Faculty of Technology
The Degree Program of Chemical Technology
Master’s Thesis
Flow phenomena, heat and mass transfer in microchannel reactors
Examiners: Professor Ilkka Turunen
D.Sc. Jukka Koskinen
Supervisors: M.Sc. Isto Eilos
M.Sc. Steven Gust
D.Sc. Azita Soleymani
Lappeenranta 12.09.2007
Warin Ratchananusorn Liesharjunkatu 5 C10 53850 Lappeenranta Finland
Tel. +358 50 9365742
I am exceptionally grateful to Professor Ilkka Turunen for giving me an opportunity to complete this Master’s Thesis with Neste Oil Oyj and for his valuable advice.
I would like to thank Jukka Koskinen, Isto Eilos, and Steven Gust for their suggestions during my stay at Neste Oil Oyj. Also, I would like to thank Azita Soleymani who greatly helps me on the simulation work. Without her guidance, I would not have been able to complete my Master’s Thesis.
Finally, I would like to thank to my family and all my friends who help me get through two years of study.
Lappeenranta University of Technology Faculty of Technology
Author: Warin Ratchananusorn
Title: Flow phenomena, heat and mass transfer in microchannel reactors
Year: 2007
The studies of flow phenomena, heat and mass transfer in microchannel reactors are beneficial to estimate and evaluate the ability of microchannel reactors to be operated for a given process reaction such as Fischer-Tropsch synthesis. The flow phenomena, for example, the flow regimes and flow patterns in microchannel reactors for both single phase and multiphase flow are affected by the configuration of the flow channel. The reviews of the previous works about the analysis of related parameters that affect the flow phenomena are shown in this report. In order to predict the phenomena of Fischer-Tropsch synthesis in microchannel reactors, the 3-dimensional computational fluid dynamic simulation with commercial software package FLUENT was done to study the flow phenomena and heat transfer for gas phase Fischer-Tropsch products flow in rectangular microchannel with hydraulic diameter 500 µm and length 15 cm. Numerical solution with slip boundary condition was used in the simulation and the flow phenomena and heat transfer were determined.
Examiner: Professor Ilkka Turunen
D.Sc. Jukka Koskinen
Keywords: Rectangular microchannel; Slip flow; Heat transfer coefficient;
Fischer-Tropsch synthesis
Nomenclature ...1
1. Introduction ...5
1.1 Microchannel reactor...5
1.2 Fischer-Tropsch synthesis ...6
1.3 Goals...7
2. Flow phenomena in microchannel reactors...8
2.1 Single phase flow ...9
2.2 Two-phase flow...10
2.2.1 Contacting principles...10
2.2.2 Flow regimes ...11
2.2.2.1 Stratified flow regime...11
2.2.2.2 Intermittent flow regime...12
2.2.2.3 Annular flow regime ...13
2.2.2.4 Dispersed flow regime...13
2.3 Trickle bed flow ...15
2.3.1 Flow regimes ...16
2.3.2 Liquid distribution...19
2.4 Slip flow ...19
2.4.1 Knudsen number...21
2.4.2 Numerical models for gas phase slip flow ...23
2.5 Friction factor and pressure drop...27
3. Heat transfer in microchannel reactors...31
3.1 Effects of the geometry of the flow channels...32
3.2 Thermal entrance length...35
3.2 Heat transfer coefficient in rectangular microchannels...36
3.2.1 Effect of the Reynolds number...38
3.2.2 Heat transfer coefficient in microchannel ...40
3.3 Heat transfer in two-phase flow ...42
4. Mass transfer in microchannel reactors...44
4.1 Sherwood correlation ...45
4.2 Mass transfer limitation in microchannels ...47
5. Fischer-Tropsch synthesis ...51
5.1 Influence of process conditions on the selectivity...53
5.1.1 Temperature...53
5.1.2 Partial pressure of H2 and CO ...54
5.1.3 Space velocity...55
5.1.4 Time on stream...55
5.2 Anderson-Schulz-Flory distribution...56
6. Simulation of the flow in microchannel ...59
6.1 ASPEN PLUS simulation on Fischer-Tropsch synthesis...59
6.2 CFD Simulation of Fischer-Tropsch products ...60
6.2.1 Domain and grid...62
6.2.2 Boundary conditions...63
6.2.3 Model consideration ...64
6.2.4 Results and discussion...66
6.2.4.1 Flow phenomena in microchannel ...66
6.2.4.2 Heat transfer in microchannel ...70
6.3 Conclusion...76
References ...78
Appendices ...83
Nomenclature
A convection heat transfer area [m2]
AC heat transfer area at control channel [m2] AR heat transfer area at reaction channel [m2] ARC heat transfer area between two flows [m2]
a constant value, 0.2332 [-]
b constant value, 0.6330 [-]
C1 first-order slip coefficient [-]
C2 second-order slip coefficient [-]
Cf friction coefficient [-]
cp specific heat of the fluid [J kg-1 K-1] Da Dämkohler number,
V
K [-]
Dh hydraulic diameter [m]
Dv volumetric diffusivity [m2 s-1]
d molecular diameter [m]
e roughness parameter [m]
Fv
external body forces [kg m-2 s-2]
f Darcy friction factor [-]
G1 non-dimensional constant [-]
G2 non-dimensional constant [-]
Gz Graetz number, L
RePrDh [-]
H height of the channel [m]
h convective heat transfer coefficient [W m-2 K-1] hc convective heat transfer coefficient of the controlling fluid [W m-2 K-1]
J mass flux [kg m-2 s-1]
Jvj
diffusion flux of species j [kg m-2 s-1]
L pipe length [m]
Kn Knudsen number, Dh
λ [-]
k thermal conductivity of the flow channel wall [W m-1 K-1] kB Boltzmann constant, 1.3806503E-23 [m2 kg s-2 K-1] kc thermal conductivity of coated catalyst layer [W m-1 K-1] keff effective conductivity [W m-1 K-1]
1
kHC constant value, 1.22E-05 [-]
5
kHC constant value, 1.05E-06 [-]
6
kHC constant value, 2.36E-06 [-]
km mass transfer coefficient [m s-1]
kp propagation rate [m s-1]
kt termination rate [m s-1]
kw thermal conductivity of the wall [W m-1 K-1] Nu Nusselt number,
k hDh
[-]
NuG Nusselt number for gas flow [-]
NuL Nusselt number for liquid flow [-]
n number of carbon atom [-]
PCO partial pressure of carbon monoxide [Pa]
H2
P partial pressure of hydrogen [Pa]
Pe Peclet number, RePr [-]
Pem mass Peclet number, kc
V [-]
Pr Prandt number, k cpμ
[-]
p Pressure [Pa]
p0 Pressure at reference position [Pa]
pz pressure at streamwise coordinate [Pa]
Δp pressure difference [Pa]
R specific gas constant [J mol-1 K-1] Re Reynolds number,
μ ρumDh
[-]
Sh heat from heat sources [W m-1 s-1]
Sm the mass added to the continuous phase [kg m-3 s-1] Sc Schmidt number,
Dv
ρ
μ [-]
Sh Sherwood number,
v h c
D D
k [-]
s central distance between reaction channels [m]
sc central distance between control channels [m]
T temperature [K]
TC temperature of control channel [K]
Tg temperature of gas [K]
TM temperature of the main fluid [K]
TR temperature of reaction channel [K]
Tw temperature at the wall [K]
U overall heat transfer coefficient [W m-2 K-1]
ug slip velocity [m s-1]
um mean fluid velocity [m s-1]
u cross-sectional average velocity [m s-1]
V axial mass convection rate [m s-1]
W width of the channel [m]
Wc center-to-center distance of microchannels [m]
Wn weight fraction [-]
x distance from the entrance [m]
x* dimensionless axial distance [-]
y lateral coordinate [m]
y gas mass fraction [-]
yB bulk mass fraction [-]
yCO fraction of carbon monoxide [-]
H2
y fraction of hydrogen [-]
yW wall mass fraction [-]
z Distance from reference position [m]
z0 Distance at reference position [m]
Greek symbol
α aspect ratio of the flow channel [-]
α chain growth probability factor [-]
αv momentum accommodation coefficient [-]
αT temperature accommodation coefficient [-]
β non-dimensional constant [m]
δ channel-plate thickness [m]
λ mean free path [m]
μ fluid viscosity [Pa s]
ρ fluid density [kg m-3]
τ stress tensor [kg m-2 s-2]
χ non-dimensional constant [m]
1. Introduction
1.1 Microchannel reactor
Microchannel reactors are increasingly used in many fields of industry due to the capabilities exceeding those of traditional macroscale reactors. The high heat and mass transfer rates in microchannel reactors allow the reaction to be performed under more aggressive conditions with higher yields (Jensen, 2001). The surface to volume ratio can be higher up to 10,000-50,000 m2 m-3 (Minsker and Renken, 2005) providing drastically higher heat and mass transfer rates than the traditional chemical reactors. Thus, microreactors can remove heat much more efficiently than traditional chemical reactors and can perform safely for highly exothermic or endothermic reaction. Therefore, high reaction temperature is possible with microchannel reactors and leads to reduced reactor volumes. Moreover, less amount of catalyst used improves the energy efficiency and reduces the operational costs.
Another benefit of microchannel reactors is that if the system fails, the amount of accidentally released chemicals is rather small and it could be easily controlled. The integrated sensor and control units could allow the failed reactor to be isolated and replaced while other parallel units continued production (Jensen, 2001).
Besides the benefits of the microchannel reactors, some problems still remain in the system. Clogging is one of the main problems occuring in the particle containing processes, for example, catalytic reactions. Clogging has been identified as the biggest problem, however, the particle containing processes also cause rather high pressure drop through the flow channel. These two main factors have to be considered while operating with microchannel reactor.
Fischer-Tropsch synthesis has been developed continuously since the invention of the process by Franz Fischer and Hans Tropsch in the 1920s. It became more interesting after the oil crisis in the 1970s while the rising oil prices and global warming are also important issues in promotion of synthetic fuels. The development increased significantly supported by both the United States and the European Union fund for research and development. This resulted in extensive studies of catalysts, kinetics, mechanisms, and reactors for FTS.
Fischer-Tropsch synthesis is a catalyzed chemical reaction of synthesis gas or syngas comprising of hydrogen and carbon monoxide. The syngas is fed to the reactor which is typically fixed-fluidized-bed or fixed-slurry-bed reactor and the catalytic reaction occurs. The conversion of syngas to product depends on many factors, for example, operating temperature, the distribution of syngas in the reactor, types and efficiencies of catalysts, heat transfer and heat distribution in the reactor, and mass transfer between phases. Therefore, the phenomena occurring in the reactor is important to be studied in order to understand the important factors that affect to the reaction.
Fischer-Tropsch synthesis is normally operated with macroscale reactor. The application of microchannel reactors for Fischer-Tropsch synthesis is novel. It has been intensively developed from past decades. The trend of applying microchannel reactors to the chemical processes is growing because replacing a large reactor with many smaller microchannel reactors that provide improved control of important operating conditions is the purpose for intensifying the process to obtain the benefits from microchannel reactors. However, the limitations of microchannel reactors and the characteristics of the reaction have to be considered carefully.
This Master’s Thesis has a purpose to study the flow phenomena, heat and mass transfer in microchannel as well as the Fischer-Tropsch synthesis. The Fischer- Tropsch synthesis in rectangular microchannel will be investigated with the commercial software package FLUENT. The simulation includes only gas phase Fischer-Tropsch products flow in microchannel. The flow phenomena and heat transfer in microchannel will be investigated.
2. Flow phenomena in microchannel reactors
The flow phenomena in microchannel have been studied extensively during past decades. Currently, the flow phenomena are analyzed using the Navier-Stokes equations but a number of publications have shown that the small hydraulic diameter makes the flows in microscale different from the one in macroscale and the Navier- Stokes equations are incapable for describing the occurring phenomena (Papautsky et al., 2001). Some neglected parameters in macroscale flows may be significant in the microscale. The observation by Ma and Gerner (1993) shows that the molecular effect on the momentum transfer in directions other than the streamwise direction can increase significantly when the length of the flow channels are reduced and the continuum assumption becomes invalid. The variation in fluid properties (e.g., fluid viscosity) can occur by the variation of temperature in the transport fluid flowing in microscale which causes invalidation of constant properties assumption (Pfahler et al., 1991).
Flow in microchannel is predominantly laminar (Jensen, 2001) because of the hydraulic diameter that makes Reynold numbers very small. Molecular effects also become more significant in microchannel when the characteristic length decreases to the point that the continuum assumption becomes invalid (Alfadhel and Kothare, 2005). In macroscale flows with large diameter flow channel, the flow pattern is mainly dominated by the influence of the gravitational force while the flows in microscale with small diameter flow channel, the flow pattern is mainly a function of the interfacial tension, the wall friction force, and the viscosity of the fluid (Waelchli and Rohr, 2006)
The traditional non-dimensional parameters used to characterize fluid flow phenomena are the Reynolds number and the Darcy friction factor (Papautsky et al., 2001). The Reynolds number which is normally used to describe the flow phenomena depend on four quantities: the diameter of the flow channel and the viscosity, density, and average linear velocity of the fluid. The Reynolds number is defined by equation 2.1
μ ρumDh
Re= (2.1)
where Re Reynolds number [-]
ρ fluid density [kg m-3] um mean fluid velocity [m s-1] Dh hydraulic diameter [m]
μ fluid viscosity [Pa s]
Peng and Peterson (1996) described the hydraulic diameter of rectangular channel by the following equation
H W Dh WH
= 2+
(2.2)
where W width of the channel [m]
H height of the channel [m]
2.1 Single phase flow
For the single phase flow in rectangular microchannel, the flow phenomena can be assumed to be layered laminar flow and fully-developed because of small hydraulic diameter resulting in low Reynolds number. However, transition region or even turbulence can be developed in the corrugated flow channel giving large Reynolds number. The rarefaction effect can occur at normal pressure in microchannel which results on the deviation from continuum flow behavior. Three-dimensional Navier- Stokes equation with slip boundary condition has to be applied for the flow in microchannel.
Many works on single phase flow in microchannel were studied. The phenomena occurring in microchannel are similar to the flow in macroscale but the interactions between the fluid and the surface properties of the wall for liquid flow and the rarefaction effect for gas flow should be taken into account as a correct boundary condition for the flow in macrochannel. These boundary conditions result on slip flow which is considered, as presented in a number of previous works, that it causes the significant reduction on the pressure drop.
For the flow in rectangular microchannel, slip boundary condition relates to the friction factor which is a function of channel aspect ratio. The aspect ratio of the flow channel is expressed by the following equation
W
= H
α (2.3)
where α aspect ratio of the flow channel [-]
The slip boundary condition will be discussed in detail in the following section.
2.2 Two-phase flow
The two-phase flow phenomena are more complicate than the single phase one. Many parameters of two-phase flow affect to the flow regimes and flow pattern in the flow channel. One of the most significant parameters is the ratio of the flow velocities of each phase which is mostly critical to the flow regimes.
2.2.1 Contacting principles
The flow phenomena in two-phase flow depend on the contacting of the two phases.
There are basically two principles for the contacting of two phases: continuous-phase contact, which is to keep both phases continuous and form an interface between them
and disphersed-phase contact, which is to disperse one phase into the other using an appropriate micromixer upstream (Hessel et al., 2005).
The two-phase flow in microchannel reactors is typically gas-liquid flow. In continuous-phase contact, two phases are fed separately and form two streams. The streams are also withdrawn separately at the reactor outlet. The main function of such microreactor is to form the interfaces without intermixing between the phases so the pressures of the two phases have to be carefully controlled to avoid phase intermixing.
In dispersed-phase contact, the dispersion is created by an inlet which induces merging of the gas and liquid streams. The feed is split fed by a multiple feed structure to split the phases in thin lamellae and form dispersion in a mixing section.
2.2.2 Flow regimes
Coleman and Garimella (1999) presented the flow regimes in two-phase flow. They have done the experiment to characterize the two-phase flow in small rectangular tubes with hydraulic diameter 5.36 mm by using water as a continuous phase and air as a dispersed phase. The results were presented as four major flow regimes, stratified, intermittent, annular, and dispersed flow regime.
2.2.2.1 Stratified flow regime
The stratified flow regime is characterized by a complete separation of the gas and liquid phases flowing in tube. The flow pattern is called stratified (stratified smooth) when both of the gas and liquid flows are laminar and have no fluctuations at the flow interface. As the gas mass flow rate is increased, instabilities can form at the gas-liquid interface due to the interfacial velocity differential. The instabilities cause the formation of small interfacial waves and this flow pattern is called wavy flow (stratified wavy). In larger diameter tubes, these waves can increase and are easier to detect. Moreover, the wave height can be large enough to allow the waves to break in
large diameter tubes. In small diameter tubes, the flow is more formational and large breaking waves were typically not observed.
Figure 2.1 Visualization of stratified flow regime (Coleman and Garimella, 1999)
2.2.2.2 Intermittent flow regime
The intermittent flow regime is characterized by discontinuities of the liquid and gas flow in tube. Elongated bubble flow pattern (plug flow) is characterized by a stream of vapor plugs flowing in the continuous liquid stream which is considered as a continuous phase. There is a thin film of liquid coats the tube wall and surrounds the gas phase plug. Small disturbances may exist around the vapor plugs, but as a whole the plugs remain uniform. As the gas mass flow rate is increased, these disturbances increase until some portion of the plug breaks apart into smaller bubbles, and then the flow pattern becomes slug flow. These smaller bubbles become trapped in the liquid flow and impact the front of the following slug causing disturbances in the front flow profile.
Figure 2.2 Visualization of intermittent flow regime (Coleman and Garimella, 1999)
2.2.2.3 Annular flow regime
The annular flow regime is characterized by a nearly complete separation of the gas and liquid along the circumference of the tube wall. The first form of annular flow occurs when the surfaces of waves in wavy flow increase to the extent that they touch the top of the tube wall. This flow pattern is known as wavy-annular flow (pseudo-slug flow). As the gas mass flow rate is increases, the liquid is pushed up around the circumference of the channel wall by the increasing gas momentum and falls downward by the gravity in the form of annular waves. When the liquid forms an annular ring coating the tube wall completely and the gas flows through the core of the tube, the flow pattern is known as annular flow. In annular flow, there could also be small droplets entrained in the gas core. This flow is also known as annular mist flow.
Figure 2.3 Visualization of annular flow regime (Coleman and Garimella, 1999)
2.2.2.4 Dispersed flow regime
Dispersed flow occurs when the liquid flow is turbulent and the gas phase is in laminar or turbulent flow. When the gas flow is laminar, small bubbles are driven by buoyancy forces and flow primarily in the top half of the tube. This pattern is known as bubble flow. As the Reynolds number of the gas increases, keeping other variables constant, the bubble size decreases and the bubbles begin to disperse across the entire tube cross section. This flow pattern is known as dispersed bubble or dispersed flow.
Figure 2.4 Visualization of dispersed flow regime (Coleman and Garimella, 1999) Coleman and Garimella (1999) also proposed the results from their flow visualization experiment to study the effect of the gas and liquid phase flow superficial velocities to the flow regimes. Gas and liquid superficial velocities ranged from 0.1 to 100 m/s and 0.01 to 10 m/s respectively. The tube aspect ratio (α) was 0.725 and hydraulic diameter was 5.36 mm. The results are shown as a flow regime map in Figure 2.5
Figure 2.5 Flow regime map for two-phase flow of water and air in rectangular tube with hydraulic diameter 5.36 mm (Coleman and Garimella, 1999) One of the most critical parameters that affect to the flow phenomena is the ratio of the gas-to-liquid flow rates. The results from Coleman and Garimella’s experiment can be concluded that at low values of the gas-to-liquid flow rate ratio, the dispersed flow
regime is obtained. As the ratio increases, the intermittent flow regime and the annular flow regime are obtained. At very high ratio, the stratified flow regime is obtained.
Serizawa et al. (2001) presented the flow regime map as shown in Figure 2.6 for the air-water flow in circular microchannel with 20 μm diameter and also compared to the previous work done by Mandhane et al. (1974) for the flow in macroscale.
Figure 2.6 Flow regime map for the flow of water and air in circular tube with hydraulic diameter 20 µm (Serizawa et al., 2002)
The flow regime map presented by Serizawa, et al. (2002)is quite similar to the flow in rectangular minichannel (5.36 mm) observed by Coleman and Garimella (1999) when considering the gas-to-liquid flow rate ratio, for example, slug flow occurs at the gas- to-liquid flow rate ratio about 1 for both flow regime maps while for the slug flow in macroscale presented as hatched lines in Figure 2.6 occurs at higher gas-to-liquid flow rate ratio.
2.3 Trickle bed flow
Trickle-bed reactors are widely used in the petrochemical and oil processing industry.
In trickle-bed flow, the gas and liquid phases usually flow through the fixed bed of stationary solid catalysts particle cocurrently downward because the downward flow
has much less pressure drop than upward flow (Binardi and Baldi, 1999) and does not present the flooding. The schematic diagram of this flow is shown in Figure 2.7
Figure 2.7 Trickle-bed cocurrent downward flow (Dekker, 2004)
2.3.1 Flow regimes
The flow regimes of trickle-bed flow also has gas-to-liquid flow ratio as a significant parameter. Biardi and Baldi (1999) presented the flow regimes in trickle-bed flow. At low liquid and gas flow rate, there is low interaction between two phases and it is considered as trickling flow regime. In this flow regime, the liquid trickles down over the catalyst packing as a laminar film or rivulets. The pressure drop has low significance in this type of flow regime and can be considered as the same for gas flow through a dry packing. As the liquid flow rate increases with constant gas flow rate, the pulsing flow regime is obtained. The pulsing flow regime is characterized by the formation of liquid slugs and gas slugs flowing through the packing. The pressure drop increases remarkably and the fluctuation occurs.
Bubbling flow regime is obtained when the liquid flow rate is further increased while the gas flow rate is kept at low level. The liquid phase becomes a continuous phase distinctly with the gas phase flow inside as bubbles. On the other hand, if the liquid
flow rate is kept low and gas flow rate increases, the spray flow regime which the droplet of liquid disperse in the gas plug occurs. Figure 2.8 presents the possible flow regimes for trickle-bed flow.
Figure 2.8 Flow regimes in trickle-bed flow (Ng and Chu, 1987).
The effect of gas flow rate and liquid-to-gas flow ratio on the flow regimes was studied by Losey et al. (2001). The experiment was done with 400 μm channels packed-bed microchannel reactor filled with sieved catalyst with particle size lower than 75 μm.
Cyclohexene and hydrogen gas were used as flowing fluid in the experiment. The flow regime map was presented as the relation of gas superficial mass velocity versus liquid-to-gas superficial mass velocity ratio as shown in Figure 2.9
Figure 2.9 Flow transition point versus superficial mass velocities for the flow of cyclohexene and hydrogen in packed-bed reactor with 400 μm flow channels (Losey et al., 2001)
Figure 2.9 shows the comparison of the effect of gas superficial mass velocity and liquid-to-gas superficial mass velocity ratio in conventional trickle-flow studied by Charpentier (1975) and packed-bed flow in microchannel reactor. The solid line shows the relations in macroscale trickle-bed flow and the dash line shows the correlations in microscale flow. In microscale, the pulsing flow can be obtained at lower liquid-to-gas ratio for low gas superficial mass velocity but when the gas superficial mass velocity increases, the liquid-to-gas ratio has to be kept higher when compare to the macroscale one in order to obtain the pulsing flow regime.
The preferred flow regime for trickle-bed flow is pulsing flow because of higher performance. Thin film of liquid coats the catalyst particles and separates the gas bubble from the catalyst providing high mass transfer rate. Stankiewicz and Moulijn (2004) gave two reasons to explain that the mass transfer rate in this type of flow is large because:
• The liquid layer coating the catalyst is thin enough for gas phase to diffuse through the liquid film
• The liquid slugs show an internal circulation during traveling through a channel. Consequently, the improvement of radial mass transfer is obtained.
2.3.2 Liquid distribution
The liquid distribution is another important parameter for trickle-bed flow. Although the flow in microscale trickle-bed is more uniform than a macroscale one (Tsochatzidis et al., 2002), a good design for the flow distributor is required to obtain a proper distribution of the fluid flow stream. Besides the flow distributor, characteristics of the packing also affect to the flow distribution. Anisotropic packing of the catalyst may leads to a high pressure drop and large non-wetted region of the packed-bed (Souadnia and Latifi, 2001). The consequences are
• Low performance (Conversion rate, selectivity, yield, etc.) of the reaction because maldistribution and non-wetted region cause the absence of reaction in some regions of the packed-bed catalyst.
• Hot spots can occur in highly exothermic reaction because maldistribution of liquid causes low heat transfer ability.
2.4 Slip flow
According to the laminar flow fluid theory, friction factor is independent of wall surface roughness for the flow in macroscale but in microscale, molecular interaction with the walls increases relatively to intermolecular interaction when compared to macroscale flows. Two boundary conditions are usually applied to characterize the flow phenomena in smooth channel in macroscale. No-slip boundary condition is the most common type of wall boundary condition implementation which can be applied for the assumption that the fluid next to the wall has the velocity as the wall which is normally zero. Another boundary condition is slip boundary condition where the shear stress at the wall is zero and the velocity of the fluid near the wall is not retarded by wall friction effects.
Figure 2.10 No-slip and slip boundary flow (Shui et al., 2007)
Slip flow is an important consideration because it may allow a significant reduction in the friction pressure drop and thus the pumping power required for microchannel reactors is less. While the underlying physical cause of slip is still not fully understood, what is clearly known is that an apparent slip occurs more readily on the surface which has opposite wet ability to the fluid, on rough surfaces, and at high shear rates. Slip may therefore become important in microchannel, as surface modification can be designed and assembled into microchannel reactors, surface roughness becomes significant relative to the channel size and due to their small dimensions, it is possible to obtain large shear rates (Rosengarten et al., 2006).
Recently, many researchers have suggested that the well-accepted no-slip boundary condition may not be suitable to predict the flow in microscale. Fluid slip may develop from soluble and entrained gases which form a gap near the wall. The effect of wall slip was studied by Tretheway et al. (2002). It has been proved that no-slip boundary condition cannot be applied for all cases of flow in microchannels. By the experiment which was done to measure the velocity of water flowing in rectangular microchannel with dimension 30 μm height and 300 μm width at different positions from the wall with no-slip (water and hydrophilic wall) and slip (water and hydrophobic wall) boundary condition. The comparison of water velocity profiles from the experiment with no-slip and slip boundary condition show in Figure 2.11
(a) (b) Figure 2.11 Velocity profiles of water flowing in microchannel with no-slip (a) and
slip (b) boundary condition (Tretheway et al., 2002)
Figure 2.11 (a) shows that no slip boundary condition can be used with the water flows in hydrophilic wall condition can as can be seen from the velocity profile at the wall position which approaches to zero while the velocity profile in Figure 2.11 (b) at the wall is larger than zero so the slip boundary condition can be applied. It can be concluded that no-slip boundary condition is not always valid and may not be accurate for fluid flow in microchannel but it depends on the interactions between the fluid and the properties of the wall surface.
2.4.1 Knudsen number
The validity of slip flow boundary condition for gas phase can be proved by Knudsen number (Kn). Knudsen number determines the degree of rarefaction of fluid and the degree of validity of the Navier-Stokes model. Knudsen number is expressed by the following equation.
Dh
Kn= λ
(2.4)
where Kn Knudsen number [-]
λ mean free path [m]
The mean free path of a particle, such as a molecule, is the average distance the particle travels between collisions with other particles. The mean free path is a function of temperature and pressure and is defined as follows
2 pd2
T kB
λ= π (2.5)
where kB Boltzmann constant, 1.3806503E-23 [m2 kg s-2 K-1] T Temperature [K]
p Pressure [Pa]
d molecular diameter [m]
Figure 2.12 Knudsen number regimes (Gad-el-Hak, 2006)
Figure 2.12 shows the regimes at different value of Knudsen number. As the Knudsen number approaches zero, the transport terms in the continuum momentum and energy equations are negligible, and the Navier-Stokes equations then are reduced by neglecting the heat conduction and viscous diffusion and dissipation. The flow is then approximately isentropic. The flow in this condition is called continuum flow. As Knudsen number increases, rarefaction effects become more important and continuum approach breaks down. Some parameters in Navier-Stokes equations cannot be neglected as in the continuum flow. The summations are as follows:
• Euler equations (neglect molecular diffusion) Kn→o
• Navier-Stokes equations with no-slip Kn<10−3 boundary condition
• Navier-Stokes equations with slip flow 10−3 ≤Kn<10−1 boundary condition
• Transition regime 10−1≤Kn<10
• Free-molecular flow Kn≥10
2.4.2 Numerical models for gas phase slip flow
Gas flow under standard condition where the molecular mean free path is typically 70 nm are often encountered in microchannels (Renksizbulut et al., 2006). The experimental data of the gas flow in microchannels from many previous studies including Wang and Li (2004), Renksizbulut et al. (2006), and Dongari et al. (2007) strongly support the applicability of the combination approach of continuum assumption and slip boundary conditions to model the flow in microchannel. The equation for determination of the slip velocity for gas phase is
2 2 1
2 2 1
8 2
6 / 1
8 2
Kn C Kn C
Kn C Kn u C
ug
+ +
= + (2.6)
where ug slip velocity [m s-1]
u cross-sectional average velocity [m s-1] C1 first-order slip coefficient [-]
C2 second-order slip coefficient [-]
The above equation was proposed by Dongari et al. (2007) with the following assumptions
• The flow is steady, two-dimensional and locally fully developed
• The flow conditions are isothermal.
• The channel is long, and the entry and exit effects are negligible
• The viscous compressive stresses are negligible
The slip coefficients C1 and C2 were determined by several researchers. The Navier- Stokes differential momentum equation with slip boundary condition was solved and the slip coefficients were given as shown in Table 2.1
Table 2.1 Value of slip coefficients (Dongari et al., 2006)
Source C1 C2 Remarks
Maxwell (1879) Schamberg (1947)
Chapman and Cowling (1952) Albertoni et al. (1963)
Deissler (1964) Cercignani (1964) Sreekanth (1969)
Hsia and Domoto (1983) Mitsuya (1993)
Pan et al. (1999) Dongari et al. (2007)
1 1 ≈1 1.1466 1 1.1466 1.1466 1 1 1.125 1.1466
0 5π/12
≈0.5 0 1.6875 0.9756 0.14 0.5 2/9 0 0.9756
Theoretical Theoretical Theoretical Theoretical Theoretical Theoretical Experimental N/A
N/A
Simulations (DSMC) N/A
The mean velocity at any position in streamwise coordinate direction (z) in the flow channel can be solved from the correlation of Reynolds number and ideal gas law and be given as
h zD p
RT u = Reμ
(2.7)
where R specific gas constant [J mol-1 K-1] pz pressure at streamwise coordinate [Pa]
The longitudinal velocities at different position from the wall and streamwise coordinate in the flow channel then can be solved by the following equation
⎟⎟⎠
⎜⎜ ⎞
⎝
⎛
+ +
+ +
= − 2
2 1
2 2 1
2
8 2
6 / 1
8 2
) / ( ) /
,
( CKn C Kn
Kn C Kn C H
y H y D p
RT z Re
y u
h z
μ (2.8)
where y lateral coordinate [m]
The expression of the variation of pressure in terms of pressure (p0) at some reference position z0 and the corresponding Knudsen number (Kn0) is shown in the following equation
⎟⎟⎠
⎜⎜ ⎞
⎝ + ⎛
⎟⎟⎠
⎜⎜ ⎞
⎝
⎛ −
+
⎟⎟ −
⎠
⎜⎜ ⎞
⎝
⎛
0 2
0 2 0
0 1 2
0
log 96
1 24
1 p
Kn p p C
Kn p p C
p
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎟⎟⎠
⎜⎜ ⎞
⎝
− ⎛
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡ ⎟⎟⎠ −
⎜⎜ ⎞
⎝ + ⎛
⎟⎟⎠
⎜⎜ ⎞
⎝
⎛ −
+
0 2
0 2 0 1 0
0 1
2 12 1 24 1 log
2 p
p p
Kn p p C
Kn p C Re βχ
Dh
z Re z 0
96 −
−
= β (2.9)
where p0 Pressure at reference position [Pa]
z0 Distance at reference position [m]
z Distance from reference position [m]
β non-dimensional constant [m] (see equation 2.11) χ non-dimensional constant [m] (see equation 2.12)
β and χ can be obtain from the following equations
2 2 0 2
Dh
p μ RT
β = (2.11)
( )
⎥⎥⎦
⎤
⎢⎢
⎣
⎡
+ +
+ +
+ +
= + 2
2 2 1
4 2 2 3
2 1 2
2 1 2 2 38 1 32
8 2
6 / 1
64 32
4 30
/ 1
Kn C Kn C
Kn C Kn
C C Kn
C Kn C Kn
χ C (2.12)
The value χ can be easily verified that it lies between 1 and 1.17 over the range of Knudsen number from 0.001 to 10 with the maximum difference in pressure at any position less than 10−3% (Dongari et al., 2007).
The simulation by Dongari et al. (2007) of the slip flow velocity profile and variation of pressure in the microchannels from the previous equations was done with the air flow under standard conditions
(
λ≈70nm)
through a 5 μm channel which gives a Knudsen number at the slip flow regime. The Knudsen number in the simulation is varied along the length of the channel which, in this simulation, is equal to 100Dh.The variation of longitudinal velocities is presented in Figure 2.13Figure 2.13 Variation of longitudinal velocities of air flows in 5 μm rectangular microchannel (Dongari et al., 2007)
The velocity profile was assumed to be fully developed through the flow channel so the velocity profile is uniform since the entrance at z/Dh = 0. The velocity profile has the highest value at the center of the flow channel and radically decrease to the wall since the fully developed assumption was used. The slip velocity at the wall, as shown in Figure 2.13, is larger than zero giving a contradiction to the continuum assumption of the flow in macroscale which the velocity at the wall is zero. As the distance in the coordinate direction z from the entrance increases, the longitudinal velocity increases and approaches the highest value at the exit as a result of pressure drop.
Figure 2.14 shows the variation of pressure along the flow channel which was obtained from equation 2.9. The comparison against the experimental data of Pong (1994) is presented in the figure and shows a good matching between the model and the experimental data which used nitrogen gas with the outlet Knudsen number 0.059.
Figure 2.14 Variation of the pressure of air flows in 5 μm rectangular microchannel comparing with the experimental data of nitrogen proposed by Pong et al. (1994) (Dongari et al., 2007)
2.5 Friction factor and pressure drop
From Darcy friction factor and Hagen-Poiseuille equation, when the hydraulic diameter decreases, the pressure difference increases by order of two while Reynolds number decreases. Consequently, the friction factor increases because the increasing of pressure difference is more significant.
The Darcy friction factor relates friction effects to pressure drop in pipes, is given by equation 2.13
2
2
m h
Lu p f D
ρ
= Δ (2.10)
where f Darcy friction factor [-]
L pipe length [m]
∆p pressure difference [Pa]
The pressure drop can be presented as a function of friction factor. Equation 2.10 becomes
h m f h
m
ReD u L C D
u p fL
2 2
2 ρ 2
Δ = ρ = (2.11)
fRe
Cf = (2.12)
where Cf friction coefficient [-]
Friction coefficient is a constant and dependent only on the geometry of the flow channel for fully developed laminar flow. For rectangular channel with different aspect ratio, the friction coefficients are listed in Table 2.1
Table 2.1 Friction coefficients for rectangular channels with different aspect ratios (Hu, 2001)
α 1 1.33 2 2.5 4 6 8 10 20 ∞
Cf 56.91 57.89 62.19 65.47 72.93 78.81 82.34 84.68 89.91 96 The friction coefficients presented in Table 2.1 are obtained from the equation presented by Papautsky (1999). The data in Table 2.1 can be fit with a polynomial equation with R2=0.9995.
(
1 1.3553 1.9467 2 1.7012 3 0.9564 4 0.2537 5)
96 − α+ α − α + α − α
f =
C (2.13)
The pressure difference for laminar flow in channels can be expressed by Hagen- Poiseuille equation showing in equation 2.13
2
32
h m
D p= Lu μ
Δ (2.14)
The frictional force for laminar flow in microchannel is caused by the interaction with the channel wall so it is necessary to express the wall shear stress through the channel.
Celata et al. (2007), Papautsky et al. (1999), and Peng and Peterson (1996) expressed the wall shear stress in term of flow properties and the friction factor, f
8
2 w m
u fρ
τ = (2.15)
Some studies clearly indicate that as the tube diameter decreases, the friction factor increases for the same Reynolds number (Kandlikar, 2005). The roughness of the microchannel wall plays a role in the friction factor as well as in the transition to turbulence flow. Increase in microchannel surface roughness, increases the friction factor so the pressure drop along the flow channel increases. The roughness can be characterized by relative roughness parameter (McCabe et al., 2001) defined by equation 2.17
Dh
roughness e
relative = (2.16)
where e roughness parameter
Kandlikar (2005) presented the effects of different relative roughness and Reynolds number on the pressure drop of the water flow in a 0.62 mm diameter stainless steel tube. Figure 2.15 shows the results that the higher relative roughness and Reynolds number, the higher pressure drop along the tube. Increasing of the Reynolds number causes the pressure drop to be increased corresponding to Darcy friction factor showing in equation 2.10. The experiment was also done with the higher diameter tube but it did not exhibit any roughness dependency. The studies show that the roughness effects are expected to be higher in microchannels.
Figure 2.15 Effect of relative roughness and Re to pressure drop value (Kandlikar, 2005)
Therefore, fabrication of microchannel reactors has a significant role to the flow phenomena in microchannels. Relative roughness of the flow channels directly depends on the roughness of the wall and gives a consequence on high pressure drop and friction factor when the degree of roughness increases. Good fabrication methods in manufacturing of microchannel reactors are needed to fabricate smooth microchannel reactors in order to obtain uniform flow with low pressure drop.
3. Heat transfer in microchannel reactors
Microchannel reactors are usually multifunctional reactors, for instance, microchannel reactors which are integrated with heat exchangers. Heat transfer in microchannel reactors is very effective because of high heat transfer area which is characterized by surface to volume ratio. This function makes the microchannel reactors be able to be operated with short residence time so they are suitable for the exothermic reaction which requires contact times in the order of milliseconds (Male et al., 2004).
The heat transfer coefficient is usually presented as a dimensionless Nusselt number defined by the following equation
k
Nu= hDh (3.1)
where Nu Nusselt number [-]
h convective heat transfer coefficient [W m-2 K-1]
k thermal conductivity of the flow channel wall [W m-1 K-1]
The investigations of convection heat transfer in microchannels were done and the heat transfer models for those experiments were developed and presented as a function of The Nusselt number. The Nusselt number is usually presented as a correlation of Reynolds number and Prandt number (Pr).
Prandt number is the ratio of the diffusivity of momentum to the thermal diffusivity.
k Pr cpμ
= (3.2)
where Pr Prandt number [-]
cp specific heat of the fluid [J kg-1 K-1]
Many of heat transfer models were developed. However, those models have the limitation on the validity of the models. They are valid for the specific flow channel geometry and flow regime within a certain range of Reynolds number.
3.1 Effects of the geometry of the flow channels
There are many factors influencing on the heat transfer characteristics in microchannels. One of the most important characteristics for the flow in rectangular microchannels is the cross sectional shape of a channel which is typically defined by the aspect ratio presented in chapter 2. Peng and Peterson (1995) studied the convective heat transfer of water flowing in rectangular microchannels with hydraulic diameter ranged from 150 μm to 267 μm and found that the aspect ratio had significant influence on the convective heat transfer.
The heat transfer in rectangular microchannels is also characterized by the distance between the flow channels where the heat transfer medium flows. Peng and Peterson (1995) proposed the correlation of Nusselt number and the geometry of the flow channel as shown in equation 3.3 which was obtained from the results of their experiments showing in Figure 3.1
3 / 1 62 . 0 79 . 81 0
. 0
1165 .
0 Re Pe
W H W
Nu D
c h
−
⎟⎠
⎜ ⎞
⎝
⎟⎟ ⎛
⎠
⎜⎜ ⎞
⎝
= ⎛ (3.3)
where Wc center-to-center distance of microchannels [m]
(a) (b) Figure 3.1 Effect of (a)Dh/Wc and (b) H/W on laminar heat transfer of water
flowing in rectangular microchannel (Peng and Peterson, 1995)
From Figure 3.1 (a), the laminar heat transfer can be enhanced by enlarging the hydraulic diameter or decreasing the center-to-center distance which result on the increasing of Nusselt number. In the other hand, the Nusselt number can be decreased by increasing H/W ratio (decreasing the width or increasing the height) as shown in Figure 3.1 (b).
However, Renksizbulut et al. (2006) has proved that Nusselt number is independent of the geometry at the entrance region from the simulation of gas flow in microchannel.
The results are shown in Figure 3.2
Figure 3.2 Axial variation of Nusselt number at Re = 1 and Kn = 0.1 for gas flows in rectangular channel with different aspect ratios (Renksizbulut et al., 2006)
It can be concluded from Figure 3.2 that for all channel aspect ratios, the Nusselt number approaches a constant value. At the entrance region, the Nusselt number starts from the same value and decreases as the distance from the entrance increases. But in the fully developed region, there is a clear distinction between different channels with the highest Nusselt number for flow between parallel plates (aspect ratio deviates from unity) and the lowest for square duct (aspect ratio approaches unity).
Figure 3.3 Variation of fully developed Nu as a function of Kn and channel aspect ratio for gas flow at Re = 10 (Renksizbulut et al., 2006)
Figure 3.3 shows the variation of Nusselt number as a function of Knudsen number for different aspect ratio. As the aspect ratio decreases towards the parallel plates flow, stronger dependence on the Knudsen number is observed. The Nusselt number can be decrease about 40% from the Knudsen number at no-slip cast at 0 to slip case at 0.1 while this value is much less (about 12%) for a square duct at the same condition. This effect can be explained considering that the temperature gradients at the wall are uniform; therefore the slip effect is peripherally uniform for parallel plates. In the other hand, for square channel, the corner regions show a strong non-uniformity and behave closer to the no-slip condition due to much lower velocity and temperature gradients there, and therefore, the Knudsen number effects are weaker.
3.2 Thermal entrance length
Thermal entrance length is a transitory region where the flow at uniform temperature develops under influence of uniformly heated wall to a steady state profile of temperature that is dependent on fluid velocity and conductivity (Celata et al., 2006).
This flow region is called thermal developing flow.
Fluid with fully developed velocity profile enters the heated section of a pipe with a uniform temperature and as the heat at the pipe wall transfers to the fluid, the temperature of the fluid at the wall starts rising. The temperature profile of the fluid then develops to thermal fully developed profile presented in Figure 3.4 at the position x=xt. Thermal entrance length is the length from x=0 to x=xt.
Figure 3.4 The principle of thermally developing flow (Celata et al., 2006)
The thermal fully developed profile can be determined by Graetz number. When Graetz number gets lower than 10, the thermal fully developed profile is achieved (Celata et al., 2006). Graetz number is shown in the following equation
x RePrD
Gz= h (3.4)
where Gz Graetz number [-]
x distance from the entrance [m]
3.2 Heat transfer coefficient in rectangular microchannels
Lee et al. (2005) has investigated and compared the developed models from previous experimental works with the conventional heat transfer theory. The Nusselt number for laminar flows in rectangular microchannels is usually higher than the conventional correlation for the flows in macroscale and is remarkably higher for turbulent flows.
The heat transfer coefficient increases with decreasing flow channel size at a given flow rate.
Lee et al. (2006) presented the experimental result showing the variation of heat transfer coefficient in 200 μm size microchannel. The fluid flowing in microchannel is water with fully developed velocity profile at 300 K. Only flow rates in the laminar regimes were considered. An axially constant wall heat flux of 50 W/cm2 with constant wall temperature was applied on all four walls. The computational fluid dynamics solutions were done with FLUENT with different mesh sizes. The results are presented in Figure 3.5
Figure 3.5 Variation of Nusselt number with axial distance from the entrance at different grid sizes for water flowing in rectangular channel with 200 μm width (Lee et al., 2006)
The heat transfer coefficient for the flow in microchannels has a remarkably high value at the entrance and drops as the axial distance from the entrance increase until the
thermally developed flow, where the heat transfer coefficient becomes constant, is obtained.
Sometimes the distance from the entrance is presented as a dimensionless axial distance which is an inversion of Graetz number.
RePr D Gz x x
h
=
= −1
* (3.5)
where x* dimensionless axial distance [-]
The variation of convective heat transfer coefficient along the dimensionless axial distance from the inlet for laminar flow of water in rectangular microchannels with different hydraulic diameter was also studied by Lee et al. (2005). The flow channels size vary from 318 μm to 902 μm. Figure 3.6 shows a comparison of the results obtained from the experiment.
Figure 3.6 Dependence of heat transfer coefficient on dimensionless axial distance for laminar flow of water in microchannels with different hydraulic diameters (Lee et al., 2005)
The results from Figure 3.6 show that heat transfer coefficient enhancements can be obtained by decreasing size of a micro channel. For example, the heat transfer coefficients for the smallest channel in Figure 3.4 (318 μm) are almost three times the
values for the largest microchannel (902 μm). It shows clearly that the heat transfer coefficient initially decreases sharply as the dimensionless axial distance increases.
The higher heat transfer coefficients at lower dimensionless axial distance are caused by the thinner boundary layers in the developing region (Lee et al., 2005).
3.2.1 Effect of the Reynolds number
Lee et al. (2005) also investigated the Nusselt number from the experiment with different size of flow channel and flow region and compared the experimental results with the previous studies and the results from numerical solution. The experiment was done with deionized water with the flow rate from 0.1 to 2.2 l/min, which corresponds to the Reynolds number range of 300-3500, flowing in microchannel with hydraulic diameter ranges from 318 to 903 μm. The flow channel was heated at the bottom and the sides with heat flux 45 W/cm2.
The Nusselt number from the experiment can be calculated from equation 3.1 while the average heat transfer coefficient from the experiment was determined from
( )
[
ATw Tm]
q
h= / − (3.6)
where q heat flow rate [W]
A convective heat transfer area [m2] Tw temperature at the wall [K]
Tm temperature of the main fluid [K]
The numerical analysis was done with the software package FLUENT. The variation of the Nusselt number at different Reynolds number and size of flow channel are shown in the following figure comparing with the results from the numerical solution and the results from previous studies presented in the literature.
In Figure 3.7, the numerical solution by FLUENT commercial software shows good results agreed to the experiments at low Reynolds number in the laminar range. In
additional, the numerical analysis showed better prediction at higher Reynolds number in larger flow channel than the smaller one. As the Reynolds number increases, the Nusselt number increases. The average deviation between the experimental and the numerical results was about 5%. The good agreement between experimental and numerical results suggests that a conventional computational analysis approach can adequately predict the heat transfer behavior in microchannels. However, the deviations between the numerical solutions and experimental results might attributable mainly to experimental uncertainties.
(a) (b)
(c) (d)
Figure 3.7 Variation of Nusselts Number at different Reynolds Numbers for (a) 229 μm, (b) 300 μm, (c) 339 μm, and (d) 534 μm wide microchannels (Lee et al., 2005)
3.2.2 Heat transfer coefficient in microchannel
Heat transfer coefficient in microchannel can be determined from the basic heat transfer calculation presented in equation 3.6 which is also presented by Hu (2001) as a heat transfer model for Fischer-Tropsch synthesis in microchannel reactors. The model is based on the assumption that the flow is thermally fully developed and the temperature of the reaction channels is maintained at a constant value throughout the channel length. Assuming the similar specific heats and densities of gases in both reaction and control channels, the differences in temperature of the control channel fluid thus produce temperature differences of similar magnitude in the reactor channels, assessment of the control channel temperature differences can be used as a measure of reactor channel temperature differences, i.e. bulk temperature variation in a layer of stack.
The microchannel reactor configuration for the model developed by Hu (2001) is shown in Figure 3.8
Figure 3.8 Microchannel reactor configuration (Hu, 2001)
The heat transfer was assumed to generate only between two adjacent substrates or two flow streams with thermal steady state conditions, i.e. no heat transfer between
channels on the same substrates. The heat transfer between two flows involves the total heat transfer coefficient as follows
) ( R C
RC T T
hA
q= − (3.7)
where ARC heat transfer area between two flows [m2] (see equation 3.14) TR temperature of reaction channel [K]
TC temperature of control channel [K]
Overall heat transfer coefficient comprises of the heat conduction through the wall between reaction channel and control channel, heat conduction through the coated catalyst layer, and heat convection of the controlling fluid. Total heat transfer coefficient is defined by equation 3.11
C RC c w
h R
RC
C A
A h k
D A
A
U k − + 1
+
= θ δ
(3.8)
where U Overall heat transfer coefficient [W m-2 K-1] θ thickness of the coated catalyst [m]
kc thermal conductivity of coated catalyst layer [W m-1 K-1] AR heat transfer area at reaction channel [m2] (see equation 3.12) AC heat transfer area at control channel [m2] (see equation 3.13) kw thermal conductivity of the wall [W m-1 K-1]
δ channel-plate thickness [m]
hc heat convection of the controlling fluid [W m-2 K-1]
The heat transfer area, AR, AC, and ARC can be determined from
( )
cR W D s
A =2 + (3.9)
(
W D)
sAc =2 c + c (3.10)
c
RC ss
A =2 (3.11)
where s central distance between reaction channels [m] (see Figure 3.8) sc central distance between control channels [m] (see Figure 3.8)
For the flow with the same size and configuration of reaction channel and control channel, AR equals to AC and ARC/AR and ARC/AC approximately equal to unity. Thus the equation 3.11 becomes
f w
h
C k h
D
h k 1
− + +
= θ δ
(3.12)
3.3 Heat transfer in two-phase flow
The previous studies on two-phase heat transfer in microchannels were mostly concentrated on the boiling of water. The heat transfer coefficient for two-phase flow is characterized by thermodynamic vapor quality which is the fraction of vapor divided by total, on a molar basis (Sobierska et al., 2006). For example, the vapor at dew point has a thermodynamic vapor quality of unity and the liquid at bubble point has a thermodynamic vapor quality of zero.
The experiment of boiling deionized water in a channel with hydraulic diameter of 1.2 mm was done by Sobierska et al. (2006). The results show that the heat transfer coefficient decreases with increasing thermodynamic vapor quality and increases with increasing heat flux. There are possibly two potential reasons that the heat transfer coefficient decreases with increasing vapor quality. The first reason may be partial dry out occurring when the liquid film between the bubble and the wall may completely evaporate at some positions, forming there intermittent local dryout areas. The second reason may be that high pressure gradients in microchannels exist and as the pressure drops, one can expect that the heat transfer coefficient decreases too.
Figure 3.9 Heat transfer coefficient at different thermodynamic vapour quality value (Sobierska et al., 2006)
In Fischer-Tropsch synthesis, reactants in gas phase are fed to the microchannel reactor. The liquid phase product generates along the flow channel so the variation of heat transfer could be opposite from the boiling case. Although the heat transfer coefficient may increase along the flow channel because of the generation of liquid product which enhances heat transfer, the thermodynamic properties, for example, high temperature and pressure may influence on the phase change in Fischer-Tropsch synthesis which may result on high thermodynamic vapor quality. However, the heat transfer coefficient also depends on the two-phase flow pattern and for two-phase flow it also depends on the liquid film thickness (Thonon and Tochon, 2004).
