

(see Table 5 in the supplementary material). Here, the relative errors of

the mass and energy conservation were determined by: Em 

m o u t  m in m in

(38)

and Page 20 of 33

Ee 

m H m H out

out

out

Q H 

 m in H in out

 m in

(39)

in

where the heat transfer rate ( Q ) is determined by Eq. (21). 5. Conclusions A CFD analysis of the moment and heat transfer characteristics of a non-Newtonian fluid flowing through a PHE with and without baffles over a wide Reynolds number range (0.2 < Re < 140 for configurations with baffles and 0.2 < Re < 1800 for configurations without baffles) using different plates number and distance between plates, and at constant wall temperature was presented. For all Reynolds numbers investigated in heat exchanger with two plates, CFD data were correctly fitted to an empirical correlation of the friction factor as a function of the generalized Reynolds number and the ratio between the friction characteristic length and the flow trajectory length, which is a result of an adequate definition of the characteristic lengths. Also, it was found that the dependence of the Nusselt number on the Peclet number obeys a similar relation to the modified Sieder-Tate equation: N u   (P e , L H

1/3 )  1 .8 6 P e  L H 



1/3

 

where

 (P e , L H

)

is a correction factor dependent of the

geometrical configuration and the Peclet number. For the PHEs with baffles, all CFD data of heat transfer rate and pumping power can be practically described by a unique relation: the heat transfer rate follows a power-law behavior as a function of pumping power, which is independent of the distance between plates and the plates number. Besides, it was found than the PHEs without baffles exhibits a better thermo-hydraulic performance than the PHEs with baffles only at low pumping power. For all configurations investigated in this work, it was found that at high flow-rate, the higher the number of plates used, the lower the ratio between the heat transfer rate and the pumping power. This is because there is a non-homogeneous average-inlet velocity distribution, which is caused by inlet effects. This negative effect is more pronounced in the PHE without baffles. Finally, the results showed that CFD can be used as a powerful tool in PHE design and analysis, and can be considered as a proper tool for testing different geometrical configurations of the PHE. Acknowledgement

Page 21 of 33

The authors acknowledge the financial support by the Universidad Autónoma de Sinaloa (grant number PROFAPI 2014/086).

Nomenclature

A

flow cross-sectional area, m2

b

characteristic friction length, m

bi

proportionality constant in Eq. (29), dimesionless

ci

proportionality constant in Eq. (30), dimensioless

cp

specific heat, J kg-1 K-1

d

inlet or outlet diameter, m

D

deformation rate tensor, s-1

E

total energy per mass unit, J kg-1

Ee

relative error for the energy conservation, dimensionless

Em

relative error for the mass conservation, dimensionless

f

Fanning friction factor, dimensionless

g

gravitational acceleration, m s-2

h

coefficient of local convective heat transfer, W m-2 K-1

H

enthalpy per mass unit, J kg-1

I

unitary tensor, dimensionless

k

consistency index, kg sn-2 m-1; index, dimensionless

kt

thermal conductivity, W m-1 K-1 square plate size, m

b

baffle length, m

L

characteristic length, m

M

plates number, dimensionless

n

behavior index, dimensionless

nb

baffles number, dimensionless

np

plates number, dimensionless

n

unitary normal vector, dimensionless

m

mass flow-rate kg s-1

Page 22 of 33

Nu

Nusselt number, dimensionless

p

pressure, Pa

P

wetted perimeter, m

Pe

Peclet number, dimensionless

Pr

Prandtl number, dimensionless

Q

heat transfer rate, W 2

R

correlation coefficient, dimensionless

Re

generalized Reynolds number, dimensionless

s

distance between baffles, m

Sh

heating surface area, m2

S

vectorial surface, m2

t

time, s; temperature, °C; distance between plates, m

T

temperature, K

U

average velocity, m s-1

Uk

average-inlet velocity for flow region k , m s-1

Uk

average of U k values for a PHE, m s-1

V

volumetric flow rate, m3 s-1

v

velocity, m s-1

W

pumping power, W

x

position vector, m

Greek Symbols 

shear rate, s-1

p

pressure drop, Pa

 tm

logarithmic mean temperature difference, °C



non-Newtonian viscosity, Pa s



characteristic non-Newtonian viscosity, Pa s



Newtonian viscosity, Pa s



density, kg m-3



U

standard deviation of distribution of normalized inlet velocity for each flow region, dimensionless

τ

stress tensor, Pa



vector differential operator    x ,   y ,   z  , m-1

Subscripts Page 23 of 33

h

thermal

H

hydraulic

in

inlet

out

outlet

w

wall

References [1] V. Kumar, A.K. Tiwari, S.K. Ghosh, Application of nanofluids in plate heat exchanger: A review, Energy Conv. Manag., 105 (2015) 1017-1036. [2] M.M. Abu-Khader, Plate heat exchangers: Recent advances, Renew. Sust. Energ. Rev., 16 (2012) 1883-1891. [3] C.S. Fernandes, R.P. Dias, J.M. Nóbrega, I.M. Afonso, L.F. Melo, J.M. Maia, Thermal behaviour of stirred yoghurt during cooling in plate heat exchangers, J. Food Eng., 76 (2006) 433-439. [4] I.M. Afonso, L. Hes, J.M. Maia, L.F. Melo, Heat transfer and rheology of stirred yoghurt during cooling in plate heat exchangers, J. Food Eng., 57 (2003) 179-187. [5] R. Cabral, J. Gut, V. Telis, J. Telis-Romero, Non-newtonian flow and pressure drop of pineapple juice in a plate heat exchanger, Braz. J. Chem. Eng., 27 (2010) 563-571. [6] T. Khan, M. Khan, M.-C. Chyu, Z. Ayub, Experimental investigation of single phase convective heat transfer coefficient in a corrugated plate heat exchanger for multiple plate configurations, Appl. Therm. Eng., 30 (2010) 1058-1065. [7] K. Muthamizhi, P. Kalaichelvi, Development of Nusselt number correlation using dimensional analysis for plate heat exchanger with a carboxymethyl cellulose solution, Heat Mass Transf., 51 (2014) 815-823. [8] F. Akturk, N. Sezer-Uzol, S. Aradag, S. Kakac, Experimental investigation and performance analysis of gasketed-plate heat exchangers, J. Therm. Sci. Technol., 35 (2015). [9] D. Crespí-Llorens, P. Vicente, A. Viedma, Generalized Reynolds number and viscosity definitions for nonNewtonian fluid flow in ducts of non-uniform cross-section, Exp. Therm. Fluid Sci., 64 (2015) 125-133. [10] F. Delplace, J. Leuliet, Generalized Reynolds number for the flow of power law fluids in cylindrical ducts of arbitrary cross-section, Chem. Eng. J. Biochem. Eng. J., 56 (1995) 33-37. [11] M.M.A. Bhutta, N. Hayat, M.H. Bashir, A.R. Khan, K.N. Ahmad, S. Khan, CFD applications in various heat exchangers design: A review, Appl. Therm. Eng., 32 (2012) 1-12. [12] J.A. Gut, J.M. Pinto, Modeling of plate heat exchangers with generalized configurations, Int. J. Heat Mass Transf, 46 (2003) 2571-2585. [13] C.S. Fernandes, R. Dias, J.M. Nóbrega, I.M. Afonso, L.F. Melo, J.M. Maia, Simulation of stirred yoghurt processing in plate heat exchangers, J. Food Eng., 69 (2005) 281-290. [14] A. Carezzato, M.R. Alcantara, J. Telis‐Romero, C.C. Tadini, J.A. Gut, Non‐Newtonian Heat Transfer on a Plate Heat Exchanger with Generalized Configurations, Chem. Eng. Technol., 30 (2007) 21-26. [15] C.S. Fernandes, R.P. Dias, J.M. Nóbrega, J.M. Maia, Laminar flow in chevron-type plate heat exchangers: CFD analysis of tortuosity, shape factor and friction factor, Chem. Eng. Process. Process. Intensif., 46 (2007) 825-833. [16] C.S. Fernandes, R.P. Dias, J.M. Nóbrega, J.M. Maia, Friction factors of power-law fluids in chevron-type plate heat exchangers, J. Food Eng., 89 (2008) 441-447. [17] Y.C. Tsai, F.B. Liu, P.T. Shen, Investigations of the pressure drop and flow distribution in a chevron-type plate heat exchanger, Int. Commun. Heat Mass Transf., 36 (2009) 574-578. [18] A. Kanaris, A. Mouza, S. Paras, Optimal design of a plate heat exchanger with undulated surfaces, Int. J. Therm. Sci., 48 (2009) 1184-1195.

Page 24 of 33

[19] T.A. Pimenta, J. Campos, Friction losses of Newtonian and non-Newtonian fluids flowing in laminar regime in a helical coil, Exp. Therm. Fluid Sci., 36 (2012) 194-204. [20] D. Martínez, A. García, J. Solano, A. Viedma, Heat transfer enhancement of laminar and transitional Newtonian and non-Newtonian flows in tubes with wire coil inserts, Int. J. Heat Mass Transf, 76 (2014) 540548. [21] A. Gratao, V. Silveira, J. Telis-Romero, Laminar flow of soursop juice through concentric annuli: Friction factors and rheology, J. Food Eng., 78 (2007) 1343-1354. [22] K. Asteriadou, A. Hasting, M.R. Bird, J. Melrose, Modeling heat exchanger performance for non-Newtonian fluids, J. Food Process Eng., 33 (2010) 1010-1035. [23] Fluent, Fluent 6.3 User's Guide, first ed. ed., Fluent Inc., Lebanon, New Hampshire, 2006. [24] M.C. Carrazco-Escalante, Análisis del transporte simultáneo de calor y momento para el calentamiento de puré de tomate mediante un sistema de tubos concéntricos., in: Facultad de Ciencias Químico Biológicas, Universidad Autónoma de Sinaloa, Culiacán, Sinaloa, México., 2013. [25] R.P. Chhabra, J.F. Richardson, Non-Newtonian flow and applied rheology: engineering applications, Second Ed. ed., Butterworth-Heinemann, Oxford, UK, 2011. [26] E.N. Sieder, G.E. Tate, Heat transfer and pressure drop of liquids in tubes, Ind. Eng. Chem., 28 (1936) 14291435.

Page 25 of 33

List of Figures Figure 1. (a) Heat exchanger with two heating flat plates and four baffles. (b) Heat exchanger with two heating flat plates and without baffles.

Figure 2. (a) Schematic of a U-type arrangement of a single pass in counter-flow for a PHE with 8 heating plates. (b) Heat exchanger with 16 heating flat plates and four baffles per flow passage.

Page 26 of 33

Figure 3. Behavior of normalized velocity field (

v U in

plates (a) without baffles and (b) with baffles at

t 

) over the middle plane in a PHE with two flat

4 mm and different inlet velocities.

Page 27 of 33

Figure 4. Behavior of the friction factor as funcion on the Reynolds number in heat exchangers with two flat plates with and without baffles.

Page 28 of 33

Figure 5. Behavior of the Nusselt number as a function on the Peclet number in heat exchangers with two flat plates with and without baffles.

Page 29 of 33

Figure 6. Behavior of the heat transfer rate as funcion on (a) the pumping power and (b) the volumetric flow-rate in PHEs with two flat plate.

(a)

(b)

Page 30 of 33

Figure 7. Behavior of normalized velocity field ( 12 n p without baffles and (b) with baffles at localized in each flow region ( k

t 

 1, 2 , 3 , 4

v

U in

4 mm and at

) in a PHE with eight flat plates ( n p

U in



1 2

np

-1  = 1.5 m s over the middle planes

v

U in

without baffles and (b) with baffles as function on the flow region U in



1 2

np

), (a)

).

Figure 8. Behavior of normalized average-inlet velocity ( 12 n p

values of

 8

) in a PHE with 16 flat plates (a) k

at

t 

3 mm and at different

.

Page 31 of 33

Figure 9. Behavior of standard deviation (  U ) of normalized average-inlet velocity distribution in a PHE with 4, 8 and 16 flat plates, with and without baffles, as function on the main inlet-velocity at t 

3 mm.

Page 32 of 33

Figure 10. Behavior of the heat transfer rate as funcion on the pumping power in PHEs with multiple flat plates.

Page 33 of 33