Numerical Simulation of One-phase Flow Distribution in PHE Channels

Oscar Mandersson

Department of Chemical Engineering, Lund Institute of Technology.

Abstract In this thesis FEMLAB has been used to build models of plate heat exchanger channels, with the purpose to simulate fluid flow and heat transfer characteristics. FEMLAB has also been evaluated as a suitable tool for these simulations. The thesis was done in collaboration with Alfa Laval Lund AB. Porous models have been used, which implies that small geometry objects in the channels have been replaces by momentum sink source terms in the governing momentum balances. Two dimensional models have been developed for heat transfer between two channels. They give without adjustments results that are 10 to 30 % off compared to those of the design programs used by Alfa Laval. Three dimensional double porous models were built with the purpose to make qualitative compares between different plate designs and configurations. Experiments on a one-channel rig revealed that the model needs more perfection in order to give better qualitative results. Even thou FEMLAB is not a pure CFD program, it can with modifications model fluid flow through the PHE channel satisfactory. The FEM method is in some situations however slower than e.g. finite volume method used by some CFD programs. FEMLAB also hosts possibilities for further improvements of the models, e.g. fouling kinetics. Introduction A plate heat exchanger (PHE) channel has several inherent patterns. The purpose of these patterns is to distribute the fluid even over the width of the channel as well as break up laminar flow. In the ideal case, the exit temperature of the fluid should be the same regardless the flow path taken. An uneven distribution can in some cases produce difficulties. Due to these reasons, there is a need for a predictive tool to simulate flow and heat transfer characteristics. This master thesis has the ambition to build a general model to simulate flow distribution and heat transfer for PHE channels, and to evaluate FEMLAB 3.1 as a useful tool for this. The study will be limited to two channels in 2D for modelling heat transfer, and one channel for 2D and 3D fluid flow simulations. For 3D fluid simulations, no more complex model than a double porous will be used. These studies will be performed on two different plates. These models are not to be tuned in for exact quantitative analysis. Experiments will be conducted for qualitative analysis of the 3D models.

These experiments will be limited to a one channel pack, and consist of heat pulses observed on both sides of the channel with an infra-red camera. Plate heat exchanger A PHE is built up by a pack of rectangular plates, pressed together by one fixed (the frame or head plate) and one moveable cover (pressure plate). A peripheral gasket or welding seals each channel in the pack. Generally a PHE has four ports, one inlet and one outlet for each of the two fluids. The gaskets in the ports are configured so that the service and process fluids are caused to flow up or down alternate channels and exchange heat. The plates in the pack are assembled with every other plate rotated 180 degrees. The resulting channels will have a cross-corrugated pattern which will form a double layer for the fluid to flow through. This construction principle dates back to the 1930s.

Porous media modeling Both isotropic and anisotropic porous media will be used. An isotropic medium will have the same inertial flow resistance in all directions, whilst an anisotropic medium will have different resistances in different directions. The anisotropic medium will not always be oriented parallel to the boundaries, but be angled compared to them. The medium is then subjected to an orthogonal transformation from its original parallel orientation. By media in this context, it is understood that it corresponds to a mathematical function, and NOT a medium occupying the space in the PHE channel, nor the medium flowing through it (process or service fluid). Generally, a porous medium is modelled through a momentum balance combined with the continuity equation, with the addition of a momentum sink source term (to give inertial resistance), and a porosity scaling factor to the momentum equations. Examples of real porous media are packed beds, filters etc. The momentum balance used is the Brinkman equation from the Brinkman application mode in FEMLABS Chemical Engineering module (Eq. 1).

( )( )[ ]0ˆ

ˆˆ

ˆˆˆ

=⋅∇

=∇++∇+∇⋅∇−∂∂

u

Fpuuutu T

κηηρ

(Eq. 1) Through the source term F, momentum sink or pressure drop due to flow past obstacles in the channel was taken into account.

F is in the form of2ˆ 2uCF ⋅

⋅−=ρ

.

(Eq. 2) C is a second order tensor on the form

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

zzzyzx

yzyyyx

xzxyxx

CCCCCCCCC

C . (Eq. 3)

An isotropic medium will be represented by a tensor with equal values in the diagonal elements, and zeros in the off-diagonal ones. An anisotropic media with different permeabilities in the different spatial directions will be represented by the corresponding values in the diagonal elements, and zeros in the off-diagonal elements. Both of these tensors are diagonal. This implies that the porous medium is oriented parallel to the boundaries and the pressure gradient. A flow through such a media would travel in the direction of the negative pressure gradient. This is however not always the case. A porous medium can be oriented so that the direction of the permeability is not aligned with the pressure gradient (or boundaries). The diagonal elements , , , are the inertial resistance factors in the x,y,z - directions.

xxC yyC zzC

E.g. h

xx4

Df

C xx⋅= (Eq. 4)

The off diagonal elements are obtained through the formula for change of basis

1)()( −⋅⋅= ϕϕ BCBCrot (Eq. 5)

Where , is the

rotational matrix for an angular rotation of the x–y plane (with a static z axis).

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

1000cossin0sincos

)( ϕϕϕϕ

ϕB

The resulting correlations for Fx and Fy are then implemented into FEMLAB as follows:

( ) ( )( )( )

( ) ( ) ( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅⋅⋅⋅⋅−

+

+⋅⋅⋅+⋅−

−=

vvCC

uuCCC

Fyyxx

yyyyxx

x

ρϕϕ

ρϕ

2cossin

2cos 2

(Eq. 6)

( ) ( ) ( )

( ) ( )( )( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅⋅⋅+⋅−

+

+⋅⋅⋅⋅⋅−

−=

vvCCC

uuCC

Fxxxxyy

yyxx

y

ρϕ

ρϕϕ

2cos

2cossin

2

(Eq. 7)

Energy balance Heat transfer between two channels were modelled through an energy balance (convection and conduction application mode in FEMLAB):

( ) TuCQTtTC ppts ∇⋅−=∇−⋅∇+∂∂ ˆρλρδ

(Eq. 8)

The source term Q was used as a heat source or heat sink in the channels, and calculated as follows:

h

fluid

DNu λ

α⋅

= , where correlations for Nu

were taken from experiments. (Eq. 9) α values were calculated in every point on both the hot and the cold channel. From the α values, Q was calculated in every point through the overall heat transfer coefficient.

coldplatehot

111αλα

++=h

k (Eq. 10)

( ) pdTTkQ /coldhot −⋅= (Eq. 11)

The unit of Q in FEMLAB is , therefore the calculated Q had to be divided with pd to form the right unit from .

3mW −⋅

2mW −⋅ Q will have a negative sign on the hot channel and a positive on the cold one. Results and Conclusions Simulations were performed on two different plate designs, denoted “A” and “B”. A simple validation was made against P11. The integral mean values of temperature and mass flow ratio were calculated at the outlet. The integral mean value of k and transferred power was calculated over the entire area where heat can be transferred. The model gave without adjustments result that were 10 – 30 % off 1 P1 is a program used by Alfa Laval for PHE calculations.

compared to P1. Examples of extractable results from simulations are shown in Table 1 and Figure 1-3. p∆ m& P k Model 50kPa 0.293 8360 6430 P1 50kPa 0.280 9250 6020

inhot,T outhot,T incold,T outcold,T

Model 50.0 43.0 30.0 37.0 P1 50.0 42.1 30.0 37.9 Table 1. Numerical results for a one channel pack. Plate type “A” with a high chevron angle. Properties equal to water. Temperature in Celsius, , k and P in SI units without prefix. m&

Figure 1. Temperature profiles on two channels exchanging heat (hot channel to the left). Same plate configuration as used for results in table 1.

Figure 2. Dynamic behaviour around outlet (top) and inlet (bottom) from an introduced heat step. Plate type “B”.

10,0°C

50,0°C

10

20

30

40

Figure 3. Heat step on a type “A” channel. Qualitative compareisment between 3D simulation (top) and experiment (bottom). All presented solutions have converged. The criterion for convergence is that the residuals . Computational times vary depending on DOF and the number of iterations needed.

6101 −⋅<

For a 2D model with approximately 120000 DOF it takes about 1 – 15 minutes for a 2.0 GHz 1GB RAM computer to reach convergence. Large 3D simulations with approximately 120000 DOF caused memory problems on Windows 2000 platforms, but could be solved on a 866MHz 384MB RAM Windows XP platform in 5 – 10 hours. The computational time would be reduced significantly on a faster processor with more RAM. In all simulations water has been used as medium, and the physical properties were set to those of water at about 25 °C. There are no objections against introducing the physical properties as functions of temperature, but it would require a simultaneous solution of both momentum and energy equations. This would require more detailed knowledge of numerical systems. To just solve the energy balance with properties as functions of temperature would be easier. An introduction of temperature dependent properties would in most cases increase computational time. Even thou FEMLAB is not a pure CFD program, it can with modifications model fluid flow through the PHE channel satisfactory. The FEM method is in some situations however slower than e.g. finite difference method used by some CFD programs. FEMLAB also hosts possibilities for further improvements of the models, e.g. fouling kinetics. Conclusions A three-channel model with heat transfer would give a better study of heat transfer characteristics of channels. An implementation of fouling kinetics would extend the field of usage for the model. To add a simple fouling kinetic with purpose to find “hot spots” would not be too time consuming, but to model foulant effects on flow dynamics would impose great difficulties.

It is also recommended that the double porous model is tuned in from a parameter study to give better correlation with reality regarding residence time in different theta half-planes. FEMLAB’S performance in CFD should also be compared with currently used CFD program in fields of input and post processing capabilities. Table of Symbols a = General constant (—) A = Area or denotation for a PHE channel. ( 2m ) B = Rotational matrix C = Drag coefficient tensor ( 1m− )

pC = Heat capacity ( ) 11 KkgkJ −− ⋅⋅

hD = Hydraulic diameter ( m ) f = Friction factor according to Fanning. (—) F = Source term (various) k = Overall heat transfer coefficient. ( ) 12 KmW −− ⋅⋅L = Length ( m ) m& = Mass flow rate ( ) 1skg −⋅n = Normal vector (—) p = Pressure ( ) 2mN −⋅P =Effect ( ) Wpd =Press Depth ( m )

Q = Heat flux ( ) 3mW −⋅q = Heat flux vector ( ) 3mW −⋅Re =Reynolds number (—) t =Time ( s ) T =Temperature ( K ) u =Velocity vector ( ) 1sm −⋅u = Velocity in the x direction ( ) 1sm −⋅v = Velocity in the y direction ( ) 1sm −⋅w = Velocity in the z direction ( ) 1sm −⋅

zyx ,, = Spatial dimensions (various)

α = Heat transfer coefficient ( ) 12 KmW −− ⋅⋅tsδ = Time scaling coefficient (—)

κ = Permeability ( 2m ) κ = Permeability tensor ( 2m ) λ = Thermal conductivity ( ) 11 KmW −− ⋅⋅ρ = Density ( ) 3mkg −⋅η = Dynamic viscosity ( ) sPa ⋅

θ = Theta angle in the chevron pattern of the PHE configuration (various) ϕ = Angle associated with orthogonal transformation (various) References Michael J. Nee, Heat Exchanger Engineering Techniques, The American Society of Mechanical Engineers, New York, 2003. Robert H. Perry, Don W. Green, Perry’s, Chemical Engineer’s Handbook (seventh edition), The McGraw-Hill Companies, Inc., 1997. FEMLAB 3.1, Documentation, COMSOL, 2004. FEMLAB 3.1, Chemical Engineering Module, Documentation, COMSOL, 2004. FLUENT 6.0 User’s Guide Volume 1, Fluent Inc, Lebanon, 2001. John C. Slattery, Advanced Transport Phenomena, Cambridge University Press, 1999. Gunnar Sparr, Linjär Algebra, Studentlitteratur, 1996. Aris Rutherford, Vectors, Tensors and the basic Equations of Fluid Mechanics, Prentice-Hall, INC, 1962. Jörgen Hager, Steam Drying of Porous Media, Dep. of Chemical Engineering I, Lund University, Sweden. Christi J. Geankoplis, Transport Processes and Unit Operations, Third Edition, Prentice-Hall International, Inc.